Reciprocal Condition Number of an LU-Factored Tridiagonal Matrix

Calling Sequence

CALL DGTCON	(NORM, N, DLOW, DDIAG, DUP1, DUP2, IPIVOT, DANORM, DRCOND, DWORK, IWORK2, INFO)
CALL SGTCON	(NORM, N, SLOW, SDIAG, SUP1, SUP2, IPIVOT, SANORM, SRCOND, SWORK, IWORK2, INFO)
CALL ZGTCON	(NORM, N, ZLOW, ZDIAG, ZUP1, ZUP2, IPIVOT, DANORM, DRCOND, ZWORK, INFO)
CALL CGTCON	(NORM, N, CLOW, CDIAG, CUP1, CUP2, IPIVOT, SANORM, SRCOND, CWORK, INFO)
void dgtcon	(char norm, int n, double *dlow, double *ddiag, double *dup1, double *dup2, int *ipivot, double danorm, double *drcond, int *info)
void sgtcon	(char norm, int n, float *slow, float *sdiag, float *sup1, float *sup2, int *ipivot, float sanorm, float *srcond, int *info)
void zgtcon	(char norm, int n, doublecomplex *zlow, doublecomplex *zdiag, doublecomplex *zup1, doublecomplex *zup2, int *ipivot, double danorm, double *drcond, int *info)
void cgtcon	(char norm, int n, complex *clow, complex *cdiag, complex *cup1, complex *cup2, int *ipivot, float sanorm, float *srcond, int *info)

Arguments

NORM	Indicates whether the 1-norm condition number or the -norm condition number is required. The legal values for NORM are listed below. Any values not listed below are illegal.
	'O', 'o', or '1'	1-norm condition number is required.
	'I' or 'i'	-norm condition number is required.
N	Order of the matrix A. N 0.
xLOW	The N-1 multipliers that define the matrix L from the LU factorization of A, as computed by xGTTRF.
xDIAG	The N diagonal elements of the upper triangular matrix U from the LU factorization of A, as computed by xGTTRF.
xUP1	The N-1 elements of the first superdiagonal of U from the LU factorization of A, as computed by xGTTRF.
xUP2	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.
xANORM	The 1-norm of matrix A.
xRCOND	On exit, the estimated reciprocal of the condition number of the matrix A where the reciprocal condition number of A is defined to be 1 / (ANORM × ||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.
xWORK	Scratch array with a dimension of 2 × N.
IWORK2	Scratch array with a dimension of N for real subroutines.
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, N, NRHS

PARAMETER (N = 5)

PARAMETER (LDB = N)

PARAMETER (LDIWRK = N)

PARAMETER (LDWORK = 2 * N)

PARAMETER (NRHS = 1)

C

DOUBLE PRECISION ANORM, B(LDB,NRHS), DIAG(N), DLOWER(N-1)

DOUBLE PRECISION DUP1(N-1), DUP2(N-2), RCOND, WORK(LDWORK)

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

C

EXTERNAL DGTCON, DGTTRF, DGTTRS

INTRINSIC ABS

C

C Initialize the arrays DLOWER, 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 -3 1 6

C 1 -2 1 12

C A = 1 -2 1 b = 18

C 1 -2 1 12

C 1 -1 6

C

DATA DLOWER / 1.0D0, 1.0D0, 1.0D0, 1.0D0 /

DATA DIAG / -3.0D0, -2.0D0, -2.0D0, -2.0D0, -1.0D0 /

DATA DUP1 / 1.0D0, 1.0D0, 1.0D0, 1.0D0 /

DATA B / 6.0D0, 1.2D1, 1.8D1, 1.2D1, 6.0D0 /

C

PRINT 1000

DO 100, IROW = 1, N

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

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

$ DIAG(IROW),

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

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

100 CONTINUE

PRINT 1020

PRINT 1030, B

C

CALL DGTTRF (N, DLOWER, DIAG, DUP1, DUP2, IPIVOT, INFO)

IF (INFO .NE. 0) THEN

PRINT 1040, INFO

STOP 1

END IF

ANORM = 4.0D0

CALL DGTCON ('ONE-NORM', N, DLOWER, DIAG, DUP1, DUP2,

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

IF (INFO .NE. 0) THEN

PRINT 1050, ABS (INFO)

STOP 2

END IF

PRINT 1060, 1.0D0 / RCOND

CALL DGTTRS ('NO TRANSPOSE A', N, NRHS, DLOWER, DIAG, DUP1,

$ DUP2, IPIVOT, B, LDB, INFO)

IF (INFO .NE. 0) THEN

PRINT 1070, INFO

STOP 3

END IF

PRINT 1080

PRINT 1030, B

C

1000 FORMAT (1X, 'A:')

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

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

1030 FORMAT (1X, F8.2)

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: ', F6.2)

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

1080 FORMAT (/1X, 'x:')

C

END

Sample Output

A:
-3.00 1.00 0.00 0.00 0.00
1.00 -2.00 1.00 0.00 0.00
0.00 1.00 -2.00 1.00 0.00
0.00 0.00 1.00 -2.00 1.00
0.00 0.00 0.00 1.00 -1.00
b:
6.00
12.00
18.00
12.00
6.00
Estimated condition number of A: 50.00
x:
-27.00
-75.00
-111.00
-129.00
-135.00