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

Calling Sequence

CALL DGERFS	(TRANSA, N, NRHS, DA, LDA, DAF, LDAF, IPIVOT, DB, LDB, DX, LDX, DFERR, DBERR, DWORK, IWORK2, INFO)
CALL SGERFS	(TRANSA, N, NRHS, SA, LDA, SAF, LDAF, IPIVOT, SB, LDB, SX, LDX, SFERR, SBERR, SWORK, IWORK2, INFO)
CALL ZGERFS	(TRANSA, N, NRHS, ZA, LDA, ZAF, LDAF, IPIVOT, ZB, LDB, ZX, LDX, DFERR, DBERR, ZWORK, DWORK2, INFO)
CALL CGERFS	(TRANSA, N, NRHS, CA, LDA, CAF, LDAF, IPIVOT, CB, LDB, CX, LDX, SFERR, SBERR, CWORK, SWORK2, INFO)
void dgerfs	(char trans, int n, int nrhs, double *da, int lda, double *daf, int ldaf, int *ipivot, double *db, int ldb, double *dx, int ldx, double *dferr, double *dberr, int *info)
void sgerfs	(char trans, int n, int nrhs, float *sa, int lda, float *saf, int ldaf, int *ipivot, float *sb, int ldb, float *sx, int ldx, float *sferr, float *sberr, int *info)
void zgerfs	(char trans, int n, int nrhs, doublecomplex *za, int lda, doublecomplex *zaf, int ldaf, int *ipivot, doublecomplex *zb, int ldb, doublecomplex *zx, int ldx, double *dferr, double *dberr, int *info)
void cgerfs	(char trans, int n, int nrhs, complex *ca, int lda, complex *caf, int ldaf, 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.
xA	Matrix A.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA max(1, N).
xAF	LU factorization of the matrix A as computed by xGETRF.
LDAF	Leading dimension of the array AF as specified in a dimension or type statement. LDAF max(1, N).
IPIVOT	Pivot indices as computed by xGETRF.
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 LDA, LDAF, LDB, LDIWRK, LDWORK, LDX

INTEGER N, NRHS

PARAMETER (N = 3)

PARAMETER (LDA = N)

PARAMETER (LDAF = N)

PARAMETER (LDB = N)

PARAMETER (LDIWRK = N)

PARAMETER (LDWORK = 3 * N)

PARAMETER (LDX = N)

PARAMETER (NRHS = 1)

C

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

DOUBLE PRECISION B1(LDB,NRHS), BERR(NRHS), EPSLON

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

INTEGER ICOL, INFO, IPIVOT(N), IROW

INTEGER IWORK(LDIWRK)

C

EXTERNAL DCOPY, DGEMV, DGERFS, DGETRF, DGETRS

INTRINSIC ABS, DBLE

C

EPSLON = ((((2.0D0 / 3.0D0) + 1.6D1) - 1.6D1) -

$ (2.0D0 / 3.0D0))

C

C Initialize the array A to store the matrix A shown below.

C Each of the diagonal elements is equal to the row number

C plus epsilon.

C

C 1+e 1 1

C A = 2 2+e 2

C 3 3 3+e

C

DO 110, ICOL = 1, N

DO 100, IROW = 1, N

A(IROW,ICOL) = DBLE(IROW)

100 CONTINUE

A(ICOL,ICOL) = A(ICOL,ICOL) + EPSLON

110 CONTINUE

CALL DCOPY (LDA*N, A, 1, AF, 1)

C

C Print the initial value of A.

C

PRINT 1000

DO 120, IROW = 1, N

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

120 CONTINUE

C

C Initialize the array B to store the right hand side

C vector b shown below. Each element is equal to N plus

C a small epsilon. Print the contents of the array after

C initialization.

C

C N+e

C b = N+e

C N+e

C

DO 130, IROW = 1, N

B(IROW,1) = DBLE(N) + EPSLON

130 CONTINUE

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

PRINT 1020

PRINT 1030, B

C

C LU factor A.

C

CALL DGETRF (N, N, AF, LDAF, IPIVOT, INFO)

IF (INFO .NE. 0) THEN

PRINT 1040, INFO

STOP 1

END IF

C

C Use the LU factorization to solve the system.

C

CALL DGETRS ('NO TRANSPOSE A', N, NRHS, AF, LDAF, IPIVOT,

$ X, LDX, INFO)

IF (INFO .NE. 0) THEN

PRINT 1050, INFO

STOP 2

END IF

C

C Print the solution and Ax.

C

PRINT 1060

PRINT 1070, X

CALL DGEMV ('N', N, N, 1.0D0, A, LDA, X, 1, 0.0D0, B1, 1)

PRINT 1080

PRINT 1030, B1

C

C Refine the solution, then print the refined solution and Ax

C with the refined x.

C

CALL DGERFS ('NO TRANSPOSE A', N, NRHS, A, LDA, AF, LDAF,

$ IPIVOT, B, LDB, X, LDX, FERR, BERR, WORK,

$ IWORK, INFO)

IF (INFO .NE. 0) THEN

PRINT 1090, INFO

STOP 2

END IF

PRINT 1100

PRINT 1070, X

CALL DGEMV ('N', N, N, 1.0D0, A, LDA, X, 1, 0.0D0, B1, 1)

PRINT 1110

PRINT 1030, B1

C

1000 FORMAT (1X, 'A:')

1010 FORMAT (3(3X, F22.17))

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

1030 FORMAT (1X, F22.17)

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

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

1060 FORMAT (/1X, 'Initial solution for Ax=b:')

1070 FORMAT (1X, E16.6)

1080 FORMAT (/1X, 'Ax with initial x:')

1090 FORMAT (1X, 'Error improving solution in DGERFS,

$ INFO = ', I5)

1100 FORMAT (/1X, 'Improved solution for Ax=b:')

1110 FORMAT (/1X, 'Ax with improved x:')

C

END

Sample Output

A:
1.00000000000000133 1.00000000000000000 1.00000000000000000
2.00000000000000000 2.00000000000000133 2.00000000000000000
3.00000000000000000 3.00000000000000000 3.00000000000000133
b:
3.00000000000000133
3.00000000000000133
3.00000000000000133
Initial solution for Ax=b:
0.112590E+16
0.333333E+00
-0.112590E+16
Ax with initial x:
2.75000000000000000
2.50000000000000000
2.50000000000000000
Improved solution for Ax=b:
0.115717E+16
0.625500E+14
-0.121972E+16
Ax with improved x:
3.25000000000000000
3.50000000000000000
3.00000000000000000