LU Factorization of a General Matrix in Banded Storage

Calling Sequence

CALL DGBTRF	(M, N, NSUB, NSUPER, DA, LDA, IPIVOT, INFO)
CALL SGBTRF	(M, N, NSUB, NSUPER, SA, LDA, IPIVOT, INFO)
CALL ZGBTRF	(M, N, NSUB, NSUPER, ZA, LDA, IPIVOT, INFO)
CALL CGBTRF	(M, N, NSUB, NSUPER, CA, LDA, IPIVOT, INFO)
void dgbtrf	(int m, int n, int nsub, int nsuper, double *da, int lda, int *ipivot, int *info)
void sgbtrf	(int m, int n, int nsub, int nsuper, float *sa, int lda, int *ipivot, int *info)
void zgbtrf	(int m, int n, int nsub, int nsuper, doublecomplex *za, int lda, int *ipivot, int *info)
void cgbtrf	(int m, int n, int nsub, int nsuper, complex *ca, int lda, int *ipivot, int *info)

Arguments

M	Number of rows in the matrix A. M 0.
N	Number of columns in 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	On entry, the matrix A stored in rows NSUB + 1 through 2 × NSUB + NSUPER + 1. Rows 1 through NSUB are ignored on entry and do not need to be set. On exit, an LU factorization of A.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA 2 × NSUB + NSUPER + 1.
IPIVOT	On exit, the pivot indices. During factorization, row i was exchanged with row IPIVOT(i) for 1 i min(M, N)
INFO	On exit:
	INFO = 0	Subroutine completed normally.
	INFO < 0	The ith argument, where i = |INFO|, had an illegal value.
	INFO > 0	U(i,i), where i = INFO, is exactly zero and U is therefore singular. The factorization has been completed, but division by zero will occur if this factorization is used to solve a linear system.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

INTEGER LDAB, LDAG, LDB, M, N, NRHS, NSUB, NSUPER

PARAMETER (NSUB = 1)

PARAMETER (NSUPER = 1)

PARAMETER (M = 4)

PARAMETER (N = 4)

PARAMETER (NRHS = 1)

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

PARAMETER (LDAG = M)

PARAMETER (LDB = N)

C

DOUBLE PRECISION AB(LDAB,N), AG(LDAG,N), B(LDB,NRHS)

INTEGER ICOL, INFO, IPIVOT(N), IROW, IROWB, I1, I2

INTEGER M1

C

EXTERNAL DGBTRF, DGBTRS

INTRINSIC MAX0, MIN0

C

C Initialize the array AG to store the 4x4 matrix A with one

C subdiagonal and one superdiagonal shown below. Initialize

C the array B to store the vector b shown below.

C

C 2 -1 5

C AG = -1 2 -1 b = 5

C -1 2 -1 5

C -1 2 5

C

DATA AB / 16 * 8.0D8 /

DATA AG / 2.0D0, -1.0D0, 2*0.0, -1.0D0, 2.0D0, -1.0D0,

$ 0.0D0, 0.0D0, -1.0D0, 2.0D0, -1.0D0, 2*0D0,

$ -1.0D0, 2.0D0 /

DATA B / N*5.0D0 /

C

C Copy the matrix A from the array AG to the array AB. The

C matrix is stored in general storage mode in AG and it will

C be stored in banded storage mode in AB. The code to copy

C from general to banded storage mode was written for LINPACK

C by Cleve Moler.

C

M1 = NSUB + NSUPER + 1

DO 20, ICOL = 1, N

I1 = MAX0 (1, ICOL - NSUPER)

I2 = MIN0 (N, ICOL + NSUB)

DO 10, IROW = I1, I2

IROWB = IROW - ICOL + M1

AB(IROWB,ICOL) = AG(IROW,ICOL)

10 CONTINUE

20 CONTINUE

C

C Print the initial values of the arrays.

C

PRINT 1000

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

PRINT 1020

PRINT 1010, ((AB(IROW,ICOL), ICOL = 1, N), IROW = 1, LDAB)

PRINT 1030

PRINT 1040, B

C

C Factor the matrix in banded form and print the results.

C

CALL DGBTRF (M, N, NSUB, NSUPER, AB, LDAB, IPIVOT, INFO)

IF (INFO .EQ. 0) THEN

CALL DGBTRS ('NO TRANSPOSE A', N, NSUB, NSUPER, NRHS,

$ AB, LDAB, IPIVOT, B, LDB, INFO)

IF (INFO .EQ. 0) THEN

PRINT 1050

PRINT 1040, B

ELSE

PRINT 1060, INFO

END IF

ELSE

PRINT 1070, INFO

END IF

C

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

1010 FORMAT (4(3X, F4.1))

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

1030 FORMAT (/1X, 'b:')

1040 FORMAT (1X, 2X, F4.1)

1050 FORMAT (/1X, 'x:')

1060 FORMAT (1X, 'Error in solving system, INFO = ', I4)

1070 FORMAT (1X, 'Error in factoring system, INFO = ', I4)

C

END

Sample Output

A in full form:
2.0 -1.0 0.0 0.0
-1.0 2.0 -1.0 0.0
0.0 -1.0 2.0 -1.0
0.0 0.0 -1.0 2.0
A in banded form: (* in unused elements)
**** **** **** ****
**** -1.0 -1.0 -1.0
2.0 2.0 2.0 2.0
-1.0 -1.0 -1.0 ****
b:
5.0
5.0
5.0
5.0
x:
10.0
15.0
15.0
10.0