*DECK IMTQL2 SUBROUTINE IMTQL2 (NM, N, D, E, Z, IERR) C***BEGIN PROLOGUE IMTQL2 C***PURPOSE Compute the eigenvalues and eigenvectors of a symmetric C tridiagonal matrix using the implicit QL method. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D4A5, D4C2A C***TYPE SINGLE PRECISION (IMTQL2-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 IMTQL2, 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 and eigenvectors C of a SYMMETRIC TRIDIAGONAL matrix by the implicit QL method. C The eigenvectors of a FULL SYMMETRIC matrix can also C be found if TRED2 has been used to reduce this C full matrix to tridiagonal form. C C On INPUT C C NM must be set to the row dimension of the two-dimensional C array parameter, Z, as declared in the calling program C dimension statement. NM is an INTEGER variable. C C N is the order of the matrix. N is an INTEGER variable. C N must be less than or equal to NM. 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 Z contains the transformation matrix produced in the reduction C by TRED2, if performed. This transformation matrix is C necessary if you want to obtain the eigenvectors of the full C symmetric matrix. If the eigenvectors of the symmetric C tridiagonal matrix are desired, Z must contain the identity C matrix. Z is a two-dimensional REAL array, dimensioned C Z(NM,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 but C unordered for indices 1, 2, ..., IERR-1. C C E has been destroyed. C C Z contains orthonormal eigenvectors of the full symmetric C or symmetric tridiagonal matrix, depending on what it C contained on input. If an error exit is made, Z contains C the eigenvectors associated with the stored eigenvalues. 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 and eigenvectors should be correct C for indices 1, 2, ..., IERR-1, but the eigenvalues C are not ordered. 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 IMTQL2 C INTEGER I,J,K,L,M,N,II,NM,MML,IERR REAL D(*),E(*),Z(NM,*) REAL B,C,F,G,P,R,S,S1,S2 REAL PYTHAG C C***FIRST EXECUTABLE STATEMENT IMTQL2 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 240 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 240 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 C .......... FORM VECTOR .......... DO 180 K = 1, N F = Z(K,I+1) Z(K,I+1) = S * Z(K,I) + C * F Z(K,I) = C * Z(K,I) - S * F 180 CONTINUE C 200 CONTINUE C D(L) = D(L) - P E(L) = G E(M) = 0.0E0 GO TO 105 240 CONTINUE C .......... ORDER EIGENVALUES AND EIGENVECTORS .......... DO 300 II = 2, N I = II - 1 K = I P = D(I) C DO 260 J = II, N IF (D(J) .GE. P) GO TO 260 K = J P = D(J) 260 CONTINUE C IF (K .EQ. I) GO TO 300 D(K) = D(I) D(I) = P C DO 280 J = 1, N P = Z(J,I) Z(J,I) = Z(J,K) Z(J,K) = P 280 CONTINUE C 300 CONTINUE C GO TO 1001 C .......... SET ERROR -- NO CONVERGENCE TO AN C EIGENVALUE AFTER 30 ITERATIONS .......... 1000 IERR = L 1001 RETURN END