Cholesky Factorization and Condition Number of a Symmetric Positive Definite Matrix

Calling Sequence

CALL DPOCO	(DA, LDA, N, DRCOND, DWORK, INFO)
CALL SPOCO	(SA, LDA, N, SRCOND, SWORK, INFO)
CALL ZPOCO	(ZA, LDA, N, DRCOND, ZWORK, INFO)
CALL CPOCO	(CA, LDA, N, SRCOND, CWORK, INFO)
void dpoco	(double *da, int lda, int n, double *drcond, int *info)
void spoco	(float *sa, int lda, int n, float *srcond, int *info)
void zpoco	(doublecomplex *za, int lda, int n, double *drcond, int *info)
void cpoco	(complex *ca, int lda, int n, float *srcond, int *info)

Arguments

xA	On entry, the upper triangle of the matrix A. On exit, a Cholesky factorization of the matrix A. The strict lower triangle of A is not referenced.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA max(1,N).
N	Order of the matrix A. N 0.
xRCOND	On exit, an estimate of the reciprocal condition number of A. 0.0 RCOND 1.0. As the value of RCOND gets smaller, operations with A such as solving Ax = b may become less stable. If RCOND satisfies RCOND + 1.0 = 1.0 then A may be singular to working precision.
xWORK	Scratch array with a dimension of N.
INFO	On exit:
	INFO = 0	Subroutine completed normally.
	INFO > 0	Returns a value k if the leading minor of order k is not positive definite.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

INTEGER LDA, N

PARAMETER (N = 4)

PARAMETER (LDA = N)

C

DOUBLE PRECISION A(LDA,N), B(N), RCOND, WORK(N)

INTEGER ICOL, INFO, IROW

C

EXTERNAL DPOCO, DPOSL

C

C Initialize the array A to store in symmetric storage mode

C the matrix A shown below. Initialize the array B to store

C the vector B shown below.

C

C 2 -1 0 0 40

C A = -1 2 -1 0 b = 30

C 0 -1 2 -1 20

C 0 0 -1 2 10

C

DATA A / 2.0D0, 3*8D8, -1.0D0, 2.0D0, 2*8D8, 0.0D0, -1.0D0,

$ 2.0D0, -1.0D0, 0.0D0, 0.0D0, -1.0D0, 2.0D0 /

DATA B / 4.0D0, 3.0D0, 2.0D0, 1.0D0 /

C

PRINT 1000

DO 100, IROW = 1, N

PRINT 1010, (A(ICOL,IROW), ICOL = 1, IROW),

$ (A(IROW,ICOL), ICOL = IROW + 1, N)

100 CONTINUE

PRINT 1020

PRINT 1010, ((A(IROW,ICOL), ICOL = 1, N), IROW = 1, N)

PRINT 1030

PRINT 1040, B

CALL DPOCO (A, LDA, N, RCOND, WORK, INFO)

IF (INFO .EQ. 0) THEN

IF ((RCOND + 1.0D0) .EQ. 1.0D0) THEN

PRINT 1070

END IF

CALL DPOSL (A, LDA, N, B)

PRINT 1050

PRINT 1040, B

PRINT 1060, RCOND

ELSE

PRINT 1080

END IF

C

1000 FORMAT (1X, 'A in full form:')

1010 FORMAT (4(3X, F7.3))

1020 FORMAT (/1X, 'A in symmetric form: (* in unused entries)')

1030 FORMAT (/1X, 'b:')

1040 FORMAT (3X, F7.3)

1050 FORMAT (/1X, 'A**(-1) * b:')

1060 FORMAT (/1X, 'Reciprocal condition number of A:', F5.1)

1070 FORMAT (1X, 'A may be singular to working precision.')

1080 FORMAT (1X, 'A is not positive definite.')

C

END

Sample Output

A in full form:
2.000 -1.000 0.000 0.000
-1.000 2.000 -1.000 0.000
0.000 -1.000 2.000 -1.000
0.000 0.000 -1.000 2.000
A in symmetric form: (* in unused entries)
2.000 -1.000 0.000 0.000
******* 2.000 -1.000 0.000
******* ******* 2.000 -1.000
******* ******* -1.000 2.000
b:
4.000
3.000
2.000
1.000
A**(-1) * b:
6.000
8.000
7.000
4.000
Reciprocal condition number of A: 0.1