*DECK IMTQLV
SUBROUTINE IMTQLV (N, D, E, E2, W, IND, IERR, RV1)
C***BEGIN PROLOGUE IMTQLV
C***PURPOSE Compute the eigenvalues of a symmetric tridiagonal matrix
C using the implicit QL method. Eigenvectors may be computed
C later.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4A5, D4C2A
C***TYPE SINGLE PRECISION (IMTQLV-S)
C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is a variant of IMTQL1 which is a translation of
C ALGOL procedure IMTQL1, NUM. MATH. 12, 377-383(1968) by Martin and
C Wilkinson, 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 TRIDIAGONAL
C matrix by the implicit QL method and associates with them
C their corresponding submatrix indices.
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 E2 contains the squares of the corresponding elements of E in
C its last N-1 positions. E2(1) is arbitrary. E2 is a one-
C dimensional REAL array, dimensioned E2(N).
C
C On OUTPUT
C
C D and E are unaltered.
C
C Elements of E2, corresponding to elements of E regarded as
C negligible, have been replaced by zero causing the matrix to
C split into a direct sum of submatrices. E2(1) is also set
C to zero.
C
C W 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. W is a one-dimensional REAL array, dimensioned
C W(N).
C
C IND contains the submatrix indices associated with the
C corresponding eigenvalues in W -- 1 for eigenvalues belonging
C to the first submatrix from the top, 2 for those belonging to
C the second submatrix, etc. IND is a one-dimensional REAL
C array, dimensioned IND(N).
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 RV1 is a one-dimensional REAL array used for temporary storage,
C dimensioned RV1(N).
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 IMTQLV
C
INTEGER I,J,K,L,M,N,II,MML,TAG,IERR
REAL D(*),E(*),E2(*),W(*),RV1(*)
REAL B,C,F,G,P,R,S,S1,S2
REAL PYTHAG
INTEGER IND(*)
C
C***FIRST EXECUTABLE STATEMENT IMTQLV
IERR = 0
K = 0
TAG = 0
C
DO 100 I = 1, N
W(I) = D(I)
IF (I .NE. 1) RV1(I-1) = E(I)
100 CONTINUE
C
E2(1) = 0.0E0
RV1(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(W(M)) + ABS(W(M+1))
S2 = S1 + ABS(RV1(M))
IF (S2 .EQ. S1) GO TO 120
C .......... GUARD AGAINST UNDERFLOWED ELEMENT OF E2 ..........
IF (E2(M+1) .EQ. 0.0E0) GO TO 125
110 CONTINUE
C
120 IF (M .LE. K) GO TO 130
IF (M .NE. N) E2(M+1) = 0.0E0
125 K = M
TAG = TAG + 1
130 P = W(L)
IF (M .EQ. L) GO TO 215
IF (J .EQ. 30) GO TO 1000
J = J + 1
C .......... FORM SHIFT ..........
G = (W(L+1) - P) / (2.0E0 * RV1(L))
R = PYTHAG(G,1.0E0)
G = W(M) - P + RV1(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 * RV1(I)
B = C * RV1(I)
IF (ABS(F) .LT. ABS(G)) GO TO 150
C = G / F
R = SQRT(C*C+1.0E0)
RV1(I+1) = F * R
S = 1.0E0 / R
C = C * S
GO TO 160
150 S = F / G
R = SQRT(S*S+1.0E0)
RV1(I+1) = G * R
C = 1.0E0 / R
S = S * C
160 G = W(I+1) - P
R = (W(I) - G) * S + 2.0E0 * C * B
P = S * R
W(I+1) = G + P
G = C * R - B
200 CONTINUE
C
W(L) = W(L) - P
RV1(L) = G
RV1(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. W(I-1)) GO TO 270
W(I) = W(I-1)
IND(I) = IND(I-1)
230 CONTINUE
C
250 I = 1
270 W(I) = P
IND(I) = TAG
290 CONTINUE
C
GO TO 1001
C .......... SET ERROR -- NO CONVERGENCE TO AN
C EIGENVALUE AFTER 30 ITERATIONS ..........
1000 IERR = L
1001 RETURN
END