*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 INTEGER LDA,N,ML,MU,IPVT(*) DOUBLE PRECISION ABE(LDA,*),Z(*) DOUBLE PRECISION RCOND C DOUBLE PRECISION DDOT,EK,T,WK,WKM DOUBLE PRECISION ANORM,S,DASUM,SM,YNORM INTEGER I,INFO,J,JU,K,KB,KP1,L,LDB,LM,LZ,M,ML1,MM,NL,NU C***FIRST EXECUTABLE STATEMENT DNBCO ML1=ML+1 LDB = LDA - 1 ANORM = 0.0D0 DO 10 J = 1, N NU = MIN(MU,J-1) NL = MIN(ML,N-J) L = 1 + NU + NL ANORM = MAX(ANORM,DASUM(L,ABE(J+NL,ML1-NL),LDB)) 10 CONTINUE C C FACTOR C CALL DNBFA(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 TRANS(A)*Y = E . C TRANS(A) IS THE TRANSPOSE OF A . THE COMPONENTS OF E ARE C CHOSEN TO CAUSE MAXIMUM LOCAL GROWTH IN THE ELEMENTS OF W WHERE C TRANS(U)*W = E . THE VECTORS ARE FREQUENTLY RESCALED TO AVOID C OVERFLOW. C C SOLVE TRANS(U)*W = E C EK = 1.0D0 DO 20 J = 1, N Z(J) = 0.0D0 20 CONTINUE M = ML + MU + 1 JU = 0 DO 100 K = 1, N IF (Z(K) .NE. 0.0D0) EK = SIGN(EK,-Z(K)) IF (ABS(EK-Z(K)) .LE. ABS(ABE(K,ML1))) GO TO 30 S = ABS(ABE(K,ML1))/ABS(EK-Z(K)) CALL DSCAL(N,S,Z,1) EK = S*EK 30 CONTINUE WK = EK - Z(K) WKM = -EK - Z(K) S = ABS(WK) SM = ABS(WKM) IF (ABE(K,ML1) .EQ. 0.0D0) GO TO 40 WK = WK/ABE(K,ML1) WKM = WKM/ABE(K,ML1) GO TO 50 40 CONTINUE WK = 1.0D0 WKM = 1.0D0 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 + ABS(Z(I)+WKM*ABE(K,MM)) Z(I) = Z(I) + WK*ABE(K,MM) S = S + ABS(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*ABE(K,MM) 70 CONTINUE 80 CONTINUE 90 CONTINUE Z(K) = WK 100 CONTINUE S = 1.0D0/DASUM(N,Z,1) CALL DSCAL(N,S,Z,1) C C SOLVE TRANS(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) + DDOT(NL,ABE(K+NL,ML1-NL),-LDB,Z(K+1) 1 ,1) IF (ABS(Z(K)) .LE. 1.0D0) GO TO 110 S = 1.0D0/ABS(Z(K)) CALL DSCAL(N,S,Z,1) 110 CONTINUE L = IPVT(K) T = Z(L) Z(L) = Z(K) Z(K) = T 120 CONTINUE S = 1.0D0/DASUM(N,Z,1) CALL DSCAL(N,S,Z,1) C YNORM = 1.0D0 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 DAXPY(NL,T,ABE(K+NL,ML1-NL),-LDB,Z(K+1),1) IF (ABS(Z(K)) .LE. 1.0D0) GO TO 130 S = 1.0D0/ABS(Z(K)) CALL DSCAL(N,S,Z,1) YNORM = S*YNORM 130 CONTINUE 140 CONTINUE S = 1.0D0/DASUM(N,Z,1) CALL DSCAL(N,S,Z,1) YNORM = S*YNORM C C SOLVE U*Z = V C DO 160 KB = 1, N K = N + 1 - KB IF (ABS(Z(K)) .LE. ABS(ABE(K,ML1))) GO TO 150 S = ABS(ABE(K,ML1))/ABS(Z(K)) CALL DSCAL(N,S,Z,1) YNORM = S*YNORM 150 CONTINUE IF (ABE(K,ML1) .NE. 0.0D0) Z(K) = Z(K)/ABE(K,ML1) IF (ABE(K,ML1) .EQ. 0.0D0) Z(K) = 1.0D0 LM = MIN(K,M) - 1 LZ = K - LM T = -Z(K) CALL DAXPY(LM,T,ABE(K-1,ML+2),-LDB,Z(LZ),1) 160 CONTINUE C MAKE ZNORM = 1.0D0 S = 1.0D0/DASUM(N,Z,1) CALL DSCAL(N,S,Z,1) YNORM = S*YNORM C IF (ANORM .NE. 0.0D0) RCOND = YNORM/ANORM IF (ANORM .EQ. 0.0D0) RCOND = 0.0D0 RETURN END