*DECK TRED3 SUBROUTINE TRED3 (N, NV, A, D, E, E2) C***BEGIN PROLOGUE TRED3 C***PURPOSE Reduce a real symmetric matrix stored in packed form to C symmetric tridiagonal matrix using orthogonal C transformations. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D4C1B1 C***TYPE SINGLE PRECISION (TRED3-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 TRED3, 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, stored as C a one-dimensional array, to a symmetric tridiagonal matrix C using orthogonal similarity transformations. C C On Input C C N is the order of the matrix A. N is an INTEGER variable. C C NV is an INTEGER variable set equal to the dimension of the C array A as specified in the calling program. NV must not C be less than N*(N+1)/2. C C A contains the lower triangle, stored row-wise, of the real C symmetric packed matrix. A is a one-dimensional REAL C array, dimensioned A(NV). C C On Output C C A contains information about the orthogonal transformations C used in the reduction in its first N*(N+1)/2 positions. 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 TRED3 C INTEGER I,J,K,L,N,II,IZ,JK,NV REAL A(*),D(*),E(*),E2(*) REAL F,G,H,HH,SCALE C C .......... FOR I=N STEP -1 UNTIL 1 DO -- .......... C***FIRST EXECUTABLE STATEMENT TRED3 DO 300 II = 1, N I = N + 1 - II L = I - 1 IZ = (I * L) / 2 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 IZ = IZ + 1 D(K) = A(IZ) SCALE = SCALE + ABS(D(K)) 120 CONTINUE 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 D(K) = D(K) / SCALE H = H + D(K) * D(K) 150 CONTINUE C E2(I) = SCALE * SCALE * H F = D(L) G = -SIGN(SQRT(H),F) E(I) = SCALE * G H = H - F * G D(L) = F - G A(IZ) = SCALE * D(L) IF (L .EQ. 1) GO TO 290 F = 0.0E0 C DO 240 J = 1, L G = 0.0E0 JK = (J * (J-1)) / 2 C .......... FORM ELEMENT OF A*U .......... DO 180 K = 1, L JK = JK + 1 IF (K .GT. J) JK = JK + K - 2 G = G + A(JK) * D(K) 180 CONTINUE C .......... FORM ELEMENT OF P .......... E(J) = G / H F = F + E(J) * D(J) 240 CONTINUE C HH = F / (H + H) JK = 0 C .......... FORM REDUCED A .......... DO 260 J = 1, L F = D(J) G = E(J) - HH * F E(J) = G C DO 260 K = 1, J JK = JK + 1 A(JK) = A(JK) - F * E(K) - G * D(K) 260 CONTINUE C 290 D(I) = A(IZ+1) A(IZ+1) = SCALE * SQRT(H) 300 CONTINUE C RETURN END