*DECK DPBCO SUBROUTINE DPBCO (ABD, LDA, N, M, RCOND, Z, INFO) C***BEGIN PROLOGUE DPBCO C***PURPOSE Factor a real symmetric positive definite matrix stored in C band form and estimate the condition number of the matrix. C***LIBRARY SLATEC (LINPACK) C***CATEGORY D2B2 C***TYPE DOUBLE PRECISION (SPBCO-S, DPBCO-D, CPBCO-C) C***KEYWORDS BANDED, CONDITION NUMBER, LINEAR ALGEBRA, LINPACK, C MATRIX FACTORIZATION, POSITIVE DEFINITE C***AUTHOR Moler, C. B., (U. of New Mexico) C***DESCRIPTION C C DPBCO factors a double precision symmetric positive definite C matrix stored in band form and estimates the condition of the C matrix. C C If RCOND is not needed, DPBFA is slightly faster. C To solve A*X = B , follow DPBCO by DPBSL. C To compute INVERSE(A)*C , follow DPBCO by DPBSL. C To compute DETERMINANT(A) , follow DPBCO by DPBDI. C C On Entry C C ABD DOUBLE PRECISION(LDA, N) C the matrix to be factored. The columns of the upper C triangle are stored in the columns of ABD and the C diagonals of the upper triangle are stored in the C rows of ABD . See the comments below for details. C C LDA INTEGER C the leading dimension of the array ABD . C LDA must be .GE. M + 1 . C C N INTEGER C the order of the matrix A . C C M INTEGER C the number of diagonals above the main diagonal. C 0 .LE. M .LT. N . C C On Return C C ABD an upper triangular matrix R , stored in band C form, so that A = TRANS(R)*R . C If INFO .NE. 0 , the factorization is not complete. 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. If INFO .NE. 0 , RCOND is unchanged. C C Z DOUBLE PRECISION(N) C a work vector whose contents are usually unimportant. C If A is singular to working precision, then Z is C an approximate null vector in the sense that C NORM(A*Z) = RCOND*NORM(A)*NORM(Z) . C If INFO .NE. 0 , Z is unchanged. C C INFO INTEGER C = 0 for normal return. C = K signals an error condition. The leading minor C of order K is not positive definite. C C Band Storage C C If A is a symmetric positive definite band matrix, C the following program segment will set up the input. C C M = (band width above diagonal) C DO 20 J = 1, N C I1 = MAX(1, J-M) C DO 10 I = I1, J C K = I-J+M+1 C ABD(K,J) = A(I,J) C 10 CONTINUE C 20 CONTINUE C C This uses M + 1 rows of A , except for the M by M C upper left triangle, which is ignored. C C Example: If the original matrix is C C 11 12 13 0 0 0 C 12 22 23 24 0 0 C 13 23 33 34 35 0 C 0 24 34 44 45 46 C 0 0 35 45 55 56 C 0 0 0 46 56 66 C C then N = 6 , M = 2 and ABD should contain C C * * 13 24 35 46 C * 12 23 34 45 56 C 11 22 33 44 55 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, DPBFA, DSCAL 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 DPBCO INTEGER LDA,N,M,INFO DOUBLE PRECISION ABD(LDA,*),Z(*) DOUBLE PRECISION RCOND C DOUBLE PRECISION DDOT,EK,T,WK,WKM DOUBLE PRECISION ANORM,S,DASUM,SM,YNORM INTEGER I,J,J2,K,KB,KP1,L,LA,LB,LM,MU C C FIND NORM OF A C C***FIRST EXECUTABLE STATEMENT DPBCO DO 30 J = 1, N L = MIN(J,M+1) MU = MAX(M+2-J,1) Z(J) = DASUM(L,ABD(MU,J),1) K = J - L IF (M .LT. MU) GO TO 20 DO 10 I = MU, M K = K + 1 Z(K) = Z(K) + ABS(ABD(I,J)) 10 CONTINUE 20 CONTINUE 30 CONTINUE ANORM = 0.0D0 DO 40 J = 1, N ANORM = MAX(ANORM,Z(J)) 40 CONTINUE C C FACTOR C CALL DPBFA(ABD,LDA,N,M,INFO) IF (INFO .NE. 0) GO TO 180 C C RCOND = 1/(NORM(A)*(ESTIMATE OF NORM(INVERSE(A)))) . C ESTIMATE = NORM(Z)/NORM(Y) WHERE A*Z = Y AND A*Y = E . C THE COMPONENTS OF E ARE CHOSEN TO CAUSE MAXIMUM LOCAL C GROWTH IN THE ELEMENTS OF W WHERE TRANS(R)*W = E . C THE VECTORS ARE FREQUENTLY RESCALED TO AVOID OVERFLOW. C C SOLVE TRANS(R)*W = E C EK = 1.0D0 DO 50 J = 1, N Z(J) = 0.0D0 50 CONTINUE DO 110 K = 1, N IF (Z(K) .NE. 0.0D0) EK = SIGN(EK,-Z(K)) IF (ABS(EK-Z(K)) .LE. ABD(M+1,K)) GO TO 60 S = ABD(M+1,K)/ABS(EK-Z(K)) CALL DSCAL(N,S,Z,1) EK = S*EK 60 CONTINUE WK = EK - Z(K) WKM = -EK - Z(K) S = ABS(WK) SM = ABS(WKM) WK = WK/ABD(M+1,K) WKM = WKM/ABD(M+1,K) KP1 = K + 1 J2 = MIN(K+M,N) I = M + 1 IF (KP1 .GT. J2) GO TO 100 DO 70 J = KP1, J2 I = I - 1 SM = SM + ABS(Z(J)+WKM*ABD(I,J)) Z(J) = Z(J) + WK*ABD(I,J) S = S + ABS(Z(J)) 70 CONTINUE IF (S .GE. SM) GO TO 90 T = WKM - WK WK = WKM I = M + 1 DO 80 J = KP1, J2 I = I - 1 Z(J) = Z(J) + T*ABD(I,J) 80 CONTINUE 90 CONTINUE 100 CONTINUE Z(K) = WK 110 CONTINUE S = 1.0D0/DASUM(N,Z,1) CALL DSCAL(N,S,Z,1) C C SOLVE R*Y = W C DO 130 KB = 1, N K = N + 1 - KB IF (ABS(Z(K)) .LE. ABD(M+1,K)) GO TO 120 S = ABD(M+1,K)/ABS(Z(K)) CALL DSCAL(N,S,Z,1) 120 CONTINUE Z(K) = Z(K)/ABD(M+1,K) LM = MIN(K-1,M) LA = M + 1 - LM LB = K - LM T = -Z(K) CALL DAXPY(LM,T,ABD(LA,K),1,Z(LB),1) 130 CONTINUE S = 1.0D0/DASUM(N,Z,1) CALL DSCAL(N,S,Z,1) C YNORM = 1.0D0 C C SOLVE TRANS(R)*V = Y C DO 150 K = 1, N LM = MIN(K-1,M) LA = M + 1 - LM LB = K - LM Z(K) = Z(K) - DDOT(LM,ABD(LA,K),1,Z(LB),1) IF (ABS(Z(K)) .LE. ABD(M+1,K)) GO TO 140 S = ABD(M+1,K)/ABS(Z(K)) CALL DSCAL(N,S,Z,1) YNORM = S*YNORM 140 CONTINUE Z(K) = Z(K)/ABD(M+1,K) 150 CONTINUE S = 1.0D0/DASUM(N,Z,1) CALL DSCAL(N,S,Z,1) YNORM = S*YNORM C C SOLVE R*Z = W C DO 170 KB = 1, N K = N + 1 - KB IF (ABS(Z(K)) .LE. ABD(M+1,K)) GO TO 160 S = ABD(M+1,K)/ABS(Z(K)) CALL DSCAL(N,S,Z,1) YNORM = S*YNORM 160 CONTINUE Z(K) = Z(K)/ABD(M+1,K) LM = MIN(K-1,M) LA = M + 1 - LM LB = K - LM T = -Z(K) CALL DAXPY(LM,T,ABD(LA,K),1,Z(LB),1) 170 CONTINUE C MAKE ZNORM = 1.0 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 180 CONTINUE RETURN END