complexGBcomputational(3)

NAME
complexGBcomputational - complex

SYNOPSIS

  Functions
subroutine cgbbrd (vect, m, n, ncc, kl, ku, ab, ldab, d, e, q, ldq, pt, ldpt, c, ldc, work, rwork, info)
CGBBRD
subroutine cgbcon (norm, n, kl, ku, ab, ldab, ipiv, anorm, rcond, work, rwork, info)
CGBCON
subroutine cgbequ (m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info)
CGBEQU
subroutine cgbequb (m, n, kl, ku, ab, ldab, r, c, rowcnd, colcnd, amax, info)
CGBEQUB
subroutine cgbrfs (trans, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
CGBRFS
subroutine cgbrfsx (trans, equed, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, r, c, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, rwork, info)
CGBRFSX
subroutine cgbtf2 (m, n, kl, ku, ab, ldab, ipiv, info)
CGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm.
subroutine cgbtrf (m, n, kl, ku, ab, ldab, ipiv, info)
CGBTRF
subroutine cgbtrs (trans, n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)
CGBTRS
subroutine cggbak (job, side, n, ilo, ihi, lscale, rscale, m, v, ldv, info)
CGGBAK
subroutine cggbal (job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)
CGGBAL
subroutine cla_gbamv (trans, m, n, kl, ku, alpha, ab, ldab, x, incx, beta, y, incy)
CLA_GBAMV performs a matrix-vector operation to calculate error bounds.
real function cla_gbrcond_c (trans, n, kl, ku, ab, ldab, afb, ldafb, ipiv, c, capply, info, work, rwork)
CLA_GBRCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for general banded matrices.
real function cla_gbrcond_x (trans, n, kl, ku, ab, ldab, afb, ldafb, ipiv, x, info, work, rwork)
CLA_GBRCOND_X computes the infinity norm condition number of op(A)*diag(x) for general banded matrices.
subroutine cla_gbrfsx_extended (prec_type, trans_type, n, kl, ku, nrhs, ab, ldab, afb, ldafb, ipiv, colequ, c, b, ldb, y, ldy, berr_out, n_norms, err_bnds_norm, err_bnds_comp, res, ayb, dy, y_tail, rcond, ithresh, rthresh, dz_ub, ignore_cwise, info)
CLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
real function cla_gbrpvgrw (n, kl, ku, ncols, ab, ldab, afb, ldafb)
CLA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix.
subroutine cungbr (vect, m, n, k, a, lda, tau, work, lwork, info)
CUNGBR

Detailed Description

This is the group of complex computational functions for GB matrices

Function Documentation

  subroutine cgbbrd (character vect, integer m, integer n, integer ncc, integer kl, integer ku, complex, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( ldpt, * ) pt, integer ldpt, complex, dimension( ldc, * ) c, integer ldc, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CGBBRD

Purpose:

     CGBBRD reduces a complex general m-by-n band matrix A to real upper
     bidiagonal form B by a unitary transformation: Q**H * A * P = B.
     The routine computes B, and optionally forms Q or P**H, or computes
     Q**H*C for a given matrix C.

Parameters
    VECT

              VECT is CHARACTER*1
              Specifies whether or not the matrices Q and P**H are to be
              formed.
              = 'N': do not form Q or P**H;
              = 'Q': form Q only;
              = 'P': form P**H only;
              = 'B': form both.

    M

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

    N

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

    NCC

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

    KL

              KL is INTEGER
              The number of subdiagonals of the matrix A. KL >= 0.

    KU

              KU is INTEGER
              The number of superdiagonals of the matrix A. KU >= 0.

    AB

              AB is COMPLEX array, dimension (LDAB,N)
              On entry, the m-by-n band matrix A, stored in rows 1 to
              KL+KU+1. The j-th column of A is stored in the j-th column of
              the array AB as follows:
              AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
              On exit, A is overwritten by values generated during the
              reduction.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array A. LDAB >= KL+KU+1.

    D

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

    E

              E is REAL array, dimension (min(M,N)-1)
              The superdiagonal elements of the bidiagonal matrix B.

    Q

              Q is COMPLEX array, dimension (LDQ,M)
              If VECT = 'Q' or 'B', the m-by-m unitary matrix Q.
              If VECT = 'N' or 'P', the array Q is not referenced.

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.
              LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.

    PT

              PT is COMPLEX array, dimension (LDPT,N)
              If VECT = 'P' or 'B', the n-by-n unitary matrix P'.
              If VECT = 'N' or 'Q', the array PT is not referenced.

    LDPT

              LDPT is INTEGER
              The leading dimension of the array PT.
              LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.

    C

              C is COMPLEX array, dimension (LDC,NCC)
              On entry, an m-by-ncc matrix C.
              On exit, C is overwritten by Q**H*C.
              C is not referenced if NCC = 0.

    LDC

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

    WORK

              WORK is COMPLEX array, dimension (max(M,N))

    RWORK

              RWORK is REAL array, dimension (max(M,N))

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 191 of file cgbbrd.f.

  subroutine cgbcon (character norm, integer n, integer kl, integer ku, complex, dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv, real anorm, real rcond, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CGBCON

Purpose:

     CGBCON estimates the reciprocal of the condition number of a complex
     general band matrix A, in either the 1-norm or the infinity-norm,
     using the LU factorization computed by CGBTRF.
     An estimate is obtained for norm(inv(A)), and the reciprocal of the
     condition number is computed as
        RCOND = 1 / ( norm(A) * norm(inv(A)) ).

Parameters
    NORM

              NORM is CHARACTER*1
              Specifies whether the 1-norm condition number or the
              infinity-norm condition number is required:
              = '1' or 'O':  1-norm;
              = 'I':         Infinity-norm.

    N

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

    KL

              KL is INTEGER
              The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
              The number of superdiagonals within the band of A.  KU >= 0.

    AB

              AB is COMPLEX array, dimension (LDAB,N)
              Details of the LU factorization of the band matrix A, as
              computed by CGBTRF.  U is stored as an upper triangular band
              matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
              the multipliers used during the factorization are stored in
              rows KL+KU+2 to 2*KL+KU+1.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

    IPIV

              IPIV is INTEGER array, dimension (N)
              The pivot indices; for 1 <= i <= N, row i of the matrix was
              interchanged with row IPIV(i).

    ANORM

              ANORM is REAL
              If NORM = '1' or 'O', the 1-norm of the original matrix A.
              If NORM = 'I', the infinity-norm of the original matrix A.

    RCOND

              RCOND is REAL
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(norm(A) * norm(inv(A))).

    WORK

              WORK is COMPLEX array, dimension (2*N)

    RWORK

              RWORK is REAL array, dimension (N)

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 145 of file cgbcon.f.

  subroutine cgbequ (integer m, integer n, integer kl, integer ku, complex, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) r, real, dimension( * ) c, real rowcnd, real colcnd, real amax, integer info)

CGBEQU

Purpose:

     CGBEQU computes row and column scalings intended to equilibrate an
     M-by-N band matrix A and reduce its condition number.  R returns the
     row scale factors and C the column scale factors, chosen to try to
     make the largest element in each row and column of the matrix B with
     elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
     R(i) and C(j) are restricted to be between SMLNUM = smallest safe
     number and BIGNUM = largest safe number.  Use of these scaling
     factors is not guaranteed to reduce the condition number of A but
     works well in practice.

Parameters
    M

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

    N

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

    KL

              KL is INTEGER
              The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
              The number of superdiagonals within the band of A.  KU >= 0.

    AB

              AB is COMPLEX array, dimension (LDAB,N)
              The band matrix A, stored in rows 1 to KL+KU+1.  The j-th
              column of A is stored in the j-th column of the array AB as
              follows:
              AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).

    LDAB

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

    R

              R is REAL array, dimension (M)
              If INFO = 0, or INFO > M, R contains the row scale factors
              for A.

    C

              C is REAL array, dimension (N)
              If INFO = 0, C contains the column scale factors for A.

    ROWCND

              ROWCND is REAL
              If INFO = 0 or INFO > M, ROWCND contains the ratio of the
              smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
              AMAX is neither too large nor too small, it is not worth
              scaling by R.

    COLCND

              COLCND is REAL
              If INFO = 0, COLCND contains the ratio of the smallest
              C(i) to the largest C(i).  If COLCND >= 0.1, it is not
              worth scaling by C.

    AMAX

              AMAX is REAL
              Absolute value of largest matrix element.  If AMAX is very
              close to overflow or very close to underflow, the matrix
              should be scaled.

    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
                    <= M:  the i-th row of A is exactly zero
                    >  M:  the (i-M)-th column of A is exactly zero

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 152 of file cgbequ.f.

  subroutine cgbequb (integer m, integer n, integer kl, integer ku, complex, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) r, real, dimension( * ) c, real rowcnd, real colcnd, real amax, integer info)

CGBEQUB

Purpose:

     CGBEQUB computes row and column scalings intended to equilibrate an
     M-by-N matrix A and reduce its condition number.  R returns the row
     scale factors and C the column scale factors, chosen to try to make
     the largest element in each row and column of the matrix B with
     elements B(i,j)=R(i)*A(i,j)*C(j) have an absolute value of at most
     the radix.
     R(i) and C(j) are restricted to be a power of the radix between
     SMLNUM = smallest safe number and BIGNUM = largest safe number.  Use
     of these scaling factors is not guaranteed to reduce the condition
     number of A but works well in practice.
     This routine differs from CGEEQU by restricting the scaling factors
     to a power of the radix.  Barring over- and underflow, scaling by
     these factors introduces no additional rounding errors.  However, the
     scaled entries' magnitudes are no longer approximately 1 but lie
     between sqrt(radix) and 1/sqrt(radix).

Parameters
    M

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

    N

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

    KL

              KL is INTEGER
              The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
              The number of superdiagonals within the band of A.  KU >= 0.

    AB

              AB is COMPLEX array, dimension (LDAB,N)
              On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
              The j-th column of A is stored in the j-th column of the
              array AB as follows:
              AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

    LDAB

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

    R

              R is REAL array, dimension (M)
              If INFO = 0 or INFO > M, R contains the row scale factors
              for A.

    C

              C is REAL array, dimension (N)
              If INFO = 0,  C contains the column scale factors for A.

    ROWCND

              ROWCND is REAL
              If INFO = 0 or INFO > M, ROWCND contains the ratio of the
              smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
              AMAX is neither too large nor too small, it is not worth
              scaling by R.

    COLCND

              COLCND is REAL
              If INFO = 0, COLCND contains the ratio of the smallest
              C(i) to the largest C(i).  If COLCND >= 0.1, it is not
              worth scaling by C.

    AMAX

              AMAX is REAL
              Absolute value of largest matrix element.  If AMAX is very
              close to overflow or very close to underflow, the matrix
              should be scaled.

    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
                    <= M:  the i-th row of A is exactly zero
                    >  M:  the (i-M)-th column of A is exactly zero

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 159 of file cgbequb.f.

  subroutine cgbrfs (character trans, integer n, integer kl, integer ku, integer nrhs, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldafb, * ) afb, integer ldafb, integer, dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real, dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CGBRFS

Purpose:

     CGBRFS improves the computed solution to a system of linear
     equations when the coefficient matrix is banded, and provides
     error bounds and backward error estimates for the solution.

Parameters
    TRANS

              TRANS is CHARACTER*1
              Specifies the form of the system of equations:
              = 'N':  A * X = B     (No transpose)
              = 'T':  A**T * X = B  (Transpose)
              = 'C':  A**H * X = B  (Conjugate transpose)

    N

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

    KL

              KL is INTEGER
              The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
              The number of superdiagonals within the band of A.  KU >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrices B and X.  NRHS >= 0.

    AB

              AB is COMPLEX array, dimension (LDAB,N)
              The original band matrix A, stored in rows 1 to KL+KU+1.
              The j-th column of A is stored in the j-th column of the
              array AB as follows:
              AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

    LDAB

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

    AFB

              AFB is COMPLEX array, dimension (LDAFB,N)
              Details of the LU factorization of the band matrix A, as
              computed by CGBTRF.  U is stored as an upper triangular band
              matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
              the multipliers used during the factorization are stored in
              rows KL+KU+2 to 2*KL+KU+1.

    LDAFB

              LDAFB is INTEGER
              The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.

    IPIV

              IPIV is INTEGER array, dimension (N)
              The pivot indices from CGBTRF; for 1<=i<=N, row i of the
              matrix was interchanged with row IPIV(i).

    B

              B is COMPLEX array, dimension (LDB,NRHS)
              The right hand side matrix B.

    LDB

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

    X

              X is COMPLEX array, dimension (LDX,NRHS)
              On entry, the solution matrix X, as computed by CGBTRS.
              On exit, the improved solution matrix X.

    LDX

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

    FERR

              FERR is REAL array, dimension (NRHS)
              The estimated forward error bound for each solution vector
              X(j) (the j-th column of the solution matrix X).
              If XTRUE is the true solution corresponding to X(j), FERR(j)
              is an estimated upper bound for the magnitude of the largest
              element in (X(j) - XTRUE) divided by the magnitude of the
              largest element in X(j).  The estimate is as reliable as
              the estimate for RCOND, and is almost always a slight
              overestimate of the true error.

    BERR

              BERR is REAL array, dimension (NRHS)
              The componentwise relative backward error of each solution
              vector X(j) (i.e., the smallest relative change in
              any element of A or B that makes X(j) an exact solution).

    WORK

              WORK is COMPLEX array, dimension (2*N)

    RWORK

              RWORK is REAL array, dimension (N)

    INFO

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

Internal Parameters:

      ITMAX is the maximum number of steps of iterative refinement.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 203 of file cgbrfs.f.

  subroutine cgbrfsx (character trans, character equed, integer n, integer kl, integer ku, integer nrhs, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldafb, * ) afb, integer ldafb, integer, dimension( * ) ipiv, real, dimension( * ) r, real, dimension( * ) c, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx , * ) x, integer ldx, real rcond, real, dimension( * ) berr, integer n_err_bnds, real, dimension( nrhs, * ) err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, integer nparams, real, dimension( * ) params, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CGBRFSX

Purpose:

        CGBRFSX improves the computed solution to a system of linear
        equations and provides error bounds and backward error estimates
        for the solution.  In addition to normwise error bound, the code
        provides maximum componentwise error bound if possible.  See
        comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the
        error bounds.
        The original system of linear equations may have been equilibrated
        before calling this routine, as described by arguments EQUED, R
        and C below. In this case, the solution and error bounds returned
        are for the original unequilibrated system.

         Some optional parameters are bundled in the PARAMS array.  These
         settings determine how refinement is performed, but often the
         defaults are acceptable.  If the defaults are acceptable, users
         can pass NPARAMS = 0 which prevents the source code from accessing
         the PARAMS argument.

Parameters
    TRANS

              TRANS is CHARACTER*1
         Specifies the form of the system of equations:
           = 'N':  A * X = B     (No transpose)
           = 'T':  A**T * X = B  (Transpose)
           = 'C':  A**H * X = B  (Conjugate transpose)

    EQUED

              EQUED is CHARACTER*1
         Specifies the form of equilibration that was done to A
         before calling this routine. This is needed to compute
         the solution and error bounds correctly.
           = 'N':  No equilibration
           = 'R':  Row equilibration, i.e., A has been premultiplied by
                   diag(R).
           = 'C':  Column equilibration, i.e., A has been postmultiplied
                   by diag(C).
           = 'B':  Both row and column equilibration, i.e., A has been
                   replaced by diag(R) * A * diag(C).
                   The right hand side B has been changed accordingly.

    N

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

    KL

              KL is INTEGER
         The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
         The number of superdiagonals within the band of A.  KU >= 0.

    NRHS

              NRHS is INTEGER
         The number of right hand sides, i.e., the number of columns
         of the matrices B and X.  NRHS >= 0.

    AB

              AB is COMPLEX array, dimension (LDAB,N)
         The original band matrix A, stored in rows 1 to KL+KU+1.
         The j-th column of A is stored in the j-th column of the
         array AB as follows:
         AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).

    LDAB

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

    AFB

              AFB is COMPLEX array, dimension (LDAFB,N)
         Details of the LU factorization of the band matrix A, as
         computed by CGBTRF.  U is stored as an upper triangular band
         matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
         the multipliers used during the factorization are stored in
         rows KL+KU+2 to 2*KL+KU+1.

    LDAFB

              LDAFB is INTEGER
         The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.

    IPIV

              IPIV is INTEGER array, dimension (N)
         The pivot indices from CGETRF; for 1<=i<=N, row i of the
         matrix was interchanged with row IPIV(i).

    R

              R is REAL array, dimension (N)
         The row scale factors for A.  If EQUED = 'R' or 'B', A is
         multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
         is not accessed.  R is an input argument if FACT = 'F';
         otherwise, R is an output argument.  If FACT = 'F' and
         EQUED = 'R' or 'B', each element of R must be positive.
         If R is output, each element of R is a power of the radix.
         If R is input, each element of R should be a power of the radix
         to ensure a reliable solution and error estimates. Scaling by
         powers of the radix does not cause rounding errors unless the
         result underflows or overflows. Rounding errors during scaling
         lead to refining with a matrix that is not equivalent to the
         input matrix, producing error estimates that may not be
         reliable.

    C

              C is REAL array, dimension (N)
         The column scale factors for A.  If EQUED = 'C' or 'B', A is
         multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
         is not accessed.  C is an input argument if FACT = 'F';
         otherwise, C is an output argument.  If FACT = 'F' and
         EQUED = 'C' or 'B', each element of C must be positive.
         If C is output, each element of C is a power of the radix.
         If C is input, each element of C should be a power of the radix
         to ensure a reliable solution and error estimates. Scaling by
         powers of the radix does not cause rounding errors unless the
         result underflows or overflows. Rounding errors during scaling
         lead to refining with a matrix that is not equivalent to the
         input matrix, producing error estimates that may not be
         reliable.

    B

              B is COMPLEX array, dimension (LDB,NRHS)
         The right hand side matrix B.

    LDB

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

    X

              X is COMPLEX array, dimension (LDX,NRHS)
         On entry, the solution matrix X, as computed by CGETRS.
         On exit, the improved solution matrix X.

    LDX

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

    RCOND

              RCOND is REAL
         Reciprocal scaled condition number.  This is an estimate of the
         reciprocal Skeel condition number of the matrix A after
         equilibration (if done).  If this is less than the machine
         precision (in particular, if it is zero), the matrix is singular
         to working precision.  Note that the error may still be small even
         if this number is very small and the matrix appears ill-
         conditioned.

    BERR

              BERR is REAL array, dimension (NRHS)
         Componentwise relative backward error.  This is the
         componentwise relative backward error of each solution vector X(j)
         (i.e., the smallest relative change in any element of A or B that
         makes X(j) an exact solution).

    N_ERR_BNDS

              N_ERR_BNDS is INTEGER
         Number of error bounds to return for each right hand side
         and each type (normwise or componentwise).  See ERR_BNDS_NORM and
         ERR_BNDS_COMP below.

    ERR_BNDS_NORM

              ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
         For each right-hand side, this array contains information about
         various error bounds and condition numbers corresponding to the
         normwise relative error, which is defined as follows:
         Normwise relative error in the ith solution vector:
                 max_j (abs(XTRUE(j,i) - X(j,i)))
                ------------------------------
                      max_j abs(X(j,i))
         The array is indexed by the type of error information as described
         below. There currently are up to three pieces of information
         returned.
         The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
         right-hand side.
         The second index in ERR_BNDS_NORM(:,err) contains the following
         three fields:
         err = 1 'Trust/don't trust' boolean. Trust the answer if the
                  reciprocal condition number is less than the threshold
                  sqrt(n) * slamch('Epsilon').
         err = 2 'Guaranteed' error bound: The estimated forward error,
                  almost certainly within a factor of 10 of the true error
                  so long as the next entry is greater than the threshold
                  sqrt(n) * slamch('Epsilon'). This error bound should only
                  be trusted if the previous boolean is true.
         err = 3  Reciprocal condition number: Estimated normwise
                  reciprocal condition number.  Compared with the threshold
                  sqrt(n) * slamch('Epsilon') to determine if the error
                  estimate is 'guaranteed'. These reciprocal condition
                  numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                  appropriately scaled matrix Z.
                  Let Z = S*A, where S scales each row by a power of the
                  radix so all absolute row sums of Z are approximately 1.
         See Lapack Working Note 165 for further details and extra
         cautions.

    ERR_BNDS_COMP

              ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
         For each right-hand side, this array contains information about
         various error bounds and condition numbers corresponding to the
         componentwise relative error, which is defined as follows:
         Componentwise relative error in the ith solution vector:
                        abs(XTRUE(j,i) - X(j,i))
                 max_j ----------------------
                             abs(X(j,i))
         The array is indexed by the right-hand side i (on which the
         componentwise relative error depends), and the type of error
         information as described below. There currently are up to three
         pieces of information returned for each right-hand side. If
         componentwise accuracy is not requested (PARAMS(3) = 0.0), then
         ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
         the first (:,N_ERR_BNDS) entries are returned.
         The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
         right-hand side.
         The second index in ERR_BNDS_COMP(:,err) contains the following
         three fields:
         err = 1 'Trust/don't trust' boolean. Trust the answer if the
                  reciprocal condition number is less than the threshold
                  sqrt(n) * slamch('Epsilon').
         err = 2 'Guaranteed' error bound: The estimated forward error,
                  almost certainly within a factor of 10 of the true error
                  so long as the next entry is greater than the threshold
                  sqrt(n) * slamch('Epsilon'). This error bound should only
                  be trusted if the previous boolean is true.
         err = 3  Reciprocal condition number: Estimated componentwise
                  reciprocal condition number.  Compared with the threshold
                  sqrt(n) * slamch('Epsilon') to determine if the error
                  estimate is 'guaranteed'. These reciprocal condition
                  numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                  appropriately scaled matrix Z.
                  Let Z = S*(A*diag(x)), where x is the solution for the
                  current right-hand side and S scales each row of
                  A*diag(x) by a power of the radix so all absolute row
                  sums of Z are approximately 1.
         See Lapack Working Note 165 for further details and extra
         cautions.

    NPARAMS

              NPARAMS is INTEGER
         Specifies the number of parameters set in PARAMS.  If <= 0, the
         PARAMS array is never referenced and default values are used.

    PARAMS

              PARAMS is REAL array, dimension NPARAMS
         Specifies algorithm parameters.  If an entry is < 0.0, then
         that entry will be filled with default value used for that
         parameter.  Only positions up to NPARAMS are accessed; defaults
         are used for higher-numbered parameters.
           PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
                refinement or not.
             Default: 1.0
                = 0.0:  No refinement is performed, and no error bounds are
                        computed.
                = 1.0:  Use the double-precision refinement algorithm,
                        possibly with doubled-single computations if the
                        compilation environment does not support DOUBLE
                        PRECISION.
                  (other values are reserved for future use)
           PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
                computations allowed for refinement.
             Default: 10
             Aggressive: Set to 100 to permit convergence using approximate
                         factorizations or factorizations other than LU. If
                         the factorization uses a technique other than
                         Gaussian elimination, the guarantees in
                         err_bnds_norm and err_bnds_comp may no longer be
                         trustworthy.
           PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
                will attempt to find a solution with small componentwise
                relative error in the double-precision algorithm.  Positive
                is true, 0.0 is false.
             Default: 1.0 (attempt componentwise convergence)

    WORK

              WORK is COMPLEX array, dimension (2*N)

    RWORK

              RWORK is REAL array, dimension (2*N)

    INFO

              INFO is INTEGER
           = 0:  Successful exit. The solution to every right-hand side is
             guaranteed.
           < 0:  If INFO = -i, the i-th argument had an illegal value
           > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
             has been completed, but the factor U is exactly singular, so
             the solution and error bounds could not be computed. RCOND = 0
             is returned.
           = N+J: The solution corresponding to the Jth right-hand side is
             not guaranteed. The solutions corresponding to other right-
             hand sides K with K > J may not be guaranteed as well, but
             only the first such right-hand side is reported. If a small
             componentwise error is not requested (PARAMS(3) = 0.0) then
             the Jth right-hand side is the first with a normwise error
             bound that is not guaranteed (the smallest J such
             that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
             the Jth right-hand side is the first with either a normwise or
             componentwise error bound that is not guaranteed (the smallest
             J such that either ERR_BNDS_NORM(J,1) = 0.0 or
             ERR_BNDS_COMP(J,1) = 0.0). See the definition of
             ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
             about all of the right-hand sides check ERR_BNDS_NORM or
             ERR_BNDS_COMP.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 435 of file cgbrfsx.f.

  subroutine cgbtf2 (integer m, integer n, integer kl, integer ku, complex, dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv, integer info)

CGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm.

Purpose:

     CGBTF2 computes an LU factorization of a complex m-by-n band matrix
     A using partial pivoting with row interchanges.
     This is the unblocked version of the algorithm, calling Level 2 BLAS.

Parameters
    M

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

    N

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

    KL

              KL is INTEGER
              The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
              The number of superdiagonals within the band of A.  KU >= 0.

    AB

              AB is COMPLEX array, dimension (LDAB,N)
              On entry, the matrix A in band storage, in rows KL+1 to
              2*KL+KU+1; rows 1 to KL of the array need not be set.
              The j-th column of A is stored in the j-th column of the
              array AB as follows:
              AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)
              On exit, details of the factorization: U is stored as an
              upper triangular band matrix with KL+KU superdiagonals in
              rows 1 to KL+KU+1, and the multipliers used during the
              factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
              See below for further details.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

    IPIV

              IPIV is INTEGER array, dimension (min(M,N))
              The pivot indices; for 1 <= i <= min(M,N), row i of the
              matrix was interchanged with row IPIV(i).

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = +i, U(i,i) is exactly zero. The factorization
                   has been completed, but the factor U is exactly
                   singular, and division by zero will occur if it is used
                   to solve a system of equations.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The band storage scheme is illustrated by the following example, when
      M = N = 6, KL = 2, KU = 1:
      On entry:                       On exit:
          *    *    *    +    +    +       *    *    *   u14  u25  u36
          *    *    +    +    +    +       *    *   u13  u24  u35  u46
          *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
         a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
         a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
         a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
      Array elements marked * are not used by the routine; elements marked
      + need not be set on entry, but are required by the routine to store
      elements of U, because of fill-in resulting from the row
      interchanges.

Definition at line 144 of file cgbtf2.f.

  subroutine cgbtrf (integer m, integer n, integer kl, integer ku, complex, dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv, integer info)

CGBTRF

Purpose:

     CGBTRF computes an LU factorization of a complex m-by-n band matrix A
     using partial pivoting with row interchanges.
     This is the blocked version of the algorithm, calling Level 3 BLAS.

Parameters
    M

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

    N

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

    KL

              KL is INTEGER
              The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
              The number of superdiagonals within the band of A.  KU >= 0.

    AB

              AB is COMPLEX array, dimension (LDAB,N)
              On entry, the matrix A in band storage, in rows KL+1 to
              2*KL+KU+1; rows 1 to KL of the array need not be set.
              The j-th column of A is stored in the j-th column of the
              array AB as follows:
              AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)
              On exit, details of the factorization: U is stored as an
              upper triangular band matrix with KL+KU superdiagonals in
              rows 1 to KL+KU+1, and the multipliers used during the
              factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
              See below for further details.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

    IPIV

              IPIV is INTEGER array, dimension (min(M,N))
              The pivot indices; for 1 <= i <= min(M,N), row i of the
              matrix was interchanged with row IPIV(i).

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value
              > 0: if INFO = +i, U(i,i) is exactly zero. The factorization
                   has been completed, but the factor U is exactly
                   singular, and division by zero will occur if it is used
                   to solve a system of equations.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      The band storage scheme is illustrated by the following example, when
      M = N = 6, KL = 2, KU = 1:
      On entry:                       On exit:
          *    *    *    +    +    +       *    *    *   u14  u25  u36
          *    *    +    +    +    +       *    *   u13  u24  u35  u46
          *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
         a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
         a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
         a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
      Array elements marked * are not used by the routine; elements marked
      + need not be set on entry, but are required by the routine to store
      elements of U because of fill-in resulting from the row interchanges.

Definition at line 143 of file cgbtrf.f.

  subroutine cgbtrs (character trans, integer n, integer kl, integer ku, integer nrhs, complex, dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb, integer info)

CGBTRS

Purpose:

     CGBTRS solves a system of linear equations
        A * X = B,  A**T * X = B,  or  A**H * X = B
     with a general band matrix A using the LU factorization computed
     by CGBTRF.

Parameters
    TRANS

              TRANS is CHARACTER*1
              Specifies the form of the system of equations.
              = 'N':  A * X = B     (No transpose)
              = 'T':  A**T * X = B  (Transpose)
              = 'C':  A**H * X = B  (Conjugate transpose)

    N

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

    KL

              KL is INTEGER
              The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
              The number of superdiagonals within the band of A.  KU >= 0.

    NRHS

              NRHS is INTEGER
              The number of right hand sides, i.e., the number of columns
              of the matrix B.  NRHS >= 0.

    AB

              AB is COMPLEX array, dimension (LDAB,N)
              Details of the LU factorization of the band matrix A, as
              computed by CGBTRF.  U is stored as an upper triangular band
              matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
              the multipliers used during the factorization are stored in
              rows KL+KU+2 to 2*KL+KU+1.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

    IPIV

              IPIV is INTEGER array, dimension (N)
              The pivot indices; for 1 <= i <= N, row i of the matrix was
              interchanged with row IPIV(i).

    B

              B is COMPLEX array, dimension (LDB,NRHS)
              On entry, the right hand side matrix B.
              On exit, the solution matrix X.

    LDB

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

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 136 of file cgbtrs.f.

  subroutine cggbak (character job, character side, integer n, integer ilo, integer ihi, real, dimension( * ) lscale, real, dimension( * ) rscale, integer m, complex, dimension( ldv, * ) v, integer ldv, integer info)

CGGBAK

Purpose:

     CGGBAK forms the right or left eigenvectors of a complex generalized
     eigenvalue problem A*x = lambda*B*x, by backward transformation on
     the computed eigenvectors of the balanced pair of matrices output by
     CGGBAL.

Parameters
    JOB

              JOB is CHARACTER*1
              Specifies the type of backward transformation required:
              = 'N':  do nothing, return immediately;
              = 'P':  do backward transformation for permutation only;
              = 'S':  do backward transformation for scaling only;
              = 'B':  do backward transformations for both permutation and
                      scaling.
              JOB must be the same as the argument JOB supplied to CGGBAL.

    SIDE

              SIDE is CHARACTER*1
              = 'R':  V contains right eigenvectors;
              = 'L':  V contains left eigenvectors.

    N

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

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
              The integers ILO and IHI determined by CGGBAL.
              1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

    LSCALE

              LSCALE is REAL array, dimension (N)
              Details of the permutations and/or scaling factors applied
              to the left side of A and B, as returned by CGGBAL.

    RSCALE

              RSCALE is REAL array, dimension (N)
              Details of the permutations and/or scaling factors applied
              to the right side of A and B, as returned by CGGBAL.

    M

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

    V

              V is COMPLEX array, dimension (LDV,M)
              On entry, the matrix of right or left eigenvectors to be
              transformed, as returned by CTGEVC.
              On exit, V is overwritten by the transformed eigenvectors.

    LDV

              LDV is INTEGER
              The leading dimension of the matrix V. LDV >= max(1,N).

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      See R.C. Ward, Balancing the generalized eigenvalue problem,
                     SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

Definition at line 146 of file cggbak.f.

  subroutine cggbal (character job, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, integer ilo, integer ihi, real, dimension( * ) lscale, real, dimension( * ) rscale, real, dimension( * ) work, integer info)

CGGBAL

Purpose:

     CGGBAL balances a pair of general complex matrices (A,B).  This
     involves, first, permuting A and B by similarity transformations to
     isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N
     elements on the diagonal; and second, applying a diagonal similarity
     transformation to rows and columns ILO to IHI to make the rows
     and columns as close in norm as possible. Both steps are optional.
     Balancing may reduce the 1-norm of the matrices, and improve the
     accuracy of the computed eigenvalues and/or eigenvectors in the
     generalized eigenvalue problem A*x = lambda*B*x.

Parameters
    JOB

              JOB is CHARACTER*1
              Specifies the operations to be performed on A and B:
              = 'N':  none:  simply set ILO = 1, IHI = N, LSCALE(I) = 1.0
                      and RSCALE(I) = 1.0 for i=1,...,N;
              = 'P':  permute only;
              = 'S':  scale only;
              = 'B':  both permute and scale.

    N

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

    A

              A is COMPLEX array, dimension (LDA,N)
              On entry, the input matrix A.
              On exit, A is overwritten by the balanced matrix.
              If JOB = 'N', A is not referenced.

    LDA

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

    B

              B is COMPLEX array, dimension (LDB,N)
              On entry, the input matrix B.
              On exit, B is overwritten by the balanced matrix.
              If JOB = 'N', B is not referenced.

    LDB

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

    ILO

              ILO is INTEGER

    IHI

              IHI is INTEGER
              ILO and IHI are set to integers such that on exit
              A(i,j) = 0 and B(i,j) = 0 if i > j and
              j = 1,...,ILO-1 or i = IHI+1,...,N.
              If JOB = 'N' or 'S', ILO = 1 and IHI = N.

    LSCALE

              LSCALE is REAL array, dimension (N)
              Details of the permutations and scaling factors applied
              to the left side of A and B.  If P(j) is the index of the
              row interchanged with row j, and D(j) is the scaling factor
              applied to row j, then
                LSCALE(j) = P(j)    for J = 1,...,ILO-1
                          = D(j)    for J = ILO,...,IHI
                          = P(j)    for J = IHI+1,...,N.
              The order in which the interchanges are made is N to IHI+1,
              then 1 to ILO-1.

    RSCALE

              RSCALE is REAL array, dimension (N)
              Details of the permutations and scaling factors applied
              to the right side of A and B.  If P(j) is the index of the
              column interchanged with column j, and D(j) is the scaling
              factor applied to column j, then
                RSCALE(j) = P(j)    for J = 1,...,ILO-1
                          = D(j)    for J = ILO,...,IHI
                          = P(j)    for J = IHI+1,...,N.
              The order in which the interchanges are made is N to IHI+1,
              then 1 to ILO-1.

    WORK

              WORK is REAL array, dimension (lwork)
              lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and
              at least 1 when JOB = 'N' or 'P'.

    INFO

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      See R.C. WARD, Balancing the generalized eigenvalue problem,
                     SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.

Definition at line 175 of file cggbal.f.

  subroutine cla_gbamv (integer trans, integer m, integer n, integer kl, integer ku, real alpha, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( * ) x, integer incx, real beta, real, dimension( * ) y, integer incy)

CLA_GBAMV performs a matrix-vector operation to calculate error bounds.

Purpose:

     CLA_GBAMV  performs one of the matrix-vector operations
             y := alpha*abs(A)*abs(x) + beta*abs(y),
        or   y := alpha*abs(A)**T*abs(x) + beta*abs(y),
     where alpha and beta are scalars, x and y are vectors and A is an
     m by n matrix.
     This function is primarily used in calculating error bounds.
     To protect against underflow during evaluation, components in
     the resulting vector are perturbed away from zero by (N+1)
     times the underflow threshold.  To prevent unnecessarily large
     errors for block-structure embedded in general matrices,
     'symbolically' zero components are not perturbed.  A zero
     entry is considered 'symbolic' if all multiplications involved
     in computing that entry have at least one zero multiplicand.

Parameters
    TRANS

              TRANS is INTEGER
               On entry, TRANS specifies the operation to be performed as
               follows:
                 BLAS_NO_TRANS      y := alpha*abs(A)*abs(x) + beta*abs(y)
                 BLAS_TRANS         y := alpha*abs(A**T)*abs(x) + beta*abs(y)
                 BLAS_CONJ_TRANS    y := alpha*abs(A**T)*abs(x) + beta*abs(y)
               Unchanged on exit.

    M

              M is INTEGER
               On entry, M specifies the number of rows of the matrix A.
               M must be at least zero.
               Unchanged on exit.

    N

              N is INTEGER
               On entry, N specifies the number of columns of the matrix A.
               N must be at least zero.
               Unchanged on exit.

    KL

              KL is INTEGER
               The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
               The number of superdiagonals within the band of A.  KU >= 0.

    ALPHA

              ALPHA is REAL
               On entry, ALPHA specifies the scalar alpha.
               Unchanged on exit.

    AB

              AB is COMPLEX array, dimension (LDAB,n)
               Before entry, the leading m by n part of the array AB must
               contain the matrix of coefficients.
               Unchanged on exit.

    LDAB

              LDAB is INTEGER
               On entry, LDAB specifies the first dimension of AB as declared
               in the calling (sub) program. LDAB must be at least
               max( 1, m ).
               Unchanged on exit.

    X

              X is COMPLEX array, dimension
               ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
               and at least
               ( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
               Before entry, the incremented array X must contain the
               vector x.
               Unchanged on exit.

    INCX

              INCX is INTEGER
               On entry, INCX specifies the increment for the elements of
               X. INCX must not be zero.
               Unchanged on exit.

    BETA

              BETA is REAL
               On entry, BETA specifies the scalar beta. When BETA is
               supplied as zero then Y need not be set on input.
               Unchanged on exit.

    Y

              Y is REAL array, dimension
               ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
               and at least
               ( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
               Before entry with BETA non-zero, the incremented array Y
               must contain the vector y. On exit, Y is overwritten by the
               updated vector y.

    INCY

              INCY is INTEGER
               On entry, INCY specifies the increment for the elements of
               Y. INCY must not be zero.
               Unchanged on exit.
      Level 2 Blas routine.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 184 of file cla_gbamv.f.

  real function cla_gbrcond_c (character trans, integer n, integer kl, integer ku, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldafb, * ) afb, integer ldafb, integer, dimension( * ) ipiv, real, dimension( * ) c, logical capply, integer info, complex, dimension( * ) work, real, dimension( * ) rwork)

CLA_GBRCOND_C computes the infinity norm condition number of op(A)*inv(diag(c)) for general banded matrices.

Purpose:

        CLA_GBRCOND_C Computes the infinity norm condition number of
        op(A) * inv(diag(C)) where C is a REAL vector.

Parameters
    TRANS

              TRANS is CHARACTER*1
         Specifies the form of the system of equations:
           = 'N':  A * X = B     (No transpose)
           = 'T':  A**T * X = B  (Transpose)
           = 'C':  A**H * X = B  (Conjugate Transpose = Transpose)

    N

              N is INTEGER
         The number of linear equations, i.e., the order of the
         matrix A.  N >= 0.

    KL

              KL is INTEGER
         The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
         The number of superdiagonals within the band of A.  KU >= 0.

    AB

              AB is COMPLEX array, dimension (LDAB,N)
         On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
         The j-th column of A is stored in the j-th column of the
         array AB as follows:
         AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

    LDAB

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

    AFB

              AFB is COMPLEX array, dimension (LDAFB,N)
         Details of the LU factorization of the band matrix A, as
         computed by CGBTRF.  U is stored as an upper triangular
         band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
         and the multipliers used during the factorization are stored
         in rows KL+KU+2 to 2*KL+KU+1.

    LDAFB

              LDAFB is INTEGER
         The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.

    IPIV

              IPIV is INTEGER array, dimension (N)
         The pivot indices from the factorization A = P*L*U
         as computed by CGBTRF; row i of the matrix was interchanged
         with row IPIV(i).

    C

              C is REAL array, dimension (N)
         The vector C in the formula op(A) * inv(diag(C)).

    CAPPLY

              CAPPLY is LOGICAL
         If .TRUE. then access the vector C in the formula above.

    INFO

              INFO is INTEGER
           = 0:  Successful exit.
         i > 0:  The ith argument is invalid.

    WORK

              WORK is COMPLEX array, dimension (2*N).
         Workspace.

    RWORK

              RWORK is REAL array, dimension (N).
         Workspace.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 158 of file cla_gbrcond_c.f.

  real function cla_gbrcond_x (character trans, integer n, integer kl, integer ku, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldafb, * ) afb, integer ldafb, integer, dimension( * ) ipiv, complex, dimension( * ) x, integer info, complex, dimension( * ) work, real, dimension( * ) rwork)

CLA_GBRCOND_X computes the infinity norm condition number of op(A)*diag(x) for general banded matrices.

Purpose:

        CLA_GBRCOND_X Computes the infinity norm condition number of
        op(A) * diag(X) where X is a COMPLEX vector.

Parameters
    TRANS

              TRANS is CHARACTER*1
         Specifies the form of the system of equations:
           = 'N':  A * X = B     (No transpose)
           = 'T':  A**T * X = B  (Transpose)
           = 'C':  A**H * X = B  (Conjugate Transpose = Transpose)

    N

              N is INTEGER
         The number of linear equations, i.e., the order of the
         matrix A.  N >= 0.

    KL

              KL is INTEGER
         The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
         The number of superdiagonals within the band of A.  KU >= 0.

    AB

              AB is COMPLEX array, dimension (LDAB,N)
         On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
         The j-th column of A is stored in the j-th column of the
         array AB as follows:
         AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

    LDAB

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

    AFB

              AFB is COMPLEX array, dimension (LDAFB,N)
         Details of the LU factorization of the band matrix A, as
         computed by CGBTRF.  U is stored as an upper triangular
         band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
         and the multipliers used during the factorization are stored
         in rows KL+KU+2 to 2*KL+KU+1.

    LDAFB

              LDAFB is INTEGER
         The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.

    IPIV

              IPIV is INTEGER array, dimension (N)
         The pivot indices from the factorization A = P*L*U
         as computed by CGBTRF; row i of the matrix was interchanged
         with row IPIV(i).

    X

              X is COMPLEX array, dimension (N)
         The vector X in the formula op(A) * diag(X).

    INFO

              INFO is INTEGER
           = 0:  Successful exit.
         i > 0:  The ith argument is invalid.

    WORK

              WORK is COMPLEX array, dimension (2*N).
         Workspace.

    RWORK

              RWORK is REAL array, dimension (N).
         Workspace.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 151 of file cla_gbrcond_x.f.

  subroutine cla_gbrfsx_extended (integer prec_type, integer trans_type, integer n, integer kl, integer ku, integer nrhs, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldafb, * ) afb, integer ldafb, integer, dimension( * ) ipiv, logical colequ, real, dimension( * ) c, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldy, * ) y, integer ldy, real, dimension( * ) berr_out, integer n_norms, real, dimension( nrhs, * ) err_bnds_norm, real, dimension( nrhs, * ) err_bnds_comp, complex, dimension( * ) res, real, dimension(*) ayb, complex, dimension( * ) dy, complex, dimension( * ) y_tail, real rcond, integer ithresh, real rthresh, real dz_ub, logical ignore_cwise, integer info)

CLA_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Purpose:

     CLA_GBRFSX_EXTENDED improves the computed solution to a system of
     linear equations by performing extra-precise iterative refinement
     and provides error bounds and backward error estimates for the solution.
     This subroutine is called by CGBRFSX to perform iterative refinement.
     In addition to normwise error bound, the code provides maximum
     componentwise error bound if possible. See comments for ERR_BNDS_NORM
     and ERR_BNDS_COMP for details of the error bounds. Note that this
     subroutine is only responsible for setting the second fields of
     ERR_BNDS_NORM and ERR_BNDS_COMP.

Parameters
    PREC_TYPE

              PREC_TYPE is INTEGER
         Specifies the intermediate precision to be used in refinement.
         The value is defined by ILAPREC(P) where P is a CHARACTER and P
              = 'S':  Single
              = 'D':  Double
              = 'I':  Indigenous
              = 'X' or 'E':  Extra

    TRANS_TYPE

              TRANS_TYPE is INTEGER
         Specifies the transposition operation on A.
         The value is defined by ILATRANS(T) where T is a CHARACTER and T
              = 'N':  No transpose
              = 'T':  Transpose
              = 'C':  Conjugate transpose

    N

              N is INTEGER
         The number of linear equations, i.e., the order of the
         matrix A.  N >= 0.

    KL

              KL is INTEGER
         The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
         The number of superdiagonals within the band of A.  KU >= 0

    NRHS

              NRHS is INTEGER
         The number of right-hand-sides, i.e., the number of columns of the
         matrix B.

    AB

              AB is COMPLEX array, dimension (LDAB,N)
         On entry, the N-by-N matrix AB.

    LDAB

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

    AFB

              AFB is COMPLEX array, dimension (LDAF,N)
         The factors L and U from the factorization
         A = P*L*U as computed by CGBTRF.

    LDAFB

              LDAFB is INTEGER
         The leading dimension of the array AF.  LDAF >= max(1,N).

    IPIV

              IPIV is INTEGER array, dimension (N)
         The pivot indices from the factorization A = P*L*U
         as computed by CGBTRF; row i of the matrix was interchanged
         with row IPIV(i).

    COLEQU

              COLEQU is LOGICAL
         If .TRUE. then column equilibration was done to A before calling
         this routine. This is needed to compute the solution and error
         bounds correctly.

    C

              C is REAL array, dimension (N)
         The column scale factors for A. If COLEQU = .FALSE., C
         is not accessed. If C is input, each element of C should be a power
         of the radix to ensure a reliable solution and error estimates.
         Scaling by powers of the radix does not cause rounding errors unless
         the result underflows or overflows. Rounding errors during scaling
         lead to refining with a matrix that is not equivalent to the
         input matrix, producing error estimates that may not be
         reliable.

    B

              B is COMPLEX array, dimension (LDB,NRHS)
         The right-hand-side matrix B.

    LDB

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

    Y

              Y is COMPLEX array, dimension (LDY,NRHS)
         On entry, the solution matrix X, as computed by CGBTRS.
         On exit, the improved solution matrix Y.

    LDY

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

    BERR_OUT

              BERR_OUT is REAL array, dimension (NRHS)
         On exit, BERR_OUT(j) contains the componentwise relative backward
         error for right-hand-side j from the formula
             max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
         where abs(Z) is the componentwise absolute value of the matrix
         or vector Z. This is computed by CLA_LIN_BERR.

    N_NORMS

              N_NORMS is INTEGER
         Determines which error bounds to return (see ERR_BNDS_NORM
         and ERR_BNDS_COMP).
         If N_NORMS >= 1 return normwise error bounds.
         If N_NORMS >= 2 return componentwise error bounds.

    ERR_BNDS_NORM

              ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
         For each right-hand side, this array contains information about
         various error bounds and condition numbers corresponding to the
         normwise relative error, which is defined as follows:
         Normwise relative error in the ith solution vector:
                 max_j (abs(XTRUE(j,i) - X(j,i)))
                ------------------------------
                      max_j abs(X(j,i))
         The array is indexed by the type of error information as described
         below. There currently are up to three pieces of information
         returned.
         The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
         right-hand side.
         The second index in ERR_BNDS_NORM(:,err) contains the following
         three fields:
         err = 1 'Trust/don't trust' boolean. Trust the answer if the
                  reciprocal condition number is less than the threshold
                  sqrt(n) * slamch('Epsilon').
         err = 2 'Guaranteed' error bound: The estimated forward error,
                  almost certainly within a factor of 10 of the true error
                  so long as the next entry is greater than the threshold
                  sqrt(n) * slamch('Epsilon'). This error bound should only
                  be trusted if the previous boolean is true.
         err = 3  Reciprocal condition number: Estimated normwise
                  reciprocal condition number.  Compared with the threshold
                  sqrt(n) * slamch('Epsilon') to determine if the error
                  estimate is 'guaranteed'. These reciprocal condition
                  numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                  appropriately scaled matrix Z.
                  Let Z = S*A, where S scales each row by a power of the
                  radix so all absolute row sums of Z are approximately 1.
         This subroutine is only responsible for setting the second field
         above.
         See Lapack Working Note 165 for further details and extra
         cautions.

    ERR_BNDS_COMP

              ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
         For each right-hand side, this array contains information about
         various error bounds and condition numbers corresponding to the
         componentwise relative error, which is defined as follows:
         Componentwise relative error in the ith solution vector:
                        abs(XTRUE(j,i) - X(j,i))
                 max_j ----------------------
                             abs(X(j,i))
         The array is indexed by the right-hand side i (on which the
         componentwise relative error depends), and the type of error
         information as described below. There currently are up to three
         pieces of information returned for each right-hand side. If
         componentwise accuracy is not requested (PARAMS(3) = 0.0), then
         ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS < 3, then at most
         the first (:,N_ERR_BNDS) entries are returned.
         The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
         right-hand side.
         The second index in ERR_BNDS_COMP(:,err) contains the following
         three fields:
         err = 1 'Trust/don't trust' boolean. Trust the answer if the
                  reciprocal condition number is less than the threshold
                  sqrt(n) * slamch('Epsilon').
         err = 2 'Guaranteed' error bound: The estimated forward error,
                  almost certainly within a factor of 10 of the true error
                  so long as the next entry is greater than the threshold
                  sqrt(n) * slamch('Epsilon'). This error bound should only
                  be trusted if the previous boolean is true.
         err = 3  Reciprocal condition number: Estimated componentwise
                  reciprocal condition number.  Compared with the threshold
                  sqrt(n) * slamch('Epsilon') to determine if the error
                  estimate is 'guaranteed'. These reciprocal condition
                  numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
                  appropriately scaled matrix Z.
                  Let Z = S*(A*diag(x)), where x is the solution for the
                  current right-hand side and S scales each row of
                  A*diag(x) by a power of the radix so all absolute row
                  sums of Z are approximately 1.
         This subroutine is only responsible for setting the second field
         above.
         See Lapack Working Note 165 for further details and extra
         cautions.

    RES

              RES is COMPLEX array, dimension (N)
         Workspace to hold the intermediate residual.

    AYB

              AYB is REAL array, dimension (N)
         Workspace.

    DY

              DY is COMPLEX array, dimension (N)
         Workspace to hold the intermediate solution.

    Y_TAIL

              Y_TAIL is COMPLEX array, dimension (N)
         Workspace to hold the trailing bits of the intermediate solution.

    RCOND

              RCOND is REAL
         Reciprocal scaled condition number.  This is an estimate of the
         reciprocal Skeel condition number of the matrix A after
         equilibration (if done).  If this is less than the machine
         precision (in particular, if it is zero), the matrix is singular
         to working precision.  Note that the error may still be small even
         if this number is very small and the matrix appears ill-
         conditioned.

    ITHRESH

              ITHRESH is INTEGER
         The maximum number of residual computations allowed for
         refinement. The default is 10. For 'aggressive' set to 100 to
         permit convergence using approximate factorizations or
         factorizations other than LU. If the factorization uses a
         technique other than Gaussian elimination, the guarantees in
         ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.

    RTHRESH

              RTHRESH is REAL
         Determines when to stop refinement if the error estimate stops
         decreasing. Refinement will stop when the next solution no longer
         satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
         the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
         default value is 0.5. For 'aggressive' set to 0.9 to permit
         convergence on extremely ill-conditioned matrices. See LAWN 165
         for more details.

    DZ_UB

              DZ_UB is REAL
         Determines when to start considering componentwise convergence.
         Componentwise convergence is only considered after each component
         of the solution Y is stable, which we define as the relative
         change in each component being less than DZ_UB. The default value
         is 0.25, requiring the first bit to be stable. See LAWN 165 for
         more details.

    IGNORE_CWISE

              IGNORE_CWISE is LOGICAL
         If .TRUE. then ignore componentwise convergence. Default value
         is .FALSE..

    INFO

              INFO is INTEGER
           = 0:  Successful exit.
           < 0:  if INFO = -i, the ith argument to CGBTRS had an illegal
                 value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 403 of file cla_gbrfsx_extended.f.

  real function cla_gbrpvgrw (integer n, integer kl, integer ku, integer ncols, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldafb, * ) afb, integer ldafb)

CLA_GBRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix.

Purpose:

     CLA_GBRPVGRW computes the reciprocal pivot growth factor
     norm(A)/norm(U). The 'max absolute element' norm is used. If this is
     much less than 1, the stability of the LU factorization of the
     (equilibrated) matrix A could be poor. This also means that the
     solution X, estimated condition numbers, and error bounds could be
     unreliable.

Parameters
    N

              N is INTEGER
         The number of linear equations, i.e., the order of the
         matrix A.  N >= 0.

    KL

              KL is INTEGER
         The number of subdiagonals within the band of A.  KL >= 0.

    KU

              KU is INTEGER
         The number of superdiagonals within the band of A.  KU >= 0.

    NCOLS

              NCOLS is INTEGER
         The number of columns of the matrix A.  NCOLS >= 0.

    AB

              AB is COMPLEX array, dimension (LDAB,N)
         On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
         The j-th column of A is stored in the j-th column of the
         array AB as follows:
         AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)

    LDAB

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

    AFB

              AFB is COMPLEX array, dimension (LDAFB,N)
         Details of the LU factorization of the band matrix A, as
         computed by CGBTRF.  U is stored as an upper triangular
         band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
         and the multipliers used during the factorization are stored
         in rows KL+KU+2 to 2*KL+KU+1.

    LDAFB

              LDAFB is INTEGER
         The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 115 of file cla_gbrpvgrw.f.

  subroutine cungbr (character vect, integer m, integer n, integer k, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) tau, complex, dimension( * ) work, integer lwork, integer info)

CUNGBR

Purpose:

     CUNGBR generates one of the complex unitary matrices Q or P**H
     determined by CGEBRD when reducing a complex matrix A to bidiagonal
     form: A = Q * B * P**H.  Q and P**H are defined as products of
     elementary reflectors H(i) or G(i) respectively.
     If VECT = 'Q', A is assumed to have been an M-by-K matrix, and Q
     is of order M:
     if m >= k, Q = H(1) H(2) . . . H(k) and CUNGBR returns the first n
     columns of Q, where m >= n >= k;
     if m < k, Q = H(1) H(2) . . . H(m-1) and CUNGBR returns Q as an
     M-by-M matrix.
     If VECT = 'P', A is assumed to have been a K-by-N matrix, and P**H
     is of order N:
     if k < n, P**H = G(k) . . . G(2) G(1) and CUNGBR returns the first m
     rows of P**H, where n >= m >= k;
     if k >= n, P**H = G(n-1) . . . G(2) G(1) and CUNGBR returns P**H as
     an N-by-N matrix.

Parameters
    VECT

              VECT is CHARACTER*1
              Specifies whether the matrix Q or the matrix P**H is
              required, as defined in the transformation applied by CGEBRD:
              = 'Q':  generate Q;
              = 'P':  generate P**H.

    M

              M is INTEGER
              The number of rows of the matrix Q or P**H to be returned.
              M >= 0.

    N

              N is INTEGER
              The number of columns of the matrix Q or P**H to be returned.
              N >= 0.
              If VECT = 'Q', M >= N >= min(M,K);
              if VECT = 'P', N >= M >= min(N,K).

    K

              K is INTEGER
              If VECT = 'Q', the number of columns in the original M-by-K
              matrix reduced by CGEBRD.
              If VECT = 'P', the number of rows in the original K-by-N
              matrix reduced by CGEBRD.
              K >= 0.

    A

              A is COMPLEX array, dimension (LDA,N)
              On entry, the vectors which define the elementary reflectors,
              as returned by CGEBRD.
              On exit, the M-by-N matrix Q or P**H.

    LDA

              LDA is INTEGER
              The leading dimension of the array A. LDA >= M.

    TAU

              TAU is COMPLEX array, dimension
                                    (min(M,K)) if VECT = 'Q'
                                    (min(N,K)) if VECT = 'P'
              TAU(i) must contain the scalar factor of the elementary
              reflector H(i) or G(i), which determines Q or P**H, as
              returned by CGEBRD in its array argument TAUQ or TAUP.

    WORK

              WORK is COMPLEX 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,min(M,N)).
              For optimum performance LWORK >= min(M,N)*NB, where NB
              is the optimal blocksize.
              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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 156 of file cungbr.f.

Author

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