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SUBROUTINE QZVAL(NM,N,A,B,ALFR,ALFI,BETA,MATZ,Z)
C
INTEGER I,J,N,EN,NA,NM,NN,ISW
DOUBLE PRECISION A(NM,N),B(NM,N),ALFR(N),ALFI(N),BETA(N),Z(NM,N)
DOUBLE PRECISION C,D,E,R,S,T,AN,A1,A2,BN,CQ,CZ,DI,DR,EI,TI,TR,U1,
X U2,V1,V2,A1I,A11,A12,A2I,A21,A22,B11,B12,B22,SQI,SQR,
X SSI,SSR,SZI,SZR,A11I,A11R,A12I,A12R,A22I,A22R,EPSB
LOGICAL MATZ
C
C THIS SUBROUTINE IS THE THIRD 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
C IN QUASI-TRIANGULAR FORM AND THE OTHER IN UPPER TRIANGULAR FORM.
C IT REDUCES THE QUASI-TRIANGULAR MATRIX FURTHER, SO THAT ANY
C REMAINING 2-BY-2 BLOCKS CORRESPOND TO PAIRS OF COMPLEX
C EIGENVALUES, AND RETURNS QUANTITIES WHOSE RATIOS GIVE THE
C GENERALIZED EIGENVALUES. IT IS USUALLY PRECEDED BY QZHES
C AND QZIT AND MAY BE FOLLOWED BY QZVEC.
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 MATZ SHOULD BE SET TO .TRUE. IF THE RIGHT HAND TRANSFORMATIONS
C ARE TO BE ACCUMULATED FOR LATER USE IN COMPUTING
C EIGENVECTORS, AND TO .FALSE. OTHERWISE.
C
C Z CONTAINS, IF MATZ HAS BEEN SET TO .TRUE., THE
C TRANSFORMATION MATRIX PRODUCED IN THE REDUCTIONS BY QZHES
C AND QZIT, IF PERFORMED, OR ELSE THE IDENTITY MATRIX.
C IF MATZ HAS BEEN SET TO .FALSE., Z IS NOT REFERENCED.
C
C ON OUTPUT
C
C A HAS BEEN REDUCED FURTHER TO A QUASI-TRIANGULAR MATRIX
C IN WHICH ALL NONZERO SUBDIAGONAL ELEMENTS CORRESPOND TO
C PAIRS OF COMPLEX EIGENVALUES.
C
C B IS STILL IN UPPER TRIANGULAR FORM, ALTHOUGH ITS ELEMENTS
C HAVE BEEN ALTERED. B(N,1) IS UNALTERED.
C
C ALFR AND ALFI CONTAIN THE REAL AND IMAGINARY PARTS OF THE
C DIAGONAL ELEMENTS OF THE TRIANGULAR MATRIX THAT WOULD BE
C OBTAINED IF A WERE REDUCED COMPLETELY TO TRIANGULAR FORM
C BY UNITARY TRANSFORMATIONS. NON-ZERO VALUES OF ALFI OCCUR
C IN PAIRS, THE FIRST MEMBER POSITIVE AND THE SECOND NEGATIVE.
C
C BETA CONTAINS THE DIAGONAL ELEMENTS OF THE CORRESPONDING B,
C NORMALIZED TO BE REAL AND NON-NEGATIVE. THE GENERALIZED
C EIGENVALUES ARE THEN THE RATIOS ((ALFR+I*ALFI)/BETA).
C
C Z CONTAINS THE PRODUCT OF THE RIGHT HAND TRANSFORMATIONS
C (FOR ALL THREE STEPS) IF MATZ HAS BEEN SET TO .TRUE.
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