Solution to a Linear System in a Symmetric Positive Definite Matrix in Banded Storage (Expert Driver)

Calling Sequence

CALL DPBSVX	(FACT, UPLO, N, NDIAG, NRHS, DA, LDA, DAF, LDAF, EQUED, DSCALE, DB, LDB, DX, LDX, DRCOND, DFERR, DBERR, DWORK, IWORK2, INFO)
CALL SPBSVX	(FACT, UPLO, N, NDIAG, NRHS, SA, LDA, SAF, LDAF, EQUED, SSCALE, SB, LDB, SX, LDX, SRCOND, SFERR, SBERR, SWORK, IWORK2, INFO)
CALL ZPBSVX	(FACT, UPLO, N, NDIAG, NRHS, ZA, LDA, ZAF, LDAF, EQUED, DSCALE, ZB, LDB, ZX, LDX, DRCOND, DFERR, DBERR, ZWORK, DWORK2, INFO)
CALL CPBSVX	(FACT, UPLO, N, NDIAG, NRHS, CA, LDA, CAF, LDAF, EQUED, SSCALE, CB, LDB, CX, LDX, SRCOND, SFERR, SBERR, CWORK, SWORK2, INFO)
void dpbsvx	(char fact, char uplo, int n, int ndiag, int nrhs, double *da, int lda, double *daf, int ldaf, char *equed, double *dscale, double *db, int ldb, double *dx, int ldx, double *drcond, double *dferr, double *dberr, int *info)
void spbsvx	(char fact, char uplo, int n, int ndiag, int nrhs, float *sa, int lda, float *saf, int ldaf, char *equed, float *sscale, float *sb, int ldb, float *sx, int ldx, float *srcond, float *sferr, float *sberr, int *info)
void zpbsvx	(char fact, char uplo, int n, int ndiag, int nrhs, doublecomplex *za, int lda, doublecomplex *zaf, int ldaf, char *equed, double *dscale, doublecomplex *zb, int ldb, doublecomplex *zx, int ldx, double *drcond, double *dferr, double *dberr, int *info)
void cpbsvx	(char fact, char uplo, int n, int ndiag, int nrhs, complex *ca, int lda, complex *caf, int ldaf, char *equed, float *sscale, complex *cb, int ldb, complex *cx, int ldx, float *srcond, float *sferr, float *sberr, int *info)

Arguments

FACT	Indicates whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A will be equilibrated before it is factored. FACT interacts with EQUED, A, and AF as shown in the table below:

FACT	EQUED entry	EQUED exit	A entry	A exit	AF entry	AF exit
'F', 'f'	'N', 'n'	Unchanged	Matrix A	Unchanged	Cholesky-factored A	Unchanged
'F', 'f'	'Y', 'y'	Unchanged	Equilibrated A	Unchanged	Cholesky-factored equilibrated A	Unchanged
'E', 'e'	Not used	'N'	Matrix A	Unchanged	Not used	Cholesky-factored A
'E', 'e'	Not used	'Y'	Matrix A	Equilibrated A	Not used	Cholesky-factored equilibrated A
'N', 'n'	Not used	'N'	Matrix A	Unchanged	Not used	Cholesky-factored A

	The above table uses the following abbreviations:
	Cholesky-factored = the UTU or LLT factored form of A as computed by xPBTRF
	Equilibrated A = diag(SCALE) × A × diag(SCALE)
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.
NDIAG	Number of superdiagonals or subdiagonals of the matrix A. N-1 NDIAG 0 but if N = 0 then NDIAG = 0.
	If UPLO = 'U' or 'u', NDIAG is the number of superdiagonals.
	If UPLO = 'L' or 'l', NDIAG is the number of subdiagonals.
NRHS	Number of right-hand sides, equal to the number of columns of the matrices B and X. NRHS 0.
xA	The contents of A vary depending on the values of FACT and EQUED. See the table for the argument FACT above.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA NDIAG + 1.
xAF	The contents of AF vary depending on the values of FACT and EQUED. See the table for the argument FACT above.
LDAF	Leading dimension of the array AF as specified in a dimension or type statement. LDAF NDIAG + 1.
EQUED	EQUED interacts with FACT as shown in the table above for the argument FACT.
	Note that one must be careful about specifying lowercase letters for the entry value of EQUED when FACT = 'F' or 'f' because ordinarily LAPACK will always return a capital letter. If the caller specifies 'n', or 'y' on entry then the lowercase letter will still be in EQUED on exit. The caller may later draw incorrect conclusions if the code contains a series of IF statements to determine whether EQUED = 'N' or 'Y'.
xSCALE	On entry, if FACT = 'F' or 'f' and EQUED = 'Y' or 'y' then SCALE contains the N scale factors that were used to equilibrate A. Each element of SCALE must be positive. If FACT or EQUED have other values, the entry values of SCALE are not used. On exit, if FACT = 'E' or 'e' and EQUED = 'Y' then SCALE contains the N scale factors that were used to equilibrate A. If FACT or EQUED have other values, SCALE is unchanged.
xB	On entry, the N×NRHS right-hand side matrix B. On exit, if EQUED = 'N', then B is not modified; if EQUED = 'Y', then B is overwritten by diag(SCALE) × B.
LDB	Leading dimension of the array B as specified in a dimension or type statement. LDB max(1, N).
xX	On exit, the N×NRHS solution matrix for the original problem. Note that if the exit value of EQUED = 'Y' then A and B are modified on exit and the solution to the equilibrated system is diag(SCALE)-1 × X.
LDX	Leading dimension of the array X as specified in a dimension or type statement. LDX max(1, N).
xRCOND	On exit, an estimate of the reciprocal condition number of the matrix A after equilibration, if any, where the reciprocal condition number is defined to be 1 / (||A|| × ||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. In this case, INFO > 0 is returned and the solution and error bounds are not computed.
xFERR	On exit, the estimated forward error bound for each solution vector X(*, j) for 1 j NRHS. If X' is the true solution corresponding to X(*, j) then FERR(j) is an upper bound on the magnitude of the largest element in X(*, j) - X' divided by the magnitude of the largest element in X(*, j).
xBERR	On exit, BERR(j) is the smallest relative change in any element of A or B(*, j) that makes X(*, j) an exact solution to AX(*, j) = B(*, j) for 1 j NRHS.
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.
	1 INFO N	The leading minor of order i of A, where i = INFO, is not positive definite. The factorization has not been completed, and the solution and error bounds could not be computed.
	INFO = N+1	RCOND is less than machine precision. The factorization has been completed, but the matrix is singular to working precision, and the solution and error bounds could not be computed.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

INTEGER LDA, LDAF, LDB, LDIWRK, LDWORK, LDX, N

INTEGER NDIAG, NRHS

PARAMETER (N = 4)

PARAMETER (NDIAG = 1)

PARAMETER (NRHS = 1)

PARAMETER (LDA = NDIAG + 1)

PARAMETER (LDAF = LDA)

PARAMETER (LDB = N)

PARAMETER (LDIWRK = N)

PARAMETER (LDX = N)

PARAMETER (LDWORK = 3 * N)

C

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

DOUBLE PRECISION BERR(NRHS), FERR(NRHS), RCOND, SCALE(N)

DOUBLE PRECISION WORK(LDWORK), X(LDX,NRHS)

INTEGER ICOL, INFO, IROW, IWORK(LDIWRK)

CHARACTER*1 EQUED

C

EXTERNAL DPBSVX

C

C Initialize the array A to store in symmetric banded form

C the 4x4 symmetric positive definite coefficient matrix A

C with one subdiagonal and one superdiagonal shown below.

C Initialize the array B to store the right hand side

C 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 / 8D8, 2.0D0, -1.0D0, 2.0D0,

$ -1.0D0, 2.0D0, -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

PRINT 1010, A(2,1), A(1,2), 0.0D0, 0.0D0

PRINT 1010, A(1,2), A(2,2), A(1,3), 0.0D0

PRINT 1010, 0.0D0, A(1,3), A(2,3), A(1,4)

PRINT 1010, 0.0D0, 0.0D0, A(1,4), A(2,4)

PRINT 1020

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

PRINT 1030

PRINT 1040, B

C

C Solve the system and print the results.

C

CALL DPBSVX ('EQUILIBRATE A', 'UPPER TRIANGLE OF A STORED',

$ N, NDIAG, NRHS, A, LDA, AF, LDAF, EQUED, SCALE,

$ B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK,

$ INFO)

IF (INFO .NE. 0) THEN

PRINT 1050, INFO

IF (INFO .EQ. (N + 1)) THEN

PRINT 1060

STOP 1

END IF

STOP 2

END IF

PRINT 1070

PRINT 1040, X

PRINT 1080, 1.0D0 / RCOND

IF (EQUED .EQ. 'N') THEN

PRINT 1090

ELSE

PRINT 1100

END IF

PRINT 1110, (IROW, BERR(IROW), IROW = 1, NRHS)

PRINT 1120, (IROW, FERR(IROW), IROW = 1, NRHS)

C

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

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

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

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

1040 FORMAT (1X, F6.2)

1050 FORMAT (1X, 'Error solving Ax=b, INFO = ', I5)

1060 FORMAT (1X, 'Matrix is singular to working precision.')

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

1080 FORMAT (/1X, 'Estimated condition number of A: ', F7.2)

1090 FORMAT (1X, 'No equilibration was required for A.')

1100 FORMAT (1X, 'Diagonal equilibration was done on A.')

1110 FORMAT (/1X, 'Backward error bound for system #', I1, ': ',

$ E12.5)

1120 FORMAT (1X, 'Forward error bound for system #', I1, ': ',

$ E12.5)

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 banded form: (* in unused elements)
****** -1.000 -1.000 -1.000
2.000 2.000 2.000 2.000
b:
6.00
12.00
12.00
6.00
x:
18.00
30.00
30.00
18.00
Estimated condition number of A: 12.00
No equilibration was required for A.
Backward error bound for system #1: 0.00000E+00
Forward error bound for system #1: 0.46185E-14