Reciprocal Condition Number of a Triangular Matrix in Banded Storage

Calling Sequence

CALL DTBCON	(NORM, UPLO, DIAG, N, NDIAG, DA, LDA, DRCOND, DWORK, IWORK2, INFO)
CALL STBCON	(NORM, UPLO, DIAG, N, NDIAG, SA, LDA, SRCOND, SWORK, IWORK2, INFO)
CALL ZTBCON	(NORM, UPLO, DIAG, N, NDIAG, ZA, LDA, DRCOND, ZWORK, DWORK2, INFO)
CALL CTBCON	(NORM, UPLO, DIAG, N, NDIAG, CA, LDA, SRCOND, CWORK, SWORK2, INFO)
void dtbcon	(char norm, char uplo, char diag, int n, int ndiag, double *da, int lda, double *drcond, int *info)
void stbcon	(char norm, char uplo, char diag, int n, int ndiag, float *sa, int lda, float *srcond, int *info)
void ztbcon	(char norm, char uplo, char diag, int n, int ndiag, doublecomplex *za, int lda, double *drcond, int *info)
void ctbcon	(char norm, char uplo, char diag, int n, int ndiag, complex *ca, int lda, 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.
UPLO	Indicates whether xA contains the upper or lower triangle of the matrix. The legal values for UPLO are listed below. Any values not listed below are illegal.
	'U' or 'u'	xA contains the upper triangle.
	'L' or 'l'	xA contains the lower triangle.
DIAG	Indicates whether or not A is unit triangular. The legal values for DIAG are listed below. Any values not listed below are illegal.
	'N' or 'n'	A is not unit triangular.
	'U' or 'u'	A is unit triangular.
N	Order of the matrix A. N 0.
NDIAG	Number of superdiagonals or subdiagonals of the triangular matrix A. N-1 NDIAG 0 but if N = 0 then NDIAG = 0.
xA	Upper or lower triangular matrix A. If DIAG = 'U' or 'u', the diagonal elements of A are assumed to be 1 and are not used.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA NDIAG + 1.
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 / (||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.
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, LDIWRK, LDWORK, N, NSUB

PARAMETER (N = 5)

PARAMETER (NSUB = 2)

PARAMETER (LDA = 2*NSUB + 1)

PARAMETER (LDIWRK = N)

PARAMETER (LDWORK = 3 * N)

C

DOUBLE PRECISION A(LDA,N), RCOND, WORK(LDWORK)

INTEGER ICOL, INFO, IROW, IWORK(LDIWRK)

C

EXTERNAL DTBCON

INTRINSIC ABS, MAX

C

C Initialize the array A to store in banded form the 5x5

C matrix A with two subdiagonals and no superdiagonals

C shown below.

C

C 1

C 1 -1

C A = 1 -2 1

C -3 3 -1

C 6 -4 1

C

DATA A / NSUB*8D8, 1.0D0, 1.0D0, 1.0D0,

$ NSUB*8D8, -1.0D0, -2.0D0, -3.0D0,

$ NSUB*8D8, 1.0D0, 3.0D0, 6.0D0,

$ NSUB*8D8, -1.0D0, 4.0D0, 8D8,

$ NSUB*8D8, 1.0D0, 8D8, 8D8 /

C

C Print the initial values of the arrays.

C

PRINT 1000

DO 100, IROW = 1, N

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

$ (A(NSUB + 1 + IROW - ICOL, ICOL),

$ ICOL = MAX(1,IROW - NSUB), IROW),

$ (0.0D0, ICOL = IROW + 1, N)

100 CONTINUE

PRINT 1020

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

C

C Compute and print the condition number of A.

C

CALL DTBCON ('ONE-NORM', 'LOWER TRIANGULAR A',

$ 'NOT UNIT DIAGONAL A', N, NSUB, A, LDA, RCOND,

$ WORK, IWORK, INFO)

IF (INFO .NE. 0) THEN

PRINT 1030, ABS(INFO)

STOP 1

END IF

PRINT 1040, 1.0D0 / RCOND

C

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

1010 FORMAT (5(3X, F4.1))

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

1030 FORMAT (1X, 'Illegal argument to DTBCON, argument #', I2)

1040 FORMAT (/1X, 'Estimated condition number of A = ', F8.4)

C

END

Sample Output

A in full form:
1.0 0.0 0.0 0.0 0.0
1.0 -1.0 0.0 0.0 0.0
1.0 -2.0 1.0 0.0 0.0
0.0 -3.0 3.0 -1.0 0.0
0.0 0.0 6.0 4.0 1.0
A in banded form: (* in unused elements)
**** **** **** **** ****
**** **** **** **** ****
1.0 -1.0 1.0 -1.0 1.0
1.0 -2.0 3.0 4.0 ****
1.0 -3.0 6.0 **** ****
Estimated condition number of A = 5.3333