*DECK TSTURM
SUBROUTINE TSTURM (NM, N, EPS1, D, E, E2, LB, UB, MM, M, W, Z,
+ IERR, RV1, RV2, RV3, RV4, RV5, RV6)
C***BEGIN PROLOGUE TSTURM
C***PURPOSE Find those eigenvalues of a symmetric tridiagonal matrix
C in a given interval and their associated eigenvectors by
C Sturm sequencing.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4A5, D4C2A
C***TYPE SINGLE PRECISION (TSTURM-S)
C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine finds those eigenvalues of a TRIDIAGONAL
C SYMMETRIC matrix which lie in a specified interval and their
C associated eigenvectors, using bisection and inverse iteration.
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 EPS1 is an absolute error tolerance for the computed eigen-
C values. It should be chosen so that the accuracy of these
C eigenvalues is commensurate with relative perturbations of
C the order of the relative machine precision in the matrix
C elements. If the input EPS1 is non-positive, it is reset
C for each submatrix to a default value, namely, minus the
C product of the relative machine precision and the 1-norm of
C the submatrix. EPS1 is a REAL 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.
C E2(1) is arbitrary. E2 is a one-dimensional REAL array,
C dimensioned E2(N).
C
C LB and UB define the interval to be searched for eigenvalues.
C If LB is not less than UB, no eigenvalues will be found.
C LB and UB are REAL variables.
C
C MM should be set to an upper bound for the number of
C eigenvalues in the interval. MM is an INTEGER variable.
C WARNING - If more than MM eigenvalues are determined to lie
C in the interval, an error return is made with no values or
C vectors found.
C
C On Output
C
C EPS1 is unaltered unless it has been reset to its
C (last) default value.
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C negligible, have been replaced by zero causing the matrix to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 M is the number of eigenvalues determined to lie in (LB,UB).
C M is an INTEGER variable.
C
C W contains the M eigenvalues in ascending order if the matrix
C does not split. If the matrix splits, the eigenvalues are
C in ascending order for each submatrix. If a vector error
C exit is made, W contains those values already found. W is a
C one-dimensional REAL array, dimensioned W(MM).
C
C Z contains the associated set of orthonormal eigenvectors.
C If an error exit is made, Z contains those vectors already
C found. Z is a one-dimensional REAL array, dimensioned
C Z(NM,MM).
C
C IERR is an INTEGER flag set to
C Zero for normal return,
C 3*N+1 if M exceeds MM no eigenvalues or eigenvectors
C are computed,
C 4*N+J if the eigenvector corresponding to the J-th
C eigenvalue fails to converge in 5 iterations, then
C the eigenvalues and eigenvectors in W and Z should
C be correct for indices 1, 2, ..., J-1.
C
C RV1, RV2, RV3, RV4, RV5, and RV6 are temporary storage arrays,
C dimensioned RV1(N), RV2(N), RV3(N), RV4(N), RV5(N), and
C RV6(N).
C
C The ALGOL procedure STURMCNT contained in TRISTURM
C appears in TSTURM in-line.
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 R1MACH
C***REVISION HISTORY (YYMMDD)
C 760101 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890531 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 TSTURM
C
INTEGER I,J,K,M,N,P,Q,R,S,II,IP,JJ,MM,M1,M2,NM,ITS
INTEGER IERR,GROUP,ISTURM
REAL D(*),E(*),E2(*),W(*),Z(NM,*)
REAL RV1(*),RV2(*),RV3(*),RV4(*),RV5(*),RV6(*)
REAL U,V,LB,T1,T2,UB,UK,XU,X0,X1,EPS1,EPS2,EPS3,EPS4
REAL NORM,MACHEP,S1,S2
LOGICAL FIRST
C
SAVE FIRST, MACHEP
DATA FIRST /.TRUE./
C***FIRST EXECUTABLE STATEMENT TSTURM
IF (FIRST) THEN
MACHEP = R1MACH(4)
ENDIF
FIRST = .FALSE.
C
IERR = 0
T1 = LB
T2 = UB
C .......... LOOK FOR SMALL SUB-DIAGONAL ENTRIES ..........
DO 40 I = 1, N
IF (I .EQ. 1) GO TO 20
S1 = ABS(D(I)) + ABS(D(I-1))
S2 = S1 + ABS(E(I))
IF (S2 .GT. S1) GO TO 40
20 E2(I) = 0.0E0
40 CONTINUE
C .......... DETERMINE THE NUMBER OF EIGENVALUES
C IN THE INTERVAL ..........
P = 1
Q = N
X1 = UB
ISTURM = 1
GO TO 320
60 M = S
X1 = LB
ISTURM = 2
GO TO 320
80 M = M - S
IF (M .GT. MM) GO TO 980
Q = 0
R = 0
C .......... ESTABLISH AND PROCESS NEXT SUBMATRIX, REFINING
C INTERVAL BY THE GERSCHGORIN BOUNDS ..........
100 IF (R .EQ. M) GO TO 1001
P = Q + 1
XU = D(P)
X0 = D(P)
U = 0.0E0
C
DO 120 Q = P, N
X1 = U
U = 0.0E0
V = 0.0E0
IF (Q .EQ. N) GO TO 110
U = ABS(E(Q+1))
V = E2(Q+1)
110 XU = MIN(D(Q)-(X1+U),XU)
X0 = MAX(D(Q)+(X1+U),X0)
IF (V .EQ. 0.0E0) GO TO 140
120 CONTINUE
C
140 X1 = MAX(ABS(XU),ABS(X0)) * MACHEP
IF (EPS1 .LE. 0.0E0) EPS1 = -X1
IF (P .NE. Q) GO TO 180
C .......... CHECK FOR ISOLATED ROOT WITHIN INTERVAL ..........
IF (T1 .GT. D(P) .OR. D(P) .GE. T2) GO TO 940
R = R + 1
C
DO 160 I = 1, N
160 Z(I,R) = 0.0E0
C
W(R) = D(P)
Z(P,R) = 1.0E0
GO TO 940
180 X1 = X1 * (Q-P+1)
LB = MAX(T1,XU-X1)
UB = MIN(T2,X0+X1)
X1 = LB
ISTURM = 3
GO TO 320
200 M1 = S + 1
X1 = UB
ISTURM = 4
GO TO 320
220 M2 = S
IF (M1 .GT. M2) GO TO 940
C .......... FIND ROOTS BY BISECTION ..........
X0 = UB
ISTURM = 5
C
DO 240 I = M1, M2
RV5(I) = UB
RV4(I) = LB
240 CONTINUE
C .......... LOOP FOR K-TH EIGENVALUE
C FOR K=M2 STEP -1 UNTIL M1 DO --
C (-DO- NOT USED TO LEGALIZE -COMPUTED GO TO-) ..........
K = M2
250 XU = LB
C .......... FOR I=K STEP -1 UNTIL M1 DO -- ..........
DO 260 II = M1, K
I = M1 + K - II
IF (XU .GE. RV4(I)) GO TO 260
XU = RV4(I)
GO TO 280
260 CONTINUE
C
280 IF (X0 .GT. RV5(K)) X0 = RV5(K)
C .......... NEXT BISECTION STEP ..........
300 X1 = (XU + X0) * 0.5E0
S1 = 2.0E0*(ABS(XU) + ABS(X0) + ABS(EPS1))
S2 = S1 + ABS(X0 - XU)
IF (S2 .EQ. S1) GO TO 420
C .......... IN-LINE PROCEDURE FOR STURM SEQUENCE ..........
320 S = P - 1
U = 1.0E0
C
DO 340 I = P, Q
IF (U .NE. 0.0E0) GO TO 325
V = ABS(E(I)) / MACHEP
IF (E2(I) .EQ. 0.0E0) V = 0.0E0
GO TO 330
325 V = E2(I) / U
330 U = D(I) - X1 - V
IF (U .LT. 0.0E0) S = S + 1
340 CONTINUE
C
GO TO (60,80,200,220,360), ISTURM
C .......... REFINE INTERVALS ..........
360 IF (S .GE. K) GO TO 400
XU = X1
IF (S .GE. M1) GO TO 380
RV4(M1) = X1
GO TO 300
380 RV4(S+1) = X1
IF (RV5(S) .GT. X1) RV5(S) = X1
GO TO 300
400 X0 = X1
GO TO 300
C .......... K-TH EIGENVALUE FOUND ..........
420 RV5(K) = X1
K = K - 1
IF (K .GE. M1) GO TO 250
C .......... FIND VECTORS BY INVERSE ITERATION ..........
NORM = ABS(D(P))
IP = P + 1
C
DO 500 I = IP, Q
500 NORM = MAX(NORM, ABS(D(I)) + ABS(E(I)))
C .......... EPS2 IS THE CRITERION FOR GROUPING,
C EPS3 REPLACES ZERO PIVOTS AND EQUAL
C ROOTS ARE MODIFIED BY EPS3,
C EPS4 IS TAKEN VERY SMALL TO AVOID OVERFLOW ..........
EPS2 = 1.0E-3 * NORM
UK = SQRT(REAL(Q-P+5))
EPS3 = UK * MACHEP * NORM
EPS4 = UK * EPS3
UK = EPS4 / SQRT(UK)
GROUP = 0
S = P
C
DO 920 K = M1, M2
R = R + 1
ITS = 1
W(R) = RV5(K)
X1 = RV5(K)
C .......... LOOK FOR CLOSE OR COINCIDENT ROOTS ..........
IF (K .EQ. M1) GO TO 520
IF (X1 - X0 .GE. EPS2) GROUP = -1
GROUP = GROUP + 1
IF (X1 .LE. X0) X1 = X0 + EPS3
C .......... ELIMINATION WITH INTERCHANGES AND
C INITIALIZATION OF VECTOR ..........
520 V = 0.0E0
C
DO 580 I = P, Q
RV6(I) = UK
IF (I .EQ. P) GO TO 560
IF (ABS(E(I)) .LT. ABS(U)) GO TO 540
XU = U / E(I)
RV4(I) = XU
RV1(I-1) = E(I)
RV2(I-1) = D(I) - X1
RV3(I-1) = 0.0E0
IF (I .NE. Q) RV3(I-1) = E(I+1)
U = V - XU * RV2(I-1)
V = -XU * RV3(I-1)
GO TO 580
540 XU = E(I) / U
RV4(I) = XU
RV1(I-1) = U
RV2(I-1) = V
RV3(I-1) = 0.0E0
560 U = D(I) - X1 - XU * V
IF (I .NE. Q) V = E(I+1)
580 CONTINUE
C
IF (U .EQ. 0.0E0) U = EPS3
RV1(Q) = U
RV2(Q) = 0.0E0
RV3(Q) = 0.0E0
C .......... BACK SUBSTITUTION
C FOR I=Q STEP -1 UNTIL P DO -- ..........
600 DO 620 II = P, Q
I = P + Q - II
RV6(I) = (RV6(I) - U * RV2(I) - V * RV3(I)) / RV1(I)
V = U
U = RV6(I)
620 CONTINUE
C .......... ORTHOGONALIZE WITH RESPECT TO PREVIOUS
C MEMBERS OF GROUP ..........
IF (GROUP .EQ. 0) GO TO 700
C
DO 680 JJ = 1, GROUP
J = R - GROUP - 1 + JJ
XU = 0.0E0
C
DO 640 I = P, Q
640 XU = XU + RV6(I) * Z(I,J)
C
DO 660 I = P, Q
660 RV6(I) = RV6(I) - XU * Z(I,J)
C
680 CONTINUE
C
700 NORM = 0.0E0
C
DO 720 I = P, Q
720 NORM = NORM + ABS(RV6(I))
C
IF (NORM .GE. 1.0E0) GO TO 840
C .......... FORWARD SUBSTITUTION ..........
IF (ITS .EQ. 5) GO TO 960
IF (NORM .NE. 0.0E0) GO TO 740
RV6(S) = EPS4
S = S + 1
IF (S .GT. Q) S = P
GO TO 780
740 XU = EPS4 / NORM
C
DO 760 I = P, Q
760 RV6(I) = RV6(I) * XU
C .......... ELIMINATION OPERATIONS ON NEXT VECTOR
C ITERATE ..........
780 DO 820 I = IP, Q
U = RV6(I)
C .......... IF RV1(I-1) .EQ. E(I), A ROW INTERCHANGE
C WAS PERFORMED EARLIER IN THE
C TRIANGULARIZATION PROCESS ..........
IF (RV1(I-1) .NE. E(I)) GO TO 800
U = RV6(I-1)
RV6(I-1) = RV6(I)
800 RV6(I) = U - RV4(I) * RV6(I-1)
820 CONTINUE
C
ITS = ITS + 1
GO TO 600
C .......... NORMALIZE SO THAT SUM OF SQUARES IS
C 1 AND EXPAND TO FULL ORDER ..........
840 U = 0.0E0
C
DO 860 I = P, Q
860 U = U + RV6(I)**2
C
XU = 1.0E0 / SQRT(U)
C
DO 880 I = 1, N
880 Z(I,R) = 0.0E0
C
DO 900 I = P, Q
900 Z(I,R) = RV6(I) * XU
C
X0 = X1
920 CONTINUE
C
940 IF (Q .LT. N) GO TO 100
GO TO 1001
C .......... SET ERROR -- NON-CONVERGED EIGENVECTOR ..........
960 IERR = 4 * N + R
GO TO 1001
C .......... SET ERROR -- UNDERESTIMATE OF NUMBER OF
C EIGENVALUES IN INTERVAL ..........
980 IERR = 3 * N + 1
1001 LB = T1
UB = T2
RETURN
END