Reciprocal Condition Number of a Cholesky-Factored Symmetric Positive Definite Matrix

Calling Sequence

CALL DPOCON	(UPLO, N, DA, LDA, DANORM, DRCOND, DWORK, IWORK2, INFO)
CALL SPOCON	(UPLO, N, SA, LDA, SANORM, SRCOND, SWORK, IWORK2, INFO)
CALL ZPOCON	(UPLO, N, ZA, LDA, DANORM, DRCOND, ZWORK, DWORK2, INFO)
CALL CPOCON	(UPLO, N, CA, LDA, SANORM, SRCOND, CWORK, SWORK2, INFO)
void dpocon	(char uplo, int n, double *da, int lda, double danorm, double *drcond, int *info)
void spocon	(char uplo, int n, float *sa, int lda, float sanorm, float *srcond, int *info)
void zpocon	(char uplo, int n, doublecomplex *za, int lda, double danorm, double *drcond, int *info)
void cpocon	(char uplo, int n, complex *ca, int lda, float sanorm, float *srcond, int *info)

Arguments

UPLO	Indicates whether xA contains the upper or lower triangle of the matrix. The legal values for UPLO are listed below. Any values not listed below are illegal.
	'U' or 'u'	xA contains the upper triangle.
	'L' or 'l'	xA contains the lower triangle.
N	Order of the matrix A. N 0.
xA	Cholesky factorization of the matrix A as computed by xPOTRF.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA max(1, N).
xANORM	The 1-norm or -norm of matrix A.
xRCOND	On exit, the estimated reciprocal of the condition number of the matrix A where the reciprocal condition number of A is defined to be 1 / (ANORM × ||A-1||). The reciprocal of the condition number is estimated instead of the condition number itself to avoid overflow or division by zero. If RCOND is less than machine precision (in particular, if RCOND = 0) then A is singular to working precision.
xWORK	Scratch array with a dimension of 3 × N for real subroutines or 2 × N for complex subroutines.
xWORK2	Scratch array with a dimension of N.
INFO	On exit:
	INFO = 0	Subroutine completed normally.
	INFO < 0	The ith argument, where i = |INFO|, had an illegal value.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

INTEGER LDA, LDIWRK, LDWORK, N

PARAMETER (N = 4)

PARAMETER (LDA = N)

PARAMETER (LDIWRK = N)

PARAMETER (LDWORK = 3 * N)

C

DOUBLE PRECISION A(LDA,N), ANORM, RCOND, WORK(LDWORK)

INTEGER ICOL, INFO, IROW, IWORK(LDIWRK)

C

EXTERNAL DPOCON, DPOTRF

INTRINSIC ABS

C

C Initialize the array A to store in symmetric form the 4x4

C symmetric positive definite matrix A shown below.

C

C 2 -1 0 0

C A = -1 2 -1 0

C 0 -1 2 -1

C 0 0 -1 2

C

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

$ 2.0D0, 8D8, 0.0D0, 0.0D0, -1.0D0, 2.0D0 /

C

C Print the initial values of the arrays.

C

PRINT 1000

DO 100, IROW = 1, N

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

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

100 CONTINUE

PRINT 1020

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

C

C Cholesky factor A.

C

CALL DPOTRF ('UPPER TRIANGLE OF A STORED', N, A, LDA, INFO)

IF (INFO .NE. 0) THEN

PRINT 1030, INFO

STOP 1

END IF

C

C Compute and print the condition number of A.

C

ANORM = 3.0D0

CALL DPOCON ('UPPER TRIANGLE FACTOR STORED', N, A, LDA,

$ ANORM, RCOND, WORK, IWORK, INFO)

IF (INFO .NE. 0) THEN

PRINT 1040, ABS(INFO)

STOP 2

END IF

PRINT 1050, 1.0D0 / RCOND

C

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

1010 FORMAT (4(3X, F6.3))

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

1030 FORMAT (1X, 'Error factoring A, INFO = ', I5)

1040 FORMAT (1X, 'Illegal argument to DPOCON, argument #', I2)

1050 FORMAT (/1X, 'Estimated condition number of A = ', F8.4)

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 elements)
2.000 -1.000 0.000 0.000
****** 2.000 -1.000 0.000
****** ****** 2.000 -1.000
****** ****** ****** 2.000
Estimated condition number of A = 9.0000