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SUBROUTINE RATQR(N,EPS1,D,E,E2,M,W,IND,BD,TYPE,IDEF,IERR)
C
INTEGER I,J,K,M,N,II,JJ,K1,IDEF,IERR,JDEF
DOUBLE PRECISION D(N),E(N),E2(N),W(N),BD(N)
DOUBLE PRECISION F,P,Q,R,S,EP,QP,ERR,TOT,EPS1,DELTA,EPSLON
INTEGER IND(N)
LOGICAL TYPE
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.
C
C EPS1 IS A THEORETICAL ABSOLUTE ERROR TOLERANCE FOR THE
C COMPUTED EIGENVALUES. IF THE INPUT EPS1 IS NON-POSITIVE,
C OR INDEED SMALLER THAN ITS DEFAULT VALUE, IT IS RESET
C AT EACH ITERATION TO THE RESPECTIVE DEFAULT VALUE,
C NAMELY, THE PRODUCT OF THE RELATIVE MACHINE PRECISION
C AND THE MAGNITUDE OF THE CURRENT EIGENVALUE ITERATE.
C THE THEORETICAL ABSOLUTE ERROR IN THE K-TH EIGENVALUE
C IS USUALLY NOT GREATER THAN K TIMES EPS1.
C
C D CONTAINS THE DIAGONAL ELEMENTS OF THE INPUT MATRIX.
C
C E CONTAINS THE SUBDIAGONAL ELEMENTS OF THE INPUT MATRIX
C IN ITS LAST N-1 POSITIONS. E(1) IS ARBITRARY.
C
C E2 CONTAINS THE SQUARES OF THE CORRESPONDING ELEMENTS OF E.
C E2(1) IS ARBITRARY.
C
C M IS THE NUMBER OF EIGENVALUES TO BE FOUND.
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.
C
C TYPE SHOULD BE SET TO .TRUE. IF THE SMALLEST EIGENVALUES
C ARE TO BE FOUND, AND TO .FALSE. IF THE LARGEST EIGENVALUES
C ARE TO BE 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 (UNLESS W OVERWRITES D).
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C AS NEGLIGIBLE, HAVE BEEN REPLACED BY ZERO CAUSING THE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 SET TO 0.0D0 IF THE SMALLEST EIGENVALUES HAVE BEEN
C FOUND, AND TO 2.0D0 IF THE LARGEST EIGENVALUES HAVE BEEN
C FOUND. E2 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
C DESCENDING ORDER. IF AN ERROR EXIT IS MADE BECAUSE OF
C AN INCORRECT SPECIFICATION OF IDEF, NO EIGENVALUES
C ARE FOUND. IF THE NEWTON ITERATES FOR A PARTICULAR
C EIGENVALUE ARE NOT MONOTONE, THE BEST ESTIMATE OBTAINED
C IS RETURNED AND IERR IS SET. W MAY COINCIDE WITH 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
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 MAY COINCIDE
C WITH E2.
C
C IERR IS 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 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 BURTON S. GARBOW,
C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C THIS VERSION DATED AUGUST 1983.
C
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C