*DECK DSPCO
SUBROUTINE DSPCO (AP, N, KPVT, RCOND, Z)
C***BEGIN PROLOGUE DSPCO
C***PURPOSE Factor a real symmetric matrix stored in packed form
C by elimination with symmetric pivoting and estimate the
C condition number of the matrix.
C***LIBRARY SLATEC (LINPACK)
C***CATEGORY D2B1A
C***TYPE DOUBLE PRECISION (SSPCO-S, DSPCO-D, CHPCO-C, CSPCO-C)
C***KEYWORDS CONDITION NUMBER, LINEAR ALGEBRA, LINPACK,
C MATRIX FACTORIZATION, PACKED, SYMMETRIC
C***AUTHOR Moler, C. B., (U. of New Mexico)
C***DESCRIPTION
C
C DSPCO factors a double precision symmetric matrix stored in
C packed form by elimination with symmetric pivoting and estimates
C the condition of the matrix.
C
C IF RCOND is not needed, DSPFA is slightly faster.
C To solve A*X = B , follow DSPCO by DSPSL.
C To compute INVERSE(A)*C , follow DSPCO by DSPSL.
C To compute INVERSE(A) , follow DSPCO by DSPDI.
C To compute DETERMINANT(A) , follow DSPCO by DSPDI.
C To compute INERTIA(A), follow DSPCO by DSPDI.
C
C On Entry
C
C AP DOUBLE PRECISION (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 Output
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 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 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 DASUM, DAXPY, DDOT, DSCAL, DSPFA
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 DSPCO
INTEGER N,KPVT(*)
DOUBLE PRECISION AP(*),Z(*)
DOUBLE PRECISION RCOND
C
DOUBLE PRECISION AK,AKM1,BK,BKM1,DDOT,DENOM,EK,T
DOUBLE PRECISION ANORM,S,DASUM,YNORM
INTEGER I,IJ,IK,IKM1,IKP1,INFO,J,JM1,J1
INTEGER K,KK,KM1K,KM1KM1,KP,KPS,KS
C
C FIND NORM OF A USING ONLY UPPER HALF
C
C***FIRST EXECUTABLE STATEMENT DSPCO
J1 = 1
DO 30 J = 1, N
Z(J) = DASUM(J,AP(J1),1)
IJ = J1
J1 = J1 + J
JM1 = J - 1
IF (JM1 .LT. 1) GO TO 20
DO 10 I = 1, JM1
Z(I) = Z(I) + ABS(AP(IJ))
IJ = IJ + 1
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 DSPFA(AP,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
IK = (N*(N - 1))/2
60 IF (K .EQ. 0) GO TO 120
KK = IK + K
IKM1 = IK - (K - 1)
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),AP(IK+1),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),AP(IKM1+1),1,Z(1),1)
80 CONTINUE
IF (KS .EQ. 2) GO TO 100
IF (ABS(Z(K)) .LE. ABS(AP(KK))) GO TO 90
S = ABS(AP(KK))/ABS(Z(K))
CALL DSCAL(N,S,Z,1)
EK = S*EK
90 CONTINUE
IF (AP(KK) .NE. 0.0D0) Z(K) = Z(K)/AP(KK)
IF (AP(KK) .EQ. 0.0D0) Z(K) = 1.0D0
GO TO 110
100 CONTINUE
KM1K = IK + K - 1
KM1KM1 = IKM1 + K - 1
AK = AP(KK)/AP(KM1K)
AKM1 = AP(KM1KM1)/AP(KM1K)
BK = Z(K)/AP(KM1K)
BKM1 = Z(K-1)/AP(KM1K)
DENOM = AK*AKM1 - 1.0D0
Z(K) = (AKM1*BK - BKM1)/DENOM
Z(K-1) = (AK*BKM1 - BK)/DENOM
110 CONTINUE
K = K - KS
IK = IK - K
IF (KS .EQ. 2) IK = IK - (K + 1)
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
IK = 0
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,AP(IK+1),1,Z(1),1)
IKP1 = IK + K
IF (KS .EQ. 2)
1 Z(K+1) = Z(K+1) + DDOT(K-1,AP(IKP1+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
IK = IK + K
IF (KS .EQ. 2) IK = IK + (K + 1)
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
IK = N*(N - 1)/2
170 IF (K .EQ. 0) GO TO 230
KK = IK + K
IKM1 = IK - (K - 1)
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),AP(IK+1),1,Z(1),1)
IF (KS .EQ. 2) CALL DAXPY(K-KS,Z(K-1),AP(IKM1+1),1,Z(1),1)
190 CONTINUE
IF (KS .EQ. 2) GO TO 210
IF (ABS(Z(K)) .LE. ABS(AP(KK))) GO TO 200
S = ABS(AP(KK))/ABS(Z(K))
CALL DSCAL(N,S,Z,1)
YNORM = S*YNORM
200 CONTINUE
IF (AP(KK) .NE. 0.0D0) Z(K) = Z(K)/AP(KK)
IF (AP(KK) .EQ. 0.0D0) Z(K) = 1.0D0
GO TO 220
210 CONTINUE
KM1K = IK + K - 1
KM1KM1 = IKM1 + K - 1
AK = AP(KK)/AP(KM1K)
AKM1 = AP(KM1KM1)/AP(KM1K)
BK = Z(K)/AP(KM1K)
BKM1 = Z(K-1)/AP(KM1K)
DENOM = AK*AKM1 - 1.0D0
Z(K) = (AKM1*BK - BKM1)/DENOM
Z(K-1) = (AK*BKM1 - BK)/DENOM
220 CONTINUE
K = K - KS
IK = IK - K
IF (KS .EQ. 2) IK = IK - (K + 1)
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
IK = 0
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,AP(IK+1),1,Z(1),1)
IKP1 = IK + K
IF (KS .EQ. 2)
1 Z(K+1) = Z(K+1) + DDOT(K-1,AP(IKP1+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
IK = IK + K
IF (KS .EQ. 2) IK = IK + (K + 1)
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