*DECK CSPFA SUBROUTINE CSPFA (AP, N, KPVT, INFO) C***BEGIN PROLOGUE CSPFA C***PURPOSE Factor a complex symmetric matrix stored in packed form by C elimination with symmetric pivoting. C***LIBRARY SLATEC (LINPACK) C***CATEGORY D2C1 C***TYPE COMPLEX (SSPFA-S, DSPFA-D, CHPFA-C, CSPFA-C) C***KEYWORDS LINEAR ALGEBRA, LINPACK, MATRIX FACTORIZATION, PACKED, C SYMMETRIC C***AUTHOR Bunch, J., (UCSD) C***DESCRIPTION C C CSPFA factors a complex symmetric matrix stored in C packed form by elimination with symmetric pivoting. C C To solve A*X = B , follow CSPFA by CSPSL. C To compute INVERSE(A)*C , follow CSPFA by CSPSL. C To compute DETERMINANT(A) , follow CSPFA by CSPDI. C To compute INVERSE(A) , follow CSPFA by CSPDI. C C On Entry C C AP COMPLEX (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 a block diagonal matrix and the multipliers which C were used to obtain it stored in packed form. 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 KVPT INTEGER(N) C an integer vector of pivot indices. C C INFO INTEGER C = 0 normal value. C = K if the K-th pivot block is singular. This is C not an error condition for this subroutine, C but it does indicate that CSPSL or CSPDI may C divide by zero if called. 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 CAXPY, CSWAP, ICAMAX 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 Corrected category and modified routine equivalence C 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 CSPFA INTEGER N,KPVT(*),INFO COMPLEX AP(*) C COMPLEX AK,AKM1,BK,BKM1,DENOM,MULK,MULKM1,T REAL ABSAKK,ALPHA,COLMAX,ROWMAX INTEGER ICAMAX,IJ,IK,IKM1,IM,IMAX,IMAXP1,IMIM,IMJ,IMK INTEGER J,JJ,JK,JKM1,JMAX,JMIM,K,KK,KM1,KM1K,KM1KM1,KM2,KSTEP LOGICAL SWAP COMPLEX ZDUM REAL CABS1 CABS1(ZDUM) = ABS(REAL(ZDUM)) + ABS(AIMAG(ZDUM)) C***FIRST EXECUTABLE STATEMENT CSPFA C C INITIALIZE C C ALPHA IS USED IN CHOOSING PIVOT BLOCK SIZE. C ALPHA = (1.0E0 + SQRT(17.0E0))/8.0E0 C INFO = 0 C C MAIN LOOP ON K, WHICH GOES FROM N TO 1. C K = N IK = (N*(N - 1))/2 10 CONTINUE C C LEAVE THE LOOP IF K=0 OR K=1. C IF (K .EQ. 0) GO TO 200 IF (K .GT. 1) GO TO 20 KPVT(1) = 1 IF (CABS1(AP(1)) .EQ. 0.0E0) INFO = 1 GO TO 200 20 CONTINUE C C THIS SECTION OF CODE DETERMINES THE KIND OF C ELIMINATION TO BE PERFORMED. WHEN IT IS COMPLETED, C KSTEP WILL BE SET TO THE SIZE OF THE PIVOT BLOCK, AND C SWAP WILL BE SET TO .TRUE. IF AN INTERCHANGE IS C REQUIRED. C KM1 = K - 1 KK = IK + K ABSAKK = CABS1(AP(KK)) C C DETERMINE THE LARGEST OFF-DIAGONAL ELEMENT IN C COLUMN K. C IMAX = ICAMAX(K-1,AP(IK+1),1) IMK = IK + IMAX COLMAX = CABS1(AP(IMK)) IF (ABSAKK .LT. ALPHA*COLMAX) GO TO 30 KSTEP = 1 SWAP = .FALSE. GO TO 90 30 CONTINUE C C DETERMINE THE LARGEST OFF-DIAGONAL ELEMENT IN C ROW IMAX. C ROWMAX = 0.0E0 IMAXP1 = IMAX + 1 IM = IMAX*(IMAX - 1)/2 IMJ = IM + 2*IMAX DO 40 J = IMAXP1, K ROWMAX = MAX(ROWMAX,CABS1(AP(IMJ))) IMJ = IMJ + J 40 CONTINUE IF (IMAX .EQ. 1) GO TO 50 JMAX = ICAMAX(IMAX-1,AP(IM+1),1) JMIM = JMAX + IM ROWMAX = MAX(ROWMAX,CABS1(AP(JMIM))) 50 CONTINUE IMIM = IMAX + IM IF (CABS1(AP(IMIM)) .LT. ALPHA*ROWMAX) GO TO 60 KSTEP = 1 SWAP = .TRUE. GO TO 80 60 CONTINUE IF (ABSAKK .LT. ALPHA*COLMAX*(COLMAX/ROWMAX)) GO TO 70 KSTEP = 1 SWAP = .FALSE. GO TO 80 70 CONTINUE KSTEP = 2 SWAP = IMAX .NE. KM1 80 CONTINUE 90 CONTINUE IF (MAX(ABSAKK,COLMAX) .NE. 0.0E0) GO TO 100 C C COLUMN K IS ZERO. SET INFO AND ITERATE THE LOOP. C KPVT(K) = K INFO = K GO TO 190 100 CONTINUE IF (KSTEP .EQ. 2) GO TO 140 C C 1 X 1 PIVOT BLOCK. C IF (.NOT.SWAP) GO TO 120 C C PERFORM AN INTERCHANGE. C CALL CSWAP(IMAX,AP(IM+1),1,AP(IK+1),1) IMJ = IK + IMAX DO 110 JJ = IMAX, K J = K + IMAX - JJ JK = IK + J T = AP(JK) AP(JK) = AP(IMJ) AP(IMJ) = T IMJ = IMJ - (J - 1) 110 CONTINUE 120 CONTINUE C C PERFORM THE ELIMINATION. C IJ = IK - (K - 1) DO 130 JJ = 1, KM1 J = K - JJ JK = IK + J MULK = -AP(JK)/AP(KK) T = MULK CALL CAXPY(J,T,AP(IK+1),1,AP(IJ+1),1) AP(JK) = MULK IJ = IJ - (J - 1) 130 CONTINUE C C SET THE PIVOT ARRAY. C KPVT(K) = K IF (SWAP) KPVT(K) = IMAX GO TO 190 140 CONTINUE C C 2 X 2 PIVOT BLOCK. C KM1K = IK + K - 1 IKM1 = IK - (K - 1) IF (.NOT.SWAP) GO TO 160 C C PERFORM AN INTERCHANGE. C CALL CSWAP(IMAX,AP(IM+1),1,AP(IKM1+1),1) IMJ = IKM1 + IMAX DO 150 JJ = IMAX, KM1 J = KM1 + IMAX - JJ JKM1 = IKM1 + J T = AP(JKM1) AP(JKM1) = AP(IMJ) AP(IMJ) = T IMJ = IMJ - (J - 1) 150 CONTINUE T = AP(KM1K) AP(KM1K) = AP(IMK) AP(IMK) = T 160 CONTINUE C C PERFORM THE ELIMINATION. C KM2 = K - 2 IF (KM2 .EQ. 0) GO TO 180 AK = AP(KK)/AP(KM1K) KM1KM1 = IKM1 + K - 1 AKM1 = AP(KM1KM1)/AP(KM1K) DENOM = 1.0E0 - AK*AKM1 IJ = IK - (K - 1) - (K - 2) DO 170 JJ = 1, KM2 J = KM1 - JJ JK = IK + J BK = AP(JK)/AP(KM1K) JKM1 = IKM1 + J BKM1 = AP(JKM1)/AP(KM1K) MULK = (AKM1*BK - BKM1)/DENOM MULKM1 = (AK*BKM1 - BK)/DENOM T = MULK CALL CAXPY(J,T,AP(IK+1),1,AP(IJ+1),1) T = MULKM1 CALL CAXPY(J,T,AP(IKM1+1),1,AP(IJ+1),1) AP(JK) = MULK AP(JKM1) = MULKM1 IJ = IJ - (J - 1) 170 CONTINUE 180 CONTINUE C C SET THE PIVOT ARRAY. C KPVT(K) = 1 - K IF (SWAP) KPVT(K) = -IMAX KPVT(K-1) = KPVT(K) 190 CONTINUE IK = IK - (K - 1) IF (KSTEP .EQ. 2) IK = IK - (K - 2) K = K - KSTEP GO TO 10 200 CONTINUE RETURN END