ggbal(3)

NAME
ggbal - ggbal: balance matrix

SYNOPSIS

  Functions

subroutine cggbal (job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)
CGGBAL
subroutine dggbal (job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)
DGGBAL
subroutine sggbal (job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)
SGGBAL
subroutine zggbal (job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)
ZGGBAL

Detailed Description

Function Documentation

  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.
    !>

  subroutine dggbal (character job, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, integer ilo, integer ihi, double precision, dimension( * ) lscale, double precision, dimension( * ) rscale, double precision, dimension( * ) work, integer info)

DGGBAL

Purpose:

    !>
    !> DGGBAL balances a pair of general real 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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
    !>            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.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION 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.
    !>

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

SGGBAL

Purpose:

    !>
    !> SGGBAL balances a pair of general real 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 REAL 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 REAL 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
    !>            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.
    !>

    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.
    !>

  subroutine zggbal (character job, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, integer ilo, integer ihi, double precision, dimension( * ) lscale, double precision, dimension( * ) rscale, double precision, dimension( * ) work, integer info)

ZGGBAL

Purpose:

    !>
    !> ZGGBAL 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*16 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*16 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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.
    !>

Author

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