UDU or LDL Factorization of a Symmetric Matrix

Calling Sequence

CALL DSYTRF	(UPLO, N, DA, LDA, IPIVOT, DWORK, LDWORK, INFO)
CALL SSYTRF	(UPLO, N, SA, LDA, IPIVOT, SWORK, LDWORK, INFO)
CALL ZSYTRF	(UPLO, N, ZA, LDA, IPIVOT, ZWORK, LDWORK, INFO)
CALL CSYTRF	(UPLO, N, CA, LDA, IPIVOT, CWORK, LDWORK, INFO)
void dsytrf	(char uplo, int n, double *da, int lda, int *ipivot, int *info)
void ssytrf	(char uplo, int n, float *sa, int lda, int *ipivot, int *info)
void zsytrf	(char uplo, int n, doublecomplex *za, int lda, int *ipivot, int *info)
void csytrf	(char uplo, int n, complex *ca, int lda, int *ipivot, 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 UDU or LDL factorization of A.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA max(1, N).
IPIVOT	On exit, an N-element array containing details of the row interchanges and block structure of D.
xWORK	Scratch array with a dimension of LDWORK.
LDWORK	Leading dimension of the array WORK as specified in a dimension or type statement. LDWORK 1.
INFO	On exit:
	INFO = 0	Subroutine completed normally.
	INFO < 0	The ith argument, where i = |INFO|, had an illegal value.
	INFO > 0	D(i,i), where i = INFO, is exactly zero, and D is therefore singular. The factorization is complete, but division by zero will occur if this factorization is used to solve a linear system.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

INTEGER LDA, LDB, LDWORK, N, NRHS

PARAMETER (N = 3)

PARAMETER (NRHS = 1)

PARAMETER (LDA = N)

PARAMETER (LDB = LDA)

PARAMETER (LDWORK = 1)

C

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

INTEGER ICOL, INFO, IROW, IPIVOT(N)

C

EXTERNAL DSYTRF, DSYTRS

C

C Initialize the array A to store the coefficient matrix A

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

C hand side vector b shown below.

C

C 1 2 2 15

C A = 2 1 2 b = 15

C 2 2 1 15

C

DATA A / 1.0D0, 8D8, 8D8, 2.0D0, 1.0D0, 8D8, 2.0D0, 2.0D0,

$ 1.0D0 /

DATA B / N*1.5D1 /

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

DO 110, IROW = 1, LDA

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

110 CONTINUE

PRINT 1030

PRINT 1040, B

C

C Solve the system Ax=b, then print the result.

C

CALL DSYTRF ('UPPER TRIANGLE OF A STORED', N, A, LDA,

$ IPIVOT, WORK, LDWORK, INFO)

IF (INFO .EQ. 0) THEN

CALL DSYTRS ('UPPER TRIANGULAR FACTOR', N, NRHS, A, LDA,

$ IPIVOT, B, LDB, INFO)

IF (INFO .EQ. 0) THEN

PRINT 1050

PRINT 1040, B

ELSE

PRINT 1060, INFO

END IF

ELSE

PRINT 1070, INFO

END IF

C

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

1010 FORMAT (3(3X, F4.1))

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

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

1040 FORMAT (1X, 2X, F4.1)

1050 FORMAT (/1X, 'x:')

1060 FORMAT (1X, 'Solve failed with INFO = ', I6)

1070 FORMAT (1X, 'Factorization failed with INFO = ', I6)

C

END

Sample Output

A in full form:
1.0 2.0 2.0
2.0 1.0 2.0
2.0 2.0 1.0
A in symmetric form: (* in unused elements)
1.0 2.0 2.0
**** 1.0 2.0
**** **** 1.0
b:
15.0
15.0
15.0
x:
3.0
3.0
3.0