*DECK TQLRAT SUBROUTINE TQLRAT (N, D, E2, IERR) C***BEGIN PROLOGUE TQLRAT C***PURPOSE Compute the eigenvalues of symmetric tridiagonal matrix C using a rational variant of the QL method. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D4A5, D4C2A C***TYPE SINGLE PRECISION (TQLRAT-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 TQLRAT. C C This subroutine finds the eigenvalues of a SYMMETRIC C TRIDIAGONAL matrix by the rational 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 E2 contains the squares of the subdiagonal elements of the C symmetric tridiagonal matrix in its last N-1 positions. C E2(1) is arbitrary. E2 is a one-dimensional REAL array, C dimensioned E2(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 E2 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 C. H. Reinsch, Eigenvalues of a real, symmetric, tri- C diagonal matrix, Algorithm 464, Communications of the C ACM 16, 11 (November 1973), pp. 689. C***ROUTINES CALLED PYTHAG, R1MACH 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 TQLRAT C INTEGER I,J,L,M,N,II,L1,MML,IERR REAL D(*),E2(*) REAL B,C,F,G,H,P,R,S,MACHEP REAL PYTHAG LOGICAL FIRST C SAVE FIRST, MACHEP DATA FIRST /.TRUE./ C***FIRST EXECUTABLE STATEMENT TQLRAT IF (FIRST) THEN MACHEP = R1MACH(4) ENDIF FIRST = .FALSE. C IERR = 0 IF (N .EQ. 1) GO TO 1001 C DO 100 I = 2, N 100 E2(I-1) = E2(I) C F = 0.0E0 B = 0.0E0 E2(N) = 0.0E0 C DO 290 L = 1, N J = 0 H = MACHEP * (ABS(D(L)) + SQRT(E2(L))) IF (B .GT. H) GO TO 105 B = H C = B * B C .......... LOOK FOR SMALL SQUARED SUB-DIAGONAL ELEMENT .......... 105 DO 110 M = L, N IF (E2(M) .LE. C) GO TO 120 C .......... E2(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 S = SQRT(E2(L)) G = D(L) P = (D(L1) - G) / (2.0E0 * S) R = PYTHAG(P,1.0E0) D(L) = S / (P + SIGN(R,P)) H = G - D(L) C DO 140 I = L1, N 140 D(I) = D(I) - H C F = F + H C .......... RATIONAL QL TRANSFORMATION .......... G = D(M) IF (G .EQ. 0.0E0) G = B H = G S = 0.0E0 MML = M - L C .......... FOR I=M-1 STEP -1 UNTIL L DO -- .......... DO 200 II = 1, MML I = M - II P = G * H R = P + E2(I) E2(I+1) = S * R S = E2(I) / R D(I+1) = H + S * (H + D(I)) G = D(I) - E2(I) / G IF (G .EQ. 0.0E0) G = B H = G * P / R 200 CONTINUE C E2(L) = S * G D(L) = H C .......... GUARD AGAINST UNDERFLOW IN CONVERGENCE TEST .......... IF (H .EQ. 0.0E0) GO TO 210 IF (ABS(E2(L)) .LE. ABS(C/H)) GO TO 210 E2(L) = H * E2(L) IF (E2(L) .NE. 0.0E0) 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