Downdate an Augmented Cholesky Decomposition

Calling Sequence

CALL DCHDD	(DA, LDA, N, DX, DZ, LDZ, NZ, DY, DRHO, DCOS, DSIN, INFO)
CALL SCHDD	(SA, LDA, N, SX, SZ, LDZ, NZ, SY, SRHO, SCOS, SSIN, INFO)
CALL ZCHDD	(ZA, LDA, N, ZX, ZZ, LDZ, NZ, ZY, DRHO, DCOS, DSIN, INFO)
CALL CCHDD	(CA, LDA, N, CX, CZ, LDZ, NZ, CY, SRHO, SCOS, SSIN, INFO)
void dchdd	(double *da, int lda, int n, double *dx, double *dz, int ldz, int nz, double *dy, double *drho, double *dcos, double *dsin, int *info)
void schdd	(float *sa, int lda, int n, float *sx, float *sz, int ldz, int nz, float *sy, float *srho, float *scos, float *ssin, int *info)
void zchdd	(doublecomplex *za, int lda, int n, doublecomplex *zx, doublecomplex *zz, int ldz, int nz, doublecomplex *zy, double *drho, double *dcos, double *dsin, int *info)
void cchdd	(complex *ca, int lda, int n, complex *cx, complex *cz, int ldz, int nz, complex *cy, double *drho, double *dcos, double *dsin, int *info)

Arguments

xA	On entry, the upper triangular matrix A. On exit, A has been downdated. 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.
xX	Row to be added to A.
xZ	Vectors to be downdated with A.
LDZ	Leading dimension on the array Z as specified in a dimension or type statement. LDZ max(1,N).
NZ	Number of vectors to be downdated with A. NZ 0. If NZ = 0 then Z, Y, and RHO are not used.
xY	Scalars for downdating the vectors in Z.
xRHO	On entry, the norms of the residual vectors that are to be downdated. On exit, RHO has been downdated. If RHO(i) is negative on entry then it is not changed.
xCOS	Cosines of the transforming rotations.
xSIN	Sines of the transforming rotations.
INFO	On exit:
	INFO = 0	Subroutine completed normally.
	INFO = -1	A could not be downdated; all values are left unchanged.
	INFO = 1	Some RHOs could not be downdated; all RHOs that could not be downdated are changed to -1.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

INTEGER LDA, N, NOPIV, NZ

PARAMETER (N = 4)

PARAMETER (LDA = N)

PARAMETER (NOPIV = 0)

PARAMETER (NZ = 0)

C

DOUBLE PRECISION A(LDA,N), ANULL, C(N), S(N), WORK(N), X(N)

INTEGER I, INFO, IPIVOT(N), J, JOB, NULL

C

EXTERNAL DCHDC, DCHDD

C

C Initialize the arrays A and Z to store the matrices A and Z

C shown below and initialize X and Y to store the vectors x

C and y shown below.

C

C 4 3 2 1 1

C A = 3 4 3 2 x = 1

C 2 3 4 3 1

C 1 2 3 4 1

C

DATA A / 4.0D0, 3*8D8, 3.0D0, 4.0D0, 2*8D8, 2.0D0, 3.0D0,

$ 4.0D0, 8D8, 1.0D0, 2.0D0, 3.0D0, 4.0D0 /

C

PRINT 1000

DO 100, I = 1, N

PRINT 1010, (A(J,I), J = 1, I), (A(I,J), J = I + 1, N)

100 CONTINUE

PRINT 1020

PRINT 1010, ((A(I,J), J = 1, N), I = 1, N)

JOB = NOPIV

CALL DCHDC (A, LDA, N, WORK, IPIVOT, JOB, INFO)

IF (INFO .EQ. N) THEN

PRINT 1030

PRINT 1010, A(1,1), A(1,2), A(1,3), A(1,4)

PRINT 1040, A(2,2), A(2,3), A(2,4)

PRINT 1050, A(3,3), A(3,4)

PRINT 1060, A(4,4)

ANULL = 0.0D0

NULL = 1

CALL DCHDD (A, LDA, N, X, ANULL, NULL, NZ, ANULL, ANULL,

$ C, S, INFO)

IF (INFO .EQ. 0) THEN

PRINT 1070

PRINT 1080, (C(I), S(I), I = 1, N)

ELSE

PRINT 1090

END IF

ELSE

PRINT 1100

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, 'Upper Cholesky factor:')

1040 FORMAT (10X, 3(3X, F7.3))

1050 FORMAT (20X, 2(3X, F7.3))

1060 FORMAT (30X, 1(3X, F7.3))

1070 FORMAT (/1X, 'Cosine', 3X, ' Sine')

1080 FORMAT (1X, F6.3, 3X, F6.3)

1090 FORMAT (/1X, 'A cannot be downdated.')

1100 FORMAT (/1X, 'A is not positive definite.')

C

END

Sample Output

A in full form:
4.000 3.000 2.000 1.000
3.000 4.000 3.000 2.000
2.000 3.000 4.000 3.000
1.000 2.000 3.000 4.000
A in symmetric form (* in unused entries)
4.000 3.000 2.000 1.000
******* 4.000 3.000 2.000
******* ******* 4.000 3.000
******* ******* ******* 4.000
Upper Cholesky factor:
2.000 1.500 1.000 0.500
1.323 1.134 0.945
1.309 1.091
1.291
Cosine Sine
1.000 0.000
1.000 0.000
1.000 0.000
1.000 0.000