Reciprocal Condition Number of an LU-Factored General Matrix in Banded Storage

Calling Sequence

CALL DGBCON	(NORM, N, NSUB, NSUPER, DA, LDA, IPIVOT, DANORM, DRCOND, DWORK, IWORK2, INFO)
CALL SGBCON	(NORM, N, NSUB, NSUPER, SA, LDA, IPIVOT, SANORM, SRCOND, SWORK, IWORK2, INFO)
CALL ZGBCON	(NORM, N, NSUB, NSUPER, ZA, LDA, IPIVOT, DANORM, DRCOND, ZWORK, DWORK2, INFO)
CALL CGBCON	(NORM, N, NSUB, NSUPER, CA, LDA, IPIVOT, SANORM, SRCOND, CWORK, SWORK2, INFO)
void dgbcon	(char norm, int n, int nsub, int nsuper, double *da, int lda, int *ipivot, double danorm, double *drcond, int *info)
void sgbcon	(char norm, int n, int nsub, int nsuper, float *sa, int lda, int *ipivot, float sanorm, float *srcond, int *info)
void zgbcon	(char norm, int n, int nsub, int nsuper, doublecomplex *za, int lda, int *ipivot, double danorm, double *drcond, int *info)
void cgbcon	(char norm, int n, int nsub, int nsuper, complex *ca, int lda, 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.
NSUB	Number of subdiagonals of A. N-1 NSUB 0 but if N = 0 then NSUB = 0.
NSUPER	Number of superdiagonals of A. N-1 NSUPER 0 but if N = 0 then NSUPER = 0.
xA	LU factorization of the matrix A, as computed by xGBTRF.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA 2 × NSUB + NSUPER + 1.
IPIVOT	Pivot indices as computed by xGBTRF.
xANORM	On entry, if NORM = 'O', 'o', or '1', then ANORM contains the 1-norm of the original matrix A; if NORM = 'I' or 'i,' then ANORM contains the -norm of the original 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 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, LDWORK, N, NSUB, NSUPER

PARAMETER (N = 5)

PARAMETER (NSUB = 4)

PARAMETER (NSUPER = 0)

PARAMETER (LDA = 2*NSUB + 1 + NSUPER)

PARAMETER (LDWORK = 3 * N)

C

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

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

C

EXTERNAL DGBCON, DGBTRF

INTRINSIC ABS

C

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

C matrix A with four subdiagonals and no superdiagonals

C shown below.

C

C 1

C 1 -1

C A = 1 -2 1

C 1 -3 3 -1

C 1 -4 6 -4 1

C

DATA A / NSUB*8D8, 5*1.0D0,

$ NSUB*8D8, -1.0D0, -2.0D0, -3.0D0, -4.0D0, 8D8,

$ NSUB*8D8, 1.0D0, 3.0D0, 6.0D0, 2*8D8,

$ NSUB*8D8, -1.0D0, 4.0D0, 3*8D8,

$ NSUB*8D8, 1.0D0, 4*8D8 /

C

C Print the initial values of the arrays.

C

PRINT 1000

DO 100, IROW = NSUB + 1, LDA

PRINT 1010,

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

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

100 CONTINUE

PRINT 1020

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

C

C LU factor A.

C

CALL DGBTRF (N, N, NSUB, NSUPER, A, LDA, IPIVOT, INFO)

IF (INFO .NE. 0) THEN

PRINT 1030, INFO

STOP 1

END IF

C

C Compute and print the condition number of A.

C

ANORM = 1.0D1

CALL DGBCON ('ONE NORM', N, NSUB, NSUPER, A, LDA, IPIVOT,

$ ANORM, RCOND, WORK, IWORK, INFO)

IF (INFO .NE. 0) THEN

PRINT 1040, ABS(INFO)

STOP 2

END IF

PRINT 1050, 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, 'Error factoring A, INFO = ', I5)

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

1050 FORMAT (/1X, 'Estimated condition number of A = ', F8.4)

C

END

Sample Output

A in full form:

1.0

1.0 -1.0

1.0 -2.0 1.0

1.0 -3.0 3.0 -1.0

1.0 -4.0 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 **** ****

1.0 -4.0 **** **** ****

1.0 **** **** **** ****

Estimated condition number of A = 260.0000