*DECK IMTQL1 SUBROUTINE IMTQL1 (N, D, E, IERR) C***BEGIN PROLOGUE IMTQL1 C***PURPOSE Compute the eigenvalues of a symmetric tridiagonal matrix C using the implicit QL method. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D4A5, D4C2A C***TYPE SINGLE PRECISION (IMTQL1-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 IMTQL1, C NUM. MATH. 12, 377-383(1968) by Martin and Wilkinson, C as modified in NUM. MATH. 15, 450(1970) by Dubrulle. C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 241-248(1971). C C This subroutine finds the eigenvalues of a SYMMETRIC C TRIDIAGONAL matrix by the implicit QL method. C C On INPUT C C N is the order of the matrix. N is an INTEGER variable. 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 C arbitrary. E is a one-dimensional REAL array, dimensioned C E(N). C C On OUTPUT C C D contains the eigenvalues in ascending order. If an error C exit is made, the eigenvalues are correct and ordered for C indices 1, 2, ..., IERR-1, but may not be the smallest C eigenvalues. C C E has been destroyed. C C IERR is an INTEGER flag set to C Zero for normal return, C J if the J-th eigenvalue has not been C determined after 30 iterations. C The eigenvalues should be correct for indices C 1, 2, ..., IERR-1. These eigenvalues are C ordered, but are not necessarily the smallest. C C Calls PYTHAG(A,B) for sqrt(A**2 + B**2). 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 PYTHAG 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 IMTQL1 C INTEGER I,J,L,M,N,II,MML,IERR REAL D(*),E(*) REAL B,C,F,G,P,R,S,S1,S2 REAL PYTHAG C C***FIRST EXECUTABLE STATEMENT IMTQL1 IERR = 0 IF (N .EQ. 1) GO TO 1001 C DO 100 I = 2, N 100 E(I-1) = E(I) C E(N) = 0.0E0 C DO 290 L = 1, N J = 0 C .......... LOOK FOR SMALL SUB-DIAGONAL ELEMENT .......... 105 DO 110 M = L, N IF (M .EQ. N) GO TO 120 S1 = ABS(D(M)) + ABS(D(M+1)) S2 = S1 + ABS(E(M)) IF (S2 .EQ. S1) GO TO 120 110 CONTINUE C 120 P = D(L) IF (M .EQ. L) GO TO 215 IF (J .EQ. 30) GO TO 1000 J = J + 1 C .......... FORM SHIFT .......... G = (D(L+1) - P) / (2.0E0 * E(L)) R = PYTHAG(G,1.0E0) G = D(M) - P + E(L) / (G + SIGN(R,G)) S = 1.0E0 C = 1.0E0 P = 0.0E0 MML = M - L C .......... FOR I=M-1 STEP -1 UNTIL L DO -- .......... DO 200 II = 1, MML I = M - II F = S * E(I) B = C * E(I) IF (ABS(F) .LT. ABS(G)) GO TO 150 C = G / F R = SQRT(C*C+1.0E0) E(I+1) = F * R S = 1.0E0 / R C = C * S GO TO 160 150 S = F / G R = SQRT(S*S+1.0E0) E(I+1) = G * R C = 1.0E0 / R S = S * C 160 G = D(I+1) - P R = (D(I) - G) * S + 2.0E0 * C * B P = S * R D(I+1) = G + P G = C * R - B 200 CONTINUE C D(L) = D(L) - P E(L) = G E(M) = 0.0E0 GO TO 105 C .......... ORDER EIGENVALUES .......... 215 IF (L .EQ. 1) GO TO 250 C .......... FOR I=L STEP -1 UNTIL 2 DO -- .......... DO 230 II = 2, L I = L + 2 - II IF (P .GE. D(I-1)) GO TO 270 D(I) = D(I-1) 230 CONTINUE C 250 I = 1 270 D(I) = P 290 CONTINUE C GO TO 1001 C .......... SET ERROR -- NO CONVERGENCE TO AN C EIGENVALUE AFTER 30 ITERATIONS .......... 1000 IERR = L 1001 RETURN END