UDU or LDL Factorization of a Symmetric Positive Definite Tridiagonal Matrix

Calling Sequence

CALL DPTTRF	(N, DDIAG, DOFFD, INFO)
CALL SPTTRF	(N, SDIAG, SOFFD, INFO)
CALL ZPTTRF	(N, DDIAG, ZOFFD, INFO)
CALL CPTTRF	(N, SDIAG, COFFD, INFO)
void dpttrf	(int n, double *ddiag, double *doffd, int *info)
void spttrf	(int n, float *sdiag, float *soffd, int *info)
void zpttrf	(int n, double *ddiag, doublecomplex *zoffd, int *info)
void cpttrf	(int n, float *sdiag, complex *coffd, int *info)

Arguments

N	Order of the matrix A. N 0.
xDIAG	On entry, the N diagonal elements of the matrix A. On exit, the N diagonal elements of the diagonal matrix D from the UDU or LDL factorization of A.
xOFFD	On entry, the N-1 off-diagonal elements of the matrix A. On exit, the N-1 off-diagonal elements of the unit bidiagonal factor U or L from the factorization of A.
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 LDB, N, NRHS

PARAMETER (N = 4)

PARAMETER (NRHS = 1)

PARAMETER (LDB = N)

C

DOUBLE PRECISION B(LDB,NRHS), DIAG(N), DLOWER(N-1)

INTEGER ICOL, INFO, IROW

C

EXTERNAL DPTTRF, DPTTRS

INTRINSIC ABS

C

C Initialize the arrays DIAG and DLOWER to store in symmetric

C tridiagonal form the 4x4 symmetric positive definite

C coefficient matrix A shown below. Initialize the array B

C to store the right 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 DIAG / 2.0D0, 2.0D0, 2.0D0, 2.0D0 /

DATA DLOWER / -1.0D0, -1.0D0, -1.0D0 /

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, (0.0D0, ICOL = 1, IROW - 2),

$ (DLOWER(ICOL + 1), ICOL = ABS(IROW - 2), IROW - 2),

$ DIAG(IROW),

$ (DLOWER(IROW), ICOL = 1, MIN(1, N - IROW)),

$ (0.0D0, ICOL = IROW + 2, N)

100 CONTINUE

PRINT 1020

PRINT 1030, B

C

C LDL factor A.

C

CALL DPTTRF (N, DIAG, DLOWER, INFO)

IF (INFO .NE. 0) THEN

PRINT 1040, INFO

STOP 1

END IF

C

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

C the result.

C

CALL DPTTRS (N, NRHS, DIAG, DLOWER, B, LDB, INFO)

IF (INFO .NE. 0) THEN

PRINT 1050, ABS(INFO)

STOP 2

END IF

PRINT 1060

PRINT 1030, B

C

1000 FORMAT (1X, 'A:')

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

1020 FORMAT (/1X, 'b:')

1030 FORMAT (1X, F6.2)

1040 FORMAT (1X, 'Error factoring A, INFO = ', I5)

1050 FORMAT (1X, 'Illegal argument to DPTTRS, argument #', I1)

1060 FORMAT (/1X, 'x:')

C

END

Sample Output

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