Cholesky Factorization of a Symmetric Positive Definite Matrix

Calling Sequence

CALL DPOTRF	(UPLO, N, DA, LDA, INFO)
CALL SPOTRF	(UPLO, N, SA, LDA, INFO)
CALL ZPOTRF	(UPLO, N, ZA, LDA, INFO)
CALL CPOTRF	(UPLO, N, CA, LDA, INFO)
void dpotrf	(char uplo, int n, double *da, int lda, int *info)
void spotrf	(char uplo, int n, float *sa, int lda, int *info)
void zpotrf	(char uplo, int n, doublecomplex *za, int lda, int *info)
void cpotrf	(char uplo, int n, complex *ca, int lda, 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	On entry, the upper or lower triangle of the matrix A. On exit, a Cholesky factorization of the matrix A.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA max(1, N).
INFO	On exit:
	INFO = 0	Subroutine completed normally.
	INFO < 0	The ith argument, where i = |INFO|, had an illegal value.
	INFO > 0	The leading minor of order i of A, where i = INFO, is not positive definite. The factorization could not be completed.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

INTEGER LDA, LDB, N, NRHS

PARAMETER (N = 4)

PARAMETER (NRHS = 1)

PARAMETER (LDA = N)

PARAMETER (LDB = N)

C

DOUBLE PRECISION A(LDA,N), B(LDB,NRHS)

INTEGER ICOL, INFO, IROW

C

EXTERNAL DPOTRF, DPOTRS

INTRINSIC ABS

C

C Initialize the array A to store in symmetric form the

C 4x4 symmetric positive definite coefficient matrix A

C shown below. Initialize the array B to store the right

C hand side vector b shown below.

C

C 2 -1 0 0 6

C A = -1 2 -1 0 b = 12

C 0 -1 2 -1 12

C 0 0 -1 2 6

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 /

DATA B / 6.0D0, 1.2D1, 1.2D1, 6.0D0 /

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)

PRINT 1030

PRINT 1040, B

C

C Cholesky factor A.

C

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

IF (INFO .NE. 0) THEN

PRINT 1050, INFO

STOP 1

END IF

C

C Use the factored form of A to solve Ax=b then print

C the result.

C

CALL DPOTRS ('UPPER TRIANGLE OF A STORED', N, NRHS, A, LDA,

$ B, LDB, INFO)

IF (INFO .NE. 0) THEN

PRINT 1060, ABS(INFO)

STOP 1

END IF

PRINT 1070

PRINT 1040, B

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, 'b:')

1040 FORMAT (1X, F6.2)

1050 FORMAT (1X, 'Error factoring A, INFO = ', I5)

1060 FORMAT (1X, 'Illegal argument to DPOTRS, argument #', I1)

1070 FORMAT (/1X, 'x:')

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
b:
6.00
12.00
12.00
6.00
x:
18.00
30.00
30.00
18.00