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265 lines
8.5 KiB
Fortran
265 lines
8.5 KiB
Fortran
*DECK CSICO
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SUBROUTINE CSICO (A, LDA, N, KPVT, RCOND, Z)
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C***BEGIN PROLOGUE CSICO
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C***PURPOSE Factor a complex symmetric matrix by elimination with
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C symmetric pivoting and estimate the condition number of the
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C matrix.
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C***LIBRARY SLATEC (LINPACK)
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C***CATEGORY D2C1
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C***TYPE COMPLEX (SSICO-S, DSICO-D, CHICO-C, CSICO-C)
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C***KEYWORDS CONDITION NUMBER, LINEAR ALGEBRA, LINPACK,
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C MATRIX FACTORIZATION, SYMMETRIC
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C***AUTHOR Moler, C. B., (U. of New Mexico)
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C***DESCRIPTION
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C
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C CSICO factors a complex symmetric matrix by elimination with
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C symmetric pivoting and estimates the condition of the matrix.
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C
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C If RCOND is not needed, CSIFA is slightly faster.
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C To solve A*X = B , follow CSICO by CSISL.
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C To compute INVERSE(A)*C , follow CSICO by CSISL.
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C To compute INVERSE(A) , follow CSICO by CSIDI.
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C To compute DETERMINANT(A) , follow CSICO by CSIDI.
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C
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C On Entry
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C
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C A COMPLEX(LDA, N)
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C the symmetric matrix to be factored.
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C Only the diagonal and upper triangle are used.
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C
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C LDA INTEGER
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C the leading dimension of the array A .
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C
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C N INTEGER
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C the order of the matrix A .
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C
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C On Return
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C
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C A a block diagonal matrix and the multipliers which
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C were used to obtain it.
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C The factorization can be written A = U*D*TRANS(U)
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C where U is a product of permutation and unit
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C upper triangular matrices , TRANS(U) is the
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C transpose of U , and D is block diagonal
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C with 1 by 1 and 2 by 2 blocks.
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C
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C KVPT INTEGER(N)
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C an integer vector of pivot indices.
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C
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C RCOND REAL
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C an estimate of the reciprocal condition of A .
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C For the system A*X = B , relative perturbations
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C in A and B of size EPSILON may cause
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C relative perturbations in X of size EPSILON/RCOND .
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C If RCOND is so small that the logical expression
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C 1.0 + RCOND .EQ. 1.0
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C is true, then A may be singular to working
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C precision. In particular, RCOND is zero if
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C exact singularity is detected or the estimate
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C underflows.
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C
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C Z COMPLEX(N)
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C a work vector whose contents are usually unimportant.
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C If A is close to a singular matrix, then Z is
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C an approximate null vector in the sense that
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C NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
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C
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C***REFERENCES J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W.
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C Stewart, LINPACK Users' Guide, SIAM, 1979.
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C***ROUTINES CALLED CAXPY, CDOTU, CSIFA, CSSCAL, SCASUM
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C***REVISION HISTORY (YYMMDD)
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C 780814 DATE WRITTEN
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C 890531 Changed all specific intrinsics to generic. (WRB)
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C 890831 Modified array declarations. (WRB)
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C 891107 Corrected category and modified routine equivalence
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C list. (WRB)
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C 891107 REVISION DATE from Version 3.2
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C 891214 Prologue converted to Version 4.0 format. (BAB)
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C 900326 Removed duplicate information from DESCRIPTION section.
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C (WRB)
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C 920501 Reformatted the REFERENCES section. (WRB)
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C***END PROLOGUE CSICO
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INTEGER LDA,N,KPVT(*)
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COMPLEX A(LDA,*),Z(*)
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REAL RCOND
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C
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COMPLEX AK,AKM1,BK,BKM1,CDOTU,DENOM,EK,T
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REAL ANORM,S,SCASUM,YNORM
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INTEGER I,INFO,J,JM1,K,KP,KPS,KS
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COMPLEX ZDUM,ZDUM2,CSIGN1
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REAL CABS1
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CABS1(ZDUM) = ABS(REAL(ZDUM)) + ABS(AIMAG(ZDUM))
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CSIGN1(ZDUM,ZDUM2) = CABS1(ZDUM)*(ZDUM2/CABS1(ZDUM2))
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C
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C FIND NORM OF A USING ONLY UPPER HALF
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C
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C***FIRST EXECUTABLE STATEMENT CSICO
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DO 30 J = 1, N
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Z(J) = CMPLX(SCASUM(J,A(1,J),1),0.0E0)
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JM1 = J - 1
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IF (JM1 .LT. 1) GO TO 20
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DO 10 I = 1, JM1
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Z(I) = CMPLX(REAL(Z(I))+CABS1(A(I,J)),0.0E0)
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10 CONTINUE
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20 CONTINUE
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30 CONTINUE
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ANORM = 0.0E0
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DO 40 J = 1, N
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ANORM = MAX(ANORM,REAL(Z(J)))
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40 CONTINUE
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C
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C FACTOR
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C
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CALL CSIFA(A,LDA,N,KPVT,INFO)
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C
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C RCOND = 1/(NORM(A)*(ESTIMATE OF NORM(INVERSE(A)))) .
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C ESTIMATE = NORM(Z)/NORM(Y) WHERE A*Z = Y AND A*Y = E .
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C THE COMPONENTS OF E ARE CHOSEN TO CAUSE MAXIMUM LOCAL
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C GROWTH IN THE ELEMENTS OF W WHERE U*D*W = E .
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C THE VECTORS ARE FREQUENTLY RESCALED TO AVOID OVERFLOW.
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C
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C SOLVE U*D*W = E
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C
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EK = (1.0E0,0.0E0)
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DO 50 J = 1, N
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Z(J) = (0.0E0,0.0E0)
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50 CONTINUE
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K = N
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60 IF (K .EQ. 0) GO TO 120
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KS = 1
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IF (KPVT(K) .LT. 0) KS = 2
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KP = ABS(KPVT(K))
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KPS = K + 1 - KS
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IF (KP .EQ. KPS) GO TO 70
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T = Z(KPS)
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Z(KPS) = Z(KP)
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Z(KP) = T
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70 CONTINUE
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IF (CABS1(Z(K)) .NE. 0.0E0) EK = CSIGN1(EK,Z(K))
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Z(K) = Z(K) + EK
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CALL CAXPY(K-KS,Z(K),A(1,K),1,Z(1),1)
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IF (KS .EQ. 1) GO TO 80
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IF (CABS1(Z(K-1)) .NE. 0.0E0) EK = CSIGN1(EK,Z(K-1))
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Z(K-1) = Z(K-1) + EK
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CALL CAXPY(K-KS,Z(K-1),A(1,K-1),1,Z(1),1)
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80 CONTINUE
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IF (KS .EQ. 2) GO TO 100
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IF (CABS1(Z(K)) .LE. CABS1(A(K,K))) GO TO 90
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S = CABS1(A(K,K))/CABS1(Z(K))
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CALL CSSCAL(N,S,Z,1)
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EK = CMPLX(S,0.0E0)*EK
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90 CONTINUE
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IF (CABS1(A(K,K)) .NE. 0.0E0) Z(K) = Z(K)/A(K,K)
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IF (CABS1(A(K,K)) .EQ. 0.0E0) Z(K) = (1.0E0,0.0E0)
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GO TO 110
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100 CONTINUE
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AK = A(K,K)/A(K-1,K)
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AKM1 = A(K-1,K-1)/A(K-1,K)
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BK = Z(K)/A(K-1,K)
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BKM1 = Z(K-1)/A(K-1,K)
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DENOM = AK*AKM1 - 1.0E0
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Z(K) = (AKM1*BK - BKM1)/DENOM
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Z(K-1) = (AK*BKM1 - BK)/DENOM
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110 CONTINUE
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K = K - KS
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GO TO 60
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120 CONTINUE
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S = 1.0E0/SCASUM(N,Z,1)
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CALL CSSCAL(N,S,Z,1)
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C
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C SOLVE TRANS(U)*Y = W
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C
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K = 1
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130 IF (K .GT. N) GO TO 160
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KS = 1
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IF (KPVT(K) .LT. 0) KS = 2
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IF (K .EQ. 1) GO TO 150
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Z(K) = Z(K) + CDOTU(K-1,A(1,K),1,Z(1),1)
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IF (KS .EQ. 2)
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1 Z(K+1) = Z(K+1) + CDOTU(K-1,A(1,K+1),1,Z(1),1)
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KP = ABS(KPVT(K))
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IF (KP .EQ. K) GO TO 140
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T = Z(K)
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Z(K) = Z(KP)
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Z(KP) = T
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140 CONTINUE
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150 CONTINUE
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K = K + KS
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GO TO 130
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160 CONTINUE
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S = 1.0E0/SCASUM(N,Z,1)
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CALL CSSCAL(N,S,Z,1)
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C
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YNORM = 1.0E0
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C
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C SOLVE U*D*V = Y
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C
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K = N
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170 IF (K .EQ. 0) GO TO 230
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KS = 1
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IF (KPVT(K) .LT. 0) KS = 2
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IF (K .EQ. KS) GO TO 190
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KP = ABS(KPVT(K))
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KPS = K + 1 - KS
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IF (KP .EQ. KPS) GO TO 180
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T = Z(KPS)
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Z(KPS) = Z(KP)
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Z(KP) = T
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180 CONTINUE
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CALL CAXPY(K-KS,Z(K),A(1,K),1,Z(1),1)
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IF (KS .EQ. 2) CALL CAXPY(K-KS,Z(K-1),A(1,K-1),1,Z(1),1)
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190 CONTINUE
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IF (KS .EQ. 2) GO TO 210
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IF (CABS1(Z(K)) .LE. CABS1(A(K,K))) GO TO 200
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S = CABS1(A(K,K))/CABS1(Z(K))
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CALL CSSCAL(N,S,Z,1)
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YNORM = S*YNORM
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200 CONTINUE
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IF (CABS1(A(K,K)) .NE. 0.0E0) Z(K) = Z(K)/A(K,K)
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IF (CABS1(A(K,K)) .EQ. 0.0E0) Z(K) = (1.0E0,0.0E0)
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GO TO 220
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210 CONTINUE
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AK = A(K,K)/A(K-1,K)
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AKM1 = A(K-1,K-1)/A(K-1,K)
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BK = Z(K)/A(K-1,K)
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BKM1 = Z(K-1)/A(K-1,K)
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DENOM = AK*AKM1 - 1.0E0
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Z(K) = (AKM1*BK - BKM1)/DENOM
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Z(K-1) = (AK*BKM1 - BK)/DENOM
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220 CONTINUE
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K = K - KS
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GO TO 170
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230 CONTINUE
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S = 1.0E0/SCASUM(N,Z,1)
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CALL CSSCAL(N,S,Z,1)
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YNORM = S*YNORM
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C
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C SOLVE TRANS(U)*Z = V
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C
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K = 1
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240 IF (K .GT. N) GO TO 270
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KS = 1
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IF (KPVT(K) .LT. 0) KS = 2
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IF (K .EQ. 1) GO TO 260
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Z(K) = Z(K) + CDOTU(K-1,A(1,K),1,Z(1),1)
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IF (KS .EQ. 2)
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1 Z(K+1) = Z(K+1) + CDOTU(K-1,A(1,K+1),1,Z(1),1)
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KP = ABS(KPVT(K))
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IF (KP .EQ. K) GO TO 250
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T = Z(K)
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Z(K) = Z(KP)
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Z(KP) = T
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250 CONTINUE
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260 CONTINUE
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K = K + KS
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GO TO 240
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270 CONTINUE
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C MAKE ZNORM = 1.0
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S = 1.0E0/SCASUM(N,Z,1)
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CALL CSSCAL(N,S,Z,1)
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YNORM = S*YNORM
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C
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IF (ANORM .NE. 0.0E0) RCOND = YNORM/ANORM
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IF (ANORM .EQ. 0.0E0) RCOND = 0.0E0
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RETURN
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END
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