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

Calling Sequence

CALL DGTSVX	(FACT, TRANSA, N, NRHS, DLOW, DDIAG, DUP, DLOWF, DDIAGF, DUPF1, DUPF2, IPIVOT, DB, LDB, DX, LDX, DRCOND, DFERR, DBERR, DWORK, IWORK2, INFO)
CALL SGTSVX	(FACT, TRANSA, N, NRHS, SLOW, SDIAG, SUP, SLOWF, SDIAGF, SUPF1, SUPF2, IPIVOT, SB, LDB, SX, LDX, SRCOND, SFERR, SBERR, SWORK, IWORK2, INFO)
CALL ZGTSVX	(FACT, TRANSA, N, NRHS, ZLOW, ZDIAG, ZUP, ZLOWF, ZDIAGF, ZUPF1, ZUPF2, IPIVOT, ZB, LDB, ZX, LDX, DRCOND, DFERR, DBERR, ZWORK, DWORK2, INFO)
CALL CGTSVX	(FACT, TRANSA, N, NRHS, CLOW, CDIAG, CUP, CLOWF, CDIAGF, CUPF1, CUPF2, IPIVOT, CB, LDB, CX, LDX, SRCOND, SFERR, SBERR, CWORK, SWORK2, INFO)
void dgtsvx	(char fact, char trans, int n, int nrhs, double *dlow, double *ddiag, double *dup, double *dlowf, double *ddiagf, double *dupf1, double *dupf2, int *ipivot, double *db, int ldb, double *dx, int ldx, double *drcond, double *dferr, double *dberr, int *info)
void sgtsvx	(char fact, char trans, int n, int nrhs, float *slow, float *sdiag, float *sup, float *slowf, float *sdiagf, float *supf1, float *supf2, int *ipivot, float *sb, int ldb, float *sx, int ldx, float *srcond, float *sferr, float *sberr, int *info)
void zgtsvx	(char fact, char trans, int n, int nrhs, doublecomplex *zlow, doublecomplex *zdiag, doublecomplex *zup, doublecomplex *zlowf, doublecomplex *zdiagf, doublecomplex *zupf1, doublecomplex *zupf2, int *ipivot, doublecomplex *zb, int ldb, doublecomplex *zx, int ldx, double *drcond, double *dferr, double *dberr, int *info)
void cgtsvx	(char fact, char trans, int n, int nrhs, complex *clow, complex *cdiag, complex *cup, complex *clowf, complex *cdiag, complex *cupf1, complex *cupf2, int *ipivot, 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 A has been supplied on entry.
	'F' or 'f'	On entry, LOWF, DIAGF, UPF, UPF2, and IPIVOT contain the factored form of A. These arguments, as well as LOW, DIAG, and UP, will not be modified.
	'N' or 'n'	The matrix A will be copied to LOWF, DIAGF, and UPF and factored.
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.
NRHS	Number of right-hand sides, equal to the number of columns of the matrix B. NRHS 0.
xLOW	The N-1 subdiagonal elements of A.
xDIAG	The N diagonal elements of A.
xUP	The N-1 superdiagonal elements of A.
xLOWF	On entry, if FACT = 'F' or 'f', then LOWF contains the N-1 multipliers that define the matrix L from the LU factorization of A, as computed by xGTTRF. On exit, if FACT = 'N', then LOWF contains the N-1 multipliers that define the matrix L from the LU factorization of A, as computed by xGTTRF.
xDIAGF	On entry, if FACT = 'F' or 'f', then DIAGF contains the N diagonal elements of the upper triangular matrix U from the LU factorization of A as computed by xGTTRF. On exit, if FACT = 'N', then DIAGF contains the N diagonal elements of the upper triangular matrix U from the LU factorization of A, as computed by xGTTRF.
xUPF1	On entry, if FACT = 'F' or 'f', then UPF1 contains the N-1 elements of the first superdiagonal of U. On exit, if FACT = 'N', then UPF1 contains the N-1 elements of the first superdiagonal of U.
xUPF2	On entry, if FACT = 'F' or 'f', then UPF2 contains the N-2 elements of the second superdiagonal of U. On exit, if FACT = 'N', then UPF2 contains the N-2 elements of the second superdiagonal of U.
IPIVOT	On entry, if FACT = 'F' or 'f', then IPIVOT contains the pivot indices from the LU factorization of A, as computed by xGTTRF. On exit, if FACT = 'N', then IPIVOT contains the pivot indices from the LU factorization of A, as computed by xGTTRF.
xB	The N×NRHS right-hand side matrix 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 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, 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 for complex subroutines.
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, U is therefore singular. The factorization has not been completed, unless i = N, 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 LDB, LDIWRK, LDWORK, LDX, N, NRHS

PARAMETER (N = 4)

PARAMETER (LDB = N)

PARAMETER (LDIWRK = N)

PARAMETER (LDWORK = 3 * N)

PARAMETER (LDX = N)

PARAMETER (NRHS = 1)

C

DOUBLE PRECISION B(LDB,NRHS), BERR(NRHS), DIAG(N), DIAGF(N)

DOUBLE PRECISION DLOW(N-1), DLOWF(N-1), DUP(N-1), DUPF(N-1)

DOUBLE PRECISION DUP2F(N-1), FERR(NRHS), RCOND

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

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

C

EXTERNAL DGTSVX

INTRINSIC ABS, MAX

C

C Initialize the arrays DLOWER, DIAG, and DUPPER to store

C the first subdiagonal, the diagonal, and the first

C superdiagonal of the coefficient matrix A shown below.

C Initialize the array B to store the right hand side

C matrix b shown below.

C

C 2 -1 0

C A = 1 2 -1 b = 2

C 1 2 -1 4

C 1 2 11

C

DATA DLOW / 1.0D0, 1.0D0, 1.0D0 /

DATA DIAG / 2.0D0, 2.0D0, 2.0D0, 2.0D0 /

DATA DUP / -1.0D0, -1.0D0, -1.0D0 /

DATA B / 0.0D0, 2.0D0, 4.0D0, 1.1D1 /

C

C Print the initial value of the arrays.

C

PRINT 1000

DO 100, IROW = 1, N

PRINT 1010, (0.0D0, ICOL = 1, IROW - 2),

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

$ DIAG(IROW),

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

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

100 CONTINUE

PRINT 1020

PRINT 1030, B

C

C Solve the system Ax=b. Print the solution and error

C information.

C

CALL DGTSVX ('NOT FACTORED ON ENTRY', 'NO TRANSPOSE A', N,

$ NRHS, DLOW, DIAG, DUP, DLOWF, DIAGF, DUPF,

$ DUP2F, IPIVOT, B, LDB, X, LDX, RCOND, FERR,

$ BERR, WORK, IWORK, INFO)

IF (INFO .LT. 0) THEN

PRINT 1040, INFO

STOP 1

ELSE IF (INFO .GT. 0) THEN

PRINT 1050

STOP 2

END IF

PRINT 1060

PRINT 1030, X

PRINT 1070, BERR(1)

PRINT 1080, FERR(1)

C

1000 FORMAT (1X, 'A:')

1010 FORMAT (5(3X, F5.2))

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

1030 FORMAT (1X, F8.2)

1040 FORMAT (1X, 'Illegal argument to DGTSVX, argument #', I2)

1050 FORMAT (1X, 'A is singular to working precision.')

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

1070 FORMAT (/1X, 'Backward error: ', E14.8)

1080 FORMAT (1X, 'Forward error: ', E14.8)

C

END

Sample Output

A:
2.00 -1.00 0.00 0.00
1.00 2.00 -1.00 0.00
0.00 1.00 2.00 -1.00
0.00 0.00 1.00 2.00
b:
0.00
2.00
4.00
11.00
x:
1.00
2.00
3.00
4.00
Backward error: 0.00000000E+00
Forward error: 0.12174170E-14