cgebal(3S)

NAME
     CGEBAL - balance a general complex matrix A

SYNOPSIS
     SUBROUTINE CGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )

         CHARACTER      JOB

         INTEGER        IHI, ILO, INFO, LDA, N

         REAL           SCALE( * )

         COMPLEX        A( LDA, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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.


ARGUMENTS
     JOB     (input) 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       (input) INTEGER
             The order of the matrix A.  N >= 0.
     A       (input/output) 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     (input) INTEGER The leading dimension of the
             array A.  LDA >= max(1,N).

     ILO     (output) INTEGER
             IHI     (output) 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   (output) 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    (output) INTEGER
             = 0:  successful exit.
             < 0:  if INFO = -i, the i-th argument had an illegal value.

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
SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

     This man page is available only online.