*DECK TRED2 SUBROUTINE TRED2 (NM, N, A, D, E, Z) C***BEGIN PROLOGUE TRED2 C***PURPOSE Reduce a real symmetric matrix to a symmetric tridiagonal C matrix using and accumulating orthogonal transformations. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D4C1B1 C***TYPE SINGLE PRECISION (TRED2-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 TRED2, C NUM. MATH. 11, 181-195(1968) by Martin, Reinsch, and Wilkinson. C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971). C C This subroutine reduces a REAL SYMMETRIC matrix to a C symmetric tridiagonal matrix using and accumulating C orthogonal similarity transformations. C C On Input C C NM must be set to the row dimension of the two-dimensional C array parameters, A and Z, as declared in the calling C program dimension statement. NM is an INTEGER variable. C C N is the order of the matrix A. N is an INTEGER variable. C N must be less than or equal to NM. C C A contains the real symmetric input matrix. Only the lower C triangle of the matrix need be supplied. A is a two- C dimensional REAL array, dimensioned A(NM,N). C C On Output C C D contains the diagonal elements of the symmetric tridiagonal C matrix. D is a one-dimensional REAL array, dimensioned D(N). C C E contains the subdiagonal elements of the symmetric C tridiagonal matrix in its last N-1 positions. E(1) is set C to zero. E is a one-dimensional REAL array, dimensioned C E(N). C C Z contains the orthogonal transformation matrix produced in C the reduction. Z is a two-dimensional REAL array, C dimensioned Z(NM,N). C C A and Z may coincide. If distinct, A is unaltered. 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 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 TRED2 C INTEGER I,J,K,L,N,II,NM,JP1 REAL A(NM,*),D(*),E(*),Z(NM,*) REAL F,G,H,HH,SCALE C C***FIRST EXECUTABLE STATEMENT TRED2 DO 100 I = 1, N C DO 100 J = 1, I Z(I,J) = A(I,J) 100 CONTINUE C IF (N .EQ. 1) GO TO 320 C .......... FOR I=N STEP -1 UNTIL 2 DO -- .......... DO 300 II = 2, N I = N + 2 - II L = I - 1 H = 0.0E0 SCALE = 0.0E0 IF (L .LT. 2) GO TO 130 C .......... SCALE ROW (ALGOL TOL THEN NOT NEEDED) .......... DO 120 K = 1, L 120 SCALE = SCALE + ABS(Z(I,K)) C IF (SCALE .NE. 0.0E0) GO TO 140 130 E(I) = Z(I,L) GO TO 290 C 140 DO 150 K = 1, L Z(I,K) = Z(I,K) / SCALE H = H + Z(I,K) * Z(I,K) 150 CONTINUE C F = Z(I,L) G = -SIGN(SQRT(H),F) E(I) = SCALE * G H = H - F * G Z(I,L) = F - G F = 0.0E0 C DO 240 J = 1, L Z(J,I) = Z(I,J) / H G = 0.0E0 C .......... FORM ELEMENT OF A*U .......... DO 180 K = 1, J 180 G = G + Z(J,K) * Z(I,K) C JP1 = J + 1 IF (L .LT. JP1) GO TO 220 C DO 200 K = JP1, L 200 G = G + Z(K,J) * Z(I,K) C .......... FORM ELEMENT OF P .......... 220 E(J) = G / H F = F + E(J) * Z(I,J) 240 CONTINUE C HH = F / (H + H) C .......... FORM REDUCED A .......... DO 260 J = 1, L F = Z(I,J) G = E(J) - HH * F E(J) = G C DO 260 K = 1, J Z(J,K) = Z(J,K) - F * E(K) - G * Z(I,K) 260 CONTINUE C 290 D(I) = H 300 CONTINUE C 320 D(1) = 0.0E0 E(1) = 0.0E0 C .......... ACCUMULATION OF TRANSFORMATION MATRICES .......... DO 500 I = 1, N L = I - 1 IF (D(I) .EQ. 0.0E0) GO TO 380 C DO 360 J = 1, L G = 0.0E0 C DO 340 K = 1, L 340 G = G + Z(I,K) * Z(K,J) C DO 360 K = 1, L Z(K,J) = Z(K,J) - G * Z(K,I) 360 CONTINUE C 380 D(I) = Z(I,I) Z(I,I) = 1.0E0 IF (L .LT. 1) GO TO 500 C DO 400 J = 1, L Z(I,J) = 0.0E0 Z(J,I) = 0.0E0 400 CONTINUE C 500 CONTINUE C RETURN END