SLATEC Routines --- DNBCO ---
*DECK DNBCO
SUBROUTINE DNBCO (ABE, LDA, N, ML, MU, IPVT, RCOND, Z)
C***BEGIN PROLOGUE DNBCO
C***PURPOSE Factor a band matrix using Gaussian elimination and
C estimate the condition number.
C***LIBRARY SLATEC
C***CATEGORY D2A2
C***TYPE DOUBLE PRECISION (SNBCO-S, DNBCO-D, CNBCO-C)
C***KEYWORDS BANDED, LINEAR EQUATIONS, MATRIX FACTORIZATION,
C NONSYMMETRIC
C***AUTHOR Voorhees, E. A., (LANL)
C***DESCRIPTION
C
C DNBCO factors a double precision band matrix by Gaussian
C elimination and estimates the condition of the matrix.
C
C If RCOND is not needed, DNBFA is slightly faster.
C To solve A*X = B , follow DNBCO by DNBSL.
C To compute INVERSE(A)*C , follow DNBCO by DNBSL.
C To compute DETERMINANT(A) , follow DNBCO by DNBDI.
C
C On Entry
C
C ABE DOUBLE PRECISION(LDA, NC)
C contains the matrix in band storage. The rows
C of the original matrix are stored in the rows
C of ABE and the diagonals of the original matrix
C are stored in columns 1 through ML+MU+1 of ABE.
C NC must be .GE. 2*ML+MU+1 .
C See the comments below for details.
C
C LDA INTEGER
C the leading dimension of the array ABE.
C LDA must be .GE. N .
C
C N INTEGER
C the order of the original matrix.
C
C ML INTEGER
C number of diagonals below the main diagonal.
C 0 .LE. ML .LT. N .
C
C MU INTEGER
C number of diagonals above the main diagonal.
C 0 .LE. MU .LT. N .
C More efficient if ML .LE. MU .
C
C On Return
C
C ABE an upper triangular matrix in band storage
C and the multipliers which were used to obtain it.
C The factorization can be written A = L*U where
C L is a product of permutation and unit lower
C triangular matrices and U is upper triangular.
C
C IPVT INTEGER(N)
C an integer vector of pivot indices.
C
C RCOND DOUBLE PRECISION
C an estimate of the reciprocal condition of A .
C For the system A*X = B , relative perturbations
C in A and B of size EPSILON may cause
C relative perturbations in X of size EPSILON/RCOND .
C If RCOND is so small that the logical expression
C 1.0 + RCOND .EQ. 1.0
C is true, then A may be singular to working
C precision. In particular, RCOND is zero if
C exact singularity is detected or the estimate
C underflows.
C
C Z DOUBLE PRECISION(N)
C a work vector whose contents are usually unimportant.
C If A is close to a singular matrix, then Z is
C an approximate null vector in the sense that
C NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
C
C Band Storage
C
C If A is a band matrix, the following program segment
C will set up the input.
C
C ML = (band width below the diagonal)
C MU = (band width above the diagonal)
C DO 20 I = 1, N
C J1 = MAX(1, I-ML)
C J2 = MIN(N, I+MU)
C DO 10 J = J1, J2
C K = J - I + ML + 1
C ABE(I,K) = A(I,J)
C 10 CONTINUE
C 20 CONTINUE
C
C This uses columns 1 through ML+MU+1 of ABE .
C Furthermore, ML additional columns are needed in
C ABE starting with column ML+MU+2 for elements
C generated during the triangularization. The total
C number of columns needed in ABE is 2*ML+MU+1 .
C
C Example: If the original matrix is
C
C 11 12 13 0 0 0
C 21 22 23 24 0 0
C 0 32 33 34 35 0
C 0 0 43 44 45 46
C 0 0 0 54 55 56
C 0 0 0 0 65 66
C
C then N = 6, ML = 1, MU = 2, LDA .GE. 5 and ABE should contain
C
C * 11 12 13 + , * = not used
C 21 22 23 24 + , + = used for pivoting
C 32 33 34 35 +
C 43 44 45 46 +
C 54 55 56 * +
C 65 66 * * +
C
C***REFERENCES J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W.
C Stewart, LINPACK Users' Guide, SIAM, 1979.
C***ROUTINES CALLED DASUM, DAXPY, DDOT, DNBFA, DSCAL
C***REVISION HISTORY (YYMMDD)
C 800728 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890831 Modified array declarations. (WRB)
C 890831 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE DNBCO