*DECK SGECO SUBROUTINE SGECO (A, LDA, N, IPVT, RCOND, Z) C***BEGIN PROLOGUE SGECO C***PURPOSE Factor a matrix using Gaussian elimination and estimate C the condition number of the matrix. C***LIBRARY SLATEC (LINPACK) C***CATEGORY D2A1 C***TYPE SINGLE PRECISION (SGECO-S, DGECO-D, CGECO-C) C***KEYWORDS CONDITION NUMBER, GENERAL MATRIX, LINEAR ALGEBRA, LINPACK, C MATRIX FACTORIZATION C***AUTHOR Moler, C. B., (U. of New Mexico) C***DESCRIPTION C C SGECO factors a real matrix by Gaussian elimination C and estimates the condition of the matrix. C C If RCOND is not needed, SGEFA is slightly faster. C To solve A*X = B , follow SGECO by SGESL. C To compute INVERSE(A)*C , follow SGECO by SGESL. C To compute DETERMINANT(A) , follow SGECO by SGEDI. C To compute INVERSE(A) , follow SGECO by SGEDI. C C On Entry C C A REAL(LDA, N) C the matrix to be factored. 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 On Return C C A an upper triangular matrix and the multipliers C 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 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***REFERENCES J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. C Stewart, LINPACK Users' Guide, SIAM, 1979. C***ROUTINES CALLED SASUM, SAXPY, SDOT, SGEFA, SSCAL 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 SGECO INTEGER LDA,N,IPVT(*) REAL A(LDA,*),Z(*) REAL RCOND C REAL SDOT,EK,T,WK,WKM REAL ANORM,S,SASUM,SM,YNORM INTEGER INFO,J,K,KB,KP1,L C C COMPUTE 1-NORM OF A C C***FIRST EXECUTABLE STATEMENT SGECO ANORM = 0.0E0 DO 10 J = 1, N ANORM = MAX(ANORM,SASUM(N,A(1,J),1)) 10 CONTINUE C C FACTOR C CALL SGEFA(A,LDA,N,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 = 1.0E0 DO 20 J = 1, N Z(J) = 0.0E0 20 CONTINUE DO 100 K = 1, N IF (Z(K) .NE. 0.0E0) EK = SIGN(EK,-Z(K)) IF (ABS(EK-Z(K)) .LE. ABS(A(K,K))) GO TO 30 S = ABS(A(K,K))/ABS(EK-Z(K)) CALL SSCAL(N,S,Z,1) EK = S*EK 30 CONTINUE WK = EK - Z(K) WKM = -EK - Z(K) S = ABS(WK) SM = ABS(WKM) IF (A(K,K) .EQ. 0.0E0) GO TO 40 WK = WK/A(K,K) WKM = WKM/A(K,K) GO TO 50 40 CONTINUE WK = 1.0E0 WKM = 1.0E0 50 CONTINUE KP1 = K + 1 IF (KP1 .GT. N) GO TO 90 DO 60 J = KP1, N SM = SM + ABS(Z(J)+WKM*A(K,J)) Z(J) = Z(J) + WK*A(K,J) S = S + ABS(Z(J)) 60 CONTINUE IF (S .GE. SM) GO TO 80 T = WKM - WK WK = WKM DO 70 J = KP1, N Z(J) = Z(J) + T*A(K,J) 70 CONTINUE 80 CONTINUE 90 CONTINUE Z(K) = WK 100 CONTINUE S = 1.0E0/SASUM(N,Z,1) CALL SSCAL(N,S,Z,1) C C SOLVE TRANS(L)*Y = W C DO 120 KB = 1, N K = N + 1 - KB IF (K .LT. N) Z(K) = Z(K) + SDOT(N-K,A(K+1,K),1,Z(K+1),1) IF (ABS(Z(K)) .LE. 1.0E0) GO TO 110 S = 1.0E0/ABS(Z(K)) CALL SSCAL(N,S,Z,1) 110 CONTINUE L = IPVT(K) T = Z(L) Z(L) = Z(K) Z(K) = T 120 CONTINUE S = 1.0E0/SASUM(N,Z,1) CALL SSCAL(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 IF (K .LT. N) CALL SAXPY(N-K,T,A(K+1,K),1,Z(K+1),1) IF (ABS(Z(K)) .LE. 1.0E0) GO TO 130 S = 1.0E0/ABS(Z(K)) CALL SSCAL(N,S,Z,1) YNORM = S*YNORM 130 CONTINUE 140 CONTINUE S = 1.0E0/SASUM(N,Z,1) CALL SSCAL(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(A(K,K))) GO TO 150 S = ABS(A(K,K))/ABS(Z(K)) CALL SSCAL(N,S,Z,1) YNORM = S*YNORM 150 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 T = -Z(K) CALL SAXPY(K-1,T,A(1,K),1,Z(1),1) 160 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