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165 lines
5.1 KiB
Fortran
165 lines
5.1 KiB
Fortran
*DECK TQLRAT
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SUBROUTINE TQLRAT (N, D, E2, IERR)
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C***BEGIN PROLOGUE TQLRAT
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C***PURPOSE Compute the eigenvalues of symmetric tridiagonal matrix
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C using a rational variant of the QL method.
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C***LIBRARY SLATEC (EISPACK)
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C***CATEGORY D4A5, D4C2A
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C***TYPE SINGLE PRECISION (TQLRAT-S)
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C***KEYWORDS EIGENVALUES OF A SYMMETRIC TRIDIAGONAL MATRIX, EISPACK,
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C QL METHOD
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C***AUTHOR Smith, B. T., et al.
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C***DESCRIPTION
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C
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C This subroutine is a translation of the ALGOL procedure TQLRAT.
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C
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C This subroutine finds the eigenvalues of a SYMMETRIC
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C TRIDIAGONAL matrix by the rational QL method.
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C
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C On Input
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C
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C N is the order of the matrix. N is an INTEGER variable.
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C
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C D contains the diagonal elements of the symmetric tridiagonal
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C matrix. D is a one-dimensional REAL array, dimensioned D(N).
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C
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C E2 contains the squares of the subdiagonal elements of the
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C symmetric tridiagonal matrix in its last N-1 positions.
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C E2(1) is arbitrary. E2 is a one-dimensional REAL array,
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C dimensioned E2(N).
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C
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C On Output
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C
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C D contains the eigenvalues in ascending order. If an
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C error exit is made, the eigenvalues are correct and
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C ordered for indices 1, 2, ..., IERR-1, but may not be
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C the smallest eigenvalues.
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C
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C E2 has been destroyed.
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C
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C IERR is an INTEGER flag set to
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C Zero for normal return,
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C J if the J-th eigenvalue has not been
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C determined after 30 iterations.
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C
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C Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
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C
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C Questions and comments should be directed to B. S. Garbow,
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C APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
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C ------------------------------------------------------------------
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C
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C***REFERENCES B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
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C Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
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C system Routines - EISPACK Guide, Springer-Verlag,
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C 1976.
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C C. H. Reinsch, Eigenvalues of a real, symmetric, tri-
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C diagonal matrix, Algorithm 464, Communications of the
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C ACM 16, 11 (November 1973), pp. 689.
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C***ROUTINES CALLED PYTHAG, R1MACH
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C***REVISION HISTORY (YYMMDD)
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C 760101 DATE WRITTEN
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C 890831 Modified array declarations. (WRB)
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C 890831 REVISION DATE from Version 3.2
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C 891214 Prologue converted to Version 4.0 format. (BAB)
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C 920501 Reformatted the REFERENCES section. (WRB)
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C***END PROLOGUE TQLRAT
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C
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INTEGER I,J,L,M,N,II,L1,MML,IERR
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REAL D(*),E2(*)
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REAL B,C,F,G,H,P,R,S,MACHEP
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REAL PYTHAG
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LOGICAL FIRST
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C
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SAVE FIRST, MACHEP
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DATA FIRST /.TRUE./
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C***FIRST EXECUTABLE STATEMENT TQLRAT
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IF (FIRST) THEN
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MACHEP = R1MACH(4)
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ENDIF
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FIRST = .FALSE.
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C
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IERR = 0
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IF (N .EQ. 1) GO TO 1001
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C
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DO 100 I = 2, N
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100 E2(I-1) = E2(I)
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C
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F = 0.0E0
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B = 0.0E0
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E2(N) = 0.0E0
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C
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DO 290 L = 1, N
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J = 0
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H = MACHEP * (ABS(D(L)) + SQRT(E2(L)))
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IF (B .GT. H) GO TO 105
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B = H
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C = B * B
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C .......... LOOK FOR SMALL SQUARED SUB-DIAGONAL ELEMENT ..........
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105 DO 110 M = L, N
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IF (E2(M) .LE. C) GO TO 120
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C .......... E2(N) IS ALWAYS ZERO, SO THERE IS NO EXIT
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C THROUGH THE BOTTOM OF THE LOOP ..........
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110 CONTINUE
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C
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120 IF (M .EQ. L) GO TO 210
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130 IF (J .EQ. 30) GO TO 1000
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J = J + 1
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C .......... FORM SHIFT ..........
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L1 = L + 1
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S = SQRT(E2(L))
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G = D(L)
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P = (D(L1) - G) / (2.0E0 * S)
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R = PYTHAG(P,1.0E0)
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D(L) = S / (P + SIGN(R,P))
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H = G - D(L)
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C
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DO 140 I = L1, N
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140 D(I) = D(I) - H
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C
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F = F + H
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C .......... RATIONAL QL TRANSFORMATION ..........
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G = D(M)
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IF (G .EQ. 0.0E0) G = B
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H = G
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S = 0.0E0
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MML = M - L
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C .......... FOR I=M-1 STEP -1 UNTIL L DO -- ..........
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DO 200 II = 1, MML
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I = M - II
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P = G * H
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R = P + E2(I)
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E2(I+1) = S * R
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S = E2(I) / R
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D(I+1) = H + S * (H + D(I))
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G = D(I) - E2(I) / G
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IF (G .EQ. 0.0E0) G = B
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H = G * P / R
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200 CONTINUE
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C
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E2(L) = S * G
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D(L) = H
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C .......... GUARD AGAINST UNDERFLOW IN CONVERGENCE TEST ..........
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IF (H .EQ. 0.0E0) GO TO 210
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IF (ABS(E2(L)) .LE. ABS(C/H)) GO TO 210
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E2(L) = H * E2(L)
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IF (E2(L) .NE. 0.0E0) GO TO 130
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210 P = D(L) + F
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C .......... ORDER EIGENVALUES ..........
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IF (L .EQ. 1) GO TO 250
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C .......... FOR I=L STEP -1 UNTIL 2 DO -- ..........
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DO 230 II = 2, L
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I = L + 2 - II
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IF (P .GE. D(I-1)) GO TO 270
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D(I) = D(I-1)
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230 CONTINUE
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C
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250 I = 1
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270 D(I) = P
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290 CONTINUE
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C
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GO TO 1001
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C .......... SET ERROR -- NO CONVERGENCE TO AN
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C EIGENVALUE AFTER 30 ITERATIONS ..........
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1000 IERR = L
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1001 RETURN
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END
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