mirror of
https://git.planet-casio.com/Lephenixnoir/OpenLibm.git
synced 2025-01-01 06:23:39 +01:00
264 lines
8.5 KiB
Fortran
264 lines
8.5 KiB
Fortran
*DECK CHICO
|
|
SUBROUTINE CHICO (A, LDA, N, KPVT, RCOND, Z)
|
|
C***BEGIN PROLOGUE CHICO
|
|
C***PURPOSE Factor a complex Hermitian matrix by elimination with sym-
|
|
C metric pivoting and estimate the condition of the matrix.
|
|
C***LIBRARY SLATEC (LINPACK)
|
|
C***CATEGORY D2D1A
|
|
C***TYPE COMPLEX (SSICO-S, DSICO-D, CHICO-C, CSICO-C)
|
|
C***KEYWORDS CONDITION NUMBER, HERMITIAN, LINEAR ALGEBRA, LINPACK,
|
|
C MATRIX FACTORIZATION
|
|
C***AUTHOR Moler, C. B., (U. of New Mexico)
|
|
C***DESCRIPTION
|
|
C
|
|
C CHICO factors a complex Hermitian matrix by elimination with
|
|
C symmetric pivoting and estimates the condition of the matrix.
|
|
C
|
|
C If RCOND is not needed, CHIFA is slightly faster.
|
|
C To solve A*X = B , follow CHICO by CHISL.
|
|
C To compute INVERSE(A)*C , follow CHICO by CHISL.
|
|
C To compute INVERSE(A) , follow CHICO by CHIDI.
|
|
C To compute DETERMINANT(A) , follow CHICO by CHIDI.
|
|
C To compute INERTIA(A), follow CHICO by CHIDI.
|
|
C
|
|
C On Entry
|
|
C
|
|
C A COMPLEX(LDA, N)
|
|
C the Hermitian 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*CTRANS(U)
|
|
C where U is a product of permutation and unit
|
|
C upper triangular matrices , CTRANS(U) is the
|
|
C conjugate transpose of U , and D is block diagonal
|
|
C with 1 by 1 and 2 by 2 blocks.
|
|
C
|
|
C KVPT 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 COMPLEX(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 CAXPY, CDOTC, CHIFA, CSSCAL, SCASUM
|
|
C***REVISION HISTORY (YYMMDD)
|
|
C 780814 DATE WRITTEN
|
|
C 890531 Changed all specific intrinsics to generic. (WRB)
|
|
C 890831 Modified array declarations. (WRB)
|
|
C 891107 Modified routine equivalence list. (WRB)
|
|
C 891107 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 CHICO
|
|
INTEGER LDA,N,KPVT(*)
|
|
COMPLEX A(LDA,*),Z(*)
|
|
REAL RCOND
|
|
C
|
|
COMPLEX AK,AKM1,BK,BKM1,CDOTC,DENOM,EK,T
|
|
REAL ANORM,S,SCASUM,YNORM
|
|
INTEGER I,INFO,J,JM1,K,KP,KPS,KS
|
|
COMPLEX ZDUM,ZDUM2,CSIGN1
|
|
REAL CABS1
|
|
CABS1(ZDUM) = ABS(REAL(ZDUM)) + ABS(AIMAG(ZDUM))
|
|
CSIGN1(ZDUM,ZDUM2) = CABS1(ZDUM)*(ZDUM2/CABS1(ZDUM2))
|
|
C
|
|
C FIND NORM OF A USING ONLY UPPER HALF
|
|
C
|
|
C***FIRST EXECUTABLE STATEMENT CHICO
|
|
DO 30 J = 1, N
|
|
Z(J) = CMPLX(SCASUM(J,A(1,J),1),0.0E0)
|
|
JM1 = J - 1
|
|
IF (JM1 .LT. 1) GO TO 20
|
|
DO 10 I = 1, JM1
|
|
Z(I) = CMPLX(REAL(Z(I))+CABS1(A(I,J)),0.0E0)
|
|
10 CONTINUE
|
|
20 CONTINUE
|
|
30 CONTINUE
|
|
ANORM = 0.0E0
|
|
DO 40 J = 1, N
|
|
ANORM = MAX(ANORM,REAL(Z(J)))
|
|
40 CONTINUE
|
|
C
|
|
C FACTOR
|
|
C
|
|
CALL CHIFA(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 = (1.0E0,0.0E0)
|
|
DO 50 J = 1, N
|
|
Z(J) = (0.0E0,0.0E0)
|
|
50 CONTINUE
|
|
K = N
|
|
60 IF (K .EQ. 0) GO TO 120
|
|
KS = 1
|
|
IF (KPVT(K) .LT. 0) KS = 2
|
|
KP = ABS(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 (CABS1(Z(K)) .NE. 0.0E0) EK = CSIGN1(EK,Z(K))
|
|
Z(K) = Z(K) + EK
|
|
CALL CAXPY(K-KS,Z(K),A(1,K),1,Z(1),1)
|
|
IF (KS .EQ. 1) GO TO 80
|
|
IF (CABS1(Z(K-1)) .NE. 0.0E0) EK = CSIGN1(EK,Z(K-1))
|
|
Z(K-1) = Z(K-1) + EK
|
|
CALL CAXPY(K-KS,Z(K-1),A(1,K-1),1,Z(1),1)
|
|
80 CONTINUE
|
|
IF (KS .EQ. 2) GO TO 100
|
|
IF (CABS1(Z(K)) .LE. CABS1(A(K,K))) GO TO 90
|
|
S = CABS1(A(K,K))/CABS1(Z(K))
|
|
CALL CSSCAL(N,S,Z,1)
|
|
EK = CMPLX(S,0.0E0)*EK
|
|
90 CONTINUE
|
|
IF (CABS1(A(K,K)) .NE. 0.0E0) Z(K) = Z(K)/A(K,K)
|
|
IF (CABS1(A(K,K)) .EQ. 0.0E0) Z(K) = (1.0E0,0.0E0)
|
|
GO TO 110
|
|
100 CONTINUE
|
|
AK = A(K,K)/CONJG(A(K-1,K))
|
|
AKM1 = A(K-1,K-1)/A(K-1,K)
|
|
BK = Z(K)/CONJG(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/SCASUM(N,Z,1)
|
|
CALL CSSCAL(N,S,Z,1)
|
|
C
|
|
C SOLVE CTRANS(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) + CDOTC(K-1,A(1,K),1,Z(1),1)
|
|
IF (KS .EQ. 2)
|
|
1 Z(K+1) = Z(K+1) + CDOTC(K-1,A(1,K+1),1,Z(1),1)
|
|
KP = ABS(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/SCASUM(N,Z,1)
|
|
CALL CSSCAL(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 = ABS(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 CAXPY(K-KS,Z(K),A(1,K),1,Z(1),1)
|
|
IF (KS .EQ. 2) CALL CAXPY(K-KS,Z(K-1),A(1,K-1),1,Z(1),1)
|
|
190 CONTINUE
|
|
IF (KS .EQ. 2) GO TO 210
|
|
IF (CABS1(Z(K)) .LE. CABS1(A(K,K))) GO TO 200
|
|
S = CABS1(A(K,K))/CABS1(Z(K))
|
|
CALL CSSCAL(N,S,Z,1)
|
|
YNORM = S*YNORM
|
|
200 CONTINUE
|
|
IF (CABS1(A(K,K)) .NE. 0.0E0) Z(K) = Z(K)/A(K,K)
|
|
IF (CABS1(A(K,K)) .EQ. 0.0E0) Z(K) = (1.0E0,0.0E0)
|
|
GO TO 220
|
|
210 CONTINUE
|
|
AK = A(K,K)/CONJG(A(K-1,K))
|
|
AKM1 = A(K-1,K-1)/A(K-1,K)
|
|
BK = Z(K)/CONJG(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/SCASUM(N,Z,1)
|
|
CALL CSSCAL(N,S,Z,1)
|
|
YNORM = S*YNORM
|
|
C
|
|
C SOLVE CTRANS(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) + CDOTC(K-1,A(1,K),1,Z(1),1)
|
|
IF (KS .EQ. 2)
|
|
1 Z(K+1) = Z(K+1) + CDOTC(K-1,A(1,K+1),1,Z(1),1)
|
|
KP = ABS(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/SCASUM(N,Z,1)
|
|
CALL CSSCAL(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
|