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*DECK DGECO
SUBROUTINE DGECO (A, LDA, N, IPVT, RCOND, Z)
C***BEGIN PROLOGUE DGECO
C***PURPOSE Factor a matrix using Gaussian elimination and estimate
C the condition number of the matrix.
C***LIBRARY SLATEC (LINPACK)
C***CATEGORY D2A1
C***TYPE DOUBLE PRECISION (SGECO-S, DGECO-D, CGECO-C)
C***KEYWORDS CONDITION NUMBER, GENERAL MATRIX, LINEAR ALGEBRA, LINPACK,
C MATRIX FACTORIZATION
C***AUTHOR Moler, C. B., (U. of New Mexico)
C***DESCRIPTION
C
C DGECO factors a double precision matrix by Gaussian elimination
C and estimates the condition of the matrix.
C
C If RCOND is not needed, DGEFA is slightly faster.
C To solve A*X = B , follow DGECO by DGESL.
C To compute INVERSE(A)*C , follow DGECO by DGESL.
C To compute DETERMINANT(A) , follow DGECO by DGEDI.
C To compute INVERSE(A) , follow DGECO by DGEDI.
C
C On Entry
C
C A DOUBLE PRECISION(LDA, N)
C the matrix to be factored.
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 and the multipliers
C which were used to obtain it.
C The factorization can be written A = L*U where
C L is a product of permutation and unit lower
C triangular matrices and U is upper triangular.
C
C IPVT 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, DGEFA, 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 DGECO
INTEGER LDA,N,IPVT(*)
DOUBLE PRECISION A(LDA,*),Z(*)
DOUBLE PRECISION RCOND
C
DOUBLE PRECISION DDOT,EK,T,WK,WKM
DOUBLE PRECISION ANORM,S,DASUM,SM,YNORM
INTEGER INFO,J,K,KB,KP1,L
C
C COMPUTE 1-NORM OF A
C
C***FIRST EXECUTABLE STATEMENT DGECO
ANORM = 0.0D0
DO 10 J = 1, N
ANORM = MAX(ANORM,DASUM(N,A(1,J),1))
10 CONTINUE
C
C FACTOR
C
CALL DGEFA(A,LDA,N,IPVT,INFO)
C
C RCOND = 1/(NORM(A)*(ESTIMATE OF NORM(INVERSE(A)))) .
C ESTIMATE = NORM(Z)/NORM(Y) WHERE A*Z = Y AND TRANS(A)*Y = E .
C TRANS(A) IS THE TRANSPOSE OF A . THE COMPONENTS OF E ARE
C CHOSEN TO CAUSE MAXIMUM LOCAL GROWTH IN THE ELEMENTS OF W WHERE
C TRANS(U)*W = E . THE VECTORS ARE FREQUENTLY RESCALED TO AVOID
C OVERFLOW.
C
C SOLVE TRANS(U)*W = E
C
EK = 1.0D0
DO 20 J = 1, N
Z(J) = 0.0D0
20 CONTINUE
DO 100 K = 1, N
IF (Z(K) .NE. 0.0D0) EK = SIGN(EK,-Z(K))
IF (ABS(EK-Z(K)) .LE. ABS(A(K,K))) GO TO 30
S = ABS(A(K,K))/ABS(EK-Z(K))
CALL DSCAL(N,S,Z,1)
EK = S*EK
30 CONTINUE
WK = EK - Z(K)
WKM = -EK - Z(K)
S = ABS(WK)
SM = ABS(WKM)
IF (A(K,K) .EQ. 0.0D0) GO TO 40
WK = WK/A(K,K)
WKM = WKM/A(K,K)
GO TO 50
40 CONTINUE
WK = 1.0D0
WKM = 1.0D0
50 CONTINUE
KP1 = K + 1
IF (KP1 .GT. N) GO TO 90
DO 60 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))
60 CONTINUE
IF (S .GE. SM) GO TO 80
T = WKM - WK
WK = WKM
DO 70 J = KP1, N
Z(J) = Z(J) + T*A(K,J)
70 CONTINUE
80 CONTINUE
90 CONTINUE
Z(K) = WK
100 CONTINUE
S = 1.0D0/DASUM(N,Z,1)
CALL DSCAL(N,S,Z,1)
C
C SOLVE TRANS(L)*Y = W
C
DO 120 KB = 1, N
K = N + 1 - KB
IF (K .LT. N) Z(K) = Z(K) + DDOT(N-K,A(K+1,K),1,Z(K+1),1)
IF (ABS(Z(K)) .LE. 1.0D0) GO TO 110
S = 1.0D0/ABS(Z(K))
CALL DSCAL(N,S,Z,1)
110 CONTINUE
L = IPVT(K)
T = Z(L)
Z(L) = Z(K)
Z(K) = T
120 CONTINUE
S = 1.0D0/DASUM(N,Z,1)
CALL DSCAL(N,S,Z,1)
C
YNORM = 1.0D0
C
C SOLVE L*V = Y
C
DO 140 K = 1, N
L = IPVT(K)
T = Z(L)
Z(L) = Z(K)
Z(K) = T
IF (K .LT. N) CALL DAXPY(N-K,T,A(K+1,K),1,Z(K+1),1)
IF (ABS(Z(K)) .LE. 1.0D0) GO TO 130
S = 1.0D0/ABS(Z(K))
CALL DSCAL(N,S,Z,1)
YNORM = S*YNORM
130 CONTINUE
140 CONTINUE
S = 1.0D0/DASUM(N,Z,1)
CALL DSCAL(N,S,Z,1)
YNORM = S*YNORM
C
C SOLVE U*Z = V
C
DO 160 KB = 1, N
K = N + 1 - KB
IF (ABS(Z(K)) .LE. ABS(A(K,K))) GO TO 150
S = ABS(A(K,K))/ABS(Z(K))
CALL DSCAL(N,S,Z,1)
YNORM = S*YNORM
150 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
T = -Z(K)
CALL DAXPY(K-1,T,A(1,K),1,Z(1),1)
160 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
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