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SUBROUTINE HQR(NM,N,LOW,IGH,H,WR,WI,IERR)
C
INTEGER I,J,K,L,M,N,EN,LL,MM,NA,NM,IGH,ITN,ITS,LOW,MP2,ENM2,IERR
DOUBLE PRECISION H(NM,N),WR(N),WI(N)
DOUBLE PRECISION P,Q,R,S,T,W,X,Y,ZZ,NORM,TST1,TST2
LOGICAL NOTLAS
C
C THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE HQR,
C NUM. MATH. 14, 219-231(1970) BY MARTIN, PETERS, AND WILKINSON.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 359-371(1971).
C
C THIS SUBROUTINE FINDS THE EIGENVALUES OF A REAL
C UPPER HESSENBERG MATRIX BY THE QR METHOD.
C
C ON INPUT
C
C NM MUST BE SET TO THE ROW DIMENSION OF TWO-DIMENSIONAL
C ARRAY PARAMETERS AS DECLARED IN THE CALLING PROGRAM
C DIMENSION STATEMENT.
C
C N IS THE ORDER OF THE MATRIX.
C
C LOW AND IGH ARE INTEGERS DETERMINED BY THE BALANCING
C SUBROUTINE BALANC. IF BALANC HAS NOT BEEN USED,
C SET LOW=1, IGH=N.
C
C H CONTAINS THE UPPER HESSENBERG MATRIX. INFORMATION ABOUT
C THE TRANSFORMATIONS USED IN THE REDUCTION TO HESSENBERG
C FORM BY ELMHES OR ORTHES, IF PERFORMED, IS STORED
C IN THE REMAINING TRIANGLE UNDER THE HESSENBERG MATRIX.
C
C ON OUTPUT
C
C H HAS BEEN DESTROYED. THEREFORE, IT MUST BE SAVED
C BEFORE CALLING HQR IF SUBSEQUENT CALCULATION AND
C BACK TRANSFORMATION OF EIGENVECTORS IS TO BE PERFORMED.
C
C WR AND WI CONTAIN THE REAL AND IMAGINARY PARTS,
C RESPECTIVELY, OF THE EIGENVALUES. THE EIGENVALUES
C ARE UNORDERED EXCEPT THAT COMPLEX CONJUGATE PAIRS
C OF VALUES APPEAR CONSECUTIVELY WITH THE EIGENVALUE
C HAVING THE POSITIVE IMAGINARY PART FIRST. IF AN
C ERROR EXIT IS MADE, THE EIGENVALUES SHOULD BE CORRECT
C FOR INDICES IERR+1,...,N.
C
C IERR IS SET TO
C ZERO FOR NORMAL RETURN,
C J IF THE LIMIT OF 30*N ITERATIONS IS EXHAUSTED
C WHILE THE J-TH EIGENVALUE IS BEING SOUGHT.
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