SUBROUTINE CNBCO (ABE, LDA, N, ML, MU, IPVT, RCOND, Z) C***BEGIN PROLOGUE CNBCO C***PURPOSE Factor a band matrix using Gaussian elimination and C estimate the condition number. C***LIBRARY SLATEC C***CATEGORY D2C2 C***TYPE COMPLEX (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 CNBCO factors a complex band matrix by Gaussian C elimination and estimates the condition of the matrix. C C If RCOND is not needed, CNBFA is slightly faster. C To solve A*X = B , follow CNBCO by CNBSL. C To compute INVERSE(A)*C , follow CNBCO by CNBSL. C To compute DETERMINANT(A) , follow CNBCO by CNBDI. C C On Entry C C ABE COMPLEX(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 REAL 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 COMPLEX(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 CAXPY, CDOTC, CNBFA, CSSCAL, SCASUM C***REVISION HISTORY (YYMMDD) C 800730 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 CNBCO