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SUBROUTINE HQR2(NM,N,LOW,IGH,H,WR,WI,Z,IERR)
C
INTEGER I,J,K,L,M,N,EN,II,JJ,LL,MM,NA,NM,NN,
X IGH,ITN,ITS,LOW,MP2,ENM2,IERR
REAL H(NM,N),WR(N),WI(N),Z(NM,N)
REAL P,Q,R,S,T,W,X,Y,RA,SA,VI,VR,ZZ,NORM,TST1,TST2
LOGICAL NOTLAS
C
C THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE HQR2,
C NUM. MATH. 16, 181-204(1970) BY PETERS AND WILKINSON.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 372-395(1971).
C
C THIS SUBROUTINE FINDS THE EIGENVALUES AND EIGENVECTORS
C OF A REAL UPPER HESSENBERG MATRIX BY THE QR METHOD. THE
C EIGENVECTORS OF A REAL GENERAL MATRIX CAN ALSO BE FOUND
C IF ELMHES AND ELTRAN OR ORTHES AND ORTRAN HAVE
C BEEN USED TO REDUCE THIS GENERAL MATRIX TO HESSENBERG FORM
C AND TO ACCUMULATE THE SIMILARITY TRANSFORMATIONS.
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.
C
C Z CONTAINS THE TRANSFORMATION MATRIX PRODUCED BY ELTRAN
C AFTER THE REDUCTION BY ELMHES, OR BY ORTRAN AFTER THE
C REDUCTION BY ORTHES, IF PERFORMED. IF THE EIGENVECTORS
C OF THE HESSENBERG MATRIX ARE DESIRED, Z MUST CONTAIN THE
C IDENTITY MATRIX.
C
C ON OUTPUT
C
C H HAS BEEN DESTROYED.
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 Z CONTAINS THE REAL AND IMAGINARY PARTS OF THE EIGENVECTORS.
C IF THE I-TH EIGENVALUE IS REAL, THE I-TH COLUMN OF Z
C CONTAINS ITS EIGENVECTOR. IF THE I-TH EIGENVALUE IS COMPLEX
C WITH POSITIVE IMAGINARY PART, THE I-TH AND (I+1)-TH
C COLUMNS OF Z CONTAIN THE REAL AND IMAGINARY PARTS OF ITS
C EIGENVECTOR. THE EIGENVECTORS ARE UNNORMALIZED. IF AN
C ERROR EXIT IS MADE, NONE OF THE EIGENVECTORS HAS BEEN FOUND.
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 CALLS CDIV FOR COMPLEX DIVISION.
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