*DECK TQL1 SUBROUTINE TQL1 (N, D, E, IERR) C***BEGIN PROLOGUE TQL1 C***PURPOSE Compute the eigenvalues of symmetric tridiagonal matrix by C the QL method. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D4A5, D4C2A C***TYPE SINGLE PRECISION (TQL1-S) C***KEYWORDS EIGENVALUES OF A SYMMETRIC TRIDIAGONAL MATRIX, EISPACK, C QL METHOD C***AUTHOR Smith, B. T., et al. C***DESCRIPTION C C This subroutine is a translation of the ALGOL procedure TQL1, C NUM. MATH. 11, 293-306(1968) by Bowdler, Martin, Reinsch, and C Wilkinson. C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 227-240(1971). C C This subroutine finds the eigenvalues of a SYMMETRIC C TRIDIAGONAL matrix by the 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 C error exit is made, the eigenvalues are correct and C ordered for indices 1, 2, ..., IERR-1, but may not be C the smallest 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 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 TQL1 C INTEGER I,J,L,M,N,II,L1,L2,MML,IERR REAL D(*),E(*) REAL B,C,C2,C3,DL1,EL1,F,G,H,P,R,S,S2 REAL PYTHAG C C***FIRST EXECUTABLE STATEMENT TQL1 IERR = 0 IF (N .EQ. 1) GO TO 1001 C DO 100 I = 2, N 100 E(I-1) = E(I) C F = 0.0E0 B = 0.0E0 E(N) = 0.0E0 C DO 290 L = 1, N J = 0 H = ABS(D(L)) + ABS(E(L)) IF (B .LT. H) B = H C .......... LOOK FOR SMALL SUB-DIAGONAL ELEMENT .......... DO 110 M = L, N IF (B + ABS(E(M)) .EQ. B) GO TO 120 C .......... E(N) IS ALWAYS ZERO, SO THERE IS NO EXIT C THROUGH THE BOTTOM OF THE LOOP .......... 110 CONTINUE C 120 IF (M .EQ. L) GO TO 210 130 IF (J .EQ. 30) GO TO 1000 J = J + 1 C .......... FORM SHIFT .......... L1 = L + 1 L2 = L1 + 1 G = D(L) P = (D(L1) - G) / (2.0E0 * E(L)) R = PYTHAG(P,1.0E0) D(L) = E(L) / (P + SIGN(R,P)) D(L1) = E(L) * (P + SIGN(R,P)) DL1 = D(L1) H = G - D(L) IF (L2 .GT. N) GO TO 145 C DO 140 I = L2, N 140 D(I) = D(I) - H C 145 F = F + H C .......... QL TRANSFORMATION .......... P = D(M) C = 1.0E0 C2 = C EL1 = E(L1) S = 0.0E0 MML = M - L C .......... FOR I=M-1 STEP -1 UNTIL L DO -- .......... DO 200 II = 1, MML C3 = C2 C2 = C S2 = S I = M - II G = C * E(I) H = C * P IF (ABS(P) .LT. ABS(E(I))) GO TO 150 C = E(I) / P R = SQRT(C*C+1.0E0) E(I+1) = S * P * R S = C / R C = 1.0E0 / R GO TO 160 150 C = P / E(I) R = SQRT(C*C+1.0E0) E(I+1) = S * E(I) * R S = 1.0E0 / R C = C * S 160 P = C * D(I) - S * G D(I+1) = H + S * (C * G + S * D(I)) 200 CONTINUE C P = -S * S2 * C3 * EL1 * E(L) / DL1 E(L) = S * P D(L) = C * P IF (B + ABS(E(L)) .GT. B) GO TO 130 210 P = D(L) + F C .......... ORDER EIGENVALUES .......... 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