*DECK RATQR
SUBROUTINE RATQR (N, EPS1, D, E, E2, M, W, IND, BD, TYPE, IDEF,
+ IERR)
C***BEGIN PROLOGUE RATQR
C***PURPOSE Compute the largest or smallest eigenvalues of a symmetric
C tridiagonal matrix using the rational QR method with Newton
C correction.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4A5, D4C2A
C***TYPE SINGLE PRECISION (RATQR-S)
C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is a translation of the ALGOL procedure RATQR,
C NUM. MATH. 11, 264-272(1968) by REINSCH and BAUER.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 257-265(1971).
C
C This subroutine finds the algebraically smallest or largest
C eigenvalues of a SYMMETRIC TRIDIAGONAL matrix by the
C rational QR method with Newton corrections.
C
C On Input
C
C N is the order of the matrix. N is an INTEGER variable.
C
C EPS1 is a theoretical absolute error tolerance for the
C computed eigenvalues. If the input EPS1 is non-positive, or
C indeed smaller than its default value, it is reset at each
C iteration to the respective default value, namely, the
C product of the relative machine precision and the magnitude
C of the current eigenvalue iterate. The theoretical absolute
C error in the K-th eigenvalue is usually not greater than
C K times EPS1. 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 in
C its last N-1 positions. E2(1) is arbitrary. E2 is a one-
C dimensional REAL array, dimensioned E2(N).
C
C M is the number of eigenvalues to be found. M is an INTEGER
C variable.
C
C IDEF should be set to 1 if the input matrix is known to be
C positive definite, to -1 if the input matrix is known to
C be negative definite, and to 0 otherwise. IDEF is an
C INTEGER variable.
C
C TYPE should be set to .TRUE. if the smallest eigenvalues are
C to be found, and to .FALSE. if the largest eigenvalues are
C to be found. TYPE is a LOGICAL 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 (unless W overwrites D).
C
C Elements of E2, corresponding to elements of E regarded as
C negligible, have been replaced by zero causing the matrix
C to split into a direct sum of submatrices. E2(1) is set
C to 0.0e0 if the smallest eigenvalues have been found, and
C to 2.0e0 if the largest eigenvalues have been found. E2
C is otherwise unaltered (unless overwritten by BD).
C
C W contains the M algebraically smallest eigenvalues in
C ascending order, or the M largest eigenvalues in descending
C order. If an error exit is made because of an incorrect
C specification of IDEF, no eigenvalues are found. If the
C Newton iterates for a particular eigenvalue are not monotone,
C the best estimate obtained is returned and IERR is set.
C W is a one-dimensional REAL array, dimensioned W(N). W need
C not be distinct from D.
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(N).
C
C BD contains refined bounds for the theoretical errors of the
C corresponding eigenvalues in W. These bounds are usually
C within the tolerance specified by EPS1. BD is a one-
C dimensional REAL array, dimensioned BD(N). BD need not be
C distinct from E2.
C
C IERR is an INTEGER flag set to
C Zero for normal return,
C 6*N+1 if IDEF is set to 1 and TYPE to .TRUE.
C when the matrix is NOT positive definite, or
C if IDEF is set to -1 and TYPE to .FALSE.
C when the matrix is NOT negative definite,
C no eigenvalues are computed, or
C M is greater than N,
C 5*N+K if successive iterates to the K-th eigenvalue
C are NOT monotone increasing, where K refers
C to the last such occurrence.
C
C Note that subroutine TRIDIB is generally faster and more
C accurate than RATQR if the eigenvalues are clustered.
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 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 RATQR
C
INTEGER I,J,K,M,N,II,JJ,K1,IDEF,IERR,JDEF
REAL D(*),E(*),E2(*),W(*),BD(*)
REAL F,P,Q,R,S,EP,QP,ERR,TOT,EPS1,DELTA,MACHEP
INTEGER IND(*)
LOGICAL FIRST, TYPE
C
SAVE FIRST, MACHEP
DATA FIRST /.TRUE./
C***FIRST EXECUTABLE STATEMENT RATQR
IF (FIRST) THEN
MACHEP = R1MACH(4)
ENDIF
FIRST = .FALSE.
C
IERR = 0
JDEF = IDEF
C .......... COPY D ARRAY INTO W ..........
DO 20 I = 1, N
20 W(I) = D(I)
C
IF (TYPE) GO TO 40
J = 1
GO TO 400
40 ERR = 0.0E0
S = 0.0E0
C .......... LOOK FOR SMALL SUB-DIAGONAL ENTRIES AND DEFINE
C INITIAL SHIFT FROM LOWER GERSCHGORIN BOUND.
C COPY E2 ARRAY INTO BD ..........
TOT = W(1)
Q = 0.0E0
J = 0
C
DO 100 I = 1, N
P = Q
IF (I .EQ. 1) GO TO 60
IF (P .GT. MACHEP * (ABS(D(I)) + ABS(D(I-1)))) GO TO 80
60 E2(I) = 0.0E0
80 BD(I) = E2(I)
C .......... COUNT ALSO IF ELEMENT OF E2 HAS UNDERFLOWED ..........
IF (E2(I) .EQ. 0.0E0) J = J + 1
IND(I) = J
Q = 0.0E0
IF (I .NE. N) Q = ABS(E(I+1))
TOT = MIN(W(I)-P-Q,TOT)
100 CONTINUE
C
IF (JDEF .EQ. 1 .AND. TOT .LT. 0.0E0) GO TO 140
C
DO 110 I = 1, N
110 W(I) = W(I) - TOT
C
GO TO 160
140 TOT = 0.0E0
C
160 DO 360 K = 1, M
C .......... NEXT QR TRANSFORMATION ..........
180 TOT = TOT + S
DELTA = W(N) - S
I = N
F = ABS(MACHEP*TOT)
IF (EPS1 .LT. F) EPS1 = F
IF (DELTA .GT. EPS1) GO TO 190
IF (DELTA .LT. (-EPS1)) GO TO 1000
GO TO 300
C .......... REPLACE SMALL SUB-DIAGONAL SQUARES BY ZERO
C TO REDUCE THE INCIDENCE OF UNDERFLOWS ..........
190 IF (K .EQ. N) GO TO 210
K1 = K + 1
DO 200 J = K1, N
IF (BD(J) .LE. (MACHEP*(W(J)+W(J-1))) ** 2) BD(J) = 0.0E0
200 CONTINUE
C
210 F = BD(N) / DELTA
QP = DELTA + F
P = 1.0E0
IF (K .EQ. N) GO TO 260
K1 = N - K
C .......... FOR I=N-1 STEP -1 UNTIL K DO -- ..........
DO 240 II = 1, K1
I = N - II
Q = W(I) - S - F
R = Q / QP
P = P * R + 1.0E0
EP = F * R
W(I+1) = QP + EP
DELTA = Q - EP
IF (DELTA .GT. EPS1) GO TO 220
IF (DELTA .LT. (-EPS1)) GO TO 1000
GO TO 300
220 F = BD(I) / Q
QP = DELTA + F
BD(I+1) = QP * EP
240 CONTINUE
C
260 W(K) = QP
S = QP / P
IF (TOT + S .GT. TOT) GO TO 180
C .......... SET ERROR -- IRREGULAR END OF ITERATION.
C DEFLATE MINIMUM DIAGONAL ELEMENT ..........
IERR = 5 * N + K
S = 0.0E0
DELTA = QP
C
DO 280 J = K, N
IF (W(J) .GT. DELTA) GO TO 280
I = J
DELTA = W(J)
280 CONTINUE
C .......... CONVERGENCE ..........
300 IF (I .LT. N) BD(I+1) = BD(I) * F / QP
II = IND(I)
IF (I .EQ. K) GO TO 340
K1 = I - K
C .......... FOR J=I-1 STEP -1 UNTIL K DO -- ..........
DO 320 JJ = 1, K1
J = I - JJ
W(J+1) = W(J) - S
BD(J+1) = BD(J)
IND(J+1) = IND(J)
320 CONTINUE
C
340 W(K) = TOT
ERR = ERR + ABS(DELTA)
BD(K) = ERR
IND(K) = II
360 CONTINUE
C
IF (TYPE) GO TO 1001
F = BD(1)
E2(1) = 2.0E0
BD(1) = F
J = 2
C .......... NEGATE ELEMENTS OF W FOR LARGEST VALUES ..........
400 DO 500 I = 1, N
500 W(I) = -W(I)
C
JDEF = -JDEF
GO TO (40,1001), J
C .......... SET ERROR -- IDEF SPECIFIED INCORRECTLY ..........
1000 IERR = 6 * N + 1
1001 RETURN
END