dgbco(l) — SunSoft Performance Library

NAME

dgbco - compute the LU factorization and condition number of a general matrix A in banded storage. If the condition number is not needed then xGBFA is slightly faster. It is typical to follow a call to xGBCO with a call to xGBSL to solve Ax = b or to xGBDI to compute the determinant of A.

SYNOPSIS

CALL DGBCO (DA, LDA, N, NSUB, NSUPER, IPIVOT, DRCOND, DWORK)

CALL SGBCO (SA, LDA, N, NSUB, NSUPER, IPIVOT, SRCOND, SWORK)

CALL ZGBCO (ZA, LDA, N, NSUB, NSUPER, IPIVOT, DRCOND, ZWORK)

CALL CGBCO (CA, LDA, N, NSUB, NSUPER, IPIVOT, SRCOND, CWORK)

ARGUMENTS

xAOn entry, the matrix A.
On exit, an LU factorization of the matrix A.

LDALeading dimension of the array A as specified in a dimension or type statement.

LDA ∗ 2 ∗ NSUB + NSUPER + 1.

NOrder of the matrix A. N >= 0.

NSUBNumber of subdiagonals of A. N-1 >= NSUB >= 0 but if N = 0 then
NSUB = 0.

NSUPERNumber of superdiagonals of A. N-1 >= NSUPER >= 0 but if N = 0
then NSUPER = 0.

IPIVOTOn exit, a vector of pivot indices.

xRCONDOn exit, an estimate of the reciprocal condition number of A.
0.0 <= RCOND <= 1.0. As the value of RCOND gets smaller, operations with A such as solving Ax = b may become less stable. If RCOND satisfies RCOND + 1.0 = 1.0 then A may be singular to working precision.

xWORKScratch array with a dimension of N.

SAMPLE PROGRAM

      PROGRAM TEST
      IMPLICIT NONE
C
      INTEGER           IAXEQB, LDA, LDAB, N, NDIAG, NSUB, NSUPER
      PARAMETER        (IAXEQB = 0)
      PARAMETER        (N = 4)
      PARAMETER        (LDA = N)
      PARAMETER        (NSUB = 1)
      PARAMETER        (NSUPER = 1)
      PARAMETER        (NDIAG = NSUB + 1 + NSUPER)
      PARAMETER        (LDAB = 2 ∗ NSUB + 1 + NSUPER)
C
      DOUBLE PRECISION AB(LDAB,N), AG(LDA,N), B(N), RCOND, WORK(N)
      INTEGER           ICOL, IPIVOT(N), IROW, IROWB, I1, I2, JOB
C
      EXTERNAL          DGBCO, DGBSL
      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∗8D8 /
      DATA AG / 2.0D0, -1.0D0, 2∗0D0, -1.0D0, 2.0D0, -1.0D0,
     $           2∗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 is taken from the
C     comment block in the original DGBFA by Cleve Moler.
C
      DO 10, ICOL = 1, N
        I1 = MAX0 (1, ICOL - NSUPER)
        I2 = MIN0 (N, ICOL + NSUB)
        DO 10, IROW = I1, I2
          IROWB = IROW - ICOL + NDIAG
          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 = 2 ∗ NSUB, 2 ∗ NSUB + 1 + NSUPER)
      PRINT 1030
      PRINT 1040, B
C
C     Factor the matrix in banded form.
C
      CALL DGBCO (AB, LDA, N, NSUB, NSUPER, IPIVOT, RCOND, WORK)
      PRINT 1050, RCOND
      IF ((RCOND + 1.0D0) .EQ. 1.0D0) THEN
        PRINT 1070
      END IF
      JOB = IAXEQB
      CALL DGBSL (AB, LDA, N, NSUB, NSUPER, IPIVOT, B, JOB)
      PRINT 1060
      PRINT 1040, B
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 (3X, F4.1)
1050 FORMAT (/1X, ’Reciprocal of the condition number: ’, F5.2)
1060 FORMAT (/1X, ’A∗∗(-1) ∗ b:’)
1070 FORMAT (1X, ’A may be singular to working precision.’)
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

Reciprocal of the condition number: 0.08

A∗∗(-1) ∗ b:
   10.0
   15.0
   15.0
   10.0