*DECK TRED1 SUBROUTINE TRED1 (NM, N, A, D, E, E2) C***BEGIN PROLOGUE TRED1 C***PURPOSE Reduce a real symmetric matrix to symmetric tridiagonal C matrix using orthogonal similarity transformations. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D4C1B1 C***TYPE SINGLE PRECISION (TRED1-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 TRED1, 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 C to a symmetric tridiagonal matrix using C orthogonal similarity transformations. C C On Input C C NM must be set to the row dimension of the two-dimensional C array parameter, A, as declared in the calling program C 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 A contains information about the orthogonal transformations C used in the reduction in its strict lower triangle. The C full upper triangle of A is unaltered. 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 E2 contains the squares of the corresponding elements of E. C E2 may coincide with E if the squares are not needed. C E2 is a one-dimensional REAL array, dimensioned E2(N). 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 TRED1 C INTEGER I,J,K,L,N,II,NM,JP1 REAL A(NM,*),D(*),E(*),E2(*) REAL F,G,H,SCALE C C***FIRST EXECUTABLE STATEMENT TRED1 DO 100 I = 1, N 100 D(I) = A(I,I) C .......... FOR I=N STEP -1 UNTIL 1 DO -- .......... DO 300 II = 1, N I = N + 1 - II L = I - 1 H = 0.0E0 SCALE = 0.0E0 IF (L .LT. 1) GO TO 130 C .......... SCALE ROW (ALGOL TOL THEN NOT NEEDED) .......... DO 120 K = 1, L 120 SCALE = SCALE + ABS(A(I,K)) C IF (SCALE .NE. 0.0E0) GO TO 140 130 E(I) = 0.0E0 E2(I) = 0.0E0 GO TO 290 C 140 DO 150 K = 1, L A(I,K) = A(I,K) / SCALE H = H + A(I,K) * A(I,K) 150 CONTINUE C E2(I) = SCALE * SCALE * H F = A(I,L) G = -SIGN(SQRT(H),F) E(I) = SCALE * G H = H - F * G A(I,L) = F - G IF (L .EQ. 1) GO TO 270 F = 0.0E0 C DO 240 J = 1, L G = 0.0E0 C .......... FORM ELEMENT OF A*U .......... DO 180 K = 1, J 180 G = G + A(J,K) * A(I,K) C JP1 = J + 1 IF (L .LT. JP1) GO TO 220 C DO 200 K = JP1, L 200 G = G + A(K,J) * A(I,K) C .......... FORM ELEMENT OF P .......... 220 E(J) = G / H F = F + E(J) * A(I,J) 240 CONTINUE C H = F / (H + H) C .......... FORM REDUCED A .......... DO 260 J = 1, L F = A(I,J) G = E(J) - H * F E(J) = G C DO 260 K = 1, J A(J,K) = A(J,K) - F * E(K) - G * A(I,K) 260 CONTINUE C 270 DO 280 K = 1, L 280 A(I,K) = SCALE * A(I,K) C 290 H = D(I) D(I) = A(I,I) A(I,I) = H 300 CONTINUE C RETURN END