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SUBROUTINE QZVEC(NM,N,A,B,ALFR,ALFI,BETA,Z)
C
INTEGER I,J,K,M,N,EN,II,JJ,NA,NM,NN,ISW,ENM2
DOUBLE PRECISION A(NM,N),B(NM,N),ALFR(N),ALFI(N),BETA(N),Z(NM,N)
DOUBLE PRECISION D,Q,R,S,T,W,X,Y,DI,DR,RA,RR,SA,TI,TR,T1,T2,W1,X1,
X ZZ,Z1,ALFM,ALMI,ALMR,BETM,EPSB
C
C THIS SUBROUTINE IS THE OPTIONAL FOURTH STEP OF THE QZ ALGORITHM
C FOR SOLVING GENERALIZED MATRIX EIGENVALUE PROBLEMS,
C SIAM J. NUMER. ANAL. 10, 241-256(1973) BY MOLER AND STEWART.
C
C THIS SUBROUTINE ACCEPTS A PAIR OF REAL MATRICES, ONE OF THEM IN
C QUASI-TRIANGULAR FORM (IN WHICH EACH 2-BY-2 BLOCK CORRESPONDS TO
C A PAIR OF COMPLEX EIGENVALUES) AND THE OTHER IN UPPER TRIANGULAR
C FORM. IT COMPUTES THE EIGENVECTORS OF THE TRIANGULAR PROBLEM AND
C TRANSFORMS THE RESULTS BACK TO THE ORIGINAL COORDINATE SYSTEM.
C IT IS USUALLY PRECEDED BY QZHES, QZIT, AND QZVAL.
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 MATRICES.
C
C A CONTAINS A REAL UPPER QUASI-TRIANGULAR MATRIX.
C
C B CONTAINS A REAL UPPER TRIANGULAR MATRIX. IN ADDITION,
C LOCATION B(N,1) CONTAINS THE TOLERANCE QUANTITY (EPSB)
C COMPUTED AND SAVED IN QZIT.
C
C ALFR, ALFI, AND BETA ARE VECTORS WITH COMPONENTS WHOSE
C RATIOS ((ALFR+I*ALFI)/BETA) ARE THE GENERALIZED
C EIGENVALUES. THEY ARE USUALLY OBTAINED FROM QZVAL.
C
C Z CONTAINS THE TRANSFORMATION MATRIX PRODUCED IN THE
C REDUCTIONS BY QZHES, QZIT, AND QZVAL, IF PERFORMED.
C IF THE EIGENVECTORS OF THE TRIANGULAR PROBLEM ARE
C DESIRED, Z MUST CONTAIN THE IDENTITY MATRIX.
C
C ON OUTPUT
C
C A IS UNALTERED. ITS SUBDIAGONAL ELEMENTS PROVIDE INFORMATION
C ABOUT THE STORAGE OF THE COMPLEX EIGENVECTORS.
C
C B HAS BEEN DESTROYED.
C
C ALFR, ALFI, AND BETA ARE UNALTERED.
C
C Z CONTAINS THE REAL AND IMAGINARY PARTS OF THE EIGENVECTORS.
C IF ALFI(I) .EQ. 0.0, THE I-TH EIGENVALUE IS REAL AND
C THE I-TH COLUMN OF Z CONTAINS ITS EIGENVECTOR.
C IF ALFI(I) .NE. 0.0, THE I-TH EIGENVALUE IS COMPLEX.
C IF ALFI(I) .GT. 0.0, THE EIGENVALUE IS THE FIRST OF
C A COMPLEX PAIR AND THE I-TH AND (I+1)-TH COLUMNS
C OF Z CONTAIN ITS EIGENVECTOR.
C IF ALFI(I) .LT. 0.0, THE EIGENVALUE IS THE SECOND OF
C A COMPLEX PAIR AND THE (I-1)-TH AND I-TH COLUMNS
C OF Z CONTAIN THE CONJUGATE OF ITS EIGENVECTOR.
C EACH EIGENVECTOR IS NORMALIZED SO THAT THE MODULUS
C OF ITS LARGEST COMPONENT IS 1.0 .
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