*DECK TRIDIB
SUBROUTINE TRIDIB (N, EPS1, D, E, E2, LB, UB, M11, M, W, IND,
+ IERR, RV4, RV5)
C***BEGIN PROLOGUE TRIDIB
C***PURPOSE Compute the eigenvalues of a symmetric tridiagonal matrix
C in a given interval using Sturm sequencing.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4A5, D4C2A
C***TYPE SINGLE PRECISION (TRIDIB-S)
C***KEYWORDS EIGENVALUES OF A REAL SYMMETRIC MATRIX, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is a translation of the ALGOL procedure BISECT,
C NUM. MATH. 9, 386-393(1967) by Barth, Martin, and Wilkinson.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 249-256(1971).
C
C This subroutine finds those eigenvalues of a TRIDIAGONAL
C SYMMETRIC matrix between specified boundary indices,
C using bisection.
C
C On Input
C
C N is the order of the matrix. N is an INTEGER variable.
C
C EPS1 is an absolute error tolerance for the computed eigen-
C values. If the input EPS1 is non-positive, it is reset for
C each submatrix to a default value, namely, minus the product
C of the relative machine precision and the 1-norm of the
C 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 M11 specifies the lower boundary index for the set of desired
C eigenvalues. M11 is an INTEGER variable.
C
C M specifies the number of eigenvalues desired. The upper
C boundary index M22 is then obtained as M22=M11+M-1.
C M is an INTEGER variable.
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
C as negligible, have been replaced by zero causing the
C matrix to split into a direct sum of submatrices.
C E2(1) is also set to zero.
C
C LB and UB define an interval containing exactly the desired
C eigenvalues. LB and UB are REAL variables.
C
C W contains, in its first M positions, the eigenvalues
C between indices M11 and M22 in ascending order.
C W is a one-dimensional REAL array, dimensioned W(M).
C
C IND contains in its first M positions the submatrix indices
C associated with the corresponding eigenvalues in W --
C 1 for eigenvalues belonging to the first submatrix from
C the top, 2 for those belonging to the second submatrix, etc.
C IND is an one-dimensional INTEGER array, dimensioned IND(M).
C
C IERR is an INTEGER flag set to
C Zero for normal return,
C 3*N+1 if multiple eigenvalues at index M11 make
C unique selection of LB impossible,
C 3*N+2 if multiple eigenvalues at index M22 make
C unique selection of UB impossible.
C
C RV4 and RV5 are one-dimensional REAL arrays used for temporary
C storage of the lower and upper bounds for the eigenvalues in
C the bisection process. RV4 and RV5 are dimensioned RV4(N)
C and RV5(N).
C
C Note that subroutine TQL1, IMTQL1, or TQLRAT is generally faster
C than TRIDIB, if more than N/4 eigenvalues are to be found.
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 TRIDIB
C
INTEGER I,J,K,L,M,N,P,Q,R,S,II,M1,M2,M11,M22,TAG,IERR,ISTURM
REAL D(*),E(*),E2(*),W(*),RV4(*),RV5(*)
REAL U,V,LB,T1,T2,UB,XU,X0,X1,EPS1,MACHEP,S1,S2
INTEGER IND(*)
LOGICAL FIRST
C
SAVE FIRST, MACHEP
DATA FIRST /.TRUE./
C***FIRST EXECUTABLE STATEMENT TRIDIB
IF (FIRST) THEN
MACHEP = R1MACH(4)
ENDIF
FIRST = .FALSE.
C
IERR = 0
TAG = 0
XU = D(1)
X0 = D(1)
U = 0.0E0
C .......... LOOK FOR SMALL SUB-DIAGONAL ENTRIES AND DETERMINE AN
C INTERVAL CONTAINING ALL THE EIGENVALUES ..........
DO 40 I = 1, N
X1 = U
U = 0.0E0
IF (I .NE. N) U = ABS(E(I+1))
XU = MIN(D(I)-(X1+U),XU)
X0 = MAX(D(I)+(X1+U),X0)
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
X1 = MAX(ABS(XU),ABS(X0)) * MACHEP * N
XU = XU - X1
T1 = XU
X0 = X0 + X1
T2 = X0
C .......... DETERMINE AN INTERVAL CONTAINING EXACTLY
C THE DESIRED EIGENVALUES ..........
P = 1
Q = N
M1 = M11 - 1
IF (M1 .EQ. 0) GO TO 75
ISTURM = 1
50 V = X1
X1 = XU + (X0 - XU) * 0.5E0
IF (X1 .EQ. V) GO TO 980
GO TO 320
60 IF (S - M1) 65, 73, 70
65 XU = X1
GO TO 50
70 X0 = X1
GO TO 50
73 XU = X1
T1 = X1
75 M22 = M1 + M
IF (M22 .EQ. N) GO TO 90
X0 = T2
ISTURM = 2
GO TO 50
80 IF (S - M22) 65, 85, 70
85 T2 = X1
90 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
TAG = TAG + 1
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
M1 = P
M2 = P
RV5(P) = D(P)
GO TO 900
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 = ABS(XU) + ABS(X0) + ABS(EPS1)
S2 = S1 + ABS(X0-XU)/2.0E0
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 .......... ORDER EIGENVALUES TAGGED WITH THEIR
C SUBMATRIX ASSOCIATIONS ..........
900 S = R
R = R + M2 - M1 + 1
J = 1
K = M1
C
DO 920 L = 1, R
IF (J .GT. S) GO TO 910
IF (K .GT. M2) GO TO 940
IF (RV5(K) .GE. W(L)) GO TO 915
C
DO 905 II = J, S
I = L + S - II
W(I+1) = W(I)
IND(I+1) = IND(I)
905 CONTINUE
C
910 W(L) = RV5(K)
IND(L) = TAG
K = K + 1
GO TO 920
915 J = J + 1
920 CONTINUE
C
940 IF (Q .LT. N) GO TO 100
GO TO 1001
C .......... SET ERROR -- INTERVAL CANNOT BE FOUND CONTAINING
C EXACTLY THE DESIRED EIGENVALUES ..........
980 IERR = 3 * N + ISTURM
1001 LB = T1
UB = T2
RETURN
END