*DECK DPOCO
SUBROUTINE DPOCO (A, LDA, N, RCOND, Z, INFO)
C***BEGIN PROLOGUE DPOCO
C***PURPOSE Factor a real symmetric positive definite matrix
C and estimate the condition of the matrix.
C***LIBRARY SLATEC (LINPACK)
C***CATEGORY D2B1B
C***TYPE DOUBLE PRECISION (SPOCO-S, DPOCO-D, CPOCO-C)
C***KEYWORDS CONDITION NUMBER, LINEAR ALGEBRA, LINPACK,
C MATRIX FACTORIZATION, POSITIVE DEFINITE
C***AUTHOR Moler, C. B., (U. of New Mexico)
C***DESCRIPTION
C
C DPOCO factors a double precision symmetric positive definite
C matrix and estimates the condition of the matrix.
C
C If RCOND is not needed, DPOFA is slightly faster.
C To solve A*X = B , follow DPOCO by DPOSL.
C To compute INVERSE(A)*C , follow DPOCO by DPOSL.
C To compute DETERMINANT(A) , follow DPOCO by DPODI.
C To compute INVERSE(A) , follow DPOCO by DPODI.
C
C On Entry
C
C A DOUBLE PRECISION(LDA, N)
C the symmetric matrix to be factored. Only the
C 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 On Return
C
C A an upper triangular matrix R so that A = TRANS(R)*R
C where TRANS(R) is the transpose.
C The strict lower triangle is unaltered.
C If INFO .NE. 0 , the factorization is not complete.
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. If INFO .NE. 0 , RCOND is unchanged.
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 If INFO .NE. 0 , Z is unchanged.
C
C INFO INTEGER
C = 0 for normal return.
C = K signals an error condition. The leading minor
C of order K is not positive definite.
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, DPOFA, DSCAL
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 DPOCO
INTEGER LDA,N,INFO
DOUBLE PRECISION A(LDA,*),Z(*)
DOUBLE PRECISION RCOND
C
DOUBLE PRECISION DDOT,EK,T,WK,WKM
DOUBLE PRECISION ANORM,S,DASUM,SM,YNORM
INTEGER I,J,JM1,K,KB,KP1
C
C FIND NORM OF A USING ONLY UPPER HALF
C
C***FIRST EXECUTABLE STATEMENT DPOCO
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 DPOFA(A,LDA,N,INFO)
IF (INFO .NE. 0) GO TO 180
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 TRANS(R)*W = E .
C THE VECTORS ARE FREQUENTLY RESCALED TO AVOID OVERFLOW.
C
C SOLVE TRANS(R)*W = E
C
EK = 1.0D0
DO 50 J = 1, N
Z(J) = 0.0D0
50 CONTINUE
DO 110 K = 1, N
IF (Z(K) .NE. 0.0D0) EK = SIGN(EK,-Z(K))
IF (ABS(EK-Z(K)) .LE. A(K,K)) GO TO 60
S = A(K,K)/ABS(EK-Z(K))
CALL DSCAL(N,S,Z,1)
EK = S*EK
60 CONTINUE
WK = EK - Z(K)
WKM = -EK - Z(K)
S = ABS(WK)
SM = ABS(WKM)
WK = WK/A(K,K)
WKM = WKM/A(K,K)
KP1 = K + 1
IF (KP1 .GT. N) GO TO 100
DO 70 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))
70 CONTINUE
IF (S .GE. SM) GO TO 90
T = WKM - WK
WK = WKM
DO 80 J = KP1, N
Z(J) = Z(J) + T*A(K,J)
80 CONTINUE
90 CONTINUE
100 CONTINUE
Z(K) = WK
110 CONTINUE
S = 1.0D0/DASUM(N,Z,1)
CALL DSCAL(N,S,Z,1)
C
C SOLVE R*Y = W
C
DO 130 KB = 1, N
K = N + 1 - KB
IF (ABS(Z(K)) .LE. A(K,K)) GO TO 120
S = A(K,K)/ABS(Z(K))
CALL DSCAL(N,S,Z,1)
120 CONTINUE
Z(K) = Z(K)/A(K,K)
T = -Z(K)
CALL DAXPY(K-1,T,A(1,K),1,Z(1),1)
130 CONTINUE
S = 1.0D0/DASUM(N,Z,1)
CALL DSCAL(N,S,Z,1)
C
YNORM = 1.0D0
C
C SOLVE TRANS(R)*V = Y
C
DO 150 K = 1, N
Z(K) = Z(K) - DDOT(K-1,A(1,K),1,Z(1),1)
IF (ABS(Z(K)) .LE. A(K,K)) GO TO 140
S = A(K,K)/ABS(Z(K))
CALL DSCAL(N,S,Z,1)
YNORM = S*YNORM
140 CONTINUE
Z(K) = Z(K)/A(K,K)
150 CONTINUE
S = 1.0D0/DASUM(N,Z,1)
CALL DSCAL(N,S,Z,1)
YNORM = S*YNORM
C
C SOLVE R*Z = V
C
DO 170 KB = 1, N
K = N + 1 - KB
IF (ABS(Z(K)) .LE. A(K,K)) GO TO 160
S = A(K,K)/ABS(Z(K))
CALL DSCAL(N,S,Z,1)
YNORM = S*YNORM
160 CONTINUE
Z(K) = Z(K)/A(K,K)
T = -Z(K)
CALL DAXPY(K-1,T,A(1,K),1,Z(1),1)
170 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
180 CONTINUE
RETURN
END