SUBROUTINE SSICO(A,LDA,N,KPVT,RCOND,Z) INTEGER LDA,N,KPVT(1) REAL A(LDA,1),Z(1) REAL RCOND C C SSICO FACTORS A REAL SYMMETRIC MATRIX BY ELIMINATION WITH C SYMMETRIC PIVOTING AND ESTIMATES THE CONDITION OF THE MATRIX. C C IF RCOND IS NOT NEEDED, SSIFA IS SLIGHTLY FASTER. C TO SOLVE A*X = B , FOLLOW SSICO BY SSISL. C TO COMPUTE INVERSE(A)*C , FOLLOW SSICO BY SSISL. C TO COMPUTE INVERSE(A) , FOLLOW SSICO BY SSIDI. C TO COMPUTE DETERMINANT(A) , FOLLOW SSICO BY SSIDI. C TO COMPUTE INERTIA(A), FOLLOW SSICO BY SSIDI. C C ON ENTRY C C A REAL(LDA, N) C THE SYMMETRIC MATRIX TO BE FACTORED. C ONLY THE DIAGONAL AND UPPER TRIANGLE ARE USED. C C LDA INTEGER C THE LEADING DIMENSION OF THE ARRAY A . C C N INTEGER C THE ORDER OF THE MATRIX A . C C OUTPUT C C A A BLOCK DIAGONAL MATRIX AND THE MULTIPLIERS WHICH C WERE USED TO OBTAIN IT. C THE FACTORIZATION CAN BE WRITTEN A = U*D*TRANS(U) C WHERE U IS A PRODUCT OF PERMUTATION AND UNIT C UPPER TRIANGULAR MATRICES , TRANS(U) IS THE C TRANSPOSE OF U , AND D IS BLOCK DIAGONAL C WITH 1 BY 1 AND 2 BY 2 BLOCKS. C C KPVT 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 REAL(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 LINPACK. THIS VERSION DATED 08/14/78 . C CLEVE MOLER, UNIVERSITY OF NEW MEXICO, ARGONNE NATIONAL LAB. C C SUBROUTINES AND FUNCTIONS C C LINPACK SSIFA C BLAS SAXPY,SDOT,SSCAL,SASUM C FORTRAN ABS,AMAX1,IABS,SIGN C C INTERNAL VARIABLES C REAL AK,AKM1,BK,BKM1,SDOT,DENOM,EK,T REAL ANORM,S,SASUM,YNORM INTEGER I,INFO,J,JM1,K,KP,KPS,KS C C C FIND NORM OF A USING ONLY UPPER HALF C DO 30 J = 1, N Z(J) = SASUM(J,A(1,J),1) JM1 = J - 1 IF (JM1 .LT. 1) GO TO 20 DO 10 I = 1, JM1 Z(I) = Z(I) + ABS(A(I,J)) 10 CONTINUE 20 CONTINUE 30 CONTINUE ANORM = 0.0E0 DO 40 J = 1, N ANORM = AMAX1(ANORM,Z(J)) 40 CONTINUE C C FACTOR C CALL SSIFA(A,LDA,N,KPVT,INFO) 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 U*D*W = E . C THE VECTORS ARE FREQUENTLY RESCALED TO AVOID OVERFLOW. C C SOLVE U*D*W = E C EK = 1.0E0 DO 50 J = 1, N Z(J) = 0.0E0 50 CONTINUE K = N 60 IF (K .EQ. 0) GO TO 120 KS = 1 IF (KPVT(K) .LT. 0) KS = 2 KP = IABS(KPVT(K)) KPS = K + 1 - KS IF (KP .EQ. KPS) GO TO 70 T = Z(KPS) Z(KPS) = Z(KP) Z(KP) = T 70 CONTINUE IF (Z(K) .NE. 0.0E0) EK = SIGN(EK,Z(K)) Z(K) = Z(K) + EK CALL SAXPY(K-KS,Z(K),A(1,K),1,Z(1),1) IF (KS .EQ. 1) GO TO 80 IF (Z(K-1) .NE. 0.0E0) EK = SIGN(EK,Z(K-1)) Z(K-1) = Z(K-1) + EK CALL SAXPY(K-KS,Z(K-1),A(1,K-1),1,Z(1),1) 80 CONTINUE IF (KS .EQ. 2) GO TO 100 IF (ABS(Z(K)) .LE. ABS(A(K,K))) GO TO 90 S = ABS(A(K,K))/ABS(Z(K)) CALL SSCAL(N,S,Z,1) EK = S*EK 90 CONTINUE IF (A(K,K) .NE. 0.0E0) Z(K) = Z(K)/A(K,K) IF (A(K,K) .EQ. 0.0E0) Z(K) = 1.0E0 GO TO 110 100 CONTINUE AK = A(K,K)/A(K-1,K) AKM1 = A(K-1,K-1)/A(K-1,K) BK = Z(K)/A(K-1,K) BKM1 = Z(K-1)/A(K-1,K) DENOM = AK*AKM1 - 1.0E0 Z(K) = (AKM1*BK - BKM1)/DENOM Z(K-1) = (AK*BKM1 - BK)/DENOM 110 CONTINUE K = K - KS GO TO 60 120 CONTINUE S = 1.0E0/SASUM(N,Z,1) CALL SSCAL(N,S,Z,1) C C SOLVE TRANS(U)*Y = W C K = 1 130 IF (K .GT. N) GO TO 160 KS = 1 IF (KPVT(K) .LT. 0) KS = 2 IF (K .EQ. 1) GO TO 150 Z(K) = Z(K) + SDOT(K-1,A(1,K),1,Z(1),1) IF (KS .EQ. 2) * Z(K+1) = Z(K+1) + SDOT(K-1,A(1,K+1),1,Z(1),1) KP = IABS(KPVT(K)) IF (KP .EQ. K) GO TO 140 T = Z(K) Z(K) = Z(KP) Z(KP) = T 140 CONTINUE 150 CONTINUE K = K + KS GO TO 130 160 CONTINUE S = 1.0E0/SASUM(N,Z,1) CALL SSCAL(N,S,Z,1) C YNORM = 1.0E0 C C SOLVE U*D*V = Y C K = N 170 IF (K .EQ. 0) GO TO 230 KS = 1 IF (KPVT(K) .LT. 0) KS = 2 IF (K .EQ. KS) GO TO 190 KP = IABS(KPVT(K)) KPS = K + 1 - KS IF (KP .EQ. KPS) GO TO 180 T = Z(KPS) Z(KPS) = Z(KP) Z(KP) = T 180 CONTINUE CALL SAXPY(K-KS,Z(K),A(1,K),1,Z(1),1) IF (KS .EQ. 2) CALL SAXPY(K-KS,Z(K-1),A(1,K-1),1,Z(1),1) 190 CONTINUE IF (KS .EQ. 2) GO TO 210 IF (ABS(Z(K)) .LE. ABS(A(K,K))) GO TO 200 S = ABS(A(K,K))/ABS(Z(K)) CALL SSCAL(N,S,Z,1) YNORM = S*YNORM 200 CONTINUE IF (A(K,K) .NE. 0.0E0) Z(K) = Z(K)/A(K,K) IF (A(K,K) .EQ. 0.0E0) Z(K) = 1.0E0 GO TO 220 210 CONTINUE AK = A(K,K)/A(K-1,K) AKM1 = A(K-1,K-1)/A(K-1,K) BK = Z(K)/A(K-1,K) BKM1 = Z(K-1)/A(K-1,K) DENOM = AK*AKM1 - 1.0E0 Z(K) = (AKM1*BK - BKM1)/DENOM Z(K-1) = (AK*BKM1 - BK)/DENOM 220 CONTINUE K = K - KS GO TO 170 230 CONTINUE S = 1.0E0/SASUM(N,Z,1) CALL SSCAL(N,S,Z,1) YNORM = S*YNORM C C SOLVE TRANS(U)*Z = V C K = 1 240 IF (K .GT. N) GO TO 270 KS = 1 IF (KPVT(K) .LT. 0) KS = 2 IF (K .EQ. 1) GO TO 260 Z(K) = Z(K) + SDOT(K-1,A(1,K),1,Z(1),1) IF (KS .EQ. 2) * Z(K+1) = Z(K+1) + SDOT(K-1,A(1,K+1),1,Z(1),1) KP = IABS(KPVT(K)) IF (KP .EQ. K) GO TO 250 T = Z(K) Z(K) = Z(KP) Z(KP) = T 250 CONTINUE 260 CONTINUE K = K + KS GO TO 240 270 CONTINUE C MAKE ZNORM = 1.0 S = 1.0E0/SASUM(N,Z,1) CALL SSCAL(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