realOTHEReigen(3)

NAME
realOTHEReigen - real

SYNOPSIS

  Functions
subroutine sbdsvdx (uplo, jobz, range, n, d, e, vl, vu, il, iu, ns, s, z, ldz, work, iwork, info)
SBDSVDX
subroutine sggglm (n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info)
SGGGLM
subroutine ssbev (jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, info)
SSBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine ssbev_2stage (jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, lwork, info)
SSBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine ssbevd (jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, lwork, iwork, liwork, info)
SSBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine ssbevd_2stage (jobz, uplo, n, kd, ab, ldab, w, z, ldz, work, lwork, iwork, liwork, info)
SSBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine ssbevx (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)
SSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine ssbevx_2stage (jobz, range, uplo, n, kd, ab, ldab, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, iwork, ifail, info)
SSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine ssbgv (jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, info)
SSBGV
subroutine ssbgvd (jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, lwork, iwork, liwork, info)
SSBGVD
subroutine ssbgvx (jobz, range, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)
SSBGVX
subroutine sspev (jobz, uplo, n, ap, w, z, ldz, work, info)
SSPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine sspevd (jobz, uplo, n, ap, w, z, ldz, work, lwork, iwork, liwork, info)
SSPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine sspevx (jobz, range, uplo, n, ap, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)
SSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine sspgv (itype, jobz, uplo, n, ap, bp, w, z, ldz, work, info)
SSPGV
subroutine sspgvd (itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork, iwork, liwork, info)
SSPGVD
subroutine sspgvx (itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)
SSPGVX
subroutine sstev (jobz, n, d, e, z, ldz, work, info)
SSTEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine sstevd (jobz, n, d, e, z, ldz, work, lwork, iwork, liwork, info)
SSTEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine sstevr (jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)
SSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices
subroutine sstevx (jobz, range, n, d, e, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)
SSTEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Detailed Description

This is the group of real Other Eigenvalue routines

Function Documentation

  subroutine sbdsvdx (character uplo, character jobz, character range, integer n, real, dimension( * ) d, real, dimension( * ) e, real vl, real vu, integer il, integer iu, integer ns, real, dimension( * ) s, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SBDSVDX

Purpose:

      SBDSVDX computes the singular value decomposition (SVD) of a real
      N-by-N (upper or lower) bidiagonal matrix B, B = U * S * VT,
      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.
      Given an upper bidiagonal B with diagonal D = [ d_1 d_2 ... d_N ]
      and superdiagonal E = [ e_1 e_2 ... e_N-1 ], SBDSVDX computes the
      singular value decompositon of B through the eigenvalues and
      eigenvectors of the N*2-by-N*2 tridiagonal matrix
            |  0  d_1                |
            | d_1  0  e_1            |
      TGK = |     e_1  0  d_2        |
            |         d_2  .   .     |
            |              .   .   . |
      If (s,u,v) is a singular triplet of B with ||u|| = ||v|| = 1, then
      (+/-s,q), ||q|| = 1, are eigenpairs of TGK, with q = P * ( u' +/-v' ) /
      sqrt(2) = ( v_1 u_1 v_2 u_2 ... v_n u_n ) / sqrt(2), and
      P = [ e_{n+1} e_{1} e_{n+2} e_{2} ... ].
      Given a TGK matrix, one can either a) compute -s,-v and change signs
      so that the singular values (and corresponding vectors) are already in
      descending order (as in SGESVD/SGESDD) or b) compute s,v and reorder
      the values (and corresponding vectors). SBDSVDX implements a) by
      calling SSTEVX (bisection plus inverse iteration, to be replaced
      with a version of the Multiple Relative Robust Representation
      algorithm. (See P. Willems and B. Lang, A framework for the MR^3
      algorithm: theory and implementation, SIAM J. Sci. Comput.,
      35:740-766, 2013.)

Parameters
    UPLO

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

    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute singular values only;
              = 'V':  Compute singular values and singular vectors.

    RANGE

              RANGE is CHARACTER*1
              = 'A': all singular values will be found.
              = 'V': all singular values in the half-open interval [VL,VU)
                     will be found.
              = 'I': the IL-th through IU-th singular values will be found.

    N

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

    D

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

    E

              E is REAL array, dimension (max(1,N-1))
              The (n-1) superdiagonal elements of the bidiagonal matrix
              B in elements 1 to N-1.

    VL

             VL is REAL
              If RANGE='V', the lower bound of the interval to
              be searched for singular values. VU > VL.
              Not referenced if RANGE = 'A' or 'I'.

    VU

             VU is REAL
              If RANGE='V', the upper bound of the interval to
              be searched for singular values. VU > VL.
              Not referenced if RANGE = 'A' or 'I'.

    IL

              IL is INTEGER
              If RANGE='I', the index of the
              smallest singular value to be returned.
              1 <= IL <= IU <= min(M,N), if min(M,N) > 0.
              Not referenced if RANGE = 'A' or 'V'.

    IU

              IU is INTEGER
              If RANGE='I', the index of the
              largest singular value to be returned.
              1 <= IL <= IU <= min(M,N), if min(M,N) > 0.
              Not referenced if RANGE = 'A' or 'V'.

    NS

              NS is INTEGER
              The total number of singular values found.  0 <= NS <= N.
              If RANGE = 'A', NS = N, and if RANGE = 'I', NS = IU-IL+1.

    S

              S is REAL array, dimension (N)
              The first NS elements contain the selected singular values in
              ascending order.

    Z

              Z is REAL array, dimension (2*N,K)
              If JOBZ = 'V', then if INFO = 0 the first NS columns of Z
              contain the singular vectors of the matrix B corresponding to
              the selected singular values, with U in rows 1 to N and V
              in rows N+1 to N*2, i.e.
              Z = [ U ]
                  [ V ]
              If JOBZ = 'N', then Z is not referenced.
              Note: The user must ensure that at least K = NS+1 columns are
              supplied in the array Z; if RANGE = 'V', the exact value of
              NS is not known in advance and an upper bound must be used.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z. LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(2,N*2).

    WORK

              WORK is REAL array, dimension (14*N)

    IWORK

              IWORK is INTEGER array, dimension (12*N)
              If JOBZ = 'V', then if INFO = 0, the first NS elements of
              IWORK are zero. If INFO > 0, then IWORK contains the indices
              of the eigenvectors that failed to converge in DSTEVX.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, then i eigenvectors failed to converge
                       in SSTEVX. The indices of the eigenvectors
                       (as returned by SSTEVX) are stored in the
                       array IWORK.
                    if INFO = N*2 + 1, an internal error occurred.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 224 of file sbdsvdx.f.

  subroutine sggglm (integer n, integer m, integer p, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) d, real, dimension( * ) x, real, dimension( * ) y, real, dimension( * ) work, integer lwork, integer info)

SGGGLM

Purpose:

     SGGGLM solves a general Gauss-Markov linear model (GLM) problem:
             minimize || y ||_2   subject to   d = A*x + B*y
                 x
     where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
     given N-vector. It is assumed that M <= N <= M+P, and
                rank(A) = M    and    rank( A B ) = N.
     Under these assumptions, the constrained equation is always
     consistent, and there is a unique solution x and a minimal 2-norm
     solution y, which is obtained using a generalized QR factorization
     of the matrices (A, B) given by
        A = Q*(R),   B = Q*T*Z.
              (0)
     In particular, if matrix B is square nonsingular, then the problem
     GLM is equivalent to the following weighted linear least squares
     problem
                  minimize || inv(B)*(d-A*x) ||_2
                      x
     where inv(B) denotes the inverse of B.

Parameters
    N

              N is INTEGER
              The number of rows of the matrices A and B.  N >= 0.

    M

              M is INTEGER
              The number of columns of the matrix A.  0 <= M <= N.

    P

              P is INTEGER
              The number of columns of the matrix B.  P >= N-M.

    A

              A is REAL array, dimension (LDA,M)
              On entry, the N-by-M matrix A.
              On exit, the upper triangular part of the array A contains
              the M-by-M upper triangular matrix R.

    LDA

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

    B

              B is REAL array, dimension (LDB,P)
              On entry, the N-by-P matrix B.
              On exit, if N <= P, the upper triangle of the subarray
              B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
              if N > P, the elements on and above the (N-P)th subdiagonal
              contain the N-by-P upper trapezoidal matrix T.

    LDB

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

    D

              D is REAL array, dimension (N)
              On entry, D is the left hand side of the GLM equation.
              On exit, D is destroyed.

    X

              X is REAL array, dimension (M)

    Y

              Y is REAL array, dimension (P)
              On exit, X and Y are the solutions of the GLM problem.

    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. LWORK >= max(1,N+M+P).
              For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
              where NB is an upper bound for the optimal blocksizes for
              SGEQRF, SGERQF, SORMQR and SORMRQ.
              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.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              = 1:  the upper triangular factor R associated with A in the
                    generalized QR factorization of the pair (A, B) is
                    singular, so that rank(A) < M; the least squares
                    solution could not be computed.
              = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
                    factor T associated with B in the generalized QR
                    factorization of the pair (A, B) is singular, so that
                    rank( A B ) < N; the least squares solution could not
                    be computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 183 of file sggglm.f.

  subroutine ssbev (character jobz, character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer info)

SSBEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

     SSBEV computes all the eigenvalues and, optionally, eigenvectors of
     a real symmetric band matrix A.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    KD

              KD is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

    AB

              AB is REAL array, dimension (LDAB, N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, AB is overwritten by values generated during the
              reduction to tridiagonal form.  If UPLO = 'U', the first
              superdiagonal and the diagonal of the tridiagonal matrix T
              are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
              the diagonal and first subdiagonal of T are returned in the
              first two rows of AB.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD + 1.

    W

              W is REAL array, dimension (N)
              If INFO = 0, the eigenvalues in ascending order.

    Z

              Z is REAL array, dimension (LDZ, N)
              If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
              eigenvectors of the matrix A, with the i-th column of Z
              holding the eigenvector associated with W(i).
              If JOBZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    WORK

              WORK is REAL array, dimension (max(1,3*N-2))

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, the algorithm failed to converge; i
                    off-diagonal elements of an intermediate tridiagonal
                    form did not converge to zero.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 144 of file ssbev.f.

  subroutine ssbev_2stage (character jobz, character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer info)

SSBEV_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

     SSBEV_2STAGE computes all the eigenvalues and, optionally, eigenvectors of
     a real symmetric band matrix A using the 2stage technique for
     the reduction to tridiagonal.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.
                      Not available in this release.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    KD

              KD is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

    AB

              AB is REAL array, dimension (LDAB, N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, AB is overwritten by values generated during the
              reduction to tridiagonal form.  If UPLO = 'U', the first
              superdiagonal and the diagonal of the tridiagonal matrix T
              are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
              the diagonal and first subdiagonal of T are returned in the
              first two rows of AB.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD + 1.

    W

              W is REAL array, dimension (N)
              If INFO = 0, the eigenvalues in ascending order.

    Z

              Z is REAL array, dimension (LDZ, N)
              If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
              eigenvectors of the matrix A, with the i-th column of Z
              holding the eigenvector associated with W(i).
              If JOBZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    WORK

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

    LWORK

              LWORK is INTEGER
              The length of the array WORK. LWORK >= 1, when N <= 1;
              otherwise
              If JOBZ = 'N' and N > 1, LWORK must be queried.
                                       LWORK = MAX(1, dimension) where
                                       dimension = (2KD+1)*N + KD*NTHREADS + N
                                       where KD is the size of the band.
                                       NTHREADS is the number of threads used when
                                       openMP compilation is enabled, otherwise =1.
              If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.
              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.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, the algorithm failed to converge; i
                    off-diagonal elements of an intermediate tridiagonal
                    form did not converge to zero.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      All details about the 2stage techniques are available in:
      Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
      Parallel reduction to condensed forms for symmetric eigenvalue problems
      using aggregated fine-grained and memory-aware kernels. In Proceedings
      of 2011 International Conference for High Performance Computing,
      Networking, Storage and Analysis (SC '11), New York, NY, USA,
      Article 8 , 11 pages.
      http://doi.acm.org/10.1145/2063384.2063394
      A. Haidar, J. Kurzak, P. Luszczek, 2013.
      An improved parallel singular value algorithm and its implementation
      for multicore hardware, In Proceedings of 2013 International Conference
      for High Performance Computing, Networking, Storage and Analysis (SC '13).
      Denver, Colorado, USA, 2013.
      Article 90, 12 pages.
      http://doi.acm.org/10.1145/2503210.2503292
      A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
      A novel hybrid CPU-GPU generalized eigensolver for electronic structure
      calculations based on fine-grained memory aware tasks.
      International Journal of High Performance Computing Applications.
      Volume 28 Issue 2, Pages 196-209, May 2014.
      http://hpc.sagepub.com/content/28/2/196

Definition at line 202 of file ssbev_2stage.f.

  subroutine ssbevd (character jobz, character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

SSBEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

     SSBEVD computes all the eigenvalues and, optionally, eigenvectors of
     a real symmetric band matrix A. If eigenvectors are desired, it uses
     a divide and conquer algorithm.
     The divide and conquer algorithm 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
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    KD

              KD is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

    AB

              AB is REAL array, dimension (LDAB, N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, AB is overwritten by values generated during the
              reduction to tridiagonal form.  If UPLO = 'U', the first
              superdiagonal and the diagonal of the tridiagonal matrix T
              are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
              the diagonal and first subdiagonal of T are returned in the
              first two rows of AB.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD + 1.

    W

              W is REAL array, dimension (N)
              If INFO = 0, the eigenvalues in ascending order.

    Z

              Z is REAL array, dimension (LDZ, N)
              If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
              eigenvectors of the matrix A, with the i-th column of Z
              holding the eigenvector associated with W(i).
              If JOBZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              IF N <= 1,                LWORK must be at least 1.
              If JOBZ  = 'N' and N > 2, LWORK must be at least 2*N.
              If JOBZ  = 'V' and N > 2, LWORK must be at least
                             ( 1 + 5*N + 2*N**2 ).
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal sizes of the WORK and IWORK
              arrays, returns these values as the first entries of the WORK
              and IWORK arrays, and no error message related to LWORK or
              LIWORK 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 JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
              If JOBZ  = 'V' and N > 2, LIWORK must be at least 3 + 5*N.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal sizes of the WORK and
              IWORK arrays, returns these values as the first entries of
              the WORK and IWORK arrays, and no error message related to
              LWORK or 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:  if INFO = i, the algorithm failed to converge; i
                    off-diagonal elements of an intermediate tridiagonal
                    form did not converge to zero.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 191 of file ssbevd.f.

  subroutine ssbevd_2stage (character jobz, character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

SSBEVD_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

     SSBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of
     a real symmetric band matrix A using the 2stage technique for
     the reduction to tridiagonal. If eigenvectors are desired, it uses
     a divide and conquer algorithm.
     The divide and conquer algorithm 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
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.
                      Not available in this release.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    KD

              KD is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

    AB

              AB is REAL array, dimension (LDAB, N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, AB is overwritten by values generated during the
              reduction to tridiagonal form.  If UPLO = 'U', the first
              superdiagonal and the diagonal of the tridiagonal matrix T
              are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
              the diagonal and first subdiagonal of T are returned in the
              first two rows of AB.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD + 1.

    W

              W is REAL array, dimension (N)
              If INFO = 0, the eigenvalues in ascending order.

    Z

              Z is REAL array, dimension (LDZ, N)
              If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
              eigenvectors of the matrix A, with the i-th column of Z
              holding the eigenvector associated with W(i).
              If JOBZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    WORK

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

    LWORK

              LWORK is INTEGER
              The length of the array WORK. LWORK >= 1, when N <= 1;
              otherwise
              If JOBZ = 'N' and N > 1, LWORK must be queried.
                                       LWORK = MAX(1, dimension) where
                                       dimension = (2KD+1)*N + KD*NTHREADS + N
                                       where KD is the size of the band.
                                       NTHREADS is the number of threads used when
                                       openMP compilation is enabled, otherwise =1.
              If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal sizes of the WORK and IWORK
              arrays, returns these values as the first entries of the WORK
              and IWORK arrays, and no error message related to LWORK or
              LIWORK 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 JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
              If JOBZ  = 'V' and N > 2, LIWORK must be at least 3 + 5*N.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal sizes of the WORK and
              IWORK arrays, returns these values as the first entries of
              the WORK and IWORK arrays, and no error message related to
              LWORK or 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:  if INFO = i, the algorithm failed to converge; i
                    off-diagonal elements of an intermediate tridiagonal
                    form did not converge to zero.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      All details about the 2stage techniques are available in:
      Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
      Parallel reduction to condensed forms for symmetric eigenvalue problems
      using aggregated fine-grained and memory-aware kernels. In Proceedings
      of 2011 International Conference for High Performance Computing,
      Networking, Storage and Analysis (SC '11), New York, NY, USA,
      Article 8 , 11 pages.
      http://doi.acm.org/10.1145/2063384.2063394
      A. Haidar, J. Kurzak, P. Luszczek, 2013.
      An improved parallel singular value algorithm and its implementation
      for multicore hardware, In Proceedings of 2013 International Conference
      for High Performance Computing, Networking, Storage and Analysis (SC '13).
      Denver, Colorado, USA, 2013.
      Article 90, 12 pages.
      http://doi.acm.org/10.1145/2503210.2503292
      A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
      A novel hybrid CPU-GPU generalized eigensolver for electronic structure
      calculations based on fine-grained memory aware tasks.
      International Journal of High Performance Computing Applications.
      Volume 28 Issue 2, Pages 196-209, May 2014.
      http://hpc.sagepub.com/content/28/2/196

Definition at line 232 of file ssbevd_2stage.f.

  subroutine ssbevx (character jobz, character range, character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldq, * ) q, integer ldq, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

SSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

     SSBEVX computes selected eigenvalues and, optionally, eigenvectors
     of a real symmetric band matrix A.  Eigenvalues and eigenvectors can
     be selected by specifying either a range of values or a range of
     indices for the desired eigenvalues.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    RANGE

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

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    KD

              KD is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

    AB

              AB is REAL array, dimension (LDAB, N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, AB is overwritten by values generated during the
              reduction to tridiagonal form.  If UPLO = 'U', the first
              superdiagonal and the diagonal of the tridiagonal matrix T
              are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
              the diagonal and first subdiagonal of T are returned in the
              first two rows of AB.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD + 1.

    Q

              Q is REAL array, dimension (LDQ, N)
              If JOBZ = 'V', the N-by-N orthogonal matrix used in the
                             reduction to tridiagonal form.
              If JOBZ = 'N', the array Q is not referenced.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.  If JOBZ = 'V', then
              LDQ >= max(1,N).

    VL

              VL is REAL
              If RANGE='V', the lower bound of the interval to
              be searched for eigenvalues. 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. 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 error tolerance for the eigenvalues.
              An approximate eigenvalue is accepted as converged
              when it is determined to lie in an interval [a,b]
              of width less than or equal to
                      ABSTOL + EPS *   max( |a|,|b| ) ,
              where EPS is the machine precision.  If ABSTOL is less than
              or equal to zero, then  EPS*|T|  will be used in its place,
              where |T| is the 1-norm of the tridiagonal matrix obtained
              by reducing AB to tridiagonal form.
              Eigenvalues will be computed most accurately when ABSTOL is
              set to twice the underflow threshold 2*SLAMCH('S'), not zero.
              If this routine returns with INFO>0, indicating that some
              eigenvectors did not converge, try setting ABSTOL to
              2*SLAMCH('S').
              See 'Computing Small Singular Values of Bidiagonal Matrices
              with Guaranteed High Relative Accuracy,' by Demmel and
              Kahan, LAPACK Working Note #3.

    M

              M is INTEGER
              The total number of eigenvalues found.  0 <= M <= N.
              If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

    W

              W is REAL array, dimension (N)
              The first M elements contain the selected eigenvalues in
              ascending order.

    Z

              Z is REAL array, dimension (LDZ, max(1,M))
              If JOBZ = 'V', then if INFO = 0, the first M columns of Z
              contain the orthonormal eigenvectors of the matrix A
              corresponding to the selected eigenvalues, with the i-th
              column of Z holding the eigenvector associated with W(i).
              If an eigenvector fails to converge, then that column of Z
              contains the latest approximation to the eigenvector, and the
              index of the eigenvector is returned in IFAIL.
              If JOBZ = 'N', then Z is not referenced.
              Note: the user must ensure that at least max(1,M) columns are
              supplied in the array Z; if RANGE = 'V', the exact value of M
              is not known in advance and an upper bound must be used.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    WORK

              WORK is REAL array, dimension (7*N)

    IWORK

              IWORK is INTEGER array, dimension (5*N)

    IFAIL

              IFAIL is INTEGER array, dimension (N)
              If JOBZ = 'V', then if INFO = 0, the first M elements of
              IFAIL are zero.  If INFO > 0, then IFAIL contains the
              indices of the eigenvectors that failed to converge.
              If JOBZ = 'N', then IFAIL is not referenced.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = i, then i eigenvectors failed to converge.
                    Their indices are stored in array IFAIL.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 262 of file ssbevx.f.

  subroutine ssbevx_2stage (character jobz, character range, character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldq, * ) q, integer ldq, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

SSBEVX_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

     SSBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors
     of a real symmetric band matrix A using the 2stage technique for
     the reduction to tridiagonal. Eigenvalues and eigenvectors can
     be selected by specifying either a range of values or a range of
     indices for the desired eigenvalues.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.
                      Not available in this release.

    RANGE

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

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    KD

              KD is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

    AB

              AB is REAL array, dimension (LDAB, N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
              On exit, AB is overwritten by values generated during the
              reduction to tridiagonal form.  If UPLO = 'U', the first
              superdiagonal and the diagonal of the tridiagonal matrix T
              are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
              the diagonal and first subdiagonal of T are returned in the
              first two rows of AB.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD + 1.

    Q

              Q is REAL array, dimension (LDQ, N)
              If JOBZ = 'V', the N-by-N orthogonal matrix used in the
                             reduction to tridiagonal form.
              If JOBZ = 'N', the array Q is not referenced.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.  If JOBZ = 'V', then
              LDQ >= max(1,N).

    VL

              VL is REAL
              If RANGE='V', the lower bound of the interval to
              be searched for eigenvalues. 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. 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 error tolerance for the eigenvalues.
              An approximate eigenvalue is accepted as converged
              when it is determined to lie in an interval [a,b]
              of width less than or equal to
                      ABSTOL + EPS *   max( |a|,|b| ) ,
              where EPS is the machine precision.  If ABSTOL is less than
              or equal to zero, then  EPS*|T|  will be used in its place,
              where |T| is the 1-norm of the tridiagonal matrix obtained
              by reducing AB to tridiagonal form.
              Eigenvalues will be computed most accurately when ABSTOL is
              set to twice the underflow threshold 2*SLAMCH('S'), not zero.
              If this routine returns with INFO>0, indicating that some
              eigenvectors did not converge, try setting ABSTOL to
              2*SLAMCH('S').
              See 'Computing Small Singular Values of Bidiagonal Matrices
              with Guaranteed High Relative Accuracy,' by Demmel and
              Kahan, LAPACK Working Note #3.

    M

              M is INTEGER
              The total number of eigenvalues found.  0 <= M <= N.
              If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

    W

              W is REAL array, dimension (N)
              The first M elements contain the selected eigenvalues in
              ascending order.

    Z

              Z is REAL array, dimension (LDZ, max(1,M))
              If JOBZ = 'V', then if INFO = 0, the first M columns of Z
              contain the orthonormal eigenvectors of the matrix A
              corresponding to the selected eigenvalues, with the i-th
              column of Z holding the eigenvector associated with W(i).
              If an eigenvector fails to converge, then that column of Z
              contains the latest approximation to the eigenvector, and the
              index of the eigenvector is returned in IFAIL.
              If JOBZ = 'N', then Z is not referenced.
              Note: the user must ensure that at least max(1,M) columns are
              supplied in the array Z; if RANGE = 'V', the exact value of M
              is not known in advance and an upper bound must be used.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    WORK

              WORK is REAL array, dimension (LWORK)

    LWORK

              LWORK is INTEGER
              The length of the array WORK. LWORK >= 1, when N <= 1;
              otherwise
              If JOBZ = 'N' and N > 1, LWORK must be queried.
                                       LWORK = MAX(1, 7*N, dimension) where
                                       dimension = (2KD+1)*N + KD*NTHREADS + 2*N
                                       where KD is the size of the band.
                                       NTHREADS is the number of threads used when
                                       openMP compilation is enabled, otherwise =1.
              If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available
              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 (5*N)

    IFAIL

              IFAIL is INTEGER array, dimension (N)
              If JOBZ = 'V', then if INFO = 0, the first M elements of
              IFAIL are zero.  If INFO > 0, then IFAIL contains the
              indices of the eigenvectors that failed to converge.
              If JOBZ = 'N', then IFAIL is not referenced.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = i, then i eigenvectors failed to converge.
                    Their indices are stored in array IFAIL.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      All details about the 2stage techniques are available in:
      Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
      Parallel reduction to condensed forms for symmetric eigenvalue problems
      using aggregated fine-grained and memory-aware kernels. In Proceedings
      of 2011 International Conference for High Performance Computing,
      Networking, Storage and Analysis (SC '11), New York, NY, USA,
      Article 8 , 11 pages.
      http://doi.acm.org/10.1145/2063384.2063394
      A. Haidar, J. Kurzak, P. Luszczek, 2013.
      An improved parallel singular value algorithm and its implementation
      for multicore hardware, In Proceedings of 2013 International Conference
      for High Performance Computing, Networking, Storage and Analysis (SC '13).
      Denver, Colorado, USA, 2013.
      Article 90, 12 pages.
      http://doi.acm.org/10.1145/2503210.2503292
      A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
      A novel hybrid CPU-GPU generalized eigensolver for electronic structure
      calculations based on fine-grained memory aware tasks.
      International Journal of High Performance Computing Applications.
      Volume 28 Issue 2, Pages 196-209, May 2014.
      http://hpc.sagepub.com/content/28/2/196

Definition at line 319 of file ssbevx_2stage.f.

  subroutine ssbgv (character jobz, character uplo, integer n, integer ka, integer kb, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldbb, * ) bb, integer ldbb, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer info)

SSBGV

Purpose:

     SSBGV computes all the eigenvalues, and optionally, the eigenvectors
     of a real generalized symmetric-definite banded eigenproblem, of
     the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric
     and banded, and B is also positive definite.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangles of A and B are stored;
              = 'L':  Lower triangles of A and B are stored.

    N

              N is INTEGER
              The order of the matrices A and B.  N >= 0.

    KA

              KA is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'. KA >= 0.

    KB

              KB is INTEGER
              The number of superdiagonals of the matrix B if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'. KB >= 0.

    AB

              AB is REAL array, dimension (LDAB, N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first ka+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
              On exit, the contents of AB are destroyed.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KA+1.

    BB

              BB is REAL array, dimension (LDBB, N)
              On entry, the upper or lower triangle of the symmetric band
              matrix B, stored in the first kb+1 rows of the array.  The
              j-th column of B is stored in the j-th column of the array BB
              as follows:
              if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
              if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).
              On exit, the factor S from the split Cholesky factorization
              B = S**T*S, as returned by SPBSTF.

    LDBB

              LDBB is INTEGER
              The leading dimension of the array BB.  LDBB >= KB+1.

    W

              W is REAL array, dimension (N)
              If INFO = 0, the eigenvalues in ascending order.

    Z

              Z is REAL array, dimension (LDZ, N)
              If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
              eigenvectors, with the i-th column of Z holding the
              eigenvector associated with W(i). The eigenvectors are
              normalized so that Z**T*B*Z = I.
              If JOBZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= N.

    WORK

              WORK is REAL array, dimension (3*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, and i is:
                 <= N:  the algorithm failed to converge:
                        i off-diagonal elements of an intermediate
                        tridiagonal form did not converge to zero;
                 > N:   if INFO = N + i, for 1 <= i <= N, then SPBSTF
                        returned INFO = i: B is not positive definite.
                        The factorization of B could not be completed and
                        no eigenvalues or eigenvectors were computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 175 of file ssbgv.f.

  subroutine ssbgvd (character jobz, character uplo, integer n, integer ka, integer kb, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldbb, * ) bb, integer ldbb, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

SSBGVD

Purpose:

     SSBGVD computes all the eigenvalues, and optionally, the eigenvectors
     of a real generalized symmetric-definite banded eigenproblem, of the
     form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric and
     banded, and B is also positive definite.  If eigenvectors are
     desired, it uses a divide and conquer algorithm.
     The divide and conquer algorithm 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
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangles of A and B are stored;
              = 'L':  Lower triangles of A and B are stored.

    N

              N is INTEGER
              The order of the matrices A and B.  N >= 0.

    KA

              KA is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KA >= 0.

    KB

              KB is INTEGER
              The number of superdiagonals of the matrix B if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KB >= 0.

    AB

              AB is REAL array, dimension (LDAB, N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first ka+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
              On exit, the contents of AB are destroyed.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KA+1.

    BB

              BB is REAL array, dimension (LDBB, N)
              On entry, the upper or lower triangle of the symmetric band
              matrix B, stored in the first kb+1 rows of the array.  The
              j-th column of B is stored in the j-th column of the array BB
              as follows:
              if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
              if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).
              On exit, the factor S from the split Cholesky factorization
              B = S**T*S, as returned by SPBSTF.

    LDBB

              LDBB is INTEGER
              The leading dimension of the array BB.  LDBB >= KB+1.

    W

              W is REAL array, dimension (N)
              If INFO = 0, the eigenvalues in ascending order.

    Z

              Z is REAL array, dimension (LDZ, N)
              If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
              eigenvectors, with the i-th column of Z holding the
              eigenvector associated with W(i).  The eigenvectors are
              normalized so Z**T*B*Z = I.
              If JOBZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', 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 N <= 1,               LWORK >= 1.
              If JOBZ = 'N' and N > 1, LWORK >= 3*N.
              If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal sizes of the WORK and IWORK
              arrays, returns these values as the first entries of the WORK
              and IWORK arrays, and no error message related to LWORK or
              LIWORK is issued by XERBLA.

    IWORK

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

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.
              If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
              If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal sizes of the WORK and
              IWORK arrays, returns these values as the first entries of
              the WORK and IWORK arrays, and no error message related to
              LWORK or 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:  if INFO = i, and i is:
                 <= N:  the algorithm failed to converge:
                        i off-diagonal elements of an intermediate
                        tridiagonal form did not converge to zero;
                 > N:   if INFO = N + i, for 1 <= i <= N, then SPBSTF
                        returned INFO = i: B is not positive definite.
                        The factorization of B could not be completed and
                        no eigenvalues or eigenvectors were computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 225 of file ssbgvd.f.

  subroutine ssbgvx (character jobz, character range, character uplo, integer n, integer ka, integer kb, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldbb, * ) bb, integer ldbb, real, dimension( ldq, * ) q, integer ldq, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

SSBGVX

Purpose:

     SSBGVX computes selected eigenvalues, and optionally, eigenvectors
     of a real generalized symmetric-definite banded eigenproblem, of
     the form A*x=(lambda)*B*x.  Here A and B are assumed to be symmetric
     and banded, and B is also positive definite.  Eigenvalues and
     eigenvectors can be selected by specifying either all eigenvalues,
     a range of values or a range of indices for the desired eigenvalues.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    RANGE

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

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangles of A and B are stored;
              = 'L':  Lower triangles of A and B are stored.

    N

              N is INTEGER
              The order of the matrices A and B.  N >= 0.

    KA

              KA is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KA >= 0.

    KB

              KB is INTEGER
              The number of superdiagonals of the matrix B if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KB >= 0.

    AB

              AB is REAL array, dimension (LDAB, N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first ka+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
              On exit, the contents of AB are destroyed.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KA+1.

    BB

              BB is REAL array, dimension (LDBB, N)
              On entry, the upper or lower triangle of the symmetric band
              matrix B, stored in the first kb+1 rows of the array.  The
              j-th column of B is stored in the j-th column of the array BB
              as follows:
              if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
              if UPLO = 'L', BB(1+i-j,j)    = B(i,j) for j<=i<=min(n,j+kb).
              On exit, the factor S from the split Cholesky factorization
              B = S**T*S, as returned by SPBSTF.

    LDBB

              LDBB is INTEGER
              The leading dimension of the array BB.  LDBB >= KB+1.

    Q

              Q is REAL array, dimension (LDQ, N)
              If JOBZ = 'V', the n-by-n matrix used in the reduction of
              A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
              and consequently C to tridiagonal form.
              If JOBZ = 'N', the array Q is not referenced.

    LDQ

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

    VL

              VL is REAL
              If RANGE='V', the lower bound of the interval to
              be searched for eigenvalues. 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. 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 error tolerance for the eigenvalues.
              An approximate eigenvalue is accepted as converged
              when it is determined to lie in an interval [a,b]
              of width less than or equal to
                      ABSTOL + EPS *   max( |a|,|b| ) ,
              where EPS is the machine precision.  If ABSTOL is less than
              or equal to zero, then  EPS*|T|  will be used in its place,
              where |T| is the 1-norm of the tridiagonal matrix obtained
              by reducing A to tridiagonal form.
              Eigenvalues will be computed most accurately when ABSTOL is
              set to twice the underflow threshold 2*SLAMCH('S'), not zero.
              If this routine returns with INFO>0, indicating that some
              eigenvectors did not converge, try setting ABSTOL to
              2*SLAMCH('S').

    M

              M is INTEGER
              The total number of eigenvalues found.  0 <= M <= N.
              If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

    W

              W is REAL array, dimension (N)
              If INFO = 0, the eigenvalues in ascending order.

    Z

              Z is REAL array, dimension (LDZ, N)
              If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
              eigenvectors, with the i-th column of Z holding the
              eigenvector associated with W(i).  The eigenvectors are
              normalized so Z**T*B*Z = I.
              If JOBZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    WORK

              WORK is REAL array, dimension (7*N)

    IWORK

              IWORK is INTEGER array, dimension (5*N)

    IFAIL

              IFAIL is INTEGER array, dimension (M)
              If JOBZ = 'V', then if INFO = 0, the first M elements of
              IFAIL are zero.  If INFO > 0, then IFAIL contains the
              indices of the eigenvalues that failed to converge.
              If JOBZ = 'N', then IFAIL is not referenced.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              <= N: if INFO = i, then i eigenvectors failed to converge.
                      Their indices are stored in IFAIL.
              > N:  SPBSTF returned an error code; i.e.,
                    if INFO = N + i, for 1 <= i <= N, then the leading
                    minor of order i of B is not positive definite.
                    The factorization of B could not be completed and
                    no eigenvalues or eigenvectors were computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 291 of file ssbgvx.f.

  subroutine sspev (character jobz, character uplo, integer n, real, dimension( * ) ap, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer info)

SSPEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

     SSPEV computes all the eigenvalues and, optionally, eigenvectors of a
     real symmetric matrix A in packed storage.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    AP

              AP is REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              On exit, AP is overwritten by values generated during the
              reduction to tridiagonal form.  If UPLO = 'U', the diagonal
              and first superdiagonal of the tridiagonal matrix T overwrite
              the corresponding elements of A, and if UPLO = 'L', the
              diagonal and first subdiagonal of T overwrite the
              corresponding elements of A.

    W

              W is REAL array, dimension (N)
              If INFO = 0, the eigenvalues in ascending order.

    Z

              Z is REAL array, dimension (LDZ, N)
              If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
              eigenvectors of the matrix A, with the i-th column of Z
              holding the eigenvector associated with W(i).
              If JOBZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    WORK

              WORK is REAL array, dimension (3*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  if INFO = i, the algorithm failed to converge; i
                    off-diagonal elements of an intermediate tridiagonal
                    form did not converge to zero.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 129 of file sspev.f.

  subroutine sspevd (character jobz, character uplo, integer n, real, dimension( * ) ap, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

SSPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

     SSPEVD computes all the eigenvalues and, optionally, eigenvectors
     of a real symmetric matrix A in packed storage. If eigenvectors are
     desired, it uses a divide and conquer algorithm.
     The divide and conquer algorithm 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
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    AP

              AP is REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              On exit, AP is overwritten by values generated during the
              reduction to tridiagonal form.  If UPLO = 'U', the diagonal
              and first superdiagonal of the tridiagonal matrix T overwrite
              the corresponding elements of A, and if UPLO = 'L', the
              diagonal and first subdiagonal of T overwrite the
              corresponding elements of A.

    W

              W is REAL array, dimension (N)
              If INFO = 0, the eigenvalues in ascending order.

    Z

              Z is REAL array, dimension (LDZ, N)
              If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
              eigenvectors of the matrix A, with the i-th column of Z
              holding the eigenvector associated with W(i).
              If JOBZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If N <= 1,               LWORK must be at least 1.
              If JOBZ = 'N' and N > 1, LWORK must be at least 2*N.
              If JOBZ = 'V' and N > 1, LWORK must be at least
                                                     1 + 6*N + N**2.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the required sizes of the WORK and IWORK
              arrays, returns these values as the first entries of the WORK
              and IWORK arrays, and no error message related to LWORK or
              LIWORK is issued by XERBLA.

    IWORK

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

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.
              If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
              If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the required sizes of the WORK and
              IWORK arrays, returns these values as the first entries of
              the WORK and IWORK arrays, and no error message related to
              LWORK or 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:  if INFO = i, the algorithm failed to converge; i
                    off-diagonal elements of an intermediate tridiagonal
                    form did not converge to zero.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 176 of file sspevd.f.

  subroutine sspevx (character jobz, character range, character uplo, integer n, real, dimension( * ) ap, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

SSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

     SSPEVX computes selected eigenvalues and, optionally, eigenvectors
     of a real symmetric matrix A in packed storage.  Eigenvalues/vectors
     can be selected by specifying either a range of values or a range of
     indices for the desired eigenvalues.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    RANGE

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

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A is stored;
              = 'L':  Lower triangle of A is stored.

    N

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

    AP

              AP is REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              On exit, AP is overwritten by values generated during the
              reduction to tridiagonal form.  If UPLO = 'U', the diagonal
              and first superdiagonal of the tridiagonal matrix T overwrite
              the corresponding elements of A, and if UPLO = 'L', the
              diagonal and first subdiagonal of T overwrite the
              corresponding elements of A.

    VL

              VL is REAL
              If RANGE='V', the lower bound of the interval to
              be searched for eigenvalues. 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. 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 error tolerance for the eigenvalues.
              An approximate eigenvalue is accepted as converged
              when it is determined to lie in an interval [a,b]
              of width less than or equal to
                      ABSTOL + EPS *   max( |a|,|b| ) ,
              where EPS is the machine precision.  If ABSTOL is less than
              or equal to zero, then  EPS*|T|  will be used in its place,
              where |T| is the 1-norm of the tridiagonal matrix obtained
              by reducing AP to tridiagonal form.
              Eigenvalues will be computed most accurately when ABSTOL is
              set to twice the underflow threshold 2*SLAMCH('S'), not zero.
              If this routine returns with INFO>0, indicating that some
              eigenvectors did not converge, try setting ABSTOL to
              2*SLAMCH('S').
              See 'Computing Small Singular Values of Bidiagonal Matrices
              with Guaranteed High Relative Accuracy,' by Demmel and
              Kahan, LAPACK Working Note #3.

    M

              M is INTEGER
              The total number of eigenvalues found.  0 <= M <= N.
              If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

    W

              W is REAL array, dimension (N)
              If INFO = 0, the selected eigenvalues in ascending order.

    Z

              Z is REAL array, dimension (LDZ, max(1,M))
              If JOBZ = 'V', then if INFO = 0, the first M columns of Z
              contain the orthonormal eigenvectors of the matrix A
              corresponding to the selected eigenvalues, with the i-th
              column of Z holding the eigenvector associated with W(i).
              If an eigenvector fails to converge, then that column of Z
              contains the latest approximation to the eigenvector, and the
              index of the eigenvector is returned in IFAIL.
              If JOBZ = 'N', then Z is not referenced.
              Note: the user must ensure that at least max(1,M) columns are
              supplied in the array Z; if RANGE = 'V', the exact value of M
              is not known in advance and an upper bound must be used.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    WORK

              WORK is REAL array, dimension (8*N)

    IWORK

              IWORK is INTEGER array, dimension (5*N)

    IFAIL

              IFAIL is INTEGER array, dimension (N)
              If JOBZ = 'V', then if INFO = 0, the first M elements of
              IFAIL are zero.  If INFO > 0, then IFAIL contains the
              indices of the eigenvectors that failed to converge.
              If JOBZ = 'N', then IFAIL is not referenced.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, then i eigenvectors failed to converge.
                    Their indices are stored in array IFAIL.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 231 of file sspevx.f.

  subroutine sspgv (integer itype, character jobz, character uplo, integer n, real, dimension( * ) ap, real, dimension( * ) bp, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer info)

SSPGV

Purpose:

     SSPGV computes all the eigenvalues and, optionally, the eigenvectors
     of a real generalized symmetric-definite eigenproblem, of the form
     A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
     Here A and B are assumed to be symmetric, stored in packed format,
     and B is also positive definite.

Parameters
    ITYPE

              ITYPE is INTEGER
              Specifies the problem type to be solved:
              = 1:  A*x = (lambda)*B*x
              = 2:  A*B*x = (lambda)*x
              = 3:  B*A*x = (lambda)*x

    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangles of A and B are stored;
              = 'L':  Lower triangles of A and B are stored.

    N

              N is INTEGER
              The order of the matrices A and B.  N >= 0.

    AP

              AP is REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              On exit, the contents of AP are destroyed.

    BP

              BP is REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              B, packed columnwise in a linear array.  The j-th column of B
              is stored in the array BP as follows:
              if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
              if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
              On exit, the triangular factor U or L from the Cholesky
              factorization B = U**T*U or B = L*L**T, in the same storage
              format as B.

    W

              W is REAL array, dimension (N)
              If INFO = 0, the eigenvalues in ascending order.

    Z

              Z is REAL array, dimension (LDZ, N)
              If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
              eigenvectors.  The eigenvectors are normalized as follows:
              if ITYPE = 1 or 2, Z**T*B*Z = I;
              if ITYPE = 3, Z**T*inv(B)*Z = I.
              If JOBZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    WORK

              WORK is REAL array, dimension (3*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  SPPTRF or SSPEV returned an error code:
                 <= N:  if INFO = i, SSPEV failed to converge;
                        i off-diagonal elements of an intermediate
                        tridiagonal form did not converge to zero.
                 > N:   if INFO = n + i, for 1 <= i <= n, then the leading
                        minor of order i of B is not positive definite.
                        The factorization of B could not be completed and
                        no eigenvalues or eigenvectors were computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 158 of file sspgv.f.

  subroutine sspgvd (integer itype, character jobz, character uplo, integer n, real, dimension( * ) ap, real, dimension( * ) bp, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

SSPGVD

Purpose:

     SSPGVD computes all the eigenvalues, and optionally, the eigenvectors
     of a real generalized symmetric-definite eigenproblem, of the form
     A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
     B are assumed to be symmetric, stored in packed format, and B is also
     positive definite.
     If eigenvectors are desired, it uses a divide and conquer algorithm.
     The divide and conquer algorithm 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
    ITYPE

              ITYPE is INTEGER
              Specifies the problem type to be solved:
              = 1:  A*x = (lambda)*B*x
              = 2:  A*B*x = (lambda)*x
              = 3:  B*A*x = (lambda)*x

    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangles of A and B are stored;
              = 'L':  Lower triangles of A and B are stored.

    N

              N is INTEGER
              The order of the matrices A and B.  N >= 0.

    AP

              AP is REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              On exit, the contents of AP are destroyed.

    BP

              BP is REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              B, packed columnwise in a linear array.  The j-th column of B
              is stored in the array BP as follows:
              if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
              if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
              On exit, the triangular factor U or L from the Cholesky
              factorization B = U**T*U or B = L*L**T, in the same storage
              format as B.

    W

              W is REAL array, dimension (N)
              If INFO = 0, the eigenvalues in ascending order.

    Z

              Z is REAL array, dimension (LDZ, N)
              If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
              eigenvectors.  The eigenvectors are normalized as follows:
              if ITYPE = 1 or 2, Z**T*B*Z = I;
              if ITYPE = 3, Z**T*inv(B)*Z = I.
              If JOBZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If N <= 1,               LWORK >= 1.
              If JOBZ = 'N' and N > 1, LWORK >= 2*N.
              If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the required sizes of the WORK and IWORK
              arrays, returns these values as the first entries of the WORK
              and IWORK arrays, and no error message related to LWORK or
              LIWORK is issued by XERBLA.

    IWORK

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

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.
              If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
              If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the required sizes of the WORK and
              IWORK arrays, returns these values as the first entries of
              the WORK and IWORK arrays, and no error message related to
              LWORK or 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:  SPPTRF or SSPEVD returned an error code:
                 <= N:  if INFO = i, SSPEVD failed to converge;
                        i off-diagonal elements of an intermediate
                        tridiagonal form did not converge to zero;
                 > N:   if INFO = N + i, for 1 <= i <= N, then the leading
                        minor of order i of B is not positive definite.
                        The factorization of B could not be completed and
                        no eigenvalues or eigenvectors were computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 208 of file sspgvd.f.

  subroutine sspgvx (integer itype, character jobz, character range, character uplo, integer n, real, dimension( * ) ap, real, dimension( * ) bp, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

SSPGVX

Purpose:

     SSPGVX computes selected eigenvalues, and optionally, eigenvectors
     of a real generalized symmetric-definite eigenproblem, of the form
     A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
     and B are assumed to be symmetric, stored in packed storage, and B
     is also positive definite.  Eigenvalues and eigenvectors can be
     selected by specifying either a range of values or a range of indices
     for the desired eigenvalues.

Parameters
    ITYPE

              ITYPE is INTEGER
              Specifies the problem type to be solved:
              = 1:  A*x = (lambda)*B*x
              = 2:  A*B*x = (lambda)*x
              = 3:  B*A*x = (lambda)*x

    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    RANGE

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

    UPLO

              UPLO is CHARACTER*1
              = 'U':  Upper triangle of A and B are stored;
              = 'L':  Lower triangle of A and B are stored.

    N

              N is INTEGER
              The order of the matrix pencil (A,B).  N >= 0.

    AP

              AP is REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              On exit, the contents of AP are destroyed.

    BP

              BP is REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              B, packed columnwise in a linear array.  The j-th column of B
              is stored in the array BP as follows:
              if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
              if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
              On exit, the triangular factor U or L from the Cholesky
              factorization B = U**T*U or B = L*L**T, in the same storage
              format as B.

    VL

              VL is REAL
              If RANGE='V', the lower bound of the interval to
              be searched for eigenvalues. 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. 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 error tolerance for the eigenvalues.
              An approximate eigenvalue is accepted as converged
              when it is determined to lie in an interval [a,b]
              of width less than or equal to
                      ABSTOL + EPS *   max( |a|,|b| ) ,
              where EPS is the machine precision.  If ABSTOL is less than
              or equal to zero, then  EPS*|T|  will be used in its place,
              where |T| is the 1-norm of the tridiagonal matrix obtained
              by reducing A to tridiagonal form.
              Eigenvalues will be computed most accurately when ABSTOL is
              set to twice the underflow threshold 2*SLAMCH('S'), not zero.
              If this routine returns with INFO>0, indicating that some
              eigenvectors did not converge, try setting ABSTOL to
              2*SLAMCH('S').

    M

              M is INTEGER
              The total number of eigenvalues found.  0 <= M <= N.
              If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

    W

              W is REAL array, dimension (N)
              On normal exit, the first M elements contain the selected
              eigenvalues in ascending order.

    Z

              Z is REAL array, dimension (LDZ, max(1,M))
              If JOBZ = 'N', then Z is not referenced.
              If JOBZ = 'V', then if INFO = 0, the first M columns of Z
              contain the orthonormal eigenvectors of the matrix A
              corresponding to the selected eigenvalues, with the i-th
              column of Z holding the eigenvector associated with W(i).
              The eigenvectors are normalized as follows:
              if ITYPE = 1 or 2, Z**T*B*Z = I;
              if ITYPE = 3, Z**T*inv(B)*Z = I.
              If an eigenvector fails to converge, then that column of Z
              contains the latest approximation to the eigenvector, and the
              index of the eigenvector is returned in IFAIL.
              Note: the user must ensure that at least max(1,M) columns are
              supplied in the array Z; if RANGE = 'V', the exact value of M
              is not known in advance and an upper bound must be used.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    WORK

              WORK is REAL array, dimension (8*N)

    IWORK

              IWORK is INTEGER array, dimension (5*N)

    IFAIL

              IFAIL is INTEGER array, dimension (N)
              If JOBZ = 'V', then if INFO = 0, the first M elements of
              IFAIL are zero.  If INFO > 0, then IFAIL contains the
              indices of the eigenvectors that failed to converge.
              If JOBZ = 'N', then IFAIL is not referenced.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  SPPTRF or SSPEVX returned an error code:
                 <= N:  if INFO = i, SSPEVX failed to converge;
                        i eigenvectors failed to converge.  Their indices
                        are stored in array IFAIL.
                 > N:   if INFO = N + i, for 1 <= i <= N, then the leading
                        minor of order i of B is not positive definite.
                        The factorization of B could not be completed and
                        no eigenvalues or eigenvectors were computed.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 269 of file sspgvx.f.

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

SSTEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

     SSTEV computes all eigenvalues and, optionally, eigenvectors of a
     real symmetric tridiagonal matrix A.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    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
              A.
              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 A, stored in elements 1 to N-1 of E.
              On exit, the contents of E are destroyed.

    Z

              Z is REAL array, dimension (LDZ, N)
              If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
              eigenvectors of the matrix A, with the i-th column of Z
              holding the eigenvector associated with D(i).
              If JOBZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    WORK

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

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, the algorithm failed to converge; i
                    off-diagonal elements of E did not converge to zero.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 115 of file sstev.f.

  subroutine sstevd (character jobz, 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)

SSTEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

     SSTEVD computes all eigenvalues and, optionally, eigenvectors of a
     real symmetric tridiagonal matrix. If eigenvectors are desired, it
     uses a divide and conquer algorithm.
     The divide and conquer algorithm 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
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    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
              A.
              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 A, stored in elements 1 to N-1 of E.
              On exit, the contents of E are destroyed.

    Z

              Z is REAL array, dimension (LDZ, N)
              If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
              eigenvectors of the matrix A, with the i-th column of Z
              holding the eigenvector associated with D(i).
              If JOBZ = 'N', then Z is not referenced.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If JOBZ  = 'N' or N <= 1 then LWORK must be at least 1.
              If JOBZ  = 'V' and N > 1 then LWORK must be at least
                             ( 1 + 4*N + N**2 ).
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal sizes of the WORK and IWORK
              arrays, returns these values as the first entries of the WORK
              and IWORK arrays, and no error message related to LWORK or
              LIWORK 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 JOBZ  = 'N' or N <= 1 then LIWORK must be at least 1.
              If JOBZ  = 'V' and N > 1 then LIWORK must be at least 3+5*N.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal sizes of the WORK and
              IWORK arrays, returns these values as the first entries of
              the WORK and IWORK arrays, and no error message related to
              LWORK or 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:  if INFO = i, the algorithm failed to converge; i
                    off-diagonal elements of E did not converge to zero.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 161 of file sstevd.f.

  subroutine sstevr (character jobz, character range, integer n, real, dimension( * ) d, real, dimension( * ) e, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

SSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

     SSTEVR computes selected eigenvalues and, optionally, eigenvectors
     of a real symmetric tridiagonal matrix T.  Eigenvalues and
     eigenvectors can be selected by specifying either a range of values
     or a range of indices for the desired eigenvalues.
     Whenever possible, SSTEVR calls SSTEMR to compute the
     eigenspectrum using Relatively Robust Representations.  SSTEMR
     computes eigenvalues by the dqds algorithm, while orthogonal
     eigenvectors are computed from various 'good' L D L^T representations
     (also known as Relatively Robust Representations). Gram-Schmidt
     orthogonalization is avoided as far as possible. More specifically,
     the various steps of the algorithm are as follows. For the i-th
     unreduced block of T,
        (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
             is a relatively robust representation,
        (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
            relative accuracy by the dqds algorithm,
        (c) If there is a cluster of close eigenvalues, 'choose' sigma_i
            close to the cluster, and go to step (a),
        (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
            compute the corresponding eigenvector by forming a
            rank-revealing twisted factorization.
     The desired accuracy of the output can be specified by the input
     parameter ABSTOL.
     For more details, see 'A new O(n^2) algorithm for the symmetric
     tridiagonal eigenvalue/eigenvector problem', by Inderjit Dhillon,
     Computer Science Division Technical Report No. UCB//CSD-97-971,
     UC Berkeley, May 1997.
     Note 1 : SSTEVR calls SSTEMR when the full spectrum is requested
     on machines which conform to the ieee-754 floating point standard.
     SSTEVR calls SSTEBZ and SSTEIN on non-ieee machines and
     when partial spectrum requests are made.
     Normal execution of SSTEMR may create NaNs and infinities and
     hence may abort due to a floating point exception in environments
     which do not handle NaNs and infinities in the ieee standard default
     manner.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    RANGE

              RANGE is CHARACTER*1
              = 'A': all eigenvalues will be found.
              = 'V': all eigenvalues in the half-open interval (VL,VU]
                     will be found.
              = 'I': the IL-th through IU-th eigenvalues will be found.
              For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and
              SSTEIN are called

    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
              A.
              On exit, D may be multiplied by a constant factor chosen
              to avoid over/underflow in computing the eigenvalues.

    E

              E is REAL array, dimension (max(1,N-1))
              On entry, the (n-1) subdiagonal elements of the tridiagonal
              matrix A in elements 1 to N-1 of E.
              On exit, E may be multiplied by a constant factor chosen
              to avoid over/underflow in computing the eigenvalues.

    VL

              VL is REAL
              If RANGE='V', the lower bound of the interval to
              be searched for eigenvalues. 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. 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 error tolerance for the eigenvalues.
              An approximate eigenvalue is accepted as converged
              when it is determined to lie in an interval [a,b]
              of width less than or equal to
                      ABSTOL + EPS *   max( |a|,|b| ) ,
              where EPS is the machine precision.  If ABSTOL is less than
              or equal to zero, then  EPS*|T|  will be used in its place,
              where |T| is the 1-norm of the tridiagonal matrix obtained
              by reducing A to tridiagonal form.
              See 'Computing Small Singular Values of Bidiagonal Matrices
              with Guaranteed High Relative Accuracy,' by Demmel and
              Kahan, LAPACK Working Note #3.
              If high relative accuracy is important, set ABSTOL to
              SLAMCH( 'Safe minimum' ).  Doing so will guarantee that
              eigenvalues are computed to high relative accuracy when
              possible in future releases.  The current code does not
              make any guarantees about high relative accuracy, but
              future releases will. See J. Barlow and J. Demmel,
              'Computing Accurate Eigensystems of Scaled Diagonally
              Dominant Matrices', LAPACK Working Note #7, for a discussion
              of which matrices define their eigenvalues to high relative
              accuracy.

    M

              M is INTEGER
              The total number of eigenvalues found.  0 <= M <= N.
              If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

    W

              W is REAL array, dimension (N)
              The first M elements contain the selected eigenvalues in
              ascending order.

    Z

              Z is REAL array, dimension (LDZ, max(1,M) )
              If JOBZ = 'V', then if INFO = 0, the first M columns of Z
              contain the orthonormal eigenvectors of the matrix A
              corresponding to the selected eigenvalues, with the i-th
              column of Z holding the eigenvector associated with W(i).
              Note: the user must ensure that at least max(1,M) columns are
              supplied in the array Z; if RANGE = 'V', the exact value of M
              is not known in advance and an upper bound must be used.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    ISUPPZ

              ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
              The support of the eigenvectors in Z, i.e., the indices
              indicating the nonzero elements in Z. The i-th eigenvector
              is nonzero only in elements ISUPPZ( 2*i-1 ) through
              ISUPPZ( 2*i ).
              Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= 20*N.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal sizes of the WORK and IWORK
              arrays, returns these values as the first entries of the WORK
              and IWORK arrays, and no error message related to LWORK or
              LIWORK is issued by XERBLA.

    IWORK

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

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.  LIWORK >= 10*N.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal sizes of the WORK and
              IWORK arrays, returns these values as the first entries of
              the WORK and IWORK arrays, and no error message related to
              LWORK or 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:  Internal error

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Inderjit Dhillon, IBM Almaden, USA
     Osni Marques, LBNL/NERSC, USA
     Ken Stanley, Computer Science Division, University of California at Berkeley, USA
     Jason Riedy, Computer Science Division, University of California at Berkeley, USA

Definition at line 303 of file sstevr.f.

  subroutine sstevx (character jobz, character range, integer n, real, dimension( * ) d, real, dimension( * ) e, real vl, real vu, integer il, integer iu, real abstol, integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info)

SSTEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices

Purpose:

     SSTEVX computes selected eigenvalues and, optionally, eigenvectors
     of a real symmetric tridiagonal matrix A.  Eigenvalues and
     eigenvectors can be selected by specifying either a range of values
     or a range of indices for the desired eigenvalues.

Parameters
    JOBZ

              JOBZ is CHARACTER*1
              = 'N':  Compute eigenvalues only;
              = 'V':  Compute eigenvalues and eigenvectors.

    RANGE

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

    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
              A.
              On exit, D may be multiplied by a constant factor chosen
              to avoid over/underflow in computing the eigenvalues.

    E

              E is REAL array, dimension (max(1,N-1))
              On entry, the (n-1) subdiagonal elements of the tridiagonal
              matrix A in elements 1 to N-1 of E.
              On exit, E may be multiplied by a constant factor chosen
              to avoid over/underflow in computing the eigenvalues.

    VL

              VL is REAL
              If RANGE='V', the lower bound of the interval to
              be searched for eigenvalues. 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. 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 error tolerance for the eigenvalues.
              An approximate eigenvalue is accepted as converged
              when it is determined to lie in an interval [a,b]
              of width less than or equal to
                      ABSTOL + EPS *   max( |a|,|b| ) ,
              where EPS is the machine precision.  If ABSTOL is less
              than or equal to zero, then  EPS*|T|  will be used in
              its place, where |T| is the 1-norm of the tridiagonal
              matrix.
              Eigenvalues will be computed most accurately when ABSTOL is
              set to twice the underflow threshold 2*SLAMCH('S'), not zero.
              If this routine returns with INFO>0, indicating that some
              eigenvectors did not converge, try setting ABSTOL to
              2*SLAMCH('S').
              See 'Computing Small Singular Values of Bidiagonal Matrices
              with Guaranteed High Relative Accuracy,' by Demmel and
              Kahan, LAPACK Working Note #3.

    M

              M is INTEGER
              The total number of eigenvalues found.  0 <= M <= N.
              If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

    W

              W is REAL array, dimension (N)
              The first M elements contain the selected eigenvalues in
              ascending order.

    Z

              Z is REAL array, dimension (LDZ, max(1,M) )
              If JOBZ = 'V', then if INFO = 0, the first M columns of Z
              contain the orthonormal eigenvectors of the matrix A
              corresponding to the selected eigenvalues, with the i-th
              column of Z holding the eigenvector associated with W(i).
              If an eigenvector fails to converge (INFO > 0), then that
              column of Z contains the latest approximation to the
              eigenvector, and the index of the eigenvector is returned
              in IFAIL.  If JOBZ = 'N', then Z is not referenced.
              Note: the user must ensure that at least max(1,M) columns are
              supplied in the array Z; if RANGE = 'V', the exact value of M
              is not known in advance and an upper bound must be used.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    WORK

              WORK is REAL array, dimension (5*N)

    IWORK

              IWORK is INTEGER array, dimension (5*N)

    IFAIL

              IFAIL is INTEGER array, dimension (N)
              If JOBZ = 'V', then if INFO = 0, the first M elements of
              IFAIL are zero.  If INFO > 0, then IFAIL contains the
              indices of the eigenvectors that failed to converge.
              If JOBZ = 'N', then IFAIL is not referenced.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  if INFO = i, then i eigenvectors failed to converge.
                    Their indices are stored in array IFAIL.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 225 of file sstevx.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.