Refined Solution to a Linear System in an LU-Factored General Tridiagonal Matrix

Calling Sequence

CALL DGTRFS	(TRANSA, N, NRHS, DLOW, DDIAG, DUP, DLOWF, DDIAGF, DUPF1, DUPF2, IPIVOT, DB, LDB, DX, LDX, DFERR, DBERR, DWORK, IWORK2, INFO)
CALL SGTRFS	(TRANSA, N, NRHS, SLOW, SDIAG, SUP, SLOWF, SDIAGF, SUPF1, SUPF2, IPIVOT, SB, LDB, SX, LDX, SFERR, SBERR, SWORK, IWORK2, INFO)
CALL ZGTRFS	(TRANSA, N, NRHS, ZLOW, ZDIAG, ZUP, ZLOWF, ZDIAGF, ZUPF1, ZUPF2, IPIVOT, ZB, LDB, ZX, LDX, DFERR, DBERR, ZWORK, DWORK2, INFO)
CALL CGTRFS	(TRANSA, N, NRHS, CLOW, CDIAG, CUP, CLOWF, CDIAGF, CUPF1, CUPF2, IPIVOT, CB, LDB, CX, LDX, SFERR, SBERR, CWORK, SWORK2, INFO)
void dgtrfs	(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 *dferr, double *dberr, int *info)
void sgtrfs	(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 *sferr, float *sberr, int *info)
void zgtrfs	(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 *dferr, double *dberr, int *info)
void cgtrfs	(char trans, int n, int nrhs, complex *clow, complex *cdiag, complex *cup, complex *clowf, complex *cdiagf, complex *cupf1, complex *cupf2, int *ipivot, complex *cb, int ldb, complex *cx, int ldx, float *sferr, float *sberr, int *info)

Arguments

TRANSA	Indicates the form of the linear system whose solution is to be refined. The legal values for TRANSA are listed below. Any values not listed below are illegal.
	'N' or 'n'	No transpose, refine AX = B.
	'T' or 't'	Transpose, refine ATX = B.
	'C' or 'c'	Conjugate transpose, refine 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 matrices B and X. NRHS 0.
xLOW	The N-1 elements of the first subdiagonal of A.
xDIAG	The N diagonal elements of A.
xUP	The N-1 elements of the first superdiagonal A.
xLOWF	The N-1 multipliers that define the matrix L from the LU factorization of A, as computed by xGTTRF.
xDIAGF	The N diagonal elements of the upper triangular matrix U from the LU factorization of A, as computed by xGTTRF.
xUPF1	The N-1 elements of the first superdiagonal of U from the LU factorization of A, as computed by xGTTRF.
xUPF2	The N-2 elements of the second superdiagonal of U from the LU factorization of A, as computed by xGTTRF.
IPIVOT	Pivot indices 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 entry, the N×NRHS solution matrix X. On exit, the refined N×NRHS solution matrix X.
LDX	Leading dimension of the array X as specified in a dimension or type statement. LDX max(1, N).
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.

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 ANORM, B(LDB,NRHS), BERR(NRHS), DIAG(N)

DOUBLE PRECISION DIAGF(N), DLOW(N-1), DLOWF(N-1), DUP1(N-1)

DOUBLE PRECISION DUP1F(N-1), DUP2F(N-2), EPSLON, FERR(NRHS)

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

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

C

EXTERNAL DCOPY, DGTCON, DGTRFS, DGTTRF, DGTTRS

INTRINSIC ABS, MAX

C

C Initialize the arrays DLOW, DIAG, and DUP1 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 0 3 5

C A = 0 0 7 b = 5

C 0.1 0 3 5

C 0 0 5

C

DATA DLOW / 0.0D0, 1.0D-1, 0.0D0 /

DATA DIAG / 0.0D0, 0.0D0, 0.0D0, 0.0D0 /

DATA DUP1 / 3.0D0, 7.0D0, 3.0D0 /

DATA B / 5.0D0, 5.0D0, 5.0D0, 5.0D0 /

C

C Add a small value to each of the elements on the diagonal

C and the first sub- and super-diagonal of A. After this

C loop, A will resemble the matrix shown below. Print A

C after adding epsilon.

C

C e 3+e

C A = e e 7+e

C 0.1+e e 3+e

C e e

C

EPSLON = ((((2.0D0 / 3.0D0) + 8.0D0) - 8.0D0) -

$ (2.0D0 / 3.0D0))

DO 100, IROW = 1, N - 1

DLOW(IROW) = DLOW(IROW) + EPSLON

DIAG(IROW) = DIAG(IROW) + EPSLON

DUP1(IROW) = DUP1(IROW) - EPSLON

100 CONTINUE

DIAG(N) = DIAG(N) + EPSLON

CALL DCOPY (N - 1, DLOW, 1, DLOWF, 1)

CALL DCOPY (N, DIAG, 1, DIAGF, 1)

CALL DCOPY (N - 1, DUP1, 1, DUP1F, 1)

C

PRINT 1000

DO 110, IROW = 1, N

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

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

$ DIAG(IROW),

$ (DUP1(IROW), ICOL = 1, MIN(1, N - IROW)),

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

110 CONTINUE

C

C Add a small value to each element of B. After this loop, B

C will resemble the matrix shown below. Print B after adding

C epsilon.

C

C 5+e

C B = 5+e

C 5+e

C 5+e

C

DO 130, ICOL = 1, NRHS

DO 120, IROW = 1, N

B(IROW,ICOL) = B(IROW,ICOL) + EPSLON

120 CONTINUE

130 CONTINUE

CALL DCOPY (N, B, 1, X, 1)

PRINT 1020

PRINT 1030, B

C

C LU factor A.

C

CALL DGTTRF (N, DLOWF, DIAGF, DUP1F, DUP2F, IPIVOT, INFO)

IF (INFO .NE. 0) THEN

PRINT 1040, INFO

STOP 1

END IF

C

C Estimate the condition number of A.

C

ANORM = 9.25D0 + (3.0D0 * EPSLON)

CALL DGTCON ('ONE-NORM', N, DLOWF, DIAGF, DUP1F, DUP2F,

$ IPIVOT, ANORM, RCOND, WORK, IWORK, INFO)

IF (INFO .NE. 0) THEN

PRINT 1050, ABS (INFO)

STOP 2

END IF

PRINT 1060, 1.0D0 / RCOND

C

C Solve Ax=b and print the solution.

C

CALL DGTTRS ('NO TRANSPOSE A', N, NRHS, DLOWF, DIAGF,

$ DUP1F, DUP2F, IPIVOT, X, LDX, INFO)

IF (INFO .NE. 0) THEN

PRINT 1070, INFO

STOP 3

END IF

PRINT 1080

PRINT 1090, X

PRINT 1100

PRINT 1030, DIAG(1) * X(1,1) + DUP1(1) *

$ X(2,1)

PRINT 1030, DLOW(1) * X(1,1) + DIAG(2) * X(2,1) + DUP1(2) *

$ X(3,1)

PRINT 1030, DLOW(2) * X(2,1) + DIAG(3) * X(3,1) + DUP1(3) *

$ X(4,1)

PRINT 1030, DLOW(3) * X(3,1) + DIAG(4) * X(4,1)

C

C Refine the solution to Ax=b and print the refined solution.

C

CALL DGTRFS ('NO TRANSPOSE A', N, NRHS, DLOW, DIAG, DUP1,

$ DLOWF, DIAGF, DUP1F, DUP2F, IPIVOT, B, LDB,

$ X, LDX, FERR, BERR, WORK, IWORK, INFO)

IF (INFO .NE. 0) THEN

PRINT 1110, ABS(INFO)

STOP 4

END IF

PRINT 1120

PRINT 1090, X

PRINT 1130

PRINT 1030, DIAG(1) * X(1,1) + DUP1(1) *

$ X(2,1)

PRINT 1030, DLOW(1) * X(1,1) + DIAG(2) * X(2,1) + DUP1(2) *

$ X(3,1)

PRINT 1030, DLOW(2) * X(2,1) + DIAG(3) * X(3,1) + DUP1(3) *

$ X(4,1)

PRINT 1030, DLOW(3) * X(3,1) + DIAG(4) * X(4,1)

PRINT 1140, BERR(1)

PRINT 1150, FERR(1)

C

1000 FORMAT (1X, 'A:')

1010 FORMAT (4(2X, F18.16))

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

1030 FORMAT (1X, F21.17)

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

1050 FORMAT (1X, 'Illegal argument to DGTCON, argument #', I2)

1060 FORMAT (/1X, 'Estimated condition number of A: ', E12.6)

1070 FORMAT (1X, 'Error solving Ax=b, INFO = ', I5)

1080 FORMAT (/1X, 'Initial solution to Ax=b:')

1090 FORMAT (1X, E25.17)

1100 FORMAT (/1X, 'Ax with the initial x:')

1110 FORMAT (1X, 'Illegal argument to DGTRFS, INFO = ', I2)

1120 FORMAT (/1X, 'Refined solution to Ax=b:')

1130 FORMAT (/1X, 'Ax with refined x:')

1140 FORMAT (/1X, 'Forward error: ', E12.4)

1150 FORMAT (1X, 'Backward error: ', E12.4)

C

END

Sample Output

A:
-.0000000000000006 3.0000000000000004 0.0000000000000000 0.0000000000000000
-.0000000000000006 -.0000000000000006 7.0000000000000009 0.0000000000000000
0.0000000000000000 0.0999999999999995 -.0000000000000006 3.0000000000000004
0.0000000000000000 0.0000000000000000 -.0000000000000006 -.0000000000000006
b:
4.99999999999999911
4.99999999999999911
4.99999999999999911
4.99999999999999911
Estimated condition number of A: 0.227847E+33
Initial solution to Ax=b:
-0.12316065590651126E+33
-0.22789299319224172E+17
-0.97668425653817900E+16
0.75964331064080138E+15
Ax with the initial x:
16.00000000000000000
24.00000000000000000
5.50000000000000000
4.99999999999999822
Refined solution to Ax=b:
-0.12316065590651124E+33
-0.22789299319224172E+17
-0.97668425653817920E+16
0.75964331064080125E+15
Ax with refined x:
0.00000000000000000
-8.00000000000000000
5.00000000000000000
4.99999999999999911
Forward error: 0.1170E-15
Backward error: 0.2384E-14