auxOTHERcomputational(3)

NAME
auxOTHERcomputational - auxiliary Computational routines

SYNOPSIS

  Functions
character *1 function chla_transtype (trans)
CHLA_TRANSTYPE
subroutine dbdsdc (uplo, compq, n, d, e, u, ldu, vt, ldvt, q, iq, work, iwork, info)
DBDSDC
subroutine dbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, work, info)
DBDSQR
subroutine ddisna (job, m, n, d, sep, info)
DDISNA
subroutine dlaed0 (icompq, qsiz, n, d, e, q, ldq, qstore, ldqs, work, iwork, info)
DLAED0 used by DSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method.
subroutine dlaed1 (n, d, q, ldq, indxq, rho, cutpnt, work, iwork, info)
DLAED1 used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.
subroutine dlaed2 (k, n, n1, d, q, ldq, indxq, rho, z, dlamda, w, q2, indx, indxc, indxp, coltyp, info)
DLAED2 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.
subroutine dlaed3 (k, n, n1, d, q, ldq, rho, dlamda, q2, indx, ctot, w, s, info)
DLAED3 used by DSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.
subroutine dlaed4 (n, i, d, z, delta, rho, dlam, info)
DLAED4 used by DSTEDC. Finds a single root of the secular equation.
subroutine dlaed5 (i, d, z, delta, rho, dlam)
DLAED5 used by DSTEDC. Solves the 2-by-2 secular equation.
subroutine dlaed6 (kniter, orgati, rho, d, z, finit, tau, info)
DLAED6 used by DSTEDC. Computes one Newton step in solution of the secular equation.
subroutine dlaed7 (icompq, n, qsiz, tlvls, curlvl, curpbm, d, q, ldq, indxq, rho, cutpnt, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, iwork, info)
DLAED7 used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.
subroutine dlaed8 (icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt, z, dlamda, q2, ldq2, w, perm, givptr, givcol, givnum, indxp, indx, info)
DLAED8 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
subroutine dlaed9 (k, kstart, kstop, n, d, q, ldq, rho, dlamda, w, s, lds, info)
DLAED9 used by DSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.
subroutine dlaeda (n, tlvls, curlvl, curpbm, prmptr, perm, givptr, givcol, givnum, q, qptr, z, ztemp, info)
DLAEDA used by DSTEDC. Computes the Z vector determining the rank-one modification of the diagonal matrix. Used when the original matrix is dense.
subroutine dlagtf (n, a, lambda, b, c, tol, d, in, info)
DLAGTF computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges.
subroutine dlamrg (n1, n2, a, dtrd1, dtrd2, index)
DLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single set sorted in ascending order.
subroutine dlartgs (x, y, sigma, cs, sn)
DLARTGS generates a plane rotation designed to introduce a bulge in implicit QR iteration for the bidiagonal SVD problem.
subroutine dlasq1 (n, d, e, work, info)
DLASQ1 computes the singular values of a real square bidiagonal matrix. Used by sbdsqr.
subroutine dlasq2 (n, z, info)
DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.
subroutine dlasq3 (i0, n0, z, pp, dmin, sigma, desig, qmax, nfail, iter, ndiv, ieee, ttype, dmin1, dmin2, dn, dn1, dn2, g, tau)
DLASQ3 checks for deflation, computes a shift and calls dqds. Used by sbdsqr.
subroutine dlasq4 (i0, n0, z, pp, n0in, dmin, dmin1, dmin2, dn, dn1, dn2, tau, ttype, g)
DLASQ4 computes an approximation to the smallest eigenvalue using values of d from the previous transform. Used by sbdsqr.
subroutine dlasq5 (i0, n0, z, pp, tau, sigma, dmin, dmin1, dmin2, dn, dnm1, dnm2, ieee, eps)
DLASQ5 computes one dqds transform in ping-pong form. Used by sbdsqr and sstegr.
subroutine dlasq6 (i0, n0, z, pp, dmin, dmin1, dmin2, dn, dnm1, dnm2)
DLASQ6 computes one dqd transform in ping-pong form. Used by sbdsqr and sstegr.
subroutine dlasrt (id, n, d, info)
DLASRT sorts numbers in increasing or decreasing order.
subroutine dstebz (range, order, n, vl, vu, il, iu, abstol, d, e, m, nsplit, w, iblock, isplit, work, iwork, info)
DSTEBZ
subroutine dstedc (compz, n, d, e, z, ldz, work, lwork, iwork, liwork, info)
DSTEDC
subroutine dsteqr (compz, n, d, e, z, ldz, work, info)
DSTEQR
subroutine dsterf (n, d, e, info)
DSTERF
integer function iladiag (diag)
ILADIAG
integer function ilaprec (prec)
ILAPREC
integer function ilatrans (trans)
ILATRANS
integer function ilauplo (uplo)
ILAUPLO
subroutine sbdsdc (uplo, compq, n, d, e, u, ldu, vt, ldvt, q, iq, work, iwork, info)
SBDSDC
subroutine sbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, work, info)
SBDSQR
subroutine sdisna (job, m, n, d, sep, info)
SDISNA
subroutine slaed0 (icompq, qsiz, n, d, e, q, ldq, qstore, ldqs, work, iwork, info)
SLAED0 used by SSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method.
subroutine slaed1 (n, d, q, ldq, indxq, rho, cutpnt, work, iwork, info)
SLAED1 used by SSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.
subroutine slaed2 (k, n, n1, d, q, ldq, indxq, rho, z, dlamda, w, q2, indx, indxc, indxp, coltyp, info)
SLAED2 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.
subroutine slaed3 (k, n, n1, d, q, ldq, rho, dlamda, q2, indx, ctot, w, s, info)
SLAED3 used by SSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.
subroutine slaed4 (n, i, d, z, delta, rho, dlam, info)
SLAED4 used by SSTEDC. Finds a single root of the secular equation.
subroutine slaed5 (i, d, z, delta, rho, dlam)
SLAED5 used by SSTEDC. Solves the 2-by-2 secular equation.
subroutine slaed6 (kniter, orgati, rho, d, z, finit, tau, info)
SLAED6 used by SSTEDC. Computes one Newton step in solution of the secular equation.
subroutine slaed7 (icompq, n, qsiz, tlvls, curlvl, curpbm, d, q, ldq, indxq, rho, cutpnt, qstore, qptr, prmptr, perm, givptr, givcol, givnum, work, iwork, info)
SLAED7 used by SSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.
subroutine slaed8 (icompq, k, n, qsiz, d, q, ldq, indxq, rho, cutpnt, z, dlamda, q2, ldq2, w, perm, givptr, givcol, givnum, indxp, indx, info)
SLAED8 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
subroutine slaed9 (k, kstart, kstop, n, d, q, ldq, rho, dlamda, w, s, lds, info)
SLAED9 used by SSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.
subroutine slaeda (n, tlvls, curlvl, curpbm, prmptr, perm, givptr, givcol, givnum, q, qptr, z, ztemp, info)
SLAEDA used by SSTEDC. Computes the Z vector determining the rank-one modification of the diagonal matrix. Used when the original matrix is dense.
subroutine slagtf (n, a, lambda, b, c, tol, d, in, info)
SLAGTF computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges.
subroutine slamrg (n1, n2, a, strd1, strd2, index)
SLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single set sorted in ascending order.
subroutine slartgs (x, y, sigma, cs, sn)
SLARTGS generates a plane rotation designed to introduce a bulge in implicit QR iteration for the bidiagonal SVD problem.
subroutine slasq1 (n, d, e, work, info)
SLASQ1 computes the singular values of a real square bidiagonal matrix. Used by sbdsqr.
subroutine slasq2 (n, z, info)
SLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.
subroutine slasq3 (i0, n0, z, pp, dmin, sigma, desig, qmax, nfail, iter, ndiv, ieee, ttype, dmin1, dmin2, dn, dn1, dn2, g, tau)
SLASQ3 checks for deflation, computes a shift and calls dqds. Used by sbdsqr.
subroutine slasq4 (i0, n0, z, pp, n0in, dmin, dmin1, dmin2, dn, dn1, dn2, tau, ttype, g)
SLASQ4 computes an approximation to the smallest eigenvalue using values of d from the previous transform. Used by sbdsqr.
subroutine slasq5 (i0, n0, z, pp, tau, sigma, dmin, dmin1, dmin2, dn, dnm1, dnm2, ieee, eps)
SLASQ5 computes one dqds transform in ping-pong form. Used by sbdsqr and sstegr.
subroutine slasq6 (i0, n0, z, pp, dmin, dmin1, dmin2, dn, dnm1, dnm2)
SLASQ6 computes one dqd transform in ping-pong form. Used by sbdsqr and sstegr.
subroutine slasrt (id, n, d, info)
SLASRT sorts numbers in increasing or decreasing order.
subroutine spttrf (n, d, e, info)
SPTTRF
subroutine sstebz (range, order, n, vl, vu, il, iu, abstol, d, e, m, nsplit, w, iblock, isplit, work, iwork, info)
SSTEBZ
subroutine sstedc (compz, n, d, e, z, ldz, work, lwork, iwork, liwork, info)
SSTEDC
subroutine ssteqr (compz, n, d, e, z, ldz, work, info)
SSTEQR
subroutine ssterf (n, d, e, info)
SSTERF

Detailed Description

This is the group of auxiliary Computational routines

Function Documentation

  character*1 function chla_transtype (integer trans)

CHLA_TRANSTYPE

Purpose:

     This subroutine translates from a BLAST-specified integer constant to
     the character string specifying a transposition operation.
     CHLA_TRANSTYPE returns an CHARACTER*1.  If CHLA_TRANSTYPE is 'X',
     then input is not an integer indicating a transposition operator.
     Otherwise CHLA_TRANSTYPE returns the constant value corresponding to
     TRANS.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 57 of file chla_transtype.f.

  subroutine dbdsdc (character uplo, character compq, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldvt, * ) vt, integer ldvt, double precision, dimension( * ) q, integer, dimension( * ) iq, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DBDSDC

Purpose:

     DBDSDC computes the singular value decomposition (SVD) of a real
     N-by-N (upper or lower) bidiagonal matrix B:  B = U * S * VT,
     using a divide and conquer method, where S is a diagonal matrix
     with non-negative diagonal elements (the singular values of B), and
     U and VT are orthogonal matrices of left and right singular vectors,
     respectively. DBDSDC can be used to compute all singular values,
     and optionally, singular vectors or singular vectors in compact form.
     This code makes very mild assumptions about floating point
     arithmetic. It will work on machines with a guard digit in
     add/subtract, or on those binary machines without guard digits
     which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
     It could conceivably fail on hexadecimal or decimal machines
     without guard digits, but we know of none.  See DLASD3 for details.
     The code currently calls DLASDQ if singular values only are desired.
     However, it can be slightly modified to compute singular values
     using the divide and conquer method.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  B is upper bidiagonal.
              = 'L':  B is lower bidiagonal.

    COMPQ

              COMPQ is CHARACTER*1
              Specifies whether singular vectors are to be computed
              as follows:
              = 'N':  Compute singular values only;
              = 'P':  Compute singular values and compute singular
                      vectors in compact form;
              = 'I':  Compute singular values and singular vectors.

    N

              N is INTEGER
              The order of the matrix B.  N >= 0.

    D

              D is DOUBLE PRECISION array, dimension (N)
              On entry, the n diagonal elements of the bidiagonal matrix B.
              On exit, if INFO=0, the singular values of B.

    E

              E is DOUBLE PRECISION array, dimension (N-1)
              On entry, the elements of E contain the offdiagonal
              elements of the bidiagonal matrix whose SVD is desired.
              On exit, E has been destroyed.

    U

              U is DOUBLE PRECISION array, dimension (LDU,N)
              If  COMPQ = 'I', then:
                 On exit, if INFO = 0, U contains the left singular vectors
                 of the bidiagonal matrix.
              For other values of COMPQ, U is not referenced.

    LDU

              LDU is INTEGER
              The leading dimension of the array U.  LDU >= 1.
              If singular vectors are desired, then LDU >= max( 1, N ).

    VT

              VT is DOUBLE PRECISION array, dimension (LDVT,N)
              If  COMPQ = 'I', then:
                 On exit, if INFO = 0, VT**T contains the right singular
                 vectors of the bidiagonal matrix.
              For other values of COMPQ, VT is not referenced.

    LDVT

              LDVT is INTEGER
              The leading dimension of the array VT.  LDVT >= 1.
              If singular vectors are desired, then LDVT >= max( 1, N ).

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ)
              If  COMPQ = 'P', then:
                 On exit, if INFO = 0, Q and IQ contain the left
                 and right singular vectors in a compact form,
                 requiring O(N log N) space instead of 2*N**2.
                 In particular, Q contains all the DOUBLE PRECISION data in
                 LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1))))
                 words of memory, where SMLSIZ is returned by ILAENV and
                 is equal to the maximum size of the subproblems at the
                 bottom of the computation tree (usually about 25).
              For other values of COMPQ, Q is not referenced.

    IQ

              IQ is INTEGER array, dimension (LDIQ)
              If  COMPQ = 'P', then:
                 On exit, if INFO = 0, Q and IQ contain the left
                 and right singular vectors in a compact form,
                 requiring O(N log N) space instead of 2*N**2.
                 In particular, IQ contains all INTEGER data in
                 LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
                 words of memory, where SMLSIZ is returned by ILAENV and
                 is equal to the maximum size of the subproblems at the
                 bottom of the computation tree (usually about 25).
              For other values of COMPQ, IQ is not referenced.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              If COMPQ = 'N' then LWORK >= (4 * N).
              If COMPQ = 'P' then LWORK >= (6 * N).
              If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).

    IWORK

              IWORK is INTEGER array, dimension (8*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  The algorithm failed to compute a singular value.
                    The update process of divide and conquer failed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 203 of file dbdsdc.f.

  subroutine dbdsqr (character uplo, integer n, integer ncvt, integer nru, integer ncc, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldvt, * ) vt, integer ldvt, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer info)

DBDSQR

Purpose:

     DBDSQR computes the singular values and, optionally, the right and/or
     left singular vectors from the singular value decomposition (SVD) of
     a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
     zero-shift QR algorithm.  The SVD of B has the form
        B = Q * S * P**T
     where S is the diagonal matrix of singular values, Q is an orthogonal
     matrix of left singular vectors, and P is an orthogonal matrix of
     right singular vectors.  If left singular vectors are requested, this
     subroutine actually returns U*Q instead of Q, and, if right singular
     vectors are requested, this subroutine returns P**T*VT instead of
     P**T, for given real input matrices U and VT.  When U and VT are the
     orthogonal matrices that reduce a general matrix A to bidiagonal
     form:  A = U*B*VT, as computed by DGEBRD, then
        A = (U*Q) * S * (P**T*VT)
     is the SVD of A.  Optionally, the subroutine may also compute Q**T*C
     for a given real input matrix C.
     See 'Computing  Small Singular Values of Bidiagonal Matrices With
     Guaranteed High Relative Accuracy,' by J. Demmel and W. Kahan,
     LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
     no. 5, pp. 873-912, Sept 1990) and
     'Accurate singular values and differential qd algorithms,' by
     B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
     Department, University of California at Berkeley, July 1992
     for a detailed description of the algorithm.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  B is upper bidiagonal;
              = 'L':  B is lower bidiagonal.

    N

              N is INTEGER
              The order of the matrix B.  N >= 0.

    NCVT

              NCVT is INTEGER
              The number of columns of the matrix VT. NCVT >= 0.

    NRU

              NRU is INTEGER
              The number of rows of the matrix U. NRU >= 0.

    NCC

              NCC is INTEGER
              The number of columns of the matrix C. NCC >= 0.

    D

              D is DOUBLE PRECISION array, dimension (N)
              On entry, the n diagonal elements of the bidiagonal matrix B.
              On exit, if INFO=0, the singular values of B in decreasing
              order.

    E

              E is DOUBLE PRECISION array, dimension (N-1)
              On entry, the N-1 offdiagonal elements of the bidiagonal
              matrix B.
              On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
              will contain the diagonal and superdiagonal elements of a
              bidiagonal matrix orthogonally equivalent to the one given
              as input.

    VT

              VT is DOUBLE PRECISION array, dimension (LDVT, NCVT)
              On entry, an N-by-NCVT matrix VT.
              On exit, VT is overwritten by P**T * VT.
              Not referenced if NCVT = 0.

    LDVT

              LDVT is INTEGER
              The leading dimension of the array VT.
              LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.

    U

              U is DOUBLE PRECISION array, dimension (LDU, N)
              On entry, an NRU-by-N matrix U.
              On exit, U is overwritten by U * Q.
              Not referenced if NRU = 0.

    LDU

              LDU is INTEGER
              The leading dimension of the array U.  LDU >= max(1,NRU).

    C

              C is DOUBLE PRECISION array, dimension (LDC, NCC)
              On entry, an N-by-NCC matrix C.
              On exit, C is overwritten by Q**T * C.
              Not referenced if NCC = 0.

    LDC

              LDC is INTEGER
              The leading dimension of the array C.
              LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.

    WORK

              WORK is DOUBLE PRECISION array, dimension (4*(N-1))

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  If INFO = -i, the i-th argument had an illegal value
              > 0:
                 if NCVT = NRU = NCC = 0,
                    = 1, a split was marked by a positive value in E
                    = 2, current block of Z not diagonalized after 30*N
                         iterations (in inner while loop)
                    = 3, termination criterion of outer while loop not met
                         (program created more than N unreduced blocks)
                 else NCVT = NRU = NCC = 0,
                       the algorithm did not converge; D and E contain the
                       elements of a bidiagonal matrix which is orthogonally
                       similar to the input matrix B;  if INFO = i, i
                       elements of E have not converged to zero.

Internal Parameters:

      TOLMUL  DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8)))
              TOLMUL controls the convergence criterion of the QR loop.
              If it is positive, TOLMUL*EPS is the desired relative
                 precision in the computed singular values.
              If it is negative, abs(TOLMUL*EPS*sigma_max) is the
                 desired absolute accuracy in the computed singular
                 values (corresponds to relative accuracy
                 abs(TOLMUL*EPS) in the largest singular value.
              abs(TOLMUL) should be between 1 and 1/EPS, and preferably
                 between 10 (for fast convergence) and .1/EPS
                 (for there to be some accuracy in the results).
              Default is to lose at either one eighth or 2 of the
                 available decimal digits in each computed singular value
                 (whichever is smaller).
      MAXITR  INTEGER, default = 6
              MAXITR controls the maximum number of passes of the
              algorithm through its inner loop. The algorithms stops
              (and so fails to converge) if the number of passes
              through the inner loop exceeds MAXITR*N**2.

Note:

      Bug report from Cezary Dendek.
      On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is
      removed since it can overflow pretty easily (for N larger or equal
      than 18,919). We instead use MAXITDIVN = MAXITR*N.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 239 of file dbdsqr.f.

  subroutine ddisna (character job, integer m, integer n, double precision, dimension( * ) d, double precision, dimension( * ) sep, integer info)

DDISNA

Purpose:

     DDISNA computes the reciprocal condition numbers for the eigenvectors
     of a real symmetric or complex Hermitian matrix or for the left or
     right singular vectors of a general m-by-n matrix. The reciprocal
     condition number is the 'gap' between the corresponding eigenvalue or
     singular value and the nearest other one.
     The bound on the error, measured by angle in radians, in the I-th
     computed vector is given by
            DLAMCH( 'E' ) * ( ANORM / SEP( I ) )
     where ANORM = 2-norm(A) = max( abs( D(j) ) ).  SEP(I) is not allowed
     to be smaller than DLAMCH( 'E' )*ANORM in order to limit the size of
     the error bound.
     DDISNA may also be used to compute error bounds for eigenvectors of
     the generalized symmetric definite eigenproblem.

Parameters
    JOB

              JOB is CHARACTER*1
              Specifies for which problem the reciprocal condition numbers
              should be computed:
              = 'E':  the eigenvectors of a symmetric/Hermitian matrix;
              = 'L':  the left singular vectors of a general matrix;
              = 'R':  the right singular vectors of a general matrix.

    M

              M is INTEGER
              The number of rows of the matrix. M >= 0.

    N

              N is INTEGER
              If JOB = 'L' or 'R', the number of columns of the matrix,
              in which case N >= 0. Ignored if JOB = 'E'.

    D

              D is DOUBLE PRECISION array, dimension (M) if JOB = 'E'
                                  dimension (min(M,N)) if JOB = 'L' or 'R'
              The eigenvalues (if JOB = 'E') or singular values (if JOB =
              'L' or 'R') of the matrix, in either increasing or decreasing
              order. If singular values, they must be non-negative.

    SEP

              SEP is DOUBLE PRECISION array, dimension (M) if JOB = 'E'
                                   dimension (min(M,N)) if JOB = 'L' or 'R'
              The reciprocal condition numbers of the vectors.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 116 of file ddisna.f.

  subroutine dlaed0 (integer icompq, integer qsiz, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( ldqs, * ) qstore, integer ldqs, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DLAED0 used by DSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method.

Purpose:

     DLAED0 computes all eigenvalues and corresponding eigenvectors of a
     symmetric tridiagonal matrix using the divide and conquer method.

Parameters
    ICOMPQ

              ICOMPQ is INTEGER
              = 0:  Compute eigenvalues only.
              = 1:  Compute eigenvectors of original dense symmetric matrix
                    also.  On entry, Q contains the orthogonal matrix used
                    to reduce the original matrix to tridiagonal form.
              = 2:  Compute eigenvalues and eigenvectors of tridiagonal
                    matrix.

    QSIZ

              QSIZ is INTEGER
             The dimension of the orthogonal matrix used to reduce
             the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

    N

              N is INTEGER
             The dimension of the symmetric tridiagonal matrix.  N >= 0.

    D

              D is DOUBLE PRECISION array, dimension (N)
             On entry, the main diagonal of the tridiagonal matrix.
             On exit, its eigenvalues.

    E

              E is DOUBLE PRECISION array, dimension (N-1)
             The off-diagonal elements of the tridiagonal matrix.
             On exit, E has been destroyed.

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ, N)
             On entry, Q must contain an N-by-N orthogonal matrix.
             If ICOMPQ = 0    Q is not referenced.
             If ICOMPQ = 1    On entry, Q is a subset of the columns of the
                              orthogonal matrix used to reduce the full
                              matrix to tridiagonal form corresponding to
                              the subset of the full matrix which is being
                              decomposed at this time.
             If ICOMPQ = 2    On entry, Q will be the identity matrix.
                              On exit, Q contains the eigenvectors of the
                              tridiagonal matrix.

    LDQ

              LDQ is INTEGER
             The leading dimension of the array Q.  If eigenvectors are
             desired, then  LDQ >= max(1,N).  In any case,  LDQ >= 1.

    QSTORE

              QSTORE is DOUBLE PRECISION array, dimension (LDQS, N)
             Referenced only when ICOMPQ = 1.  Used to store parts of
             the eigenvector matrix when the updating matrix multiplies
             take place.

    LDQS

              LDQS is INTEGER
             The leading dimension of the array QSTORE.  If ICOMPQ = 1,
             then  LDQS >= max(1,N).  In any case,  LDQS >= 1.

    WORK

              WORK is DOUBLE PRECISION array,
             If ICOMPQ = 0 or 1, the dimension of WORK must be at least
                         1 + 3*N + 2*N*lg N + 3*N**2
                         ( lg( N ) = smallest integer k
                                     such that 2^k >= N )
             If ICOMPQ = 2, the dimension of WORK must be at least
                         4*N + N**2.

    IWORK

              IWORK is INTEGER array,
             If ICOMPQ = 0 or 1, the dimension of IWORK must be at least
                            6 + 6*N + 5*N*lg N.
                            ( lg( N ) = smallest integer k
                                        such that 2^k >= N )
             If ICOMPQ = 2, the dimension of IWORK must be at least
                            3 + 5*N.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  The algorithm failed to compute an eigenvalue while
                    working on the submatrix lying in rows and columns
                    INFO/(N+1) through mod(INFO,N+1).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 170 of file dlaed0.f.

  subroutine dlaed1 (integer n, double precision, dimension( * ) d, double precision, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, double precision rho, integer cutpnt, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DLAED1 used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.

Purpose:

     DLAED1 computes the updated eigensystem of a diagonal
     matrix after modification by a rank-one symmetric matrix.  This
     routine is used only for the eigenproblem which requires all
     eigenvalues and eigenvectors of a tridiagonal matrix.  DLAED7 handles
     the case in which eigenvalues only or eigenvalues and eigenvectors
     of a full symmetric matrix (which was reduced to tridiagonal form)
     are desired.
       T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
        where Z = Q**T*u, u is a vector of length N with ones in the
        CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
        The eigenvectors of the original matrix are stored in Q, and the
        eigenvalues are in D.  The algorithm consists of three stages:
           The first stage consists of deflating the size of the problem
           when there are multiple eigenvalues or if there is a zero in
           the Z vector.  For each such occurrence the dimension of the
           secular equation problem is reduced by one.  This stage is
           performed by the routine DLAED2.
           The second stage consists of calculating the updated
           eigenvalues. This is done by finding the roots of the secular
           equation via the routine DLAED4 (as called by DLAED3).
           This routine also calculates the eigenvectors of the current
           problem.
           The final stage consists of computing the updated eigenvectors
           directly using the updated eigenvalues.  The eigenvectors for
           the current problem are multiplied with the eigenvectors from
           the overall problem.

Parameters
    N

              N is INTEGER
             The dimension of the symmetric tridiagonal matrix.  N >= 0.

    D

              D is DOUBLE PRECISION array, dimension (N)
             On entry, the eigenvalues of the rank-1-perturbed matrix.
             On exit, the eigenvalues of the repaired matrix.

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ,N)
             On entry, the eigenvectors of the rank-1-perturbed matrix.
             On exit, the eigenvectors of the repaired tridiagonal matrix.

    LDQ

              LDQ is INTEGER
             The leading dimension of the array Q.  LDQ >= max(1,N).

    INDXQ

              INDXQ is INTEGER array, dimension (N)
             On entry, the permutation which separately sorts the two
             subproblems in D into ascending order.
             On exit, the permutation which will reintegrate the
             subproblems back into sorted order,
             i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.

    RHO

              RHO is DOUBLE PRECISION
             The subdiagonal entry used to create the rank-1 modification.

    CUTPNT

              CUTPNT is INTEGER
             The location of the last eigenvalue in the leading sub-matrix.
             min(1,N) <= CUTPNT <= N/2.

    WORK

              WORK is DOUBLE PRECISION array, dimension (4*N + N**2)

    IWORK

              IWORK is INTEGER array, dimension (4*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = 1, an eigenvalue did not converge

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
     Modified by Francoise Tisseur, University of Tennessee

Definition at line 161 of file dlaed1.f.

  subroutine dlaed2 (integer k, integer n, integer n1, double precision, dimension( * ) d, double precision, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, double precision rho, double precision, dimension( * ) z, double precision, dimension( * ) dlamda, double precision, dimension( * ) w, double precision, dimension( * ) q2, integer, dimension( * ) indx, integer, dimension( * ) indxc, integer, dimension( * ) indxp, integer, dimension( * ) coltyp, integer info)

DLAED2 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.

Purpose:

     DLAED2 merges the two sets of eigenvalues together into a single
     sorted set.  Then it tries to deflate the size of the problem.
     There are two ways in which deflation can occur:  when two or more
     eigenvalues are close together or if there is a tiny entry in the
     Z vector.  For each such occurrence the order of the related secular
     equation problem is reduced by one.

Parameters
    K

              K is INTEGER
             The number of non-deflated eigenvalues, and the order of the
             related secular equation. 0 <= K <=N.

    N

              N is INTEGER
             The dimension of the symmetric tridiagonal matrix.  N >= 0.

    N1

              N1 is INTEGER
             The location of the last eigenvalue in the leading sub-matrix.
             min(1,N) <= N1 <= N/2.

    D

              D is DOUBLE PRECISION array, dimension (N)
             On entry, D contains the eigenvalues of the two submatrices to
             be combined.
             On exit, D contains the trailing (N-K) updated eigenvalues
             (those which were deflated) sorted into increasing order.

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ, N)
             On entry, Q contains the eigenvectors of two submatrices in
             the two square blocks with corners at (1,1), (N1,N1)
             and (N1+1, N1+1), (N,N).
             On exit, Q contains the trailing (N-K) updated eigenvectors
             (those which were deflated) in its last N-K columns.

    LDQ

              LDQ is INTEGER
             The leading dimension of the array Q.  LDQ >= max(1,N).

    INDXQ

              INDXQ is INTEGER array, dimension (N)
             The permutation which separately sorts the two sub-problems
             in D into ascending order.  Note that elements in the second
             half of this permutation must first have N1 added to their
             values. Destroyed on exit.

    RHO

              RHO is DOUBLE PRECISION
             On entry, the off-diagonal element associated with the rank-1
             cut which originally split the two submatrices which are now
             being recombined.
             On exit, RHO has been modified to the value required by
             DLAED3.

    Z

              Z is DOUBLE PRECISION array, dimension (N)
             On entry, Z contains the updating vector (the last
             row of the first sub-eigenvector matrix and the first row of
             the second sub-eigenvector matrix).
             On exit, the contents of Z have been destroyed by the updating
             process.

    DLAMDA

              DLAMDA is DOUBLE PRECISION array, dimension (N)
             A copy of the first K eigenvalues which will be used by
             DLAED3 to form the secular equation.

    W

              W is DOUBLE PRECISION array, dimension (N)
             The first k values of the final deflation-altered z-vector
             which will be passed to DLAED3.

    Q2

              Q2 is DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)
             A copy of the first K eigenvectors which will be used by
             DLAED3 in a matrix multiply (DGEMM) to solve for the new
             eigenvectors.

    INDX

              INDX is INTEGER array, dimension (N)
             The permutation used to sort the contents of DLAMDA into
             ascending order.

    INDXC

              INDXC is INTEGER array, dimension (N)
             The permutation used to arrange the columns of the deflated
             Q matrix into three groups:  the first group contains non-zero
             elements only at and above N1, the second contains
             non-zero elements only below N1, and the third is dense.

    INDXP

              INDXP is INTEGER array, dimension (N)
             The permutation used to place deflated values of D at the end
             of the array.  INDXP(1:K) points to the nondeflated D-values
             and INDXP(K+1:N) points to the deflated eigenvalues.

    COLTYP

              COLTYP is INTEGER array, dimension (N)
             During execution, a label which will indicate which of the
             following types a column in the Q2 matrix is:
             1 : non-zero in the upper half only;
             2 : dense;
             3 : non-zero in the lower half only;
             4 : deflated.
             On exit, COLTYP(i) is the number of columns of type i,
             for i=1 to 4 only.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
     Modified by Francoise Tisseur, University of Tennessee

Definition at line 210 of file dlaed2.f.

  subroutine dlaed3 (integer k, integer n, integer n1, double precision, dimension( * ) d, double precision, dimension( ldq, * ) q, integer ldq, double precision rho, double precision, dimension( * ) dlamda, double precision, dimension( * ) q2, integer, dimension( * ) indx, integer, dimension( * ) ctot, double precision, dimension( * ) w, double precision, dimension( * ) s, integer info)

DLAED3 used by DSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.

Purpose:

     DLAED3 finds the roots of the secular equation, as defined by the
     values in D, W, and RHO, between 1 and K.  It makes the
     appropriate calls to DLAED4 and then updates the eigenvectors by
     multiplying the matrix of eigenvectors of the pair of eigensystems
     being combined by the matrix of eigenvectors of the K-by-K system
     which is solved here.
     This code makes very mild assumptions about floating point
     arithmetic. It will work on machines with a guard digit in
     add/subtract, or on those binary machines without guard digits
     which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
     It could conceivably fail on hexadecimal or decimal machines
     without guard digits, but we know of none.

Parameters
    K

              K is INTEGER
              The number of terms in the rational function to be solved by
              DLAED4.  K >= 0.

    N

              N is INTEGER
              The number of rows and columns in the Q matrix.
              N >= K (deflation may result in N>K).

    N1

              N1 is INTEGER
              The location of the last eigenvalue in the leading submatrix.
              min(1,N) <= N1 <= N/2.

    D

              D is DOUBLE PRECISION array, dimension (N)
              D(I) contains the updated eigenvalues for
              1 <= I <= K.

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ,N)
              Initially the first K columns are used as workspace.
              On output the columns 1 to K contain
              the updated eigenvectors.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.  LDQ >= max(1,N).

    RHO

              RHO is DOUBLE PRECISION
              The value of the parameter in the rank one update equation.
              RHO >= 0 required.

    DLAMDA

              DLAMDA is DOUBLE PRECISION array, dimension (K)
              The first K elements of this array contain the old roots
              of the deflated updating problem.  These are the poles
              of the secular equation. May be changed on output by
              having lowest order bit set to zero on Cray X-MP, Cray Y-MP,
              Cray-2, or Cray C-90, as described above.

    Q2

              Q2 is DOUBLE PRECISION array, dimension (LDQ2*N)
              The first K columns of this matrix contain the non-deflated
              eigenvectors for the split problem.

    INDX

              INDX is INTEGER array, dimension (N)
              The permutation used to arrange the columns of the deflated
              Q matrix into three groups (see DLAED2).
              The rows of the eigenvectors found by DLAED4 must be likewise
              permuted before the matrix multiply can take place.

    CTOT

              CTOT is INTEGER array, dimension (4)
              A count of the total number of the various types of columns
              in Q, as described in INDX.  The fourth column type is any
              column which has been deflated.

    W

              W is DOUBLE PRECISION array, dimension (K)
              The first K elements of this array contain the components
              of the deflation-adjusted updating vector. Destroyed on
              output.

    S

              S is DOUBLE PRECISION array, dimension (N1 + 1)*K
              Will contain the eigenvectors of the repaired matrix which
              will be multiplied by the previously accumulated eigenvectors
              to update the system.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = 1, an eigenvalue did not converge

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
     Modified by Francoise Tisseur, University of Tennessee

Definition at line 183 of file dlaed3.f.

  subroutine dlaed4 (integer n, integer i, double precision, dimension( * ) d, double precision, dimension( * ) z, double precision, dimension( * ) delta, double precision rho, double precision dlam, integer info)

DLAED4 used by DSTEDC. Finds a single root of the secular equation.

Purpose:

     This subroutine computes the I-th updated eigenvalue of a symmetric
     rank-one modification to a diagonal matrix whose elements are
     given in the array d, and that
                D(i) < D(j)  for  i < j
     and that RHO > 0.  This is arranged by the calling routine, and is
     no loss in generality.  The rank-one modified system is thus
                diag( D )  +  RHO * Z * Z_transpose.
     where we assume the Euclidean norm of Z is 1.
     The method consists of approximating the rational functions in the
     secular equation by simpler interpolating rational functions.

Parameters
    N

              N is INTEGER
             The length of all arrays.

    I

              I is INTEGER
             The index of the eigenvalue to be computed.  1 <= I <= N.

    D

              D is DOUBLE PRECISION array, dimension (N)
             The original eigenvalues.  It is assumed that they are in
             order, D(I) < D(J)  for I < J.

    Z

              Z is DOUBLE PRECISION array, dimension (N)
             The components of the updating vector.

    DELTA

              DELTA is DOUBLE PRECISION array, dimension (N)
             If N > 2, DELTA contains (D(j) - lambda_I) in its  j-th
             component.  If N = 1, then DELTA(1) = 1. If N = 2, see DLAED5
             for detail. The vector DELTA contains the information necessary
             to construct the eigenvectors by DLAED3 and DLAED9.

    RHO

              RHO is DOUBLE PRECISION
             The scalar in the symmetric updating formula.

    DLAM

              DLAM is DOUBLE PRECISION
             The computed lambda_I, the I-th updated eigenvalue.

    INFO

              INFO is INTEGER
             = 0:  successful exit
             > 0:  if INFO = 1, the updating process failed.

Internal Parameters:

      Logical variable ORGATI (origin-at-i?) is used for distinguishing
      whether D(i) or D(i+1) is treated as the origin.
                ORGATI = .true.    origin at i
                ORGATI = .false.   origin at i+1
       Logical variable SWTCH3 (switch-for-3-poles?) is for noting
       if we are working with THREE poles!
       MAXIT is the maximum number of iterations allowed for each
       eigenvalue.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

Definition at line 144 of file dlaed4.f.

  subroutine dlaed5 (integer i, double precision, dimension( 2 ) d, double precision, dimension( 2 ) z, double precision, dimension( 2 ) delta, double precision rho, double precision dlam)

DLAED5 used by DSTEDC. Solves the 2-by-2 secular equation.

Purpose:

     This subroutine computes the I-th eigenvalue of a symmetric rank-one
     modification of a 2-by-2 diagonal matrix
                diag( D )  +  RHO * Z * transpose(Z) .
     The diagonal elements in the array D are assumed to satisfy
                D(i) < D(j)  for  i < j .
     We also assume RHO > 0 and that the Euclidean norm of the vector
     Z is one.

Parameters
    I

              I is INTEGER
             The index of the eigenvalue to be computed.  I = 1 or I = 2.

    D

              D is DOUBLE PRECISION array, dimension (2)
             The original eigenvalues.  We assume D(1) < D(2).

    Z

              Z is DOUBLE PRECISION array, dimension (2)
             The components of the updating vector.

    DELTA

              DELTA is DOUBLE PRECISION array, dimension (2)
             The vector DELTA contains the information necessary
             to construct the eigenvectors.

    RHO

              RHO is DOUBLE PRECISION
             The scalar in the symmetric updating formula.

    DLAM

              DLAM is DOUBLE PRECISION
             The computed lambda_I, the I-th updated eigenvalue.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

Definition at line 107 of file dlaed5.f.

  subroutine dlaed6 (integer kniter, logical orgati, double precision rho, double precision, dimension( 3 ) d, double precision, dimension( 3 ) z, double precision finit, double precision tau, integer info)

DLAED6 used by DSTEDC. Computes one Newton step in solution of the secular equation.

Purpose:

     DLAED6 computes the positive or negative root (closest to the origin)
     of
                      z(1)        z(2)        z(3)
     f(x) =   rho + --------- + ---------- + ---------
                     d(1)-x      d(2)-x      d(3)-x
     It is assumed that
           if ORGATI = .true. the root is between d(2) and d(3);
           otherwise it is between d(1) and d(2)
     This routine will be called by DLAED4 when necessary. In most cases,
     the root sought is the smallest in magnitude, though it might not be
     in some extremely rare situations.

Parameters
    KNITER

              KNITER is INTEGER
                   Refer to DLAED4 for its significance.

    ORGATI

              ORGATI is LOGICAL
                   If ORGATI is true, the needed root is between d(2) and
                   d(3); otherwise it is between d(1) and d(2).  See
                   DLAED4 for further details.

    RHO

              RHO is DOUBLE PRECISION
                   Refer to the equation f(x) above.

    D

              D is DOUBLE PRECISION array, dimension (3)
                   D satisfies d(1) < d(2) < d(3).

    Z

              Z is DOUBLE PRECISION array, dimension (3)
                   Each of the elements in z must be positive.

    FINIT

              FINIT is DOUBLE PRECISION
                   The value of f at 0. It is more accurate than the one
                   evaluated inside this routine (if someone wants to do
                   so).

    TAU

              TAU is DOUBLE PRECISION
                   The root of the equation f(x).

    INFO

              INFO is INTEGER
                   = 0: successful exit
                   > 0: if INFO = 1, failure to converge

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      10/02/03: This version has a few statements commented out for thread
      safety (machine parameters are computed on each entry). SJH.
      05/10/06: Modified from a new version of Ren-Cang Li, use
         Gragg-Thornton-Warner cubic convergent scheme for better stability.

Contributors:
    Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

Definition at line 139 of file dlaed6.f.

  subroutine dlaed7 (integer icompq, integer n, integer qsiz, integer tlvls, integer curlvl, integer curpbm, double precision, dimension( * ) d, double precision, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, double precision rho, integer cutpnt, double precision, dimension( * ) qstore, integer, dimension( * ) qptr, integer, dimension( * ) prmptr, integer, dimension( * ) perm, integer, dimension( * ) givptr, integer, dimension( 2, * ) givcol, double precision, dimension( 2, * ) givnum, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DLAED7 used by DSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

Purpose:

     DLAED7 computes the updated eigensystem of a diagonal
     matrix after modification by a rank-one symmetric matrix. This
     routine is used only for the eigenproblem which requires all
     eigenvalues and optionally eigenvectors of a dense symmetric matrix
     that has been reduced to tridiagonal form.  DLAED1 handles
     the case in which all eigenvalues and eigenvectors of a symmetric
     tridiagonal matrix are desired.
       T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
        where Z = Q**Tu, u is a vector of length N with ones in the
        CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
        The eigenvectors of the original matrix are stored in Q, and the
        eigenvalues are in D.  The algorithm consists of three stages:
           The first stage consists of deflating the size of the problem
           when there are multiple eigenvalues or if there is a zero in
           the Z vector.  For each such occurrence the dimension of the
           secular equation problem is reduced by one.  This stage is
           performed by the routine DLAED8.
           The second stage consists of calculating the updated
           eigenvalues. This is done by finding the roots of the secular
           equation via the routine DLAED4 (as called by DLAED9).
           This routine also calculates the eigenvectors of the current
           problem.
           The final stage consists of computing the updated eigenvectors
           directly using the updated eigenvalues.  The eigenvectors for
           the current problem are multiplied with the eigenvectors from
           the overall problem.

Parameters
    ICOMPQ

              ICOMPQ is INTEGER
              = 0:  Compute eigenvalues only.
              = 1:  Compute eigenvectors of original dense symmetric matrix
                    also.  On entry, Q contains the orthogonal matrix used
                    to reduce the original matrix to tridiagonal form.

    N

              N is INTEGER
             The dimension of the symmetric tridiagonal matrix.  N >= 0.

    QSIZ

              QSIZ is INTEGER
             The dimension of the orthogonal matrix used to reduce
             the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

    TLVLS

              TLVLS is INTEGER
             The total number of merging levels in the overall divide and
             conquer tree.

    CURLVL

              CURLVL is INTEGER
             The current level in the overall merge routine,
             0 <= CURLVL <= TLVLS.

    CURPBM

              CURPBM is INTEGER
             The current problem in the current level in the overall
             merge routine (counting from upper left to lower right).

    D

              D is DOUBLE PRECISION array, dimension (N)
             On entry, the eigenvalues of the rank-1-perturbed matrix.
             On exit, the eigenvalues of the repaired matrix.

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ, N)
             On entry, the eigenvectors of the rank-1-perturbed matrix.
             On exit, the eigenvectors of the repaired tridiagonal matrix.

    LDQ

              LDQ is INTEGER
             The leading dimension of the array Q.  LDQ >= max(1,N).

    INDXQ

              INDXQ is INTEGER array, dimension (N)
             The permutation which will reintegrate the subproblem just
             solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
             will be in ascending order.

    RHO

              RHO is DOUBLE PRECISION
             The subdiagonal element used to create the rank-1
             modification.

    CUTPNT

              CUTPNT is INTEGER
             Contains the location of the last eigenvalue in the leading
             sub-matrix.  min(1,N) <= CUTPNT <= N.

    QSTORE

              QSTORE is DOUBLE PRECISION array, dimension (N**2+1)
             Stores eigenvectors of submatrices encountered during
             divide and conquer, packed together. QPTR points to
             beginning of the submatrices.

    QPTR

              QPTR is INTEGER array, dimension (N+2)
             List of indices pointing to beginning of submatrices stored
             in QSTORE. The submatrices are numbered starting at the
             bottom left of the divide and conquer tree, from left to
             right and bottom to top.

    PRMPTR

              PRMPTR is INTEGER array, dimension (N lg N)
             Contains a list of pointers which indicate where in PERM a
             level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
             indicates the size of the permutation and also the size of
             the full, non-deflated problem.

    PERM

              PERM is INTEGER array, dimension (N lg N)
             Contains the permutations (from deflation and sorting) to be
             applied to each eigenblock.

    GIVPTR

              GIVPTR is INTEGER array, dimension (N lg N)
             Contains a list of pointers which indicate where in GIVCOL a
             level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
             indicates the number of Givens rotations.

    GIVCOL

              GIVCOL is INTEGER array, dimension (2, N lg N)
             Each pair of numbers indicates a pair of columns to take place
             in a Givens rotation.

    GIVNUM

              GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
             Each number indicates the S value to be used in the
             corresponding Givens rotation.

    WORK

              WORK is DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N)

    IWORK

              IWORK is INTEGER array, dimension (4*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = 1, an eigenvalue did not converge

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 256 of file dlaed7.f.

  subroutine dlaed8 (integer icompq, integer k, integer n, integer qsiz, double precision, dimension( * ) d, double precision, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, double precision rho, integer cutpnt, double precision, dimension( * ) z, double precision, dimension( * ) dlamda, double precision, dimension( ldq2, * ) q2, integer ldq2, double precision, dimension( * ) w, integer, dimension( * ) perm, integer givptr, integer, dimension( 2, * ) givcol, double precision, dimension( 2, * ) givnum, integer, dimension( * ) indxp, integer, dimension( * ) indx, integer info)

DLAED8 used by DSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

Purpose:

     DLAED8 merges the two sets of eigenvalues together into a single
     sorted set.  Then it tries to deflate the size of the problem.
     There are two ways in which deflation can occur:  when two or more
     eigenvalues are close together or if there is a tiny element in the
     Z vector.  For each such occurrence the order of the related secular
     equation problem is reduced by one.

Parameters
    ICOMPQ

              ICOMPQ is INTEGER
              = 0:  Compute eigenvalues only.
              = 1:  Compute eigenvectors of original dense symmetric matrix
                    also.  On entry, Q contains the orthogonal matrix used
                    to reduce the original matrix to tridiagonal form.

    K

              K is INTEGER
             The number of non-deflated eigenvalues, and the order of the
             related secular equation.

    N

              N is INTEGER
             The dimension of the symmetric tridiagonal matrix.  N >= 0.

    QSIZ

              QSIZ is INTEGER
             The dimension of the orthogonal matrix used to reduce
             the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

    D

              D is DOUBLE PRECISION array, dimension (N)
             On entry, the eigenvalues of the two submatrices to be
             combined.  On exit, the trailing (N-K) updated eigenvalues
             (those which were deflated) sorted into increasing order.

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ,N)
             If ICOMPQ = 0, Q is not referenced.  Otherwise,
             on entry, Q contains the eigenvectors of the partially solved
             system which has been previously updated in matrix
             multiplies with other partially solved eigensystems.
             On exit, Q contains the trailing (N-K) updated eigenvectors
             (those which were deflated) in its last N-K columns.

    LDQ

              LDQ is INTEGER
             The leading dimension of the array Q.  LDQ >= max(1,N).

    INDXQ

              INDXQ is INTEGER array, dimension (N)
             The permutation which separately sorts the two sub-problems
             in D into ascending order.  Note that elements in the second
             half of this permutation must first have CUTPNT added to
             their values in order to be accurate.

    RHO

              RHO is DOUBLE PRECISION
             On entry, the off-diagonal element associated with the rank-1
             cut which originally split the two submatrices which are now
             being recombined.
             On exit, RHO has been modified to the value required by
             DLAED3.

    CUTPNT

              CUTPNT is INTEGER
             The location of the last eigenvalue in the leading
             sub-matrix.  min(1,N) <= CUTPNT <= N.

    Z

              Z is DOUBLE PRECISION array, dimension (N)
             On entry, Z contains the updating vector (the last row of
             the first sub-eigenvector matrix and the first row of the
             second sub-eigenvector matrix).
             On exit, the contents of Z are destroyed by the updating
             process.

    DLAMDA

              DLAMDA is DOUBLE PRECISION array, dimension (N)
             A copy of the first K eigenvalues which will be used by
             DLAED3 to form the secular equation.

    Q2

              Q2 is DOUBLE PRECISION array, dimension (LDQ2,N)
             If ICOMPQ = 0, Q2 is not referenced.  Otherwise,
             a copy of the first K eigenvectors which will be used by
             DLAED7 in a matrix multiply (DGEMM) to update the new
             eigenvectors.

    LDQ2

              LDQ2 is INTEGER
             The leading dimension of the array Q2.  LDQ2 >= max(1,N).

    W

              W is DOUBLE PRECISION array, dimension (N)
             The first k values of the final deflation-altered z-vector and
             will be passed to DLAED3.

    PERM

              PERM is INTEGER array, dimension (N)
             The permutations (from deflation and sorting) to be applied
             to each eigenblock.

    GIVPTR

              GIVPTR is INTEGER
             The number of Givens rotations which took place in this
             subproblem.

    GIVCOL

              GIVCOL is INTEGER array, dimension (2, N)
             Each pair of numbers indicates a pair of columns to take place
             in a Givens rotation.

    GIVNUM

              GIVNUM is DOUBLE PRECISION array, dimension (2, N)
             Each number indicates the S value to be used in the
             corresponding Givens rotation.

    INDXP

              INDXP is INTEGER array, dimension (N)
             The permutation used to place deflated values of D at the end
             of the array.  INDXP(1:K) points to the nondeflated D-values
             and INDXP(K+1:N) points to the deflated eigenvalues.

    INDX

              INDX is INTEGER array, dimension (N)
             The permutation used to sort the contents of D into ascending
             order.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 240 of file dlaed8.f.

  subroutine dlaed9 (integer k, integer kstart, integer kstop, integer n, double precision, dimension( * ) d, double precision, dimension( ldq, * ) q, integer ldq, double precision rho, double precision, dimension( * ) dlamda, double precision, dimension( * ) w, double precision, dimension( lds, * ) s, integer lds, integer info)

DLAED9 used by DSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.

Purpose:

     DLAED9 finds the roots of the secular equation, as defined by the
     values in D, Z, and RHO, between KSTART and KSTOP.  It makes the
     appropriate calls to DLAED4 and then stores the new matrix of
     eigenvectors for use in calculating the next level of Z vectors.

Parameters
    K

              K is INTEGER
              The number of terms in the rational function to be solved by
              DLAED4.  K >= 0.

    KSTART

              KSTART is INTEGER

    KSTOP

              KSTOP is INTEGER
              The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
              are to be computed.  1 <= KSTART <= KSTOP <= K.

    N

              N is INTEGER
              The number of rows and columns in the Q matrix.
              N >= K (delation may result in N > K).

    D

              D is DOUBLE PRECISION array, dimension (N)
              D(I) contains the updated eigenvalues
              for KSTART <= I <= KSTOP.

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ,N)

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.  LDQ >= max( 1, N ).

    RHO

              RHO is DOUBLE PRECISION
              The value of the parameter in the rank one update equation.
              RHO >= 0 required.

    DLAMDA

              DLAMDA is DOUBLE PRECISION array, dimension (K)
              The first K elements of this array contain the old roots
              of the deflated updating problem.  These are the poles
              of the secular equation.

    W

              W is DOUBLE PRECISION array, dimension (K)
              The first K elements of this array contain the components
              of the deflation-adjusted updating vector.

    S

              S is DOUBLE PRECISION array, dimension (LDS, K)
              Will contain the eigenvectors of the repaired matrix which
              will be stored for subsequent Z vector calculation and
              multiplied by the previously accumulated eigenvectors
              to update the system.

    LDS

              LDS is INTEGER
              The leading dimension of S.  LDS >= max( 1, K ).

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = 1, an eigenvalue did not converge

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 154 of file dlaed9.f.

  subroutine dlaeda (integer n, integer tlvls, integer curlvl, integer curpbm, integer, dimension( * ) prmptr, integer, dimension( * ) perm, integer, dimension( * ) givptr, integer, dimension( 2, * ) givcol, double precision, dimension( 2, * ) givnum, double precision, dimension( * ) q, integer, dimension( * ) qptr, double precision, dimension( * ) z, double precision, dimension( * ) ztemp, integer info)

DLAEDA used by DSTEDC. Computes the Z vector determining the rank-one modification of the diagonal matrix. Used when the original matrix is dense.

Purpose:

     DLAEDA computes the Z vector corresponding to the merge step in the
     CURLVLth step of the merge process with TLVLS steps for the CURPBMth
     problem.

Parameters
    N

              N is INTEGER
             The dimension of the symmetric tridiagonal matrix.  N >= 0.

    TLVLS

              TLVLS is INTEGER
             The total number of merging levels in the overall divide and
             conquer tree.

    CURLVL

              CURLVL is INTEGER
             The current level in the overall merge routine,
             0 <= curlvl <= tlvls.

    CURPBM

              CURPBM is INTEGER
             The current problem in the current level in the overall
             merge routine (counting from upper left to lower right).

    PRMPTR

              PRMPTR is INTEGER array, dimension (N lg N)
             Contains a list of pointers which indicate where in PERM a
             level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
             indicates the size of the permutation and incidentally the
             size of the full, non-deflated problem.

    PERM

              PERM is INTEGER array, dimension (N lg N)
             Contains the permutations (from deflation and sorting) to be
             applied to each eigenblock.

    GIVPTR

              GIVPTR is INTEGER array, dimension (N lg N)
             Contains a list of pointers which indicate where in GIVCOL a
             level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
             indicates the number of Givens rotations.

    GIVCOL

              GIVCOL is INTEGER array, dimension (2, N lg N)
             Each pair of numbers indicates a pair of columns to take place
             in a Givens rotation.

    GIVNUM

              GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
             Each number indicates the S value to be used in the
             corresponding Givens rotation.

    Q

              Q is DOUBLE PRECISION array, dimension (N**2)
             Contains the square eigenblocks from previous levels, the
             starting positions for blocks are given by QPTR.

    QPTR

              QPTR is INTEGER array, dimension (N+2)
             Contains a list of pointers which indicate where in Q an
             eigenblock is stored.  SQRT( QPTR(i+1) - QPTR(i) ) indicates
             the size of the block.

    Z

              Z is DOUBLE PRECISION array, dimension (N)
             On output this vector contains the updating vector (the last
             row of the first sub-eigenvector matrix and the first row of
             the second sub-eigenvector matrix).

    ZTEMP

              ZTEMP is DOUBLE PRECISION array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 164 of file dlaeda.f.

  subroutine dlagtf (integer n, double precision, dimension( * ) a, double precision lambda, double precision, dimension( * ) b, double precision, dimension( * ) c, double precision tol, double precision, dimension( * ) d, integer, dimension( * ) in, integer info)

DLAGTF computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges.

Purpose:

     DLAGTF factorizes the matrix (T - lambda*I), where T is an n by n
     tridiagonal matrix and lambda is a scalar, as
        T - lambda*I = PLU,
     where P is a permutation matrix, L is a unit lower tridiagonal matrix
     with at most one non-zero sub-diagonal elements per column and U is
     an upper triangular matrix with at most two non-zero super-diagonal
     elements per column.
     The factorization is obtained by Gaussian elimination with partial
     pivoting and implicit row scaling.
     The parameter LAMBDA is included in the routine so that DLAGTF may
     be used, in conjunction with DLAGTS, to obtain eigenvectors of T by
     inverse iteration.

Parameters
    N

              N is INTEGER
              The order of the matrix T.

    A

              A is DOUBLE PRECISION array, dimension (N)
              On entry, A must contain the diagonal elements of T.
              On exit, A is overwritten by the n diagonal elements of the
              upper triangular matrix U of the factorization of T.

    LAMBDA

              LAMBDA is DOUBLE PRECISION
              On entry, the scalar lambda.

    B

              B is DOUBLE PRECISION array, dimension (N-1)
              On entry, B must contain the (n-1) super-diagonal elements of
              T.
              On exit, B is overwritten by the (n-1) super-diagonal
              elements of the matrix U of the factorization of T.

    C

              C is DOUBLE PRECISION array, dimension (N-1)
              On entry, C must contain the (n-1) sub-diagonal elements of
              T.
              On exit, C is overwritten by the (n-1) sub-diagonal elements
              of the matrix L of the factorization of T.

    TOL

              TOL is DOUBLE PRECISION
              On entry, a relative tolerance used to indicate whether or
              not the matrix (T - lambda*I) is nearly singular. TOL should
              normally be chose as approximately the largest relative error
              in the elements of T. For example, if the elements of T are
              correct to about 4 significant figures, then TOL should be
              set to about 5*10**(-4). If TOL is supplied as less than eps,
              where eps is the relative machine precision, then the value
              eps is used in place of TOL.

    D

              D is DOUBLE PRECISION array, dimension (N-2)
              On exit, D is overwritten by the (n-2) second super-diagonal
              elements of the matrix U of the factorization of T.

    IN

              IN is INTEGER array, dimension (N)
              On exit, IN contains details of the permutation matrix P. If
              an interchange occurred at the kth step of the elimination,
              then IN(k) = 1, otherwise IN(k) = 0. The element IN(n)
              returns the smallest positive integer j such that
                 abs( u(j,j) ) <= norm( (T - lambda*I)(j) )*TOL,
              where norm( A(j) ) denotes the sum of the absolute values of
              the jth row of the matrix A. If no such j exists then IN(n)
              is returned as zero. If IN(n) is returned as positive, then a
              diagonal element of U is small, indicating that
              (T - lambda*I) is singular or nearly singular,

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -k, the kth argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 155 of file dlagtf.f.

  subroutine dlamrg (integer n1, integer n2, double precision, dimension( * ) a, integer dtrd1, integer dtrd2, integer, dimension( * ) index)

DLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single set sorted in ascending order.

Purpose:

     DLAMRG will create a permutation list which will merge the elements
     of A (which is composed of two independently sorted sets) into a
     single set which is sorted in ascending order.

Parameters
    N1

              N1 is INTEGER

    N2

              N2 is INTEGER
             These arguments contain the respective lengths of the two
             sorted lists to be merged.

    A

              A is DOUBLE PRECISION array, dimension (N1+N2)
             The first N1 elements of A contain a list of numbers which
             are sorted in either ascending or descending order.  Likewise
             for the final N2 elements.

    DTRD1

              DTRD1 is INTEGER

    DTRD2

              DTRD2 is INTEGER
             These are the strides to be taken through the array A.
             Allowable strides are 1 and -1.  They indicate whether a
             subset of A is sorted in ascending (DTRDx = 1) or descending
             (DTRDx = -1) order.

    INDEX

              INDEX is INTEGER array, dimension (N1+N2)
             On exit this array will contain a permutation such that
             if B( I ) = A( INDEX( I ) ) for I=1,N1+N2, then B will be
             sorted in ascending order.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 98 of file dlamrg.f.

  subroutine dlartgs (double precision x, double precision y, double precision sigma, double precision cs, double precision sn)

DLARTGS generates a plane rotation designed to introduce a bulge in implicit QR iteration for the bidiagonal SVD problem.

Purpose:

     DLARTGS generates a plane rotation designed to introduce a bulge in
     Golub-Reinsch-style implicit QR iteration for the bidiagonal SVD
     problem. X and Y are the top-row entries, and SIGMA is the shift.
     The computed CS and SN define a plane rotation satisfying
        [  CS  SN  ]  .  [ X^2 - SIGMA ]  =  [ R ],
        [ -SN  CS  ]     [    X * Y    ]     [ 0 ]
     with R nonnegative.  If X^2 - SIGMA and X * Y are 0, then the
     rotation is by PI/2.

Parameters
    X

              X is DOUBLE PRECISION
              The (1,1) entry of an upper bidiagonal matrix.

    Y

              Y is DOUBLE PRECISION
              The (1,2) entry of an upper bidiagonal matrix.

    SIGMA

              SIGMA is DOUBLE PRECISION
              The shift.

    CS

              CS is DOUBLE PRECISION
              The cosine of the rotation.

    SN

              SN is DOUBLE PRECISION
              The sine of the rotation.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 89 of file dlartgs.f.

  subroutine dlasq1 (integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) work, integer info)

DLASQ1 computes the singular values of a real square bidiagonal matrix. Used by sbdsqr.

Purpose:

     DLASQ1 computes the singular values of a real N-by-N bidiagonal
     matrix with diagonal D and off-diagonal E. The singular values
     are computed to high relative accuracy, in the absence of
     denormalization, underflow and overflow. The algorithm was first
     presented in
     'Accurate singular values and differential qd algorithms' by K. V.
     Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
     1994,
     and the present implementation is described in 'An implementation of
     the dqds Algorithm (Positive Case)', LAPACK Working Note.

Parameters
    N

              N is INTEGER
            The number of rows and columns in the matrix. N >= 0.

    D

              D is DOUBLE PRECISION array, dimension (N)
            On entry, D contains the diagonal elements of the
            bidiagonal matrix whose SVD is desired. On normal exit,
            D contains the singular values in decreasing order.

    E

              E is DOUBLE PRECISION array, dimension (N)
            On entry, elements E(1:N-1) contain the off-diagonal elements
            of the bidiagonal matrix whose SVD is desired.
            On exit, E is overwritten.

    WORK

              WORK is DOUBLE PRECISION array, dimension (4*N)

    INFO

              INFO is INTEGER
            = 0: successful exit
            < 0: if INFO = -i, the i-th argument had an illegal value
            > 0: the algorithm failed
                 = 1, a split was marked by a positive value in E
                 = 2, current block of Z not diagonalized after 100*N
                      iterations (in inner while loop)  On exit D and E
                      represent a matrix with the same singular values
                      which the calling subroutine could use to finish the
                      computation, or even feed back into DLASQ1
                 = 3, termination criterion of outer while loop not met
                      (program created more than N unreduced blocks)

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 107 of file dlasq1.f.

  subroutine dlasq2 (integer n, double precision, dimension( * ) z, integer info)

DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.

Purpose:

     DLASQ2 computes all the eigenvalues of the symmetric positive
     definite tridiagonal matrix associated with the qd array Z to high
     relative accuracy are computed to high relative accuracy, in the
     absence of denormalization, underflow and overflow.
     To see the relation of Z to the tridiagonal matrix, let L be a
     unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
     let U be an upper bidiagonal matrix with 1's above and diagonal
     Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
     symmetric tridiagonal to which it is similar.
     Note : DLASQ2 defines a logical variable, IEEE, which is true
     on machines which follow ieee-754 floating-point standard in their
     handling of infinities and NaNs, and false otherwise. This variable
     is passed to DLASQ3.

Parameters
    N

              N is INTEGER
            The number of rows and columns in the matrix. N >= 0.

    Z

              Z is DOUBLE PRECISION array, dimension ( 4*N )
            On entry Z holds the qd array. On exit, entries 1 to N hold
            the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
            trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
            N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
            holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
            shifts that failed.

    INFO

              INFO is INTEGER
            = 0: successful exit
            < 0: if the i-th argument is a scalar and had an illegal
                 value, then INFO = -i, if the i-th argument is an
                 array and the j-entry had an illegal value, then
                 INFO = -(i*100+j)
            > 0: the algorithm failed
                  = 1, a split was marked by a positive value in E
                  = 2, current block of Z not diagonalized after 100*N
                       iterations (in inner while loop).  On exit Z holds
                       a qd array with the same eigenvalues as the given Z.
                  = 3, termination criterion of outer while loop not met
                       (program created more than N unreduced blocks)

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      Local Variables: I0:N0 defines a current unreduced segment of Z.
      The shifts are accumulated in SIGMA. Iteration count is in ITER.
      Ping-pong is controlled by PP (alternates between 0 and 1).

Definition at line 111 of file dlasq2.f.

  subroutine dlasq3 (integer i0, integer n0, double precision, dimension( * ) z, integer pp, double precision dmin, double precision sigma, double precision desig, double precision qmax, integer nfail, integer iter, integer ndiv, logical ieee, integer ttype, double precision dmin1, double precision dmin2, double precision dn, double precision dn1, double precision dn2, double precision g, double precision tau)

DLASQ3 checks for deflation, computes a shift and calls dqds. Used by sbdsqr.

Purpose:

     DLASQ3 checks for deflation, computes a shift (TAU) and calls dqds.
     In case of failure it changes shifts, and tries again until output
     is positive.

Parameters
    I0

              I0 is INTEGER
             First index.

    N0

              N0 is INTEGER
             Last index.

    Z

              Z is DOUBLE PRECISION array, dimension ( 4*N0 )
             Z holds the qd array.

    PP

              PP is INTEGER
             PP=0 for ping, PP=1 for pong.
             PP=2 indicates that flipping was applied to the Z array
             and that the initial tests for deflation should not be
             performed.

    DMIN

              DMIN is DOUBLE PRECISION
             Minimum value of d.

    SIGMA

              SIGMA is DOUBLE PRECISION
             Sum of shifts used in current segment.

    DESIG

              DESIG is DOUBLE PRECISION
             Lower order part of SIGMA

    QMAX

              QMAX is DOUBLE PRECISION
             Maximum value of q.

    NFAIL

              NFAIL is INTEGER
             Increment NFAIL by 1 each time the shift was too big.

    ITER

              ITER is INTEGER
             Increment ITER by 1 for each iteration.

    NDIV

              NDIV is INTEGER
             Increment NDIV by 1 for each division.

    IEEE

              IEEE is LOGICAL
             Flag for IEEE or non IEEE arithmetic (passed to DLASQ5).

    TTYPE

              TTYPE is INTEGER
             Shift type.

    DMIN1

              DMIN1 is DOUBLE PRECISION

    DMIN2

              DMIN2 is DOUBLE PRECISION

    DN

              DN is DOUBLE PRECISION

    DN1

              DN1 is DOUBLE PRECISION

    DN2

              DN2 is DOUBLE PRECISION

    G

              G is DOUBLE PRECISION

    TAU

              TAU is DOUBLE PRECISION
             These are passed as arguments in order to save their values
             between calls to DLASQ3.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 179 of file dlasq3.f.

  subroutine dlasq4 (integer i0, integer n0, double precision, dimension( * ) z, integer pp, integer n0in, double precision dmin, double precision dmin1, double precision dmin2, double precision dn, double precision dn1, double precision dn2, double precision tau, integer ttype, double precision g)

DLASQ4 computes an approximation to the smallest eigenvalue using values of d from the previous transform. Used by sbdsqr.

Purpose:

     DLASQ4 computes an approximation TAU to the smallest eigenvalue
     using values of d from the previous transform.

Parameters
    I0

              I0 is INTEGER
            First index.

    N0

              N0 is INTEGER
            Last index.

    Z

              Z is DOUBLE PRECISION array, dimension ( 4*N0 )
            Z holds the qd array.

    PP

              PP is INTEGER
            PP=0 for ping, PP=1 for pong.

    N0IN

              N0IN is INTEGER
            The value of N0 at start of EIGTEST.

    DMIN

              DMIN is DOUBLE PRECISION
            Minimum value of d.

    DMIN1

              DMIN1 is DOUBLE PRECISION
            Minimum value of d, excluding D( N0 ).

    DMIN2

              DMIN2 is DOUBLE PRECISION
            Minimum value of d, excluding D( N0 ) and D( N0-1 ).

    DN

              DN is DOUBLE PRECISION
            d(N)

    DN1

              DN1 is DOUBLE PRECISION
            d(N-1)

    DN2

              DN2 is DOUBLE PRECISION
            d(N-2)

    TAU

              TAU is DOUBLE PRECISION
            This is the shift.

    TTYPE

              TTYPE is INTEGER
            Shift type.

    G

              G is DOUBLE PRECISION
            G is passed as an argument in order to save its value between
            calls to DLASQ4.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      CNST1 = 9/16

Definition at line 149 of file dlasq4.f.

  subroutine dlasq5 (integer i0, integer n0, double precision, dimension( * ) z, integer pp, double precision tau, double precision sigma, double precision dmin, double precision dmin1, double precision dmin2, double precision dn, double precision dnm1, double precision dnm2, logical ieee, double precision eps)

DLASQ5 computes one dqds transform in ping-pong form. Used by sbdsqr and sstegr.

Purpose:

     DLASQ5 computes one dqds transform in ping-pong form, one
     version for IEEE machines another for non IEEE machines.

Parameters
    I0

              I0 is INTEGER
            First index.

    N0

              N0 is INTEGER
            Last index.

    Z

              Z is DOUBLE PRECISION array, dimension ( 4*N )
            Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
            an extra argument.

    PP

              PP is INTEGER
            PP=0 for ping, PP=1 for pong.

    TAU

              TAU is DOUBLE PRECISION
            This is the shift.

    SIGMA

              SIGMA is DOUBLE PRECISION
            This is the accumulated shift up to this step.

    DMIN

              DMIN is DOUBLE PRECISION
            Minimum value of d.

    DMIN1

              DMIN1 is DOUBLE PRECISION
            Minimum value of d, excluding D( N0 ).

    DMIN2

              DMIN2 is DOUBLE PRECISION
            Minimum value of d, excluding D( N0 ) and D( N0-1 ).

    DN

              DN is DOUBLE PRECISION
            d(N0), the last value of d.

    DNM1

              DNM1 is DOUBLE PRECISION
            d(N0-1).

    DNM2

              DNM2 is DOUBLE PRECISION
            d(N0-2).

    IEEE

              IEEE is LOGICAL
            Flag for IEEE or non IEEE arithmetic.

    EPS

              EPS is DOUBLE PRECISION
            This is the value of epsilon used.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 142 of file dlasq5.f.

  subroutine dlasq6 (integer i0, integer n0, double precision, dimension( * ) z, integer pp, double precision dmin, double precision dmin1, double precision dmin2, double precision dn, double precision dnm1, double precision dnm2)

DLASQ6 computes one dqd transform in ping-pong form. Used by sbdsqr and sstegr.

Purpose:

     DLASQ6 computes one dqd (shift equal to zero) transform in
     ping-pong form, with protection against underflow and overflow.

Parameters
    I0

              I0 is INTEGER
            First index.

    N0

              N0 is INTEGER
            Last index.

    Z

              Z is DOUBLE PRECISION array, dimension ( 4*N )
            Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
            an extra argument.

    PP

              PP is INTEGER
            PP=0 for ping, PP=1 for pong.

    DMIN

              DMIN is DOUBLE PRECISION
            Minimum value of d.

    DMIN1

              DMIN1 is DOUBLE PRECISION
            Minimum value of d, excluding D( N0 ).

    DMIN2

              DMIN2 is DOUBLE PRECISION
            Minimum value of d, excluding D( N0 ) and D( N0-1 ).

    DN

              DN is DOUBLE PRECISION
            d(N0), the last value of d.

    DNM1

              DNM1 is DOUBLE PRECISION
            d(N0-1).

    DNM2

              DNM2 is DOUBLE PRECISION
            d(N0-2).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 117 of file dlasq6.f.

  subroutine dlasrt (character id, integer n, double precision, dimension( * ) d, integer info)

DLASRT sorts numbers in increasing or decreasing order.

Purpose:

     Sort the numbers in D in increasing order (if ID = 'I') or
     in decreasing order (if ID = 'D' ).
     Use Quick Sort, reverting to Insertion sort on arrays of
     size <= 20. Dimension of STACK limits N to about 2**32.

Parameters
    ID

              ID is CHARACTER*1
              = 'I': sort D in increasing order;
              = 'D': sort D in decreasing order.

    N

              N is INTEGER
              The length of the array D.

    D

              D is DOUBLE PRECISION array, dimension (N)
              On entry, the array to be sorted.
              On exit, D has been sorted into increasing order
              (D(1) <= ... <= D(N) ) or into decreasing order
              (D(1) >= ... >= D(N) ), depending on ID.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 87 of file dlasrt.f.

  subroutine dstebz (character range, character order, integer n, double precision vl, double precision vu, integer il, integer iu, double precision abstol, double precision, dimension( * ) d, double precision, dimension( * ) e, integer m, integer nsplit, double precision, dimension( * ) w, integer, dimension( * ) iblock, integer, dimension( * ) isplit, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DSTEBZ

Purpose:

     DSTEBZ computes the eigenvalues of a symmetric tridiagonal
     matrix T.  The user may ask for all eigenvalues, all eigenvalues
     in the half-open interval (VL, VU], or the IL-th through IU-th
     eigenvalues.
     To avoid overflow, the matrix must be scaled so that its
     largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
     accuracy, it should not be much smaller than that.
     See W. Kahan 'Accurate Eigenvalues of a Symmetric Tridiagonal
     Matrix', Report CS41, Computer Science Dept., Stanford
     University, July 21, 1966.

Parameters
    RANGE

              RANGE is CHARACTER*1
              = 'A': ('All')   all eigenvalues will be found.
              = 'V': ('Value') all eigenvalues in the half-open interval
                               (VL, VU] will be found.
              = 'I': ('Index') the IL-th through IU-th eigenvalues (of the
                               entire matrix) will be found.

    ORDER

              ORDER is CHARACTER*1
              = 'B': ('By Block') the eigenvalues will be grouped by
                                  split-off block (see IBLOCK, ISPLIT) and
                                  ordered from smallest to largest within
                                  the block.
              = 'E': ('Entire matrix')
                                  the eigenvalues for the entire matrix
                                  will be ordered from smallest to
                                  largest.

    N

              N is INTEGER
              The order of the tridiagonal matrix T.  N >= 0.

    VL

              VL is DOUBLE PRECISION
              If RANGE='V', the lower bound of the interval to
              be searched for eigenvalues.  Eigenvalues less than or equal
              to VL, or greater than VU, will not be returned.  VL < VU.
              Not referenced if RANGE = 'A' or 'I'.

    VU

              VU is DOUBLE PRECISION
              If RANGE='V', the upper bound of the interval to
              be searched for eigenvalues.  Eigenvalues less than or equal
              to VL, or greater than VU, will not be returned.  VL < VU.
              Not referenced if RANGE = 'A' or 'I'.

    IL

              IL is INTEGER
              If RANGE='I', the index of the
              smallest eigenvalue to be returned.
              1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
              Not referenced if RANGE = 'A' or 'V'.

    IU

              IU is INTEGER
              If RANGE='I', the index of the
              largest eigenvalue to be returned.
              1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
              Not referenced if RANGE = 'A' or 'V'.

    ABSTOL

              ABSTOL is DOUBLE PRECISION
              The absolute tolerance for the eigenvalues.  An eigenvalue
              (or cluster) is considered to be located if it has been
              determined to lie in an interval whose width is ABSTOL or
              less.  If ABSTOL is less than or equal to zero, then ULP*|T|
              will be used, where |T| means the 1-norm of T.
              Eigenvalues will be computed most accurately when ABSTOL is
              set to twice the underflow threshold 2*DLAMCH('S'), not zero.

    D

              D is DOUBLE PRECISION array, dimension (N)
              The n diagonal elements of the tridiagonal matrix T.

    E

              E is DOUBLE PRECISION array, dimension (N-1)
              The (n-1) off-diagonal elements of the tridiagonal matrix T.

    M

              M is INTEGER
              The actual number of eigenvalues found. 0 <= M <= N.
              (See also the description of INFO=2,3.)

    NSPLIT

              NSPLIT is INTEGER
              The number of diagonal blocks in the matrix T.
              1 <= NSPLIT <= N.

    W

              W is DOUBLE PRECISION array, dimension (N)
              On exit, the first M elements of W will contain the
              eigenvalues.  (DSTEBZ may use the remaining N-M elements as
              workspace.)

    IBLOCK

              IBLOCK is INTEGER array, dimension (N)
              At each row/column j where E(j) is zero or small, the
              matrix T is considered to split into a block diagonal
              matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which
              block (from 1 to the number of blocks) the eigenvalue W(i)
              belongs.  (DSTEBZ may use the remaining N-M elements as
              workspace.)

    ISPLIT

              ISPLIT is INTEGER array, dimension (N)
              The splitting points, at which T breaks up into submatrices.
              The first submatrix consists of rows/columns 1 to ISPLIT(1),
              the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
              etc., and the NSPLIT-th consists of rows/columns
              ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
              (Only the first NSPLIT elements will actually be used, but
              since the user cannot know a priori what value NSPLIT will
              have, N words must be reserved for ISPLIT.)

    WORK

              WORK is DOUBLE PRECISION array, dimension (4*N)

    IWORK

              IWORK is INTEGER array, dimension (3*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  some or all of the eigenvalues failed to converge or
                    were not computed:
                    =1 or 3: Bisection failed to converge for some
                            eigenvalues; these eigenvalues are flagged by a
                            negative block number.  The effect is that the
                            eigenvalues may not be as accurate as the
                            absolute and relative tolerances.  This is
                            generally caused by unexpectedly inaccurate
                            arithmetic.
                    =2 or 3: RANGE='I' only: Not all of the eigenvalues
                            IL:IU were found.
                            Effect: M < IU+1-IL
                            Cause:  non-monotonic arithmetic, causing the
                                    Sturm sequence to be non-monotonic.
                            Cure:   recalculate, using RANGE='A', and pick
                                    out eigenvalues IL:IU.  In some cases,
                                    increasing the PARAMETER 'FUDGE' may
                                    make things work.
                    = 4:    RANGE='I', and the Gershgorin interval
                            initially used was too small.  No eigenvalues
                            were computed.
                            Probable cause: your machine has sloppy
                                            floating-point arithmetic.
                            Cure: Increase the PARAMETER 'FUDGE',
                                  recompile, and try again.

Internal Parameters:

      RELFAC  DOUBLE PRECISION, default = 2.0e0
              The relative tolerance.  An interval (a,b] lies within
              'relative tolerance' if  b-a < RELFAC*ulp*max(|a|,|b|),
              where 'ulp' is the machine precision (distance from 1 to
              the next larger floating point number.)
      FUDGE   DOUBLE PRECISION, default = 2
              A 'fudge factor' to widen the Gershgorin intervals.  Ideally,
              a value of 1 should work, but on machines with sloppy
              arithmetic, this needs to be larger.  The default for
              publicly released versions should be large enough to handle
              the worst machine around.  Note that this has no effect
              on accuracy of the solution.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 270 of file dstebz.f.

  subroutine dstedc (character compz, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

DSTEDC

Purpose:

     DSTEDC computes all eigenvalues and, optionally, eigenvectors of a
     symmetric tridiagonal matrix using the divide and conquer method.
     The eigenvectors of a full or band real symmetric matrix can also be
     found if DSYTRD or DSPTRD or DSBTRD has been used to reduce this
     matrix to tridiagonal form.
     This code makes very mild assumptions about floating point
     arithmetic. It will work on machines with a guard digit in
     add/subtract, or on those binary machines without guard digits
     which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
     It could conceivably fail on hexadecimal or decimal machines
     without guard digits, but we know of none.  See DLAED3 for details.

Parameters
    COMPZ

              COMPZ is CHARACTER*1
              = 'N':  Compute eigenvalues only.
              = 'I':  Compute eigenvectors of tridiagonal matrix also.
              = 'V':  Compute eigenvectors of original dense symmetric
                      matrix also.  On entry, Z contains the orthogonal
                      matrix used to reduce the original matrix to
                      tridiagonal form.

    N

              N is INTEGER
              The dimension of the symmetric tridiagonal matrix.  N >= 0.

    D

              D is DOUBLE PRECISION array, dimension (N)
              On entry, the diagonal elements of the tridiagonal matrix.
              On exit, if INFO = 0, the eigenvalues in ascending order.

    E

              E is DOUBLE PRECISION array, dimension (N-1)
              On entry, the subdiagonal elements of the tridiagonal matrix.
              On exit, E has been destroyed.

    Z

              Z is DOUBLE PRECISION array, dimension (LDZ,N)
              On entry, if COMPZ = 'V', then Z contains the orthogonal
              matrix used in the reduction to tridiagonal form.
              On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
              orthonormal eigenvectors of the original symmetric matrix,
              and if COMPZ = 'I', Z contains the orthonormal eigenvectors
              of the symmetric tridiagonal matrix.
              If  COMPZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1.
              If eigenvectors are desired, then LDZ >= max(1,N).

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If COMPZ = 'N' or N <= 1 then LWORK must be at least 1.
              If COMPZ = 'V' and N > 1 then LWORK must be at least
                             ( 1 + 3*N + 2*N*lg N + 4*N**2 ),
                             where lg( N ) = smallest integer k such
                             that 2**k >= N.
              If COMPZ = 'I' and N > 1 then LWORK must be at least
                             ( 1 + 4*N + N**2 ).
              Note that for COMPZ = 'I' or 'V', then if N is less than or
              equal to the minimum divide size, usually 25, then LWORK need
              only be max(1,2*(N-1)).
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (MAX(1,LIWORK))
              On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.
              If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1.
              If COMPZ = 'V' and N > 1 then LIWORK must be at least
                             ( 6 + 6*N + 5*N*lg N ).
              If COMPZ = 'I' and N > 1 then LIWORK must be at least
                             ( 3 + 5*N ).
              Note that for COMPZ = 'I' or 'V', then if N is less than or
              equal to the minimum divide size, usually 25, then LIWORK
              need only be 1.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of the IWORK array,
              returns this value as the first entry of the IWORK array, and
              no error message related to LIWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  The algorithm failed to compute an eigenvalue while
                    working on the submatrix lying in rows and columns
                    INFO/(N+1) through mod(INFO,N+1).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
     Modified by Francoise Tisseur, University of Tennessee

Definition at line 186 of file dstedc.f.

  subroutine dsteqr (character compz, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer info)

DSTEQR

Purpose:

     DSTEQR computes all eigenvalues and, optionally, eigenvectors of a
     symmetric tridiagonal matrix using the implicit QL or QR method.
     The eigenvectors of a full or band symmetric matrix can also be found
     if DSYTRD or DSPTRD or DSBTRD has been used to reduce this matrix to
     tridiagonal form.

Parameters
    COMPZ

              COMPZ is CHARACTER*1
              = 'N':  Compute eigenvalues only.
              = 'V':  Compute eigenvalues and eigenvectors of the original
                      symmetric matrix.  On entry, Z must contain the
                      orthogonal matrix used to reduce the original matrix
                      to tridiagonal form.
              = 'I':  Compute eigenvalues and eigenvectors of the
                      tridiagonal matrix.  Z is initialized to the identity
                      matrix.

    N

              N is INTEGER
              The order of the matrix.  N >= 0.

    D

              D is DOUBLE PRECISION array, dimension (N)
              On entry, the diagonal elements of the tridiagonal matrix.
              On exit, if INFO = 0, the eigenvalues in ascending order.

    E

              E is DOUBLE PRECISION array, dimension (N-1)
              On entry, the (n-1) subdiagonal elements of the tridiagonal
              matrix.
              On exit, E has been destroyed.

    Z

              Z is DOUBLE PRECISION array, dimension (LDZ, N)
              On entry, if  COMPZ = 'V', then Z contains the orthogonal
              matrix used in the reduction to tridiagonal form.
              On exit, if INFO = 0, then if  COMPZ = 'V', Z contains the
              orthonormal eigenvectors of the original symmetric matrix,
              and if COMPZ = 'I', Z contains the orthonormal eigenvectors
              of the symmetric tridiagonal matrix.
              If COMPZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              eigenvectors are desired, then  LDZ >= max(1,N).

    WORK

              WORK is DOUBLE PRECISION array, dimension (max(1,2*N-2))
              If COMPZ = 'N', then WORK is not referenced.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  the algorithm has failed to find all the eigenvalues in
                    a total of 30*N iterations; if INFO = i, then i
                    elements of E have not converged to zero; on exit, D
                    and E contain the elements of a symmetric tridiagonal
                    matrix which is orthogonally similar to the original
                    matrix.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 130 of file dsteqr.f.

  subroutine dsterf (integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, integer info)

DSTERF

Purpose:

     DSTERF computes all eigenvalues of a symmetric tridiagonal matrix
     using the Pal-Walker-Kahan variant of the QL or QR algorithm.

Parameters
    N

              N is INTEGER
              The order of the matrix.  N >= 0.

    D

              D is DOUBLE PRECISION array, dimension (N)
              On entry, the n diagonal elements of the tridiagonal matrix.
              On exit, if INFO = 0, the eigenvalues in ascending order.

    E

              E is DOUBLE PRECISION array, dimension (N-1)
              On entry, the (n-1) subdiagonal elements of the tridiagonal
              matrix.
              On exit, E has been destroyed.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  the algorithm failed to find all of the eigenvalues in
                    a total of 30*N iterations; if INFO = i, then i
                    elements of E have not converged to zero.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 85 of file dsterf.f.

  integer function iladiag (character diag)

ILADIAG

Purpose:

     This subroutine translated from a character string specifying if a
     matrix has unit diagonal or not to the relevant BLAST-specified
     integer constant.
     ILADIAG returns an INTEGER.  If ILADIAG < 0, then the input is not a
     character indicating a unit or non-unit diagonal.  Otherwise ILADIAG
     returns the constant value corresponding to DIAG.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 57 of file iladiag.f.

  integer function ilaprec (character prec)

ILAPREC

Purpose:

     This subroutine translated from a character string specifying an
     intermediate precision to the relevant BLAST-specified integer
     constant.
     ILAPREC returns an INTEGER.  If ILAPREC < 0, then the input is not a
     character indicating a supported intermediate precision.  Otherwise
     ILAPREC returns the constant value corresponding to PREC.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 57 of file ilaprec.f.

  integer function ilatrans (character trans)

ILATRANS

Purpose:

     This subroutine translates from a character string specifying a
     transposition operation to the relevant BLAST-specified integer
     constant.
     ILATRANS returns an INTEGER.  If ILATRANS < 0, then the input is not
     a character indicating a transposition operator.  Otherwise ILATRANS
     returns the constant value corresponding to TRANS.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 57 of file ilatrans.f.

  integer function ilauplo (character uplo)

ILAUPLO

Purpose:

     This subroutine translated from a character string specifying a
     upper- or lower-triangular matrix to the relevant BLAST-specified
     integer constant.
     ILAUPLO returns an INTEGER.  If ILAUPLO < 0, then the input is not
     a character indicating an upper- or lower-triangular matrix.
     Otherwise ILAUPLO returns the constant value corresponding to UPLO.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 57 of file ilauplo.f.

  subroutine sbdsdc (character uplo, character compq, integer n, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldvt, * ) vt, integer ldvt, real, dimension( * ) q, integer, dimension( * ) iq, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SBDSDC

Purpose:

     SBDSDC computes the singular value decomposition (SVD) of a real
     N-by-N (upper or lower) bidiagonal matrix B:  B = U * S * VT,
     using a divide and conquer method, where S is a diagonal matrix
     with non-negative diagonal elements (the singular values of B), and
     U and VT are orthogonal matrices of left and right singular vectors,
     respectively. SBDSDC can be used to compute all singular values,
     and optionally, singular vectors or singular vectors in compact form.
     This code makes very mild assumptions about floating point
     arithmetic. It will work on machines with a guard digit in
     add/subtract, or on those binary machines without guard digits
     which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
     It could conceivably fail on hexadecimal or decimal machines
     without guard digits, but we know of none.  See SLASD3 for details.
     The code currently calls SLASDQ if singular values only are desired.
     However, it can be slightly modified to compute singular values
     using the divide and conquer method.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  B is upper bidiagonal.
              = 'L':  B is lower bidiagonal.

    COMPQ

              COMPQ is CHARACTER*1
              Specifies whether singular vectors are to be computed
              as follows:
              = 'N':  Compute singular values only;
              = 'P':  Compute singular values and compute singular
                      vectors in compact form;
              = 'I':  Compute singular values and singular vectors.

    N

              N is INTEGER
              The order of the matrix B.  N >= 0.

    D

              D is REAL array, dimension (N)
              On entry, the n diagonal elements of the bidiagonal matrix B.
              On exit, if INFO=0, the singular values of B.

    E

              E is REAL array, dimension (N-1)
              On entry, the elements of E contain the offdiagonal
              elements of the bidiagonal matrix whose SVD is desired.
              On exit, E has been destroyed.

    U

              U is REAL array, dimension (LDU,N)
              If  COMPQ = 'I', then:
                 On exit, if INFO = 0, U contains the left singular vectors
                 of the bidiagonal matrix.
              For other values of COMPQ, U is not referenced.

    LDU

              LDU is INTEGER
              The leading dimension of the array U.  LDU >= 1.
              If singular vectors are desired, then LDU >= max( 1, N ).

    VT

              VT is REAL array, dimension (LDVT,N)
              If  COMPQ = 'I', then:
                 On exit, if INFO = 0, VT**T contains the right singular
                 vectors of the bidiagonal matrix.
              For other values of COMPQ, VT is not referenced.

    LDVT

              LDVT is INTEGER
              The leading dimension of the array VT.  LDVT >= 1.
              If singular vectors are desired, then LDVT >= max( 1, N ).

    Q

              Q is REAL array, dimension (LDQ)
              If  COMPQ = 'P', then:
                 On exit, if INFO = 0, Q and IQ contain the left
                 and right singular vectors in a compact form,
                 requiring O(N log N) space instead of 2*N**2.
                 In particular, Q contains all the REAL data in
                 LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1))))
                 words of memory, where SMLSIZ is returned by ILAENV and
                 is equal to the maximum size of the subproblems at the
                 bottom of the computation tree (usually about 25).
              For other values of COMPQ, Q is not referenced.

    IQ

              IQ is INTEGER array, dimension (LDIQ)
              If  COMPQ = 'P', then:
                 On exit, if INFO = 0, Q and IQ contain the left
                 and right singular vectors in a compact form,
                 requiring O(N log N) space instead of 2*N**2.
                 In particular, IQ contains all INTEGER data in
                 LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
                 words of memory, where SMLSIZ is returned by ILAENV and
                 is equal to the maximum size of the subproblems at the
                 bottom of the computation tree (usually about 25).
              For other values of COMPQ, IQ is not referenced.

    WORK

              WORK is REAL array, dimension (MAX(1,LWORK))
              If COMPQ = 'N' then LWORK >= (4 * N).
              If COMPQ = 'P' then LWORK >= (6 * N).
              If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).

    IWORK

              IWORK is INTEGER array, dimension (8*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  The algorithm failed to compute a singular value.
                    The update process of divide and conquer failed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 203 of file sbdsdc.f.

  subroutine sbdsqr (character uplo, integer n, integer ncvt, integer nru, integer ncc, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldvt, * ) vt, integer ldvt, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldc, * ) c, integer ldc, real, dimension( * ) work, integer info)

SBDSQR

Purpose:

     SBDSQR computes the singular values and, optionally, the right and/or
     left singular vectors from the singular value decomposition (SVD) of
     a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
     zero-shift QR algorithm.  The SVD of B has the form
        B = Q * S * P**T
     where S is the diagonal matrix of singular values, Q is an orthogonal
     matrix of left singular vectors, and P is an orthogonal matrix of
     right singular vectors.  If left singular vectors are requested, this
     subroutine actually returns U*Q instead of Q, and, if right singular
     vectors are requested, this subroutine returns P**T*VT instead of
     P**T, for given real input matrices U and VT.  When U and VT are the
     orthogonal matrices that reduce a general matrix A to bidiagonal
     form:  A = U*B*VT, as computed by SGEBRD, then
        A = (U*Q) * S * (P**T*VT)
     is the SVD of A.  Optionally, the subroutine may also compute Q**T*C
     for a given real input matrix C.
     See 'Computing  Small Singular Values of Bidiagonal Matrices With
     Guaranteed High Relative Accuracy,' by J. Demmel and W. Kahan,
     LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
     no. 5, pp. 873-912, Sept 1990) and
     'Accurate singular values and differential qd algorithms,' by
     B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
     Department, University of California at Berkeley, July 1992
     for a detailed description of the algorithm.

Parameters
    UPLO

              UPLO is CHARACTER*1
              = 'U':  B is upper bidiagonal;
              = 'L':  B is lower bidiagonal.

    N

              N is INTEGER
              The order of the matrix B.  N >= 0.

    NCVT

              NCVT is INTEGER
              The number of columns of the matrix VT. NCVT >= 0.

    NRU

              NRU is INTEGER
              The number of rows of the matrix U. NRU >= 0.

    NCC

              NCC is INTEGER
              The number of columns of the matrix C. NCC >= 0.

    D

              D is REAL array, dimension (N)
              On entry, the n diagonal elements of the bidiagonal matrix B.
              On exit, if INFO=0, the singular values of B in decreasing
              order.

    E

              E is REAL array, dimension (N-1)
              On entry, the N-1 offdiagonal elements of the bidiagonal
              matrix B.
              On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
              will contain the diagonal and superdiagonal elements of a
              bidiagonal matrix orthogonally equivalent to the one given
              as input.

    VT

              VT is REAL array, dimension (LDVT, NCVT)
              On entry, an N-by-NCVT matrix VT.
              On exit, VT is overwritten by P**T * VT.
              Not referenced if NCVT = 0.

    LDVT

              LDVT is INTEGER
              The leading dimension of the array VT.
              LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.

    U

              U is REAL array, dimension (LDU, N)
              On entry, an NRU-by-N matrix U.
              On exit, U is overwritten by U * Q.
              Not referenced if NRU = 0.

    LDU

              LDU is INTEGER
              The leading dimension of the array U.  LDU >= max(1,NRU).

    C

              C is REAL array, dimension (LDC, NCC)
              On entry, an N-by-NCC matrix C.
              On exit, C is overwritten by Q**T * C.
              Not referenced if NCC = 0.

    LDC

              LDC is INTEGER
              The leading dimension of the array C.
              LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.

    WORK

              WORK is REAL array, dimension (4*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  If INFO = -i, the i-th argument had an illegal value
              > 0:
                 if NCVT = NRU = NCC = 0,
                    = 1, a split was marked by a positive value in E
                    = 2, current block of Z not diagonalized after 30*N
                         iterations (in inner while loop)
                    = 3, termination criterion of outer while loop not met
                         (program created more than N unreduced blocks)
                 else NCVT = NRU = NCC = 0,
                       the algorithm did not converge; D and E contain the
                       elements of a bidiagonal matrix which is orthogonally
                       similar to the input matrix B;  if INFO = i, i
                       elements of E have not converged to zero.

Internal Parameters:

      TOLMUL  REAL, default = max(10,min(100,EPS**(-1/8)))
              TOLMUL controls the convergence criterion of the QR loop.
              If it is positive, TOLMUL*EPS is the desired relative
                 precision in the computed singular values.
              If it is negative, abs(TOLMUL*EPS*sigma_max) is the
                 desired absolute accuracy in the computed singular
                 values (corresponds to relative accuracy
                 abs(TOLMUL*EPS) in the largest singular value.
              abs(TOLMUL) should be between 1 and 1/EPS, and preferably
                 between 10 (for fast convergence) and .1/EPS
                 (for there to be some accuracy in the results).
              Default is to lose at either one eighth or 2 of the
                 available decimal digits in each computed singular value
                 (whichever is smaller).
      MAXITR  INTEGER, default = 6
              MAXITR controls the maximum number of passes of the
              algorithm through its inner loop. The algorithms stops
              (and so fails to converge) if the number of passes
              through the inner loop exceeds MAXITR*N**2.

Note:

      Bug report from Cezary Dendek.
      On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is
      removed since it can overflow pretty easily (for N larger or equal
      than 18,919). We instead use MAXITDIVN = MAXITR*N.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 238 of file sbdsqr.f.

  subroutine sdisna (character job, integer m, integer n, real, dimension( * ) d, real, dimension( * ) sep, integer info)

SDISNA

Purpose:

     SDISNA computes the reciprocal condition numbers for the eigenvectors
     of a real symmetric or complex Hermitian matrix or for the left or
     right singular vectors of a general m-by-n matrix. The reciprocal
     condition number is the 'gap' between the corresponding eigenvalue or
     singular value and the nearest other one.
     The bound on the error, measured by angle in radians, in the I-th
     computed vector is given by
            SLAMCH( 'E' ) * ( ANORM / SEP( I ) )
     where ANORM = 2-norm(A) = max( abs( D(j) ) ).  SEP(I) is not allowed
     to be smaller than SLAMCH( 'E' )*ANORM in order to limit the size of
     the error bound.
     SDISNA may also be used to compute error bounds for eigenvectors of
     the generalized symmetric definite eigenproblem.

Parameters
    JOB

              JOB is CHARACTER*1
              Specifies for which problem the reciprocal condition numbers
              should be computed:
              = 'E':  the eigenvectors of a symmetric/Hermitian matrix;
              = 'L':  the left singular vectors of a general matrix;
              = 'R':  the right singular vectors of a general matrix.

    M

              M is INTEGER
              The number of rows of the matrix. M >= 0.

    N

              N is INTEGER
              If JOB = 'L' or 'R', the number of columns of the matrix,
              in which case N >= 0. Ignored if JOB = 'E'.

    D

              D is REAL array, dimension (M) if JOB = 'E'
                                  dimension (min(M,N)) if JOB = 'L' or 'R'
              The eigenvalues (if JOB = 'E') or singular values (if JOB =
              'L' or 'R') of the matrix, in either increasing or decreasing
              order. If singular values, they must be non-negative.

    SEP

              SEP is REAL array, dimension (M) if JOB = 'E'
                                   dimension (min(M,N)) if JOB = 'L' or 'R'
              The reciprocal condition numbers of the vectors.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 116 of file sdisna.f.

  subroutine slaed0 (integer icompq, integer qsiz, integer n, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldq, * ) q, integer ldq, real, dimension( ldqs, * ) qstore, integer ldqs, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SLAED0 used by SSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method.

Purpose:

     SLAED0 computes all eigenvalues and corresponding eigenvectors of a
     symmetric tridiagonal matrix using the divide and conquer method.

Parameters
    ICOMPQ

              ICOMPQ is INTEGER
              = 0:  Compute eigenvalues only.
              = 1:  Compute eigenvectors of original dense symmetric matrix
                    also.  On entry, Q contains the orthogonal matrix used
                    to reduce the original matrix to tridiagonal form.
              = 2:  Compute eigenvalues and eigenvectors of tridiagonal
                    matrix.

    QSIZ

              QSIZ is INTEGER
             The dimension of the orthogonal matrix used to reduce
             the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

    N

              N is INTEGER
             The dimension of the symmetric tridiagonal matrix.  N >= 0.

    D

              D is REAL array, dimension (N)
             On entry, the main diagonal of the tridiagonal matrix.
             On exit, its eigenvalues.

    E

              E is REAL array, dimension (N-1)
             The off-diagonal elements of the tridiagonal matrix.
             On exit, E has been destroyed.

    Q

              Q is REAL array, dimension (LDQ, N)
             On entry, Q must contain an N-by-N orthogonal matrix.
             If ICOMPQ = 0    Q is not referenced.
             If ICOMPQ = 1    On entry, Q is a subset of the columns of the
                              orthogonal matrix used to reduce the full
                              matrix to tridiagonal form corresponding to
                              the subset of the full matrix which is being
                              decomposed at this time.
             If ICOMPQ = 2    On entry, Q will be the identity matrix.
                              On exit, Q contains the eigenvectors of the
                              tridiagonal matrix.

    LDQ

              LDQ is INTEGER
             The leading dimension of the array Q.  If eigenvectors are
             desired, then  LDQ >= max(1,N).  In any case,  LDQ >= 1.

    QSTORE

              QSTORE is REAL array, dimension (LDQS, N)
             Referenced only when ICOMPQ = 1.  Used to store parts of
             the eigenvector matrix when the updating matrix multiplies
             take place.

    LDQS

              LDQS is INTEGER
             The leading dimension of the array QSTORE.  If ICOMPQ = 1,
             then  LDQS >= max(1,N).  In any case,  LDQS >= 1.

    WORK

              WORK is REAL array,
             If ICOMPQ = 0 or 1, the dimension of WORK must be at least
                         1 + 3*N + 2*N*lg N + 3*N**2
                         ( lg( N ) = smallest integer k
                                     such that 2^k >= N )
             If ICOMPQ = 2, the dimension of WORK must be at least
                         4*N + N**2.

    IWORK

              IWORK is INTEGER array,
             If ICOMPQ = 0 or 1, the dimension of IWORK must be at least
                            6 + 6*N + 5*N*lg N.
                            ( lg( N ) = smallest integer k
                                        such that 2^k >= N )
             If ICOMPQ = 2, the dimension of IWORK must be at least
                            3 + 5*N.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  The algorithm failed to compute an eigenvalue while
                    working on the submatrix lying in rows and columns
                    INFO/(N+1) through mod(INFO,N+1).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 170 of file slaed0.f.

  subroutine slaed1 (integer n, real, dimension( * ) d, real, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, real rho, integer cutpnt, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SLAED1 used by SSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.

Purpose:

     SLAED1 computes the updated eigensystem of a diagonal
     matrix after modification by a rank-one symmetric matrix.  This
     routine is used only for the eigenproblem which requires all
     eigenvalues and eigenvectors of a tridiagonal matrix.  SLAED7 handles
     the case in which eigenvalues only or eigenvalues and eigenvectors
     of a full symmetric matrix (which was reduced to tridiagonal form)
     are desired.
       T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
        where Z = Q**T*u, u is a vector of length N with ones in the
        CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
        The eigenvectors of the original matrix are stored in Q, and the
        eigenvalues are in D.  The algorithm consists of three stages:
           The first stage consists of deflating the size of the problem
           when there are multiple eigenvalues or if there is a zero in
           the Z vector.  For each such occurrence the dimension of the
           secular equation problem is reduced by one.  This stage is
           performed by the routine SLAED2.
           The second stage consists of calculating the updated
           eigenvalues. This is done by finding the roots of the secular
           equation via the routine SLAED4 (as called by SLAED3).
           This routine also calculates the eigenvectors of the current
           problem.
           The final stage consists of computing the updated eigenvectors
           directly using the updated eigenvalues.  The eigenvectors for
           the current problem are multiplied with the eigenvectors from
           the overall problem.

Parameters
    N

              N is INTEGER
             The dimension of the symmetric tridiagonal matrix.  N >= 0.

    D

              D is REAL array, dimension (N)
             On entry, the eigenvalues of the rank-1-perturbed matrix.
             On exit, the eigenvalues of the repaired matrix.

    Q

              Q is REAL array, dimension (LDQ,N)
             On entry, the eigenvectors of the rank-1-perturbed matrix.
             On exit, the eigenvectors of the repaired tridiagonal matrix.

    LDQ

              LDQ is INTEGER
             The leading dimension of the array Q.  LDQ >= max(1,N).

    INDXQ

              INDXQ is INTEGER array, dimension (N)
             On entry, the permutation which separately sorts the two
             subproblems in D into ascending order.
             On exit, the permutation which will reintegrate the
             subproblems back into sorted order,
             i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.

    RHO

              RHO is REAL
             The subdiagonal entry used to create the rank-1 modification.

    CUTPNT

              CUTPNT is INTEGER
             The location of the last eigenvalue in the leading sub-matrix.
             min(1,N) <= CUTPNT <= N/2.

    WORK

              WORK is REAL array, dimension (4*N + N**2)

    IWORK

              IWORK is INTEGER array, dimension (4*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = 1, an eigenvalue did not converge

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
     Modified by Francoise Tisseur, University of Tennessee

Definition at line 161 of file slaed1.f.

  subroutine slaed2 (integer k, integer n, integer n1, real, dimension( * ) d, real, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, real rho, real, dimension( * ) z, real, dimension( * ) dlamda, real, dimension( * ) w, real, dimension( * ) q2, integer, dimension( * ) indx, integer, dimension( * ) indxc, integer, dimension( * ) indxp, integer, dimension( * ) coltyp, integer info)

SLAED2 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.

Purpose:

     SLAED2 merges the two sets of eigenvalues together into a single
     sorted set.  Then it tries to deflate the size of the problem.
     There are two ways in which deflation can occur:  when two or more
     eigenvalues are close together or if there is a tiny entry in the
     Z vector.  For each such occurrence the order of the related secular
     equation problem is reduced by one.

Parameters
    K

              K is INTEGER
             The number of non-deflated eigenvalues, and the order of the
             related secular equation. 0 <= K <=N.

    N

              N is INTEGER
             The dimension of the symmetric tridiagonal matrix.  N >= 0.

    N1

              N1 is INTEGER
             The location of the last eigenvalue in the leading sub-matrix.
             min(1,N) <= N1 <= N/2.

    D

              D is REAL array, dimension (N)
             On entry, D contains the eigenvalues of the two submatrices to
             be combined.
             On exit, D contains the trailing (N-K) updated eigenvalues
             (those which were deflated) sorted into increasing order.

    Q

              Q is REAL array, dimension (LDQ, N)
             On entry, Q contains the eigenvectors of two submatrices in
             the two square blocks with corners at (1,1), (N1,N1)
             and (N1+1, N1+1), (N,N).
             On exit, Q contains the trailing (N-K) updated eigenvectors
             (those which were deflated) in its last N-K columns.

    LDQ

              LDQ is INTEGER
             The leading dimension of the array Q.  LDQ >= max(1,N).

    INDXQ

              INDXQ is INTEGER array, dimension (N)
             The permutation which separately sorts the two sub-problems
             in D into ascending order.  Note that elements in the second
             half of this permutation must first have N1 added to their
             values. Destroyed on exit.

    RHO

              RHO is REAL
             On entry, the off-diagonal element associated with the rank-1
             cut which originally split the two submatrices which are now
             being recombined.
             On exit, RHO has been modified to the value required by
             SLAED3.

    Z

              Z is REAL array, dimension (N)
             On entry, Z contains the updating vector (the last
             row of the first sub-eigenvector matrix and the first row of
             the second sub-eigenvector matrix).
             On exit, the contents of Z have been destroyed by the updating
             process.

    DLAMDA

              DLAMDA is REAL array, dimension (N)
             A copy of the first K eigenvalues which will be used by
             SLAED3 to form the secular equation.

    W

              W is REAL array, dimension (N)
             The first k values of the final deflation-altered z-vector
             which will be passed to SLAED3.

    Q2

              Q2 is REAL array, dimension (N1**2+(N-N1)**2)
             A copy of the first K eigenvectors which will be used by
             SLAED3 in a matrix multiply (SGEMM) to solve for the new
             eigenvectors.

    INDX

              INDX is INTEGER array, dimension (N)
             The permutation used to sort the contents of DLAMDA into
             ascending order.

    INDXC

              INDXC is INTEGER array, dimension (N)
             The permutation used to arrange the columns of the deflated
             Q matrix into three groups:  the first group contains non-zero
             elements only at and above N1, the second contains
             non-zero elements only below N1, and the third is dense.

    INDXP

              INDXP is INTEGER array, dimension (N)
             The permutation used to place deflated values of D at the end
             of the array.  INDXP(1:K) points to the nondeflated D-values
             and INDXP(K+1:N) points to the deflated eigenvalues.

    COLTYP

              COLTYP is INTEGER array, dimension (N)
             During execution, a label which will indicate which of the
             following types a column in the Q2 matrix is:
             1 : non-zero in the upper half only;
             2 : dense;
             3 : non-zero in the lower half only;
             4 : deflated.
             On exit, COLTYP(i) is the number of columns of type i,
             for i=1 to 4 only.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
     Modified by Francoise Tisseur, University of Tennessee

Definition at line 210 of file slaed2.f.

  subroutine slaed3 (integer k, integer n, integer n1, real, dimension( * ) d, real, dimension( ldq, * ) q, integer ldq, real rho, real, dimension( * ) dlamda, real, dimension( * ) q2, integer, dimension( * ) indx, integer, dimension( * ) ctot, real, dimension( * ) w, real, dimension( * ) s, integer info)

SLAED3 used by SSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.

Purpose:

     SLAED3 finds the roots of the secular equation, as defined by the
     values in D, W, and RHO, between 1 and K.  It makes the
     appropriate calls to SLAED4 and then updates the eigenvectors by
     multiplying the matrix of eigenvectors of the pair of eigensystems
     being combined by the matrix of eigenvectors of the K-by-K system
     which is solved here.
     This code makes very mild assumptions about floating point
     arithmetic. It will work on machines with a guard digit in
     add/subtract, or on those binary machines without guard digits
     which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
     It could conceivably fail on hexadecimal or decimal machines
     without guard digits, but we know of none.

Parameters
    K

              K is INTEGER
              The number of terms in the rational function to be solved by
              SLAED4.  K >= 0.

    N

              N is INTEGER
              The number of rows and columns in the Q matrix.
              N >= K (deflation may result in N>K).

    N1

              N1 is INTEGER
              The location of the last eigenvalue in the leading submatrix.
              min(1,N) <= N1 <= N/2.

    D

              D is REAL array, dimension (N)
              D(I) contains the updated eigenvalues for
              1 <= I <= K.

    Q

              Q is REAL array, dimension (LDQ,N)
              Initially the first K columns are used as workspace.
              On output the columns 1 to K contain
              the updated eigenvectors.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.  LDQ >= max(1,N).

    RHO

              RHO is REAL
              The value of the parameter in the rank one update equation.
              RHO >= 0 required.

    DLAMDA

              DLAMDA is REAL array, dimension (K)
              The first K elements of this array contain the old roots
              of the deflated updating problem.  These are the poles
              of the secular equation. May be changed on output by
              having lowest order bit set to zero on Cray X-MP, Cray Y-MP,
              Cray-2, or Cray C-90, as described above.

    Q2

              Q2 is REAL array, dimension (LDQ2*N)
              The first K columns of this matrix contain the non-deflated
              eigenvectors for the split problem.

    INDX

              INDX is INTEGER array, dimension (N)
              The permutation used to arrange the columns of the deflated
              Q matrix into three groups (see SLAED2).
              The rows of the eigenvectors found by SLAED4 must be likewise
              permuted before the matrix multiply can take place.

    CTOT

              CTOT is INTEGER array, dimension (4)
              A count of the total number of the various types of columns
              in Q, as described in INDX.  The fourth column type is any
              column which has been deflated.

    W

              W is REAL array, dimension (K)
              The first K elements of this array contain the components
              of the deflation-adjusted updating vector. Destroyed on
              output.

    S

              S is REAL array, dimension (N1 + 1)*K
              Will contain the eigenvectors of the repaired matrix which
              will be multiplied by the previously accumulated eigenvectors
              to update the system.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = 1, an eigenvalue did not converge

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
     Modified by Francoise Tisseur, University of Tennessee

Definition at line 183 of file slaed3.f.

  subroutine slaed4 (integer n, integer i, real, dimension( * ) d, real, dimension( * ) z, real, dimension( * ) delta, real rho, real dlam, integer info)

SLAED4 used by SSTEDC. Finds a single root of the secular equation.

Purpose:

     This subroutine computes the I-th updated eigenvalue of a symmetric
     rank-one modification to a diagonal matrix whose elements are
     given in the array d, and that
                D(i) < D(j)  for  i < j
     and that RHO > 0.  This is arranged by the calling routine, and is
     no loss in generality.  The rank-one modified system is thus
                diag( D )  +  RHO * Z * Z_transpose.
     where we assume the Euclidean norm of Z is 1.
     The method consists of approximating the rational functions in the
     secular equation by simpler interpolating rational functions.

Parameters
    N

              N is INTEGER
             The length of all arrays.

    I

              I is INTEGER
             The index of the eigenvalue to be computed.  1 <= I <= N.

    D

              D is REAL array, dimension (N)
             The original eigenvalues.  It is assumed that they are in
             order, D(I) < D(J)  for I < J.

    Z

              Z is REAL array, dimension (N)
             The components of the updating vector.

    DELTA

              DELTA is REAL array, dimension (N)
             If N > 2, DELTA contains (D(j) - lambda_I) in its  j-th
             component.  If N = 1, then DELTA(1) = 1. If N = 2, see SLAED5
             for detail. The vector DELTA contains the information necessary
             to construct the eigenvectors by SLAED3 and SLAED9.

    RHO

              RHO is REAL
             The scalar in the symmetric updating formula.

    DLAM

              DLAM is REAL
             The computed lambda_I, the I-th updated eigenvalue.

    INFO

              INFO is INTEGER
             = 0:  successful exit
             > 0:  if INFO = 1, the updating process failed.

Internal Parameters:

      Logical variable ORGATI (origin-at-i?) is used for distinguishing
      whether D(i) or D(i+1) is treated as the origin.
                ORGATI = .true.    origin at i
                ORGATI = .false.   origin at i+1
       Logical variable SWTCH3 (switch-for-3-poles?) is for noting
       if we are working with THREE poles!
       MAXIT is the maximum number of iterations allowed for each
       eigenvalue.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

Definition at line 144 of file slaed4.f.

  subroutine slaed5 (integer i, real, dimension( 2 ) d, real, dimension( 2 ) z, real, dimension( 2 ) delta, real rho, real dlam)

SLAED5 used by SSTEDC. Solves the 2-by-2 secular equation.

Purpose:

     This subroutine computes the I-th eigenvalue of a symmetric rank-one
     modification of a 2-by-2 diagonal matrix
                diag( D )  +  RHO * Z * transpose(Z) .
     The diagonal elements in the array D are assumed to satisfy
                D(i) < D(j)  for  i < j .
     We also assume RHO > 0 and that the Euclidean norm of the vector
     Z is one.

Parameters
    I

              I is INTEGER
             The index of the eigenvalue to be computed.  I = 1 or I = 2.

    D

              D is REAL array, dimension (2)
             The original eigenvalues.  We assume D(1) < D(2).

    Z

              Z is REAL array, dimension (2)
             The components of the updating vector.

    DELTA

              DELTA is REAL array, dimension (2)
             The vector DELTA contains the information necessary
             to construct the eigenvectors.

    RHO

              RHO is REAL
             The scalar in the symmetric updating formula.

    DLAM

              DLAM is REAL
             The computed lambda_I, the I-th updated eigenvalue.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

Definition at line 107 of file slaed5.f.

  subroutine slaed6 (integer kniter, logical orgati, real rho, real, dimension( 3 ) d, real, dimension( 3 ) z, real finit, real tau, integer info)

SLAED6 used by SSTEDC. Computes one Newton step in solution of the secular equation.

Purpose:

     SLAED6 computes the positive or negative root (closest to the origin)
     of
                      z(1)        z(2)        z(3)
     f(x) =   rho + --------- + ---------- + ---------
                     d(1)-x      d(2)-x      d(3)-x
     It is assumed that
           if ORGATI = .true. the root is between d(2) and d(3);
           otherwise it is between d(1) and d(2)
     This routine will be called by SLAED4 when necessary. In most cases,
     the root sought is the smallest in magnitude, though it might not be
     in some extremely rare situations.

Parameters
    KNITER

              KNITER is INTEGER
                   Refer to SLAED4 for its significance.

    ORGATI

              ORGATI is LOGICAL
                   If ORGATI is true, the needed root is between d(2) and
                   d(3); otherwise it is between d(1) and d(2).  See
                   SLAED4 for further details.

    RHO

              RHO is REAL
                   Refer to the equation f(x) above.

    D

              D is REAL array, dimension (3)
                   D satisfies d(1) < d(2) < d(3).

    Z

              Z is REAL array, dimension (3)
                   Each of the elements in z must be positive.

    FINIT

              FINIT is REAL
                   The value of f at 0. It is more accurate than the one
                   evaluated inside this routine (if someone wants to do
                   so).

    TAU

              TAU is REAL
                   The root of the equation f(x).

    INFO

              INFO is INTEGER
                   = 0: successful exit
                   > 0: if INFO = 1, failure to converge

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      10/02/03: This version has a few statements commented out for thread
      safety (machine parameters are computed on each entry). SJH.
      05/10/06: Modified from a new version of Ren-Cang Li, use
         Gragg-Thornton-Warner cubic convergent scheme for better stability.

Contributors:
    Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

Definition at line 139 of file slaed6.f.

  subroutine slaed7 (integer icompq, integer n, integer qsiz, integer tlvls, integer curlvl, integer curpbm, real, dimension( * ) d, real, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, real rho, integer cutpnt, real, dimension( * ) qstore, integer, dimension( * ) qptr, integer, dimension( * ) prmptr, integer, dimension( * ) perm, integer, dimension( * ) givptr, integer, dimension( 2, * ) givcol, real, dimension( 2, * ) givnum, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SLAED7 used by SSTEDC. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.

Purpose:

     SLAED7 computes the updated eigensystem of a diagonal
     matrix after modification by a rank-one symmetric matrix. This
     routine is used only for the eigenproblem which requires all
     eigenvalues and optionally eigenvectors of a dense symmetric matrix
     that has been reduced to tridiagonal form.  SLAED1 handles
     the case in which all eigenvalues and eigenvectors of a symmetric
     tridiagonal matrix are desired.
       T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
        where Z = Q**Tu, u is a vector of length N with ones in the
        CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
        The eigenvectors of the original matrix are stored in Q, and the
        eigenvalues are in D.  The algorithm consists of three stages:
           The first stage consists of deflating the size of the problem
           when there are multiple eigenvalues or if there is a zero in
           the Z vector.  For each such occurrence the dimension of the
           secular equation problem is reduced by one.  This stage is
           performed by the routine SLAED8.
           The second stage consists of calculating the updated
           eigenvalues. This is done by finding the roots of the secular
           equation via the routine SLAED4 (as called by SLAED9).
           This routine also calculates the eigenvectors of the current
           problem.
           The final stage consists of computing the updated eigenvectors
           directly using the updated eigenvalues.  The eigenvectors for
           the current problem are multiplied with the eigenvectors from
           the overall problem.

Parameters
    ICOMPQ

              ICOMPQ is INTEGER
              = 0:  Compute eigenvalues only.
              = 1:  Compute eigenvectors of original dense symmetric matrix
                    also.  On entry, Q contains the orthogonal matrix used
                    to reduce the original matrix to tridiagonal form.

    N

              N is INTEGER
             The dimension of the symmetric tridiagonal matrix.  N >= 0.

    QSIZ

              QSIZ is INTEGER
             The dimension of the orthogonal matrix used to reduce
             the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

    TLVLS

              TLVLS is INTEGER
             The total number of merging levels in the overall divide and
             conquer tree.

    CURLVL

              CURLVL is INTEGER
             The current level in the overall merge routine,
             0 <= CURLVL <= TLVLS.

    CURPBM

              CURPBM is INTEGER
             The current problem in the current level in the overall
             merge routine (counting from upper left to lower right).

    D

              D is REAL array, dimension (N)
             On entry, the eigenvalues of the rank-1-perturbed matrix.
             On exit, the eigenvalues of the repaired matrix.

    Q

              Q is REAL array, dimension (LDQ, N)
             On entry, the eigenvectors of the rank-1-perturbed matrix.
             On exit, the eigenvectors of the repaired tridiagonal matrix.

    LDQ

              LDQ is INTEGER
             The leading dimension of the array Q.  LDQ >= max(1,N).

    INDXQ

              INDXQ is INTEGER array, dimension (N)
             The permutation which will reintegrate the subproblem just
             solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
             will be in ascending order.

    RHO

              RHO is REAL
             The subdiagonal element used to create the rank-1
             modification.

    CUTPNT

              CUTPNT is INTEGER
             Contains the location of the last eigenvalue in the leading
             sub-matrix.  min(1,N) <= CUTPNT <= N.

    QSTORE

              QSTORE is REAL array, dimension (N**2+1)
             Stores eigenvectors of submatrices encountered during
             divide and conquer, packed together. QPTR points to
             beginning of the submatrices.

    QPTR

              QPTR is INTEGER array, dimension (N+2)
             List of indices pointing to beginning of submatrices stored
             in QSTORE. The submatrices are numbered starting at the
             bottom left of the divide and conquer tree, from left to
             right and bottom to top.

    PRMPTR

              PRMPTR is INTEGER array, dimension (N lg N)
             Contains a list of pointers which indicate where in PERM a
             level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
             indicates the size of the permutation and also the size of
             the full, non-deflated problem.

    PERM

              PERM is INTEGER array, dimension (N lg N)
             Contains the permutations (from deflation and sorting) to be
             applied to each eigenblock.

    GIVPTR

              GIVPTR is INTEGER array, dimension (N lg N)
             Contains a list of pointers which indicate where in GIVCOL a
             level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
             indicates the number of Givens rotations.

    GIVCOL

              GIVCOL is INTEGER array, dimension (2, N lg N)
             Each pair of numbers indicates a pair of columns to take place
             in a Givens rotation.

    GIVNUM

              GIVNUM is REAL array, dimension (2, N lg N)
             Each number indicates the S value to be used in the
             corresponding Givens rotation.

    WORK

              WORK is REAL array, dimension (3*N+2*QSIZ*N)

    IWORK

              IWORK is INTEGER array, dimension (4*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = 1, an eigenvalue did not converge

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 256 of file slaed7.f.

  subroutine slaed8 (integer icompq, integer k, integer n, integer qsiz, real, dimension( * ) d, real, dimension( ldq, * ) q, integer ldq, integer, dimension( * ) indxq, real rho, integer cutpnt, real, dimension( * ) z, real, dimension( * ) dlamda, real, dimension( ldq2, * ) q2, integer ldq2, real, dimension( * ) w, integer, dimension( * ) perm, integer givptr, integer, dimension( 2, * ) givcol, real, dimension( 2, * ) givnum, integer, dimension( * ) indxp, integer, dimension( * ) indx, integer info)

SLAED8 used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.

Purpose:

     SLAED8 merges the two sets of eigenvalues together into a single
     sorted set.  Then it tries to deflate the size of the problem.
     There are two ways in which deflation can occur:  when two or more
     eigenvalues are close together or if there is a tiny element in the
     Z vector.  For each such occurrence the order of the related secular
     equation problem is reduced by one.

Parameters
    ICOMPQ

              ICOMPQ is INTEGER
              = 0:  Compute eigenvalues only.
              = 1:  Compute eigenvectors of original dense symmetric matrix
                    also.  On entry, Q contains the orthogonal matrix used
                    to reduce the original matrix to tridiagonal form.

    K

              K is INTEGER
             The number of non-deflated eigenvalues, and the order of the
             related secular equation.

    N

              N is INTEGER
             The dimension of the symmetric tridiagonal matrix.  N >= 0.

    QSIZ

              QSIZ is INTEGER
             The dimension of the orthogonal matrix used to reduce
             the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

    D

              D is REAL array, dimension (N)
             On entry, the eigenvalues of the two submatrices to be
             combined.  On exit, the trailing (N-K) updated eigenvalues
             (those which were deflated) sorted into increasing order.

    Q

              Q is REAL array, dimension (LDQ,N)
             If ICOMPQ = 0, Q is not referenced.  Otherwise,
             on entry, Q contains the eigenvectors of the partially solved
             system which has been previously updated in matrix
             multiplies with other partially solved eigensystems.
             On exit, Q contains the trailing (N-K) updated eigenvectors
             (those which were deflated) in its last N-K columns.

    LDQ

              LDQ is INTEGER
             The leading dimension of the array Q.  LDQ >= max(1,N).

    INDXQ

              INDXQ is INTEGER array, dimension (N)
             The permutation which separately sorts the two sub-problems
             in D into ascending order.  Note that elements in the second
             half of this permutation must first have CUTPNT added to
             their values in order to be accurate.

    RHO

              RHO is REAL
             On entry, the off-diagonal element associated with the rank-1
             cut which originally split the two submatrices which are now
             being recombined.
             On exit, RHO has been modified to the value required by
             SLAED3.

    CUTPNT

              CUTPNT is INTEGER
             The location of the last eigenvalue in the leading
             sub-matrix.  min(1,N) <= CUTPNT <= N.

    Z

              Z is REAL array, dimension (N)
             On entry, Z contains the updating vector (the last row of
             the first sub-eigenvector matrix and the first row of the
             second sub-eigenvector matrix).
             On exit, the contents of Z are destroyed by the updating
             process.

    DLAMDA

              DLAMDA is REAL array, dimension (N)
             A copy of the first K eigenvalues which will be used by
             SLAED3 to form the secular equation.

    Q2

              Q2 is REAL array, dimension (LDQ2,N)
             If ICOMPQ = 0, Q2 is not referenced.  Otherwise,
             a copy of the first K eigenvectors which will be used by
             SLAED7 in a matrix multiply (SGEMM) to update the new
             eigenvectors.

    LDQ2

              LDQ2 is INTEGER
             The leading dimension of the array Q2.  LDQ2 >= max(1,N).

    W

              W is REAL array, dimension (N)
             The first k values of the final deflation-altered z-vector and
             will be passed to SLAED3.

    PERM

              PERM is INTEGER array, dimension (N)
             The permutations (from deflation and sorting) to be applied
             to each eigenblock.

    GIVPTR

              GIVPTR is INTEGER
             The number of Givens rotations which took place in this
             subproblem.

    GIVCOL

              GIVCOL is INTEGER array, dimension (2, N)
             Each pair of numbers indicates a pair of columns to take place
             in a Givens rotation.

    GIVNUM

              GIVNUM is REAL array, dimension (2, N)
             Each number indicates the S value to be used in the
             corresponding Givens rotation.

    INDXP

              INDXP is INTEGER array, dimension (N)
             The permutation used to place deflated values of D at the end
             of the array.  INDXP(1:K) points to the nondeflated D-values
             and INDXP(K+1:N) points to the deflated eigenvalues.

    INDX

              INDX is INTEGER array, dimension (N)
             The permutation used to sort the contents of D into ascending
             order.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 240 of file slaed8.f.

  subroutine slaed9 (integer k, integer kstart, integer kstop, integer n, real, dimension( * ) d, real, dimension( ldq, * ) q, integer ldq, real rho, real, dimension( * ) dlamda, real, dimension( * ) w, real, dimension( lds, * ) s, integer lds, integer info)

SLAED9 used by SSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.

Purpose:

     SLAED9 finds the roots of the secular equation, as defined by the
     values in D, Z, and RHO, between KSTART and KSTOP.  It makes the
     appropriate calls to SLAED4 and then stores the new matrix of
     eigenvectors for use in calculating the next level of Z vectors.

Parameters
    K

              K is INTEGER
              The number of terms in the rational function to be solved by
              SLAED4.  K >= 0.

    KSTART

              KSTART is INTEGER

    KSTOP

              KSTOP is INTEGER
              The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
              are to be computed.  1 <= KSTART <= KSTOP <= K.

    N

              N is INTEGER
              The number of rows and columns in the Q matrix.
              N >= K (delation may result in N > K).

    D

              D is REAL array, dimension (N)
              D(I) contains the updated eigenvalues
              for KSTART <= I <= KSTOP.

    Q

              Q is REAL array, dimension (LDQ,N)

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.  LDQ >= max( 1, N ).

    RHO

              RHO is REAL
              The value of the parameter in the rank one update equation.
              RHO >= 0 required.

    DLAMDA

              DLAMDA is REAL array, dimension (K)
              The first K elements of this array contain the old roots
              of the deflated updating problem.  These are the poles
              of the secular equation.

    W

              W is REAL array, dimension (K)
              The first K elements of this array contain the components
              of the deflation-adjusted updating vector.

    S

              S is REAL array, dimension (LDS, K)
              Will contain the eigenvectors of the repaired matrix which
              will be stored for subsequent Z vector calculation and
              multiplied by the previously accumulated eigenvectors
              to update the system.

    LDS

              LDS is INTEGER
              The leading dimension of S.  LDS >= max( 1, K ).

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = 1, an eigenvalue did not converge

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 154 of file slaed9.f.

  subroutine slaeda (integer n, integer tlvls, integer curlvl, integer curpbm, integer, dimension( * ) prmptr, integer, dimension( * ) perm, integer, dimension( * ) givptr, integer, dimension( 2, * ) givcol, real, dimension( 2, * ) givnum, real, dimension( * ) q, integer, dimension( * ) qptr, real, dimension( * ) z, real, dimension( * ) ztemp, integer info)

SLAEDA used by SSTEDC. Computes the Z vector determining the rank-one modification of the diagonal matrix. Used when the original matrix is dense.

Purpose:

     SLAEDA computes the Z vector corresponding to the merge step in the
     CURLVLth step of the merge process with TLVLS steps for the CURPBMth
     problem.

Parameters
    N

              N is INTEGER
             The dimension of the symmetric tridiagonal matrix.  N >= 0.

    TLVLS

              TLVLS is INTEGER
             The total number of merging levels in the overall divide and
             conquer tree.

    CURLVL

              CURLVL is INTEGER
             The current level in the overall merge routine,
             0 <= curlvl <= tlvls.

    CURPBM

              CURPBM is INTEGER
             The current problem in the current level in the overall
             merge routine (counting from upper left to lower right).

    PRMPTR

              PRMPTR is INTEGER array, dimension (N lg N)
             Contains a list of pointers which indicate where in PERM a
             level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
             indicates the size of the permutation and incidentally the
             size of the full, non-deflated problem.

    PERM

              PERM is INTEGER array, dimension (N lg N)
             Contains the permutations (from deflation and sorting) to be
             applied to each eigenblock.

    GIVPTR

              GIVPTR is INTEGER array, dimension (N lg N)
             Contains a list of pointers which indicate where in GIVCOL a
             level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
             indicates the number of Givens rotations.

    GIVCOL

              GIVCOL is INTEGER array, dimension (2, N lg N)
             Each pair of numbers indicates a pair of columns to take place
             in a Givens rotation.

    GIVNUM

              GIVNUM is REAL array, dimension (2, N lg N)
             Each number indicates the S value to be used in the
             corresponding Givens rotation.

    Q

              Q is REAL array, dimension (N**2)
             Contains the square eigenblocks from previous levels, the
             starting positions for blocks are given by QPTR.

    QPTR

              QPTR is INTEGER array, dimension (N+2)
             Contains a list of pointers which indicate where in Q an
             eigenblock is stored.  SQRT( QPTR(i+1) - QPTR(i) ) indicates
             the size of the block.

    Z

              Z is REAL array, dimension (N)
             On output this vector contains the updating vector (the last
             row of the first sub-eigenvector matrix and the first row of
             the second sub-eigenvector matrix).

    ZTEMP

              ZTEMP is REAL array, dimension (N)

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

Definition at line 164 of file slaeda.f.

  subroutine slagtf (integer n, real, dimension( * ) a, real lambda, real, dimension( * ) b, real, dimension( * ) c, real tol, real, dimension( * ) d, integer, dimension( * ) in, integer info)

SLAGTF computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges.

Purpose:

     SLAGTF factorizes the matrix (T - lambda*I), where T is an n by n
     tridiagonal matrix and lambda is a scalar, as
        T - lambda*I = PLU,
     where P is a permutation matrix, L is a unit lower tridiagonal matrix
     with at most one non-zero sub-diagonal elements per column and U is
     an upper triangular matrix with at most two non-zero super-diagonal
     elements per column.
     The factorization is obtained by Gaussian elimination with partial
     pivoting and implicit row scaling.
     The parameter LAMBDA is included in the routine so that SLAGTF may
     be used, in conjunction with SLAGTS, to obtain eigenvectors of T by
     inverse iteration.

Parameters
    N

              N is INTEGER
              The order of the matrix T.

    A

              A is REAL array, dimension (N)
              On entry, A must contain the diagonal elements of T.
              On exit, A is overwritten by the n diagonal elements of the
              upper triangular matrix U of the factorization of T.

    LAMBDA

              LAMBDA is REAL
              On entry, the scalar lambda.

    B

              B is REAL array, dimension (N-1)
              On entry, B must contain the (n-1) super-diagonal elements of
              T.
              On exit, B is overwritten by the (n-1) super-diagonal
              elements of the matrix U of the factorization of T.

    C

              C is REAL array, dimension (N-1)
              On entry, C must contain the (n-1) sub-diagonal elements of
              T.
              On exit, C is overwritten by the (n-1) sub-diagonal elements
              of the matrix L of the factorization of T.

    TOL

              TOL is REAL
              On entry, a relative tolerance used to indicate whether or
              not the matrix (T - lambda*I) is nearly singular. TOL should
              normally be chose as approximately the largest relative error
              in the elements of T. For example, if the elements of T are
              correct to about 4 significant figures, then TOL should be
              set to about 5*10**(-4). If TOL is supplied as less than eps,
              where eps is the relative machine precision, then the value
              eps is used in place of TOL.

    D

              D is REAL array, dimension (N-2)
              On exit, D is overwritten by the (n-2) second super-diagonal
              elements of the matrix U of the factorization of T.

    IN

              IN is INTEGER array, dimension (N)
              On exit, IN contains details of the permutation matrix P. If
              an interchange occurred at the kth step of the elimination,
              then IN(k) = 1, otherwise IN(k) = 0. The element IN(n)
              returns the smallest positive integer j such that
                 abs( u(j,j) ) <= norm( (T - lambda*I)(j) )*TOL,
              where norm( A(j) ) denotes the sum of the absolute values of
              the jth row of the matrix A. If no such j exists then IN(n)
              is returned as zero. If IN(n) is returned as positive, then a
              diagonal element of U is small, indicating that
              (T - lambda*I) is singular or nearly singular,

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -k, the kth argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 155 of file slagtf.f.

  subroutine slamrg (integer n1, integer n2, real, dimension( * ) a, integer strd1, integer strd2, integer, dimension( * ) index)

SLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single set sorted in ascending order.

Purpose:

     SLAMRG will create a permutation list which will merge the elements
     of A (which is composed of two independently sorted sets) into a
     single set which is sorted in ascending order.

Parameters
    N1

              N1 is INTEGER

    N2

              N2 is INTEGER
             These arguments contain the respective lengths of the two
             sorted lists to be merged.

    A

              A is REAL array, dimension (N1+N2)
             The first N1 elements of A contain a list of numbers which
             are sorted in either ascending or descending order.  Likewise
             for the final N2 elements.

    STRD1

              STRD1 is INTEGER

    STRD2

              STRD2 is INTEGER
             These are the strides to be taken through the array A.
             Allowable strides are 1 and -1.  They indicate whether a
             subset of A is sorted in ascending (STRDx = 1) or descending
             (STRDx = -1) order.

    INDEX

              INDEX is INTEGER array, dimension (N1+N2)
             On exit this array will contain a permutation such that
             if B( I ) = A( INDEX( I ) ) for I=1,N1+N2, then B will be
             sorted in ascending order.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 98 of file slamrg.f.

  subroutine slartgs (real x, real y, real sigma, real cs, real sn)

SLARTGS generates a plane rotation designed to introduce a bulge in implicit QR iteration for the bidiagonal SVD problem.

Purpose:

     SLARTGS generates a plane rotation designed to introduce a bulge in
     Golub-Reinsch-style implicit QR iteration for the bidiagonal SVD
     problem. X and Y are the top-row entries, and SIGMA is the shift.
     The computed CS and SN define a plane rotation satisfying
        [  CS  SN  ]  .  [ X^2 - SIGMA ]  =  [ R ],
        [ -SN  CS  ]     [    X * Y    ]     [ 0 ]
     with R nonnegative.  If X^2 - SIGMA and X * Y are 0, then the
     rotation is by PI/2.

Parameters
    X

              X is REAL
              The (1,1) entry of an upper bidiagonal matrix.

    Y

              Y is REAL
              The (1,2) entry of an upper bidiagonal matrix.

    SIGMA

              SIGMA is REAL
              The shift.

    CS

              CS is REAL
              The cosine of the rotation.

    SN

              SN is REAL
              The sine of the rotation.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 89 of file slartgs.f.

  subroutine slasq1 (integer n, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) work, integer info)

SLASQ1 computes the singular values of a real square bidiagonal matrix. Used by sbdsqr.

Purpose:

     SLASQ1 computes the singular values of a real N-by-N bidiagonal
     matrix with diagonal D and off-diagonal E. The singular values
     are computed to high relative accuracy, in the absence of
     denormalization, underflow and overflow. The algorithm was first
     presented in
     'Accurate singular values and differential qd algorithms' by K. V.
     Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
     1994,
     and the present implementation is described in 'An implementation of
     the dqds Algorithm (Positive Case)', LAPACK Working Note.

Parameters
    N

              N is INTEGER
            The number of rows and columns in the matrix. N >= 0.

    D

              D is REAL array, dimension (N)
            On entry, D contains the diagonal elements of the
            bidiagonal matrix whose SVD is desired. On normal exit,
            D contains the singular values in decreasing order.

    E

              E is REAL array, dimension (N)
            On entry, elements E(1:N-1) contain the off-diagonal elements
            of the bidiagonal matrix whose SVD is desired.
            On exit, E is overwritten.

    WORK

              WORK is REAL array, dimension (4*N)

    INFO

              INFO is INTEGER
            = 0: successful exit
            < 0: if INFO = -i, the i-th argument had an illegal value
            > 0: the algorithm failed
                 = 1, a split was marked by a positive value in E
                 = 2, current block of Z not diagonalized after 100*N
                      iterations (in inner while loop)  On exit D and E
                      represent a matrix with the same singular values
                      which the calling subroutine could use to finish the
                      computation, or even feed back into SLASQ1
                 = 3, termination criterion of outer while loop not met
                      (program created more than N unreduced blocks)

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 107 of file slasq1.f.

  subroutine slasq2 (integer n, real, dimension( * ) z, integer info)

SLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.

Purpose:

     SLASQ2 computes all the eigenvalues of the symmetric positive
     definite tridiagonal matrix associated with the qd array Z to high
     relative accuracy are computed to high relative accuracy, in the
     absence of denormalization, underflow and overflow.
     To see the relation of Z to the tridiagonal matrix, let L be a
     unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
     let U be an upper bidiagonal matrix with 1's above and diagonal
     Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
     symmetric tridiagonal to which it is similar.
     Note : SLASQ2 defines a logical variable, IEEE, which is true
     on machines which follow ieee-754 floating-point standard in their
     handling of infinities and NaNs, and false otherwise. This variable
     is passed to SLASQ3.

Parameters
    N

              N is INTEGER
            The number of rows and columns in the matrix. N >= 0.

    Z

              Z is REAL array, dimension ( 4*N )
            On entry Z holds the qd array. On exit, entries 1 to N hold
            the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
            trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
            N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
            holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
            shifts that failed.

    INFO

              INFO is INTEGER
            = 0: successful exit
            < 0: if the i-th argument is a scalar and had an illegal
                 value, then INFO = -i, if the i-th argument is an
                 array and the j-entry had an illegal value, then
                 INFO = -(i*100+j)
            > 0: the algorithm failed
                  = 1, a split was marked by a positive value in E
                  = 2, current block of Z not diagonalized after 100*N
                       iterations (in inner while loop).  On exit Z holds
                       a qd array with the same eigenvalues as the given Z.
                  = 3, termination criterion of outer while loop not met
                       (program created more than N unreduced blocks)

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      Local Variables: I0:N0 defines a current unreduced segment of Z.
      The shifts are accumulated in SIGMA. Iteration count is in ITER.
      Ping-pong is controlled by PP (alternates between 0 and 1).

Definition at line 111 of file slasq2.f.

  subroutine slasq3 (integer i0, integer n0, real, dimension( * ) z, integer pp, real dmin, real sigma, real desig, real qmax, integer nfail, integer iter, integer ndiv, logical ieee, integer ttype, real dmin1, real dmin2, real dn, real dn1, real dn2, real g, real tau)

SLASQ3 checks for deflation, computes a shift and calls dqds. Used by sbdsqr.

Purpose:

     SLASQ3 checks for deflation, computes a shift (TAU) and calls dqds.
     In case of failure it changes shifts, and tries again until output
     is positive.

Parameters
    I0

              I0 is INTEGER
             First index.

    N0

              N0 is INTEGER
             Last index.

    Z

              Z is REAL array, dimension ( 4*N0 )
             Z holds the qd array.

    PP

              PP is INTEGER
             PP=0 for ping, PP=1 for pong.
             PP=2 indicates that flipping was applied to the Z array
             and that the initial tests for deflation should not be
             performed.

    DMIN

              DMIN is REAL
             Minimum value of d.

    SIGMA

              SIGMA is REAL
             Sum of shifts used in current segment.

    DESIG

              DESIG is REAL
             Lower order part of SIGMA

    QMAX

              QMAX is REAL
             Maximum value of q.

    NFAIL

              NFAIL is INTEGER
             Increment NFAIL by 1 each time the shift was too big.

    ITER

              ITER is INTEGER
             Increment ITER by 1 for each iteration.

    NDIV

              NDIV is INTEGER
             Increment NDIV by 1 for each division.

    IEEE

              IEEE is LOGICAL
             Flag for IEEE or non IEEE arithmetic (passed to SLASQ5).

    TTYPE

              TTYPE is INTEGER
             Shift type.

    DMIN1

              DMIN1 is REAL

    DMIN2

              DMIN2 is REAL

    DN

              DN is REAL

    DN1

              DN1 is REAL

    DN2

              DN2 is REAL

    G

              G is REAL

    TAU

              TAU is REAL
             These are passed as arguments in order to save their values
             between calls to SLASQ3.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 179 of file slasq3.f.

  subroutine slasq4 (integer i0, integer n0, real, dimension( * ) z, integer pp, integer n0in, real dmin, real dmin1, real dmin2, real dn, real dn1, real dn2, real tau, integer ttype, real g)

SLASQ4 computes an approximation to the smallest eigenvalue using values of d from the previous transform. Used by sbdsqr.

Purpose:

     SLASQ4 computes an approximation TAU to the smallest eigenvalue
     using values of d from the previous transform.

Parameters
    I0

              I0 is INTEGER
            First index.

    N0

              N0 is INTEGER
            Last index.

    Z

              Z is REAL array, dimension ( 4*N0 )
            Z holds the qd array.

    PP

              PP is INTEGER
            PP=0 for ping, PP=1 for pong.

    N0IN

              N0IN is INTEGER
            The value of N0 at start of EIGTEST.

    DMIN

              DMIN is REAL
            Minimum value of d.

    DMIN1

              DMIN1 is REAL
            Minimum value of d, excluding D( N0 ).

    DMIN2

              DMIN2 is REAL
            Minimum value of d, excluding D( N0 ) and D( N0-1 ).

    DN

              DN is REAL
            d(N)

    DN1

              DN1 is REAL
            d(N-1)

    DN2

              DN2 is REAL
            d(N-2)

    TAU

              TAU is REAL
            This is the shift.

    TTYPE

              TTYPE is INTEGER
            Shift type.

    G

              G is REAL
            G is passed as an argument in order to save its value between
            calls to SLASQ4.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      CNST1 = 9/16

Definition at line 149 of file slasq4.f.

  subroutine slasq5 (integer i0, integer n0, real, dimension( * ) z, integer pp, real tau, real sigma, real dmin, real dmin1, real dmin2, real dn, real dnm1, real dnm2, logical ieee, real eps)

SLASQ5 computes one dqds transform in ping-pong form. Used by sbdsqr and sstegr.

Purpose:

     SLASQ5 computes one dqds transform in ping-pong form, one
     version for IEEE machines another for non IEEE machines.

Parameters
    I0

              I0 is INTEGER
            First index.

    N0

              N0 is INTEGER
            Last index.

    Z

              Z is REAL array, dimension ( 4*N )
            Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
            an extra argument.

    PP

              PP is INTEGER
            PP=0 for ping, PP=1 for pong.

    TAU

              TAU is REAL
            This is the shift.

    SIGMA

              SIGMA is REAL
            This is the accumulated shift up to this step.

    DMIN

              DMIN is REAL
            Minimum value of d.

    DMIN1

              DMIN1 is REAL
            Minimum value of d, excluding D( N0 ).

    DMIN2

              DMIN2 is REAL
            Minimum value of d, excluding D( N0 ) and D( N0-1 ).

    DN

              DN is REAL
            d(N0), the last value of d.

    DNM1

              DNM1 is REAL
            d(N0-1).

    DNM2

              DNM2 is REAL
            d(N0-2).

    IEEE

              IEEE is LOGICAL
            Flag for IEEE or non IEEE arithmetic.

    EPS

             EPS is REAL
            This is the value of epsilon used.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 142 of file slasq5.f.

  subroutine slasq6 (integer i0, integer n0, real, dimension( * ) z, integer pp, real dmin, real dmin1, real dmin2, real dn, real dnm1, real dnm2)

SLASQ6 computes one dqd transform in ping-pong form. Used by sbdsqr and sstegr.

Purpose:

     SLASQ6 computes one dqd (shift equal to zero) transform in
     ping-pong form, with protection against underflow and overflow.

Parameters
    I0

              I0 is INTEGER
            First index.

    N0

              N0 is INTEGER
            Last index.

    Z

              Z is REAL array, dimension ( 4*N )
            Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
            an extra argument.

    PP

              PP is INTEGER
            PP=0 for ping, PP=1 for pong.

    DMIN

              DMIN is REAL
            Minimum value of d.

    DMIN1

              DMIN1 is REAL
            Minimum value of d, excluding D( N0 ).

    DMIN2

              DMIN2 is REAL
            Minimum value of d, excluding D( N0 ) and D( N0-1 ).

    DN

              DN is REAL
            d(N0), the last value of d.

    DNM1

              DNM1 is REAL
            d(N0-1).

    DNM2

              DNM2 is REAL
            d(N0-2).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 117 of file slasq6.f.

  subroutine slasrt (character id, integer n, real, dimension( * ) d, integer info)

SLASRT sorts numbers in increasing or decreasing order.

Purpose:

     Sort the numbers in D in increasing order (if ID = 'I') or
     in decreasing order (if ID = 'D' ).
     Use Quick Sort, reverting to Insertion sort on arrays of
     size <= 20. Dimension of STACK limits N to about 2**32.

Parameters
    ID

              ID is CHARACTER*1
              = 'I': sort D in increasing order;
              = 'D': sort D in decreasing order.

    N

              N is INTEGER
              The length of the array D.

    D

              D is REAL array, dimension (N)
              On entry, the array to be sorted.
              On exit, D has been sorted into increasing order
              (D(1) <= ... <= D(N) ) or into decreasing order
              (D(1) >= ... >= D(N) ), depending on ID.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 87 of file slasrt.f.

  subroutine spttrf (integer n, real, dimension( * ) d, real, dimension( * ) e, integer info)

SPTTRF

Purpose:

     SPTTRF computes the L*D*L**T factorization of a real symmetric
     positive definite tridiagonal matrix A.  The factorization may also
     be regarded as having the form A = U**T*D*U.

Parameters
    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    D

              D is REAL array, dimension (N)
              On entry, the n diagonal elements of the tridiagonal matrix
              A.  On exit, the n diagonal elements of the diagonal matrix
              D from the L*D*L**T factorization of A.

    E

              E is REAL array, dimension (N-1)
              On entry, the (n-1) subdiagonal elements of the tridiagonal
              matrix A.  On exit, the (n-1) subdiagonal elements of the
              unit bidiagonal factor L from the L*D*L**T factorization of A.
              E can also be regarded as the superdiagonal of the unit
              bidiagonal factor U from the U**T*D*U factorization of A.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -k, the k-th argument had an illegal value
              > 0: if INFO = k, the leading minor of order k is not
                   positive definite; if k < N, the factorization could not
                   be completed, while if k = N, the factorization was
                   completed, but D(N) <= 0.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 90 of file spttrf.f.

  subroutine sstebz (character range, character order, integer n, real vl, real vu, integer il, integer iu, real abstol, real, dimension( * ) d, real, dimension( * ) e, integer m, integer nsplit, real, dimension( * ) w, integer, dimension( * ) iblock, integer, dimension( * ) isplit, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SSTEBZ

Purpose:

     SSTEBZ computes the eigenvalues of a symmetric tridiagonal
     matrix T.  The user may ask for all eigenvalues, all eigenvalues
     in the half-open interval (VL, VU], or the IL-th through IU-th
     eigenvalues.
     To avoid overflow, the matrix must be scaled so that its
     largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
     accuracy, it should not be much smaller than that.
     See W. Kahan 'Accurate Eigenvalues of a Symmetric Tridiagonal
     Matrix', Report CS41, Computer Science Dept., Stanford
     University, July 21, 1966.

Parameters
    RANGE

              RANGE is CHARACTER*1
              = 'A': ('All')   all eigenvalues will be found.
              = 'V': ('Value') all eigenvalues in the half-open interval
                               (VL, VU] will be found.
              = 'I': ('Index') the IL-th through IU-th eigenvalues (of the
                               entire matrix) will be found.

    ORDER

              ORDER is CHARACTER*1
              = 'B': ('By Block') the eigenvalues will be grouped by
                                  split-off block (see IBLOCK, ISPLIT) and
                                  ordered from smallest to largest within
                                  the block.
              = 'E': ('Entire matrix')
                                  the eigenvalues for the entire matrix
                                  will be ordered from smallest to
                                  largest.

    N

              N is INTEGER
              The order of the tridiagonal matrix T.  N >= 0.

    VL

              VL is REAL
              If RANGE='V', the lower bound of the interval to
              be searched for eigenvalues.  Eigenvalues less than or equal
              to VL, or greater than VU, will not be returned.  VL < VU.
              Not referenced if RANGE = 'A' or 'I'.

    VU

              VU is REAL
              If RANGE='V', the upper bound of the interval to
              be searched for eigenvalues.  Eigenvalues less than or equal
              to VL, or greater than VU, will not be returned.  VL < VU.
              Not referenced if RANGE = 'A' or 'I'.

    IL

              IL is INTEGER
              If RANGE='I', the index of the
              smallest eigenvalue to be returned.
              1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
              Not referenced if RANGE = 'A' or 'V'.

    IU

              IU is INTEGER
              If RANGE='I', the index of the
              largest eigenvalue to be returned.
              1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
              Not referenced if RANGE = 'A' or 'V'.

    ABSTOL

              ABSTOL is REAL
              The absolute tolerance for the eigenvalues.  An eigenvalue
              (or cluster) is considered to be located if it has been
              determined to lie in an interval whose width is ABSTOL or
              less.  If ABSTOL is less than or equal to zero, then ULP*|T|
              will be used, where |T| means the 1-norm of T.
              Eigenvalues will be computed most accurately when ABSTOL is
              set to twice the underflow threshold 2*SLAMCH('S'), not zero.

    D

              D is REAL array, dimension (N)
              The n diagonal elements of the tridiagonal matrix T.

    E

              E is REAL array, dimension (N-1)
              The (n-1) off-diagonal elements of the tridiagonal matrix T.

    M

              M is INTEGER
              The actual number of eigenvalues found. 0 <= M <= N.
              (See also the description of INFO=2,3.)

    NSPLIT

              NSPLIT is INTEGER
              The number of diagonal blocks in the matrix T.
              1 <= NSPLIT <= N.

    W

              W is REAL array, dimension (N)
              On exit, the first M elements of W will contain the
              eigenvalues.  (SSTEBZ may use the remaining N-M elements as
              workspace.)

    IBLOCK

              IBLOCK is INTEGER array, dimension (N)
              At each row/column j where E(j) is zero or small, the
              matrix T is considered to split into a block diagonal
              matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which
              block (from 1 to the number of blocks) the eigenvalue W(i)
              belongs.  (SSTEBZ may use the remaining N-M elements as
              workspace.)

    ISPLIT

              ISPLIT is INTEGER array, dimension (N)
              The splitting points, at which T breaks up into submatrices.
              The first submatrix consists of rows/columns 1 to ISPLIT(1),
              the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
              etc., and the NSPLIT-th consists of rows/columns
              ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
              (Only the first NSPLIT elements will actually be used, but
              since the user cannot know a priori what value NSPLIT will
              have, N words must be reserved for ISPLIT.)

    WORK

              WORK is REAL array, dimension (4*N)

    IWORK

              IWORK is INTEGER array, dimension (3*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  some or all of the eigenvalues failed to converge or
                    were not computed:
                    =1 or 3: Bisection failed to converge for some
                            eigenvalues; these eigenvalues are flagged by a
                            negative block number.  The effect is that the
                            eigenvalues may not be as accurate as the
                            absolute and relative tolerances.  This is
                            generally caused by unexpectedly inaccurate
                            arithmetic.
                    =2 or 3: RANGE='I' only: Not all of the eigenvalues
                            IL:IU were found.
                            Effect: M < IU+1-IL
                            Cause:  non-monotonic arithmetic, causing the
                                    Sturm sequence to be non-monotonic.
                            Cure:   recalculate, using RANGE='A', and pick
                                    out eigenvalues IL:IU.  In some cases,
                                    increasing the PARAMETER 'FUDGE' may
                                    make things work.
                    = 4:    RANGE='I', and the Gershgorin interval
                            initially used was too small.  No eigenvalues
                            were computed.
                            Probable cause: your machine has sloppy
                                            floating-point arithmetic.
                            Cure: Increase the PARAMETER 'FUDGE',
                                  recompile, and try again.

Internal Parameters:

      RELFAC  REAL, default = 2.0e0
              The relative tolerance.  An interval (a,b] lies within
              'relative tolerance' if  b-a < RELFAC*ulp*max(|a|,|b|),
              where 'ulp' is the machine precision (distance from 1 to
              the next larger floating point number.)
      FUDGE   REAL, default = 2
              A 'fudge factor' to widen the Gershgorin intervals.  Ideally,
              a value of 1 should work, but on machines with sloppy
              arithmetic, this needs to be larger.  The default for
              publicly released versions should be large enough to handle
              the worst machine around.  Note that this has no effect
              on accuracy of the solution.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 270 of file sstebz.f.

  subroutine sstedc (character compz, integer n, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

SSTEDC

Purpose:

     SSTEDC computes all eigenvalues and, optionally, eigenvectors of a
     symmetric tridiagonal matrix using the divide and conquer method.
     The eigenvectors of a full or band real symmetric matrix can also be
     found if SSYTRD or SSPTRD or SSBTRD has been used to reduce this
     matrix to tridiagonal form.
     This code makes very mild assumptions about floating point
     arithmetic. It will work on machines with a guard digit in
     add/subtract, or on those binary machines without guard digits
     which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
     It could conceivably fail on hexadecimal or decimal machines
     without guard digits, but we know of none.  See SLAED3 for details.

Parameters
    COMPZ

              COMPZ is CHARACTER*1
              = 'N':  Compute eigenvalues only.
              = 'I':  Compute eigenvectors of tridiagonal matrix also.
              = 'V':  Compute eigenvectors of original dense symmetric
                      matrix also.  On entry, Z contains the orthogonal
                      matrix used to reduce the original matrix to
                      tridiagonal form.

    N

              N is INTEGER
              The dimension of the symmetric tridiagonal matrix.  N >= 0.

    D

              D is REAL array, dimension (N)
              On entry, the diagonal elements of the tridiagonal matrix.
              On exit, if INFO = 0, the eigenvalues in ascending order.

    E

              E is REAL array, dimension (N-1)
              On entry, the subdiagonal elements of the tridiagonal matrix.
              On exit, E has been destroyed.

    Z

              Z is REAL array, dimension (LDZ,N)
              On entry, if COMPZ = 'V', then Z contains the orthogonal
              matrix used in the reduction to tridiagonal form.
              On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
              orthonormal eigenvectors of the original symmetric matrix,
              and if COMPZ = 'I', Z contains the orthonormal eigenvectors
              of the symmetric tridiagonal matrix.
              If  COMPZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1.
              If eigenvectors are desired, then LDZ >= max(1,N).

    WORK

              WORK is REAL array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If COMPZ = 'N' or N <= 1 then LWORK must be at least 1.
              If COMPZ = 'V' and N > 1 then LWORK must be at least
                             ( 1 + 3*N + 2*N*lg N + 4*N**2 ),
                             where lg( N ) = smallest integer k such
                             that 2**k >= N.
              If COMPZ = 'I' and N > 1 then LWORK must be at least
                             ( 1 + 4*N + N**2 ).
              Note that for COMPZ = 'I' or 'V', then if N is less than or
              equal to the minimum divide size, usually 25, then LWORK need
              only be max(1,2*(N-1)).
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (MAX(1,LIWORK))
              On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.
              If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1.
              If COMPZ = 'V' and N > 1 then LIWORK must be at least
                             ( 6 + 6*N + 5*N*lg N ).
              If COMPZ = 'I' and N > 1 then LIWORK must be at least
                             ( 3 + 5*N ).
              Note that for COMPZ = 'I' or 'V', then if N is less than or
              equal to the minimum divide size, usually 25, then LIWORK
              need only be 1.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of the IWORK array,
              returns this value as the first entry of the IWORK array, and
              no error message related to LIWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  The algorithm failed to compute an eigenvalue while
                    working on the submatrix lying in rows and columns
                    INFO/(N+1) through mod(INFO,N+1).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Jeff Rutter, Computer Science Division, University of California at Berkeley, USA
     Modified by Francoise Tisseur, University of Tennessee

Definition at line 186 of file sstedc.f.

  subroutine ssteqr (character compz, integer n, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer info)

SSTEQR

Purpose:

     SSTEQR computes all eigenvalues and, optionally, eigenvectors of a
     symmetric tridiagonal matrix using the implicit QL or QR method.
     The eigenvectors of a full or band symmetric matrix can also be found
     if SSYTRD or SSPTRD or SSBTRD has been used to reduce this matrix to
     tridiagonal form.

Parameters
    COMPZ

              COMPZ is CHARACTER*1
              = 'N':  Compute eigenvalues only.
              = 'V':  Compute eigenvalues and eigenvectors of the original
                      symmetric matrix.  On entry, Z must contain the
                      orthogonal matrix used to reduce the original matrix
                      to tridiagonal form.
              = 'I':  Compute eigenvalues and eigenvectors of the
                      tridiagonal matrix.  Z is initialized to the identity
                      matrix.

    N

              N is INTEGER
              The order of the matrix.  N >= 0.

    D

              D is REAL array, dimension (N)
              On entry, the diagonal elements of the tridiagonal matrix.
              On exit, if INFO = 0, the eigenvalues in ascending order.

    E

              E is REAL array, dimension (N-1)
              On entry, the (n-1) subdiagonal elements of the tridiagonal
              matrix.
              On exit, E has been destroyed.

    Z

              Z is REAL array, dimension (LDZ, N)
              On entry, if  COMPZ = 'V', then Z contains the orthogonal
              matrix used in the reduction to tridiagonal form.
              On exit, if INFO = 0, then if  COMPZ = 'V', Z contains the
              orthonormal eigenvectors of the original symmetric matrix,
              and if COMPZ = 'I', Z contains the orthonormal eigenvectors
              of the symmetric tridiagonal matrix.
              If COMPZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              eigenvectors are desired, then  LDZ >= max(1,N).

    WORK

              WORK is REAL array, dimension (max(1,2*N-2))
              If COMPZ = 'N', then WORK is not referenced.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  the algorithm has failed to find all the eigenvalues in
                    a total of 30*N iterations; if INFO = i, then i
                    elements of E have not converged to zero; on exit, D
                    and E contain the elements of a symmetric tridiagonal
                    matrix which is orthogonally similar to the original
                    matrix.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 130 of file ssteqr.f.

  subroutine ssterf (integer n, real, dimension( * ) d, real, dimension( * ) e, integer info)

SSTERF

Purpose:

     SSTERF computes all eigenvalues of a symmetric tridiagonal matrix
     using the Pal-Walker-Kahan variant of the QL or QR algorithm.

Parameters
    N

              N is INTEGER
              The order of the matrix.  N >= 0.

    D

              D is REAL array, dimension (N)
              On entry, the n diagonal elements of the tridiagonal matrix.
              On exit, if INFO = 0, the eigenvalues in ascending order.

    E

              E is REAL array, dimension (N-1)
              On entry, the (n-1) subdiagonal elements of the tridiagonal
              matrix.
              On exit, E has been destroyed.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  the algorithm failed to find all of the eigenvalues in
                    a total of 30*N iterations; if INFO = i, then i
                    elements of E have not converged to zero.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 85 of file ssterf.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.