SUBROUTINE CGBCO (ABD, LDA, N, ML, MU, IPVT, RCOND, Z) C***BEGIN PROLOGUE CGBCO C***PURPOSE Factor a band matrix by Gaussian elimination and C estimate the condition number of the matrix. C***LIBRARY SLATEC (LINPACK) C***CATEGORY D2C2 C***TYPE COMPLEX (SGBCO-S, DGBCO-D, CGBCO-C) C***KEYWORDS BANDED, CONDITION NUMBER, LINEAR ALGEBRA, LINPACK, C MATRIX FACTORIZATION C***AUTHOR Moler, C. B., (U. of New Mexico) C***DESCRIPTION C C CGBCO factors a complex band matrix by Gaussian C elimination and estimates the condition of the matrix. C C If RCOND is not needed, CGBFA is slightly faster. C To solve A*X = B , follow CGBCO by CGBSL. C To compute INVERSE(A)*C , follow CGBCO by CGBSL. C To compute DETERMINANT(A) , follow CGBCO by CGBDI. C C On Entry C C ABD COMPLEX(LDA, N) C contains the matrix in band storage. The columns C of the matrix are stored in the columns of ABD and C the diagonals of the matrix are stored in rows C ML+1 through 2*ML+MU+1 of ABD . C See the comments below for details. C C LDA INTEGER C the leading dimension of the array ABD . C LDA must be .GE. 2*ML + MU + 1 . 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 ABD an upper triangular matrix in band storage and C 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 M = ML + MU + 1 C DO 20 J = 1, N C I1 = MAX(1, J-MU) C I2 = MIN(N, J+Ml) C DO 10 I = I1, I2 C K = I - J + M C ABD(K,J) = A(I,J) C 10 CONTINUE C 20 CONTINUE C C This uses rows ML+1 through 2*ML+MU+1 of ABD . C In addition, the first ML rows in ABD are used for C elements generated during the triangularization. C The total number of rows needed in ABD is 2*ML+MU+1 . C The ML+MU by ML+MU upper left triangle and the C ML by ML lower right triangle are not referenced. 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 ABD should contain C C * * * + + + , * = not used C * * 13 24 35 46 , + = used for pivoting C * 12 23 34 45 56 C 11 22 33 44 55 66 C 21 32 43 54 65 * 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, CGBFA, CSSCAL, SCASUM C***REVISION HISTORY (YYMMDD) C 780814 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 900326 Removed duplicate information from DESCRIPTION section. C (WRB) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE CGBCO