*DECK DPPCO SUBROUTINE DPPCO (AP, N, RCOND, Z, INFO) C***BEGIN PROLOGUE DPPCO C***PURPOSE Factor a symmetric positive definite matrix stored in C packed form and estimate the condition number of the C matrix. C***LIBRARY SLATEC (LINPACK) C***CATEGORY D2B1B C***TYPE DOUBLE PRECISION (SPPCO-S, DPPCO-D, CPPCO-C) C***KEYWORDS CONDITION NUMBER, LINEAR ALGEBRA, LINPACK, C MATRIX FACTORIZATION, PACKED, POSITIVE DEFINITE C***AUTHOR Moler, C. B., (U. of New Mexico) C***DESCRIPTION C C DPPCO factors a double precision symmetric positive definite C matrix stored in packed form C and estimates the condition of the matrix. C C If RCOND is not needed, DPPFA is slightly faster. C To solve A*X = B , follow DPPCO by DPPSL. C To compute INVERSE(A)*C , follow DPPCO by DPPSL. C To compute DETERMINANT(A) , follow DPPCO by DPPDI. C To compute INVERSE(A) , follow DPPCO by DPPDI. C C On Entry C C AP DOUBLE PRECISION (N*(N+1)/2) C the packed form of a symmetric matrix A . The C columns of the upper triangle are stored sequentially C in a one-dimensional array of length N*(N+1)/2 . C See comments below for details. C C N INTEGER C the order of the matrix A . C C On Return C C AP an upper triangular matrix R , stored in packed 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 Packed Storage C C The following program segment will pack the upper C triangle of a symmetric matrix. C C K = 0 C DO 20 J = 1, N C DO 10 I = 1, J C K = K + 1 C AP(K) = A(I,J) C 10 CONTINUE C 20 CONTINUE 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, DPPFA, 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 DPPCO INTEGER N,INFO DOUBLE PRECISION AP(*),Z(*) DOUBLE PRECISION RCOND C DOUBLE PRECISION DDOT,EK,T,WK,WKM DOUBLE PRECISION ANORM,S,DASUM,SM,YNORM INTEGER I,IJ,J,JM1,J1,K,KB,KJ,KK,KP1 C C FIND NORM OF A C C***FIRST EXECUTABLE STATEMENT DPPCO J1 = 1 DO 30 J = 1, N Z(J) = DASUM(J,AP(J1),1) IJ = J1 J1 = J1 + J JM1 = J - 1 IF (JM1 .LT. 1) GO TO 20 DO 10 I = 1, JM1 Z(I) = Z(I) + ABS(AP(IJ)) IJ = IJ + 1 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 DPPFA(AP,N,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 KK = 0 DO 110 K = 1, N KK = KK + K IF (Z(K) .NE. 0.0D0) EK = SIGN(EK,-Z(K)) IF (ABS(EK-Z(K)) .LE. AP(KK)) GO TO 60 S = AP(KK)/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/AP(KK) WKM = WKM/AP(KK) KP1 = K + 1 KJ = KK + K IF (KP1 .GT. N) GO TO 100 DO 70 J = KP1, N SM = SM + ABS(Z(J)+WKM*AP(KJ)) Z(J) = Z(J) + WK*AP(KJ) S = S + ABS(Z(J)) KJ = KJ + J 70 CONTINUE IF (S .GE. SM) GO TO 90 T = WKM - WK WK = WKM KJ = KK + K DO 80 J = KP1, N Z(J) = Z(J) + T*AP(KJ) KJ = KJ + 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. AP(KK)) GO TO 120 S = AP(KK)/ABS(Z(K)) CALL DSCAL(N,S,Z,1) 120 CONTINUE Z(K) = Z(K)/AP(KK) KK = KK - K T = -Z(K) CALL DAXPY(K-1,T,AP(KK+1),1,Z(1),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 Z(K) = Z(K) - DDOT(K-1,AP(KK+1),1,Z(1),1) KK = KK + K IF (ABS(Z(K)) .LE. AP(KK)) GO TO 140 S = AP(KK)/ABS(Z(K)) CALL DSCAL(N,S,Z,1) YNORM = S*YNORM 140 CONTINUE Z(K) = Z(K)/AP(KK) 150 CONTINUE S = 1.0D0/DASUM(N,Z,1) CALL DSCAL(N,S,Z,1) YNORM = S*YNORM C C SOLVE R*Z = V C DO 170 KB = 1, N K = N + 1 - KB IF (ABS(Z(K)) .LE. AP(KK)) GO TO 160 S = AP(KK)/ABS(Z(K)) CALL DSCAL(N,S,Z,1) YNORM = S*YNORM 160 CONTINUE Z(K) = Z(K)/AP(KK) KK = KK - K T = -Z(K) CALL DAXPY(K-1,T,AP(KK+1),1,Z(1),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