gebal(3)

NAME
gebal - gebal: balance matrix

SYNOPSIS

  Functions

subroutine cgebal (job, n, a, lda, ilo, ihi, scale, info)
CGEBAL
subroutine dgebal (job, n, a, lda, ilo, ihi, scale, info)
DGEBAL
subroutine sgebal (job, n, a, lda, ilo, ihi, scale, info)
SGEBAL
subroutine zgebal (job, n, a, lda, ilo, ihi, scale, info)
ZGEBAL

Detailed Description

Function Documentation

  subroutine cgebal (character job, integer n, complex, dimension( lda, * ) a, integer lda, integer ilo, integer ihi, real, dimension( * ) scale, integer info)

CGEBAL

Purpose:

    !>
    !> CGEBAL balances a general complex matrix A.  This involves, first,
    !> permuting A by a similarity transformation 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 matrix, and improve the
    !> accuracy of the computed eigenvalues and/or eigenvectors.
    !>

Parameters
    JOB

    !>          JOB is CHARACTER*1
    !>          Specifies the operations to be performed on A:
    !>          = 'N':  none:  simply set ILO = 1, IHI = N, SCALE(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 matrix A.  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.
    !>          See Further Details.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= 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 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
    !>          If JOB = 'N' or 'S', ILO = 1 and IHI = N.
    !>

    SCALE

    !>          SCALE is REAL array, dimension (N)
    !>          Details of the permutations and scaling factors applied to
    !>          A.  If P(j) is the index of the row and column interchanged
    !>          with row and column j and D(j) is the scaling factor
    !>          applied to row and column j, then
    !>          SCALE(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.
    !>

    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:

    !>
    !>  The permutations consist of row and column interchanges which put
    !>  the matrix in the form
    !>
    !>             ( T1   X   Y  )
    !>     P A P = (  0   B   Z  )
    !>             (  0   0   T2 )
    !>
    !>  where T1 and T2 are upper triangular matrices whose eigenvalues lie
    !>  along the diagonal.  The column indices ILO and IHI mark the starting
    !>  and ending columns of the submatrix B. Balancing consists of applying
    !>  a diagonal similarity transformation inv(D) * B * D to make the
    !>  1-norms of each row of B and its corresponding column nearly equal.
    !>  The output matrix is
    !>
    !>     ( T1     X*D          Y    )
    !>     (  0  inv(D)*B*D  inv(D)*Z ).
    !>     (  0      0           T2   )
    !>
    !>  Information about the permutations P and the diagonal matrix D is
    !>  returned in the vector SCALE.
    !>
    !>  This subroutine is based on the EISPACK routine CBAL.
    !>
    !>  Modified by Tzu-Yi Chen, Computer Science Division, University of
    !>    California at Berkeley, USA
    !>
    !>  Refactored by Evert Provoost, Department of Computer Science,
    !>    KU Leuven, Belgium
    !>

  subroutine dgebal (character job, integer n, double precision, dimension( lda, * ) a, integer lda, integer ilo, integer ihi, double precision, dimension( * ) scale, integer info)

DGEBAL

Purpose:

    !>
    !> DGEBAL balances a general real matrix A.  This involves, first,
    !> permuting A by a similarity transformation 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 matrix, and improve the
    !> accuracy of the computed eigenvalues and/or eigenvectors.
    !>

Parameters
    JOB

    !>          JOB is CHARACTER*1
    !>          Specifies the operations to be performed on A:
    !>          = 'N':  none:  simply set ILO = 1, IHI = N, SCALE(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 matrix A.  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.
    !>          See Further Details.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= 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 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
    !>          If JOB = 'N' or 'S', ILO = 1 and IHI = N.
    !>

    SCALE

    !>          SCALE is DOUBLE PRECISION array, dimension (N)
    !>          Details of the permutations and scaling factors applied to
    !>          A.  If P(j) is the index of the row and column interchanged
    !>          with row and column j and D(j) is the scaling factor
    !>          applied to row and column j, then
    !>          SCALE(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.
    !>

    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:

    !>
    !>  The permutations consist of row and column interchanges which put
    !>  the matrix in the form
    !>
    !>             ( T1   X   Y  )
    !>     P A P = (  0   B   Z  )
    !>             (  0   0   T2 )
    !>
    !>  where T1 and T2 are upper triangular matrices whose eigenvalues lie
    !>  along the diagonal.  The column indices ILO and IHI mark the starting
    !>  and ending columns of the submatrix B. Balancing consists of applying
    !>  a diagonal similarity transformation inv(D) * B * D to make the
    !>  1-norms of each row of B and its corresponding column nearly equal.
    !>  The output matrix is
    !>
    !>     ( T1     X*D          Y    )
    !>     (  0  inv(D)*B*D  inv(D)*Z ).
    !>     (  0      0           T2   )
    !>
    !>  Information about the permutations P and the diagonal matrix D is
    !>  returned in the vector SCALE.
    !>
    !>  This subroutine is based on the EISPACK routine BALANC.
    !>
    !>  Modified by Tzu-Yi Chen, Computer Science Division, University of
    !>    California at Berkeley, USA
    !>
    !>  Refactored by Evert Provoost, Department of Computer Science,
    !>    KU Leuven, Belgium
    !>

  subroutine sgebal (character job, integer n, real, dimension( lda, * ) a, integer lda, integer ilo, integer ihi, real, dimension( * ) scale, integer info)

SGEBAL

Purpose:

    !>
    !> SGEBAL balances a general real matrix A.  This involves, first,
    !> permuting A by a similarity transformation 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 matrix, and improve the
    !> accuracy of the computed eigenvalues and/or eigenvectors.
    !>

Parameters
    JOB

    !>          JOB is CHARACTER*1
    !>          Specifies the operations to be performed on A:
    !>          = 'N':  none:  simply set ILO = 1, IHI = N, SCALE(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 matrix A.  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.
    !>          See Further Details.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= 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 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
    !>          If JOB = 'N' or 'S', ILO = 1 and IHI = N.
    !>

    SCALE

    !>          SCALE is REAL array, dimension (N)
    !>          Details of the permutations and scaling factors applied to
    !>          A.  If P(j) is the index of the row and column interchanged
    !>          with row and column j and D(j) is the scaling factor
    !>          applied to row and column j, then
    !>          SCALE(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.
    !>

    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:

    !>
    !>  The permutations consist of row and column interchanges which put
    !>  the matrix in the form
    !>
    !>             ( T1   X   Y  )
    !>     P A P = (  0   B   Z  )
    !>             (  0   0   T2 )
    !>
    !>  where T1 and T2 are upper triangular matrices whose eigenvalues lie
    !>  along the diagonal.  The column indices ILO and IHI mark the starting
    !>  and ending columns of the submatrix B. Balancing consists of applying
    !>  a diagonal similarity transformation inv(D) * B * D to make the
    !>  1-norms of each row of B and its corresponding column nearly equal.
    !>  The output matrix is
    !>
    !>     ( T1     X*D          Y    )
    !>     (  0  inv(D)*B*D  inv(D)*Z ).
    !>     (  0      0           T2   )
    !>
    !>  Information about the permutations P and the diagonal matrix D is
    !>  returned in the vector SCALE.
    !>
    !>  This subroutine is based on the EISPACK routine BALANC.
    !>
    !>  Modified by Tzu-Yi Chen, Computer Science Division, University of
    !>    California at Berkeley, USA
    !>
    !>  Refactored by Evert Provoost, Department of Computer Science,
    !>    KU Leuven, Belgium
    !>

  subroutine zgebal (character job, integer n, complex*16, dimension( lda, * ) a, integer lda, integer ilo, integer ihi, double precision, dimension( * ) scale, integer info)

ZGEBAL

Purpose:

    !>
    !> ZGEBAL balances a general complex matrix A.  This involves, first,
    !> permuting A by a similarity transformation 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 matrix, and improve the
    !> accuracy of the computed eigenvalues and/or eigenvectors.
    !>

Parameters
    JOB

    !>          JOB is CHARACTER*1
    !>          Specifies the operations to be performed on A:
    !>          = 'N':  none:  simply set ILO = 1, IHI = N, SCALE(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 matrix A.  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.
    !>          See Further Details.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= 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 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
    !>          If JOB = 'N' or 'S', ILO = 1 and IHI = N.
    !>

    SCALE

    !>          SCALE is DOUBLE PRECISION array, dimension (N)
    !>          Details of the permutations and scaling factors applied to
    !>          A.  If P(j) is the index of the row and column interchanged
    !>          with row and column j and D(j) is the scaling factor
    !>          applied to row and column j, then
    !>          SCALE(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.
    !>

    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:

    !>
    !>  The permutations consist of row and column interchanges which put
    !>  the matrix in the form
    !>
    !>             ( T1   X   Y  )
    !>     P A P = (  0   B   Z  )
    !>             (  0   0   T2 )
    !>
    !>  where T1 and T2 are upper triangular matrices whose eigenvalues lie
    !>  along the diagonal.  The column indices ILO and IHI mark the starting
    !>  and ending columns of the submatrix B. Balancing consists of applying
    !>  a diagonal similarity transformation inv(D) * B * D to make the
    !>  1-norms of each row of B and its corresponding column nearly equal.
    !>  The output matrix is
    !>
    !>     ( T1     X*D          Y    )
    !>     (  0  inv(D)*B*D  inv(D)*Z ).
    !>     (  0      0           T2   )
    !>
    !>  Information about the permutations P and the diagonal matrix D is
    !>  returned in the vector SCALE.
    !>
    !>  This subroutine is based on the EISPACK routine CBAL.
    !>
    !>  Modified by Tzu-Yi Chen, Computer Science Division, University of
    !>    California at Berkeley, USA
    !>
    !>  Refactored by Evert Provoost, Department of Computer Science,
    !>    KU Leuven, Belgium
    !>

Author

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