Solution to a Linear System in a General Matrix in Banded Storage (Expert Driver)

Calling Sequence

CALL DGBSVX	(FACT, TRANSA, N, NSUB, NSUPER, NRHS, DA, LDA, DAF, LDAF, IPIVOT, EQUED, DROWSC, DCOLSC, DB, LDB, DX, LDX, DRCOND, DFERR, DBERR, DWORK, IWORK2, INFO)
CALL SGBSVX	(FACT, TRANSA, N, NSUB, NSUPER, NRHS, SA, LDA, SAF, LDAF,IPIVOT, EQUED, SROWSC, SCOLSC, SB, LDB, SX, LDX, SRCOND, SFERR, SBERR, SWORK, IWORK2, INFO)
CALL ZGBSVX	(FACT, TRANSA, N, NSUB, NSUPER, NRHS, ZA, LDA, ZAF, LDAF, IPIVOT, EQUED, DROWSC, DCOLSC, ZB, LDB, ZX, LDX, DRCOND, DFERR, DBERR, ZWORK, DWORK2, INFO)
CALL CGBSVX	(FACT, TRANSA, N, NSUB, NSUPER, NRHS, CA, LDA, CAF, LDAF, IPIVOT, EQUED, SROWSC, SCOLSC, CB, LDB, CX, LDX, SRCOND, SFERR, SBERR, CWORK, SWORK2, INFO)
void dgbsvx	(char fact, char trans, int n, int nsub, int nsuper, int nrhs, double *da, int lda, double *daf, int ldaf, int *ipivot, char *equed, double *drowsc, double *dcolsc, double *db, int ldb, double *dx, int ldx, double *drcond, double *dferr, double *dberr, int *info)
void sgbsvx	(char fact, char trans, int n, int nsub, int nsuper, int nrhs, float *sa, int lda, float *saf, int ldaf, int *ipivot, char *equed, float *srowsc, float *scolsc, float *sb, int ldb, float *sx, int ldx, float *srcond, float *sferr, float *sberr, int *info)
void zgbsvx	(char fact, char trans, int n, int nsub, int nsuper, int nrhs, doublecomplex *za, int lda, doublecomplex *zaf, int ldaf, int *ipivot, char *equed, double *drowsc, double *dcolsc, doublecomplex *zb, int ldb, doublecomplex *zx, int ldx, double *drcond, double *dferr, double *dberr, int *info)
void cgbsvx	(char fact, char trans, int n, int nsub, int nsuper, int nrhs, complex *ca, int lda, complex *caf, int ldaf, int *ipivot, char *equed, float *srowsc, float *scolsc, 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 this table. FACT also affects the contents of IPIVOT; see its description below.

FACT	EQUED entry	EQUED exit	A entry	A exit	AF entry	AF exit
'F', 'f'	'B', 'b'	Unchanged	Row- and col-equilibrated A	Unchanged	LU-factored row- and col-equilibrated A	Unchanged
'F', 'f'	'C','c'	Unchanged	Col-equilibrated A	Unchanged	LU-factored col-equilibrated A	Unchanged
'F', 'f'	'N', 'n'	Unchanged	Matrix A	Unchanged	LU-factored A	Unchanged
'F', 'f'	'R', 'r'	Unchanged	Row-equilibrated A	Unchanged	LU-factored row-equilibrated A	Unchanged
'E', 'e'	Not used	'B'	Matrix A	Row- and col-equilibrated A	Not used	LU-factored row- and col-equilibrated A
'E', 'e'	Not used	'C'	Matrix A	Col-equilibrated A	Not used	LU-factored col-equilibrated A
'E', 'e'	Not used	'N'	Matrix A	Unchanged	Not used	LU-factored A
'E', 'e'	Not used	'R'	Matrix A	Row-equilibrated A	Not used	LU-factored row-equilibrated A
'N', 'n'	Not used	'N'	Matrix A	Unchanged	Not used	LU-factored A

	The above table uses the following abbreviations: LU-factored = the factored form of A as computed by xGBTRF Row-equilibrated A = diag(ROWSC) × A Col-equilibrated A = A × diag(COLSC) Row- and col-equilibrated A = diag(ROWSC) × A × diag(COLSC)
TRANSA	Indicates the form of the linear system to solve. The legal values for TRANSA are listed below. Any values not listed below are illegal.
	'N' or 'n'	No transpose, solve AX = B.
	'T' or 't'	Transpose, solve ATX = B.
	'C' or 'c'	Conjugate transpose, solve AHX = B.
	Note that AT and AH are the same for real matrices.
N	Order of the matrix A. N 0.
NSUB	Number of subdiagonals of A. N-1 NSUB 0 but if N = 0 then NSUB = 0.
NSUPER	Number of superdiagonals of A. N-1 NSUPER 0 but if N = 0 then NSUPER = 0.
NRHS	Number of right-hand sides, equal to the number of columns in 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 NSUB + NSUPER + 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. LDA 2 × NSUB + NSUPER + 1.
IPIVOT	Pivot indices; values vary depending on FACT and EQUED as shown below:

FACT	EQUED Entry	EQUED Exit	IPIVOT Entry	IPIVOT Exit
'F' or 'f'	'B', 'b', 'C', 'c', 'R', 'r'	Not used	Pivot indices from LU factorization of equilibrated A as computed by xGBTRF	Unchanged
'F' or 'f'	'N' or 'n'	Not used	Pivot indices from LU factorization of A as computed by xGBTRF	Unchanged
'E' or 'e'	Not used	'B', 'C', 'R'	Not used	Pivot indices from LU factorization of equilibrated A as computed by xGBTRF
'E' or 'e'	Not used	'N'	Not used	Pivot indices from LU factorization of A as computed by xGBTRF
'N' or 'n'	Not used	'B', 'C', 'R'	Not used	Pivot indices from LU factorization of equilibrated A as computed by xGBTRF
'N' or 'n'	Not used	'N'	Not used	Pivot indices from LU factorization of A as computed by xGBTRF

EQUED	EQUED interacts with FACT and IPIVOT as shown in the tables above. 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 'b', 'c', 'n', or 'r' 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 = 'B', 'C', 'N', or 'R'.
xROWSC	On entry, if FACT = 'F' or 'f' and EQUED = 'R', 'r', 'B', or 'b', then ROWSC contains the N row scale factors that were used to equilibrate A. Each element of ROWSC must be positive. If FACT or EQUED have other values, the entry values of ROWSC are not used. On exit, if FACT = 'E' or 'e' and EQUED = 'R' or 'B', then ROWSC contains the N row scale factors that were used to equilibrate A. If FACT or EQUED have other values, ROWSC is unchanged.
xCOLSC	On entry, if FACT = 'F' or 'f' and EQUED = 'C', 'c', 'B', or 'b', then COLSC contains the N column scale factors that were used to equilibrate A. Each element of COLSC must be positive. If FACT or EQUED have other values, the entry values of COLSC are not used. On exit, if FACT = 'E' or 'e' and EQUED = 'C' or 'B', then COLSC contains the N column scale factors that were used to equilibrate A. If FACT or EQUED have other values, COLSC is unchanged.
xB	On entry, the N×NRHS right-hand side matrix B. On exit, the value of B is determined by the input value TRANSA and the output value of EQUED as shown below. For combinations of TRANSA and EQUED not shown below, B is not changed.
	TRANSA	EQUED	Output value of B
	'N', 'n'	'R' or 'B'	diag(R) × B
	'T', 't', 'C', 'c'	'C' or 'B'	diag(C) × 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 the solution to the original problem may differ from the solution to the equilibrated system for the cases shown below:
	TRANSA	EQUED	Solution to equilibrated system
	'N', 'n'	'C' or 'B'	diag(C)-1 × X
	'T', 't', 'C', 'c'	'R' or 'B'	diag(R)-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	U(i,i), where i = INFO, is exactly zero and U is therefore singular. The LU factorization of A has been completed, but the solution and error bounds could not be computed.
	INFO = N+1	RCOND is less than machine precision. The LU factorization of A has been completed but A is singular to working precision and so the solution and error bounds could not be computed.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

INTEGER LDAB, LDAFB, LDB, LDX, N, NRHS, NSUB

INTEGER NSUPER, SIZEAB, SIZEB, SIZEX

PARAMETER (NSUB = 1)

PARAMETER (NSUPER = 1)

PARAMETER (N = 4)

PARAMETER (NRHS = 1)

PARAMETER (LDAB = NSUB + NSUPER + 1)

PARAMETER (LDAFB = 2*NSUB + NSUPER + 1)

PARAMETER (LDB = N)

PARAMETER (LDX = N)

PARAMETER (SIZEAB = LDAB * N)

PARAMETER (SIZEB = LDB * NRHS)

PARAMETER (SIZEX = LDX * NRHS)

C

DOUBLE PRECISION AB(LDAB,N), AFB(LDAFB,N), B(LDB,NRHS)

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

DOUBLE PRECISION ROWSCA(N), WORK(3*N), X(LDX,NRHS)

INTEGER ICOL, INFO, IPIVOT(N), IROW, IWORK(N)

CHARACTER*1 EQUED

C

EXTERNAL DGBSVX

INTRINSIC MAX0, MIN0

C

C Initialize the array AB to store in banded storage mode the

C 4x4 matrix A with one subdiagonal and one superdiagonal

C shown below. Initialize the array B to store the vector b

C shown below.

C

C 2 -1 5

C A = -1 2 -1 b = 5

C -1 2 -1 5

C -1 2 5

C

DATA AB / 8.8D8, 2.0D0, -1.0D0, -1.0D0, 2.0D0, -1.0D0,

$ -1.0D0, 2.0D0, -1.0D0, -1.0D0, 2.0D0, 8.8D8 /

DATA B / SIZEB*5.0D0 /

DATA X / SIZEX*9.9D9 /

C

C Print the initial values of the arrays.

C

PRINT 1000

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

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

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

PRINT 1010, 0.0D0, 0.0D0, AB(3,3), AB(2,4)

PRINT 1020

PRINT 1010, ((AB(IROW,ICOL), ICOL = 1, N), IROW = 1, LDAB)

PRINT 1030

PRINT 1040, B

C

C Solve the system Ax=b.

C

CALL DGBSVX ('EQUILIBRATE A', 'NO TRANSPOSE A', N, NSUB,

$ NSUPER, NRHS, AB, LDAB, AFB, LDAFB, IPIVOT,

$ EQUED, ROWSCA, COLSCA, B, LDB, X, LDX, RCOND,

$ FERR, BERR, WORK, IWORK, INFO)

IF (INFO .EQ. 0) THEN

PRINT 1050

PRINT 1040, X

PRINT 1060, 1.0D0 / RCOND

IF (EQUED .EQ. 'N') PRINT 1070

IF (EQUED .EQ. 'R') PRINT 1080

IF (EQUED .EQ. 'C') PRINT 1090

IF (EQUED .EQ. 'B') PRINT 1100

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

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

ELSE

PRINT 1130, INFO

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

PRINT 1140

END IF

END IF

C

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

1010 FORMAT (4(3X, F4.1))

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

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

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

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

1060 FORMAT (/1X, 'Estimated condition number of A: ', F7.2)

1070 FORMAT (1X, 'No equilibration was required for A.')

1080 FORMAT (1X, 'Row equilibration was done on A.')

1090 FORMAT (1X, 'Column equilibration was done on A.')

1100 FORMAT (1X, 'Row and column equilibration was done on A.')

1110 FORMAT (/1X, 'Backward error for system #', I1, ': ', E12.6)

1120 FORMAT (1X, 'Forward error for system #', I1, ': ', E12.6)

1130 FORMAT (1X, 'Error in solving system, INFO = ', I3)

1140 FORMAT (1X, 'Matrix is singular to working precision.')

C

END

Sample Output

A in full form:
2.0 -1.0 0.0 0.0
-1.0 2.0 -1.0 0.0
0.0 -1.0 2.0 -1.0
0.0 0.0 -1.0 2.0
A in banded form: (* in unused elements)
**** -1.0 -1.0 -1.0
2.0 2.0 2.0 2.0
-1.0 -1.0 -1.0 ****
b:
5.0
5.0
5.0
5.0
x:
10.0
15.0
15.0
10.0
Estimated condition number of A: 12.00
No equilibration was required for A.
Backward error for system #1: 0.592119E-16
Forward error for system #1: 0.511591E-14