*DECK REDUC
SUBROUTINE REDUC (NM, N, A, B, DL, IERR)
C***BEGIN PROLOGUE REDUC
C***PURPOSE Reduce a generalized symmetric eigenproblem to a standard
C symmetric eigenproblem using Cholesky factorization.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C1C
C***TYPE SINGLE PRECISION (REDUC-S)
C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is a translation of the ALGOL procedure REDUC1,
C NUM. MATH. 11, 99-110(1968) by Martin and Wilkinson.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 303-314(1971).
C
C This subroutine reduces the generalized SYMMETRIC eigenproblem
C Ax=(LAMBDA)Bx, where B is POSITIVE DEFINITE, to the standard
C symmetric eigenproblem using the Cholesky factorization of B.
C
C On Input
C
C NM must be set to the row dimension of the two-dimensional
C array parameters, A and B, as declared in the calling
C program dimension statement. NM is an INTEGER variable.
C
C N is the order of the matrices A and B. If the Cholesky
C factor L of B is already available, N should be prefixed
C with a minus sign. N is an INTEGER variable.
C
C A and B contain the real symmetric input matrices. Only
C the full upper triangles of the matrices need be supplied.
C If N is negative, the strict lower triangle of B contains,
C instead, the strict lower triangle of its Cholesky factor L.
C A and B are two-dimensional REAL arrays, dimensioned A(NM,N)
C and B(NM,N).
C
C DL contains, if N is negative, the diagonal elements of L.
C DL is a one-dimensional REAL array, dimensioned DL(N).
C
C On Output
C
C A contains in its full lower triangle the full lower triangle
C of the symmetric matrix derived from the reduction to the
C standard form. The strict upper triangle of A is unaltered.
C
C B contains in its strict lower triangle the strict lower
C triangle of its Cholesky factor L. The full upper triangle
C of B is unaltered.
C
C DL contains the diagonal elements of L.
C
C IERR is an INTEGER flag set to
C Zero for normal return,
C 7*N+1 if B is not positive definite.
C
C Questions and comments should be directed to B. S. Garbow,
C APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
C ------------------------------------------------------------------
C
C***REFERENCES B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
C Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
C system Routines - EISPACK Guide, Springer-Verlag,
C 1976.
C***ROUTINES CALLED (NONE)
C***REVISION HISTORY (YYMMDD)
C 760101 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 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE REDUC
C
INTEGER I,J,K,N,I1,J1,NM,NN,IERR
REAL A(NM,*),B(NM,*),DL(*)
REAL X,Y
C
C***FIRST EXECUTABLE STATEMENT REDUC
IERR = 0
NN = ABS(N)
IF (N .LT. 0) GO TO 100
C .......... FORM L IN THE ARRAYS B AND DL ..........
DO 80 I = 1, N
I1 = I - 1
C
DO 80 J = I, N
X = B(I,J)
IF (I .EQ. 1) GO TO 40
C
DO 20 K = 1, I1
20 X = X - B(I,K) * B(J,K)
C
40 IF (J .NE. I) GO TO 60
IF (X .LE. 0.0E0) GO TO 1000
Y = SQRT(X)
DL(I) = Y
GO TO 80
60 B(J,I) = X / Y
80 CONTINUE
C .......... FORM THE TRANSPOSE OF THE UPPER TRIANGLE OF INV(L)*A
C IN THE LOWER TRIANGLE OF THE ARRAY A ..........
100 DO 200 I = 1, NN
I1 = I - 1
Y = DL(I)
C
DO 200 J = I, NN
X = A(I,J)
IF (I .EQ. 1) GO TO 180
C
DO 160 K = 1, I1
160 X = X - B(I,K) * A(J,K)
C
180 A(J,I) = X / Y
200 CONTINUE
C .......... PRE-MULTIPLY BY INV(L) AND OVERWRITE ..........
DO 300 J = 1, NN
J1 = J - 1
C
DO 300 I = J, NN
X = A(I,J)
IF (I .EQ. J) GO TO 240
I1 = I - 1
C
DO 220 K = J, I1
220 X = X - A(K,J) * B(I,K)
C
240 IF (J .EQ. 1) GO TO 280
C
DO 260 K = 1, J1
260 X = X - A(J,K) * B(I,K)
C
280 A(I,J) = X / DL(I)
300 CONTINUE
C
GO TO 1001
C .......... SET ERROR -- B IS NOT POSITIVE DEFINITE ..........
1000 IERR = 7 * N + 1
1001 RETURN
END