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RATQR(N,EPS1,D,E,E2,M,W,IND,BD,TYPE,IDEF,IERR)

       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
       REAL D(N),E(N),E2(N),W(N),BD(N)
       REAL 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 C          AS NEGLIGIBLE, HAVE BEEN REPLACED BY ZERO CAUSING THE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 0.0E0 IF THE SMALLEST EIGENVALUES HAVE BEEN
 C          FOUND, AND TO 2.0E0 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
 C     ------------------------------------------------------------------
 C