cgebal(3P) Sun Performance Library cgebal(3P)NAMEcgebal - balance a general complex matrix A
SYNOPSIS
SUBROUTINE CGEBAL(JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
CHARACTER * 1 JOB
COMPLEX A(LDA,*)
INTEGER N, LDA, ILO, IHI, INFO
REAL SCALE(*)
SUBROUTINE CGEBAL_64(JOB, N, A, LDA, ILO, IHI, SCALE, INFO)
CHARACTER * 1 JOB
COMPLEX A(LDA,*)
INTEGER*8 N, LDA, ILO, IHI, INFO
REAL SCALE(*)
F95 INTERFACE
SUBROUTINE GEBAL(JOB, [N], A, [LDA], ILO, IHI, SCALE, [INFO])
CHARACTER(LEN=1) :: JOB
COMPLEX, DIMENSION(:,:) :: A
INTEGER :: N, LDA, ILO, IHI, INFO
REAL, DIMENSION(:) :: SCALE
SUBROUTINE GEBAL_64(JOB, [N], A, [LDA], ILO, IHI, SCALE, [INFO])
CHARACTER(LEN=1) :: JOB
COMPLEX, DIMENSION(:,:) :: A
INTEGER(8) :: N, LDA, ILO, IHI, INFO
REAL, DIMENSION(:) :: SCALE
C INTERFACE
#include <sunperf.h>
void cgebal(char job, int n, complex *a, int lda, int *ilo, int *ihi,
float *scale, int *info);
void cgebal_64(char job, long n, complex *a, long lda, long *ilo, long
*ihi, float *scale, long *info);
PURPOSEcgebal balances a general complex matrix A. This involves, first, per‐
muting 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 sec‐
ond, 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. However, the diagonal
transformation step can occasionally make the norm larger and hence
degrade performance.
ARGUMENTS
JOB (input)
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) The order of the matrix A. N >= 0.
A (input/output)
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)
The leading dimension of the array A. LDA >= max(1,N).
ILO (output)
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.
IHI (output)
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)
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)
= 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
6 Mar 2009 cgebal(3P)