*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