relibc_openlibm/slatec/dsico.f

*DECK DSICO
      SUBROUTINE DSICO (A, LDA, N, KPVT, RCOND, Z)
C***BEGIN PROLOGUE  DSICO
C***PURPOSE  Factor a symmetric matrix by elimination with symmetric
C            pivoting and estimate the condition number of the matrix.
C***LIBRARY   SLATEC (LINPACK)
C***CATEGORY  D2B1A
C***TYPE      DOUBLE PRECISION (SSICO-S, DSICO-D, CHICO-C, CSICO-C)
C***KEYWORDS  CONDITION NUMBER, LINEAR ALGEBRA, LINPACK,
C             MATRIX FACTORIZATION, SYMMETRIC
C***AUTHOR  Moler, C. B., (U. of New Mexico)
C***DESCRIPTION
C
C     DSICO factors a double precision symmetric matrix by elimination
C     with symmetric pivoting and estimates the condition of the
C     matrix.
C
C     If  RCOND  is not needed, DSIFA is slightly faster.
C     To solve  A*X = B , follow DSICO by DSISL.
C     To compute  INVERSE(A)*C , follow DSICO by DSISL.
C     To compute  INVERSE(A) , follow DSICO by DSIDI.
C     To compute  DETERMINANT(A) , follow DSICO by DSIDI.
C     To compute  INERTIA(A), follow DSICO by DSIDI.
C
C     On Entry
C
C        A       DOUBLE PRECISION(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   DOUBLE PRECISION
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       DOUBLE PRECISION(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  DASUM, DAXPY, DDOT, DSCAL, DSIFA
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  DSICO
      INTEGER LDA,N,KPVT(*)
      DOUBLE PRECISION A(LDA,*),Z(*)
      DOUBLE PRECISION RCOND
C
      DOUBLE PRECISION AK,AKM1,BK,BKM1,DDOT,DENOM,EK,T
      DOUBLE PRECISION ANORM,S,DASUM,YNORM
      INTEGER I,INFO,J,JM1,K,KP,KPS,KS
C
C     FIND NORM OF A USING ONLY UPPER HALF
C
C***FIRST EXECUTABLE STATEMENT  DSICO
      DO 30 J = 1, N
         Z(J) = DASUM(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 = 0.0D0
      DO 40 J = 1, N
         ANORM = MAX(ANORM,Z(J))
   40 CONTINUE
C
C     FACTOR
C
      CALL DSIFA(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 = 1.0D0
      DO 50 J = 1, N
         Z(J) = 0.0D0
   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 (Z(K) .NE. 0.0D0) EK = SIGN(EK,Z(K))
         Z(K) = Z(K) + EK
         CALL DAXPY(K-KS,Z(K),A(1,K),1,Z(1),1)
         IF (KS .EQ. 1) GO TO 80
            IF (Z(K-1) .NE. 0.0D0) EK = SIGN(EK,Z(K-1))
            Z(K-1) = Z(K-1) + EK
            CALL DAXPY(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 DSCAL(N,S,Z,1)
               EK = S*EK
   90       CONTINUE
            IF (A(K,K) .NE. 0.0D0) Z(K) = Z(K)/A(K,K)
            IF (A(K,K) .EQ. 0.0D0) Z(K) = 1.0D0
         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 - 1.0D0
            Z(K) = (AKM1*BK - BKM1)/DENOM
            Z(K-1) = (AK*BKM1 - BK)/DENOM
  110    CONTINUE
         K = K - KS
      GO TO 60
  120 CONTINUE
      S = 1.0D0/DASUM(N,Z,1)
      CALL DSCAL(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) + DDOT(K-1,A(1,K),1,Z(1),1)
            IF (KS .EQ. 2)
     1         Z(K+1) = Z(K+1) + DDOT(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 = 1.0D0/DASUM(N,Z,1)
      CALL DSCAL(N,S,Z,1)
C
      YNORM = 1.0D0
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 DAXPY(K-KS,Z(K),A(1,K),1,Z(1),1)
            IF (KS .EQ. 2) CALL DAXPY(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 DSCAL(N,S,Z,1)
               YNORM = S*YNORM
  200       CONTINUE
            IF (A(K,K) .NE. 0.0D0) Z(K) = Z(K)/A(K,K)
            IF (A(K,K) .EQ. 0.0D0) Z(K) = 1.0D0
         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 - 1.0D0
            Z(K) = (AKM1*BK - BKM1)/DENOM
            Z(K-1) = (AK*BKM1 - BK)/DENOM
  220    CONTINUE
         K = K - KS
      GO TO 170
  230 CONTINUE
      S = 1.0D0/DASUM(N,Z,1)
      CALL DSCAL(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) + DDOT(K-1,A(1,K),1,Z(1),1)
            IF (KS .EQ. 2)
     1         Z(K+1) = Z(K+1) + DDOT(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 = 1.0D0/DASUM(N,Z,1)
      CALL DSCAL(N,S,Z,1)
      YNORM = S*YNORM
C
      IF (ANORM .NE. 0.0D0) RCOND = YNORM/ANORM
      IF (ANORM .EQ. 0.0D0) RCOND = 0.0D0
      RETURN
      END