*DECK CNBCO 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 INTEGER LDA,N,ML,MU,IPVT(*) COMPLEX ABE(LDA,*),Z(*) REAL RCOND C COMPLEX CDOTC,EK,T,WK,WKM REAL ANORM,S,SCASUM,SM,YNORM INTEGER I,INFO,J,JU,K,KB,KP1,L,LDB,LM,LZ,M,ML1,MM,NL,NU COMPLEX ZDUM,ZDUM1,ZDUM2,CSIGN1 REAL CABS1 CABS1(ZDUM) = ABS(REAL(ZDUM)) + ABS(AIMAG(ZDUM)) CSIGN1(ZDUM1,ZDUM2) = CABS1(ZDUM1)*(ZDUM2/CABS1(ZDUM2)) C C COMPUTE 1-NORM OF A C C***FIRST EXECUTABLE STATEMENT CNBCO ML1=ML+1 LDB = LDA - 1 ANORM = 0.0E0 DO 10 J = 1, N NU = MIN(MU,J-1) NL = MIN(ML,N-J) L = 1 + NU + NL ANORM = MAX(ANORM,SCASUM(L,ABE(J+NL,ML1-NL),LDB)) 10 CONTINUE C C FACTOR C CALL CNBFA(ABE,LDA,N,ML,MU,IPVT,INFO) C C RCOND = 1/(NORM(A)*(ESTIMATE OF NORM(INVERSE(A)))) . C ESTIMATE = NORM(Z)/NORM(Y) WHERE A*Z = Y AND CTRANS(A)*Y = E . C CTRANS(A) IS THE CONJUGATE TRANSPOSE OF A . C THE COMPONENTS OF E ARE CHOSEN TO CAUSE MAXIMUM LOCAL C GROWTH IN THE ELEMENTS OF W WHERE CTRANS(U)*W = E . C THE VECTORS ARE FREQUENTLY RESCALED TO AVOID OVERFLOW. C C SOLVE CTRANS(U)*W = E C EK = (1.0E0,0.0E0) DO 20 J = 1, N Z(J) = (0.0E0,0.0E0) 20 CONTINUE M = ML + MU + 1 JU = 0 DO 100 K = 1, N IF (CABS1(Z(K)) .NE. 0.0E0) EK = CSIGN1(EK,-Z(K)) IF (CABS1(EK-Z(K)) .LE. CABS1(ABE(K,ML1))) GO TO 30 S = CABS1(ABE(K,ML1))/CABS1(EK-Z(K)) CALL CSSCAL(N,S,Z,1) EK = CMPLX(S,0.0E0)*EK 30 CONTINUE WK = EK - Z(K) WKM = -EK - Z(K) S = CABS1(WK) SM = CABS1(WKM) IF (CABS1(ABE(K,ML1)) .EQ. 0.0E0) GO TO 40 WK = WK/CONJG(ABE(K,ML1)) WKM = WKM/CONJG(ABE(K,ML1)) GO TO 50 40 CONTINUE WK = (1.0E0,0.0E0) WKM = (1.0E0,0.0E0) 50 CONTINUE KP1 = K + 1 JU = MIN(MAX(JU,MU+IPVT(K)),N) MM = ML1 IF (KP1 .GT. JU) GO TO 90 DO 60 I = KP1, JU MM = MM + 1 SM = SM + CABS1(Z(I)+WKM*CONJG(ABE(K,MM))) Z(I) = Z(I) + WK*CONJG(ABE(K,MM)) S = S + CABS1(Z(I)) 60 CONTINUE IF (S .GE. SM) GO TO 80 T = WKM -WK WK = WKM MM = ML1 DO 70 I = KP1, JU MM = MM + 1 Z(I) = Z(I) + T*CONJG(ABE(K,MM)) 70 CONTINUE 80 CONTINUE 90 CONTINUE Z(K) = WK 100 CONTINUE S = 1.0E0/SCASUM(N,Z,1) CALL CSSCAL(N,S,Z,1) C C SOLVE CTRANS(L)*Y = W C DO 120 KB = 1, N K = N + 1 - KB NL = MIN(ML,N-K) IF (K .LT. N) Z(K) = Z(K) + CDOTC(NL,ABE(K+NL,ML1-NL),-LDB, 1 Z(K+1),1) IF (CABS1(Z(K)) .LE. 1.0E0) GO TO 110 S = 1.0E0/CABS1(Z(K)) CALL CSSCAL(N,S,Z,1) 110 CONTINUE L = IPVT(K) T = Z(L) Z(L) = Z(K) Z(K) = T 120 CONTINUE S = 1.0E0/SCASUM(N,Z,1) CALL CSSCAL(N,S,Z,1) C YNORM = 1.0E0 C C SOLVE L*V = Y C DO 140 K = 1, N L = IPVT(K) T = Z(L) Z(L) = Z(K) Z(K) = T NL = MIN(ML,N-K) IF (K .LT. N) CALL CAXPY(NL,T,ABE(K+NL,ML1-NL),-LDB,Z(K+1),1) IF (CABS1(Z(K)) .LE. 1.0E0) GO TO 130 S = 1.0E0/CABS1(Z(K)) CALL CSSCAL(N,S,Z,1) YNORM = S*YNORM 130 CONTINUE 140 CONTINUE S = 1.0E0/SCASUM(N,Z,1) CALL CSSCAL(N,S,Z,1) YNORM = S*YNORM C C SOLVE U*Z = V C DO 160 KB = 1, N K = N + 1 - KB IF (CABS1(Z(K)) .LE. CABS1(ABE(K,ML1))) GO TO 150 S = CABS1(ABE(K,ML1))/CABS1(Z(K)) CALL CSSCAL(N,S,Z,1) YNORM = S*YNORM 150 CONTINUE IF (CABS1(ABE(K,ML1)) .NE. 0.0E0) Z(K) = Z(K)/ABE(K,ML1) IF (CABS1(ABE(K,ML1)) .EQ. 0.0E0) Z(K) = 1.0E0 LM = MIN(K,M) - 1 LZ = K - LM T = -Z(K) CALL CAXPY(LM,T,ABE(K-1,ML+2),-LDB,Z(LZ),1) 160 CONTINUE C MAKE ZNORM = 1.0E0 S = 1.0E0/SCASUM(N,Z,1) CALL CSSCAL(N,S,Z,1) YNORM = S*YNORM C IF (ANORM .NE. 0.0E0) RCOND = YNORM/ANORM IF (ANORM .EQ. 0.0E0) RCOND = 0.0E0 RETURN END