comqr2.f
SUBROUTINE COMQR2 (NM, N, LOW, IGH, ORTR, ORTI, HR, HI, WR, WI,
+ ZR, ZI, IERR)
C***BEGIN PROLOGUE COMQR2
C***PURPOSE Compute the eigenvalues and eigenvectors of a complex upper
C Hessenberg matrix.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C2B
C***TYPE COMPLEX (HQR2-S, COMQR2-C)
C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is a translation of a unitary analogue of the
C ALGOL procedure COMLR2, NUM. MATH. 16, 181-204(1970) by Peters
C and Wilkinson.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 372-395(1971).
C The unitary analogue substitutes the QR algorithm of Francis
C (COMP. JOUR. 4, 332-345(1962)) for the LR algorithm.
C
C This subroutine finds the eigenvalues and eigenvectors
C of a COMPLEX UPPER Hessenberg matrix by the QR
C method. The eigenvectors of a COMPLEX GENERAL matrix
C can also be found if CORTH has been used to reduce
C this general matrix to Hessenberg form.
C
C On INPUT
C
C NM must be set to the row dimension of the two-dimensional
C array parameters, HR, HI, ZR, and ZI, as declared in the
C calling program dimension statement. NM is an INTEGER
C variable.
C
C N is the order of the matrix H=(HR,HI). N is an INTEGER
C variable. N must be less than or equal to NM.
C
C LOW and IGH are two INTEGER variables determined by the
C balancing subroutine CBAL. If CBAL has not been used,
C set LOW=1 and IGH equal to the order of the matrix, N.
C
C ORTR and ORTI contain information about the unitary trans-
C formations used in the reduction by CORTH, if performed.
C Only elements LOW through IGH are used. If the eigenvectors
C of the Hessenberg matrix are desired, set ORTR(J) and
C ORTI(J) to 0.0E0 for these elements. ORTR and ORTI are
C one-dimensional REAL arrays, dimensioned ORTR(IGH) and
C ORTI(IGH).
C
C HR and HI contain the real and imaginary parts, respectively,
C of the complex upper Hessenberg matrix. Their lower
C triangles below the subdiagonal contain information about
C the unitary transformations used in the reduction by CORTH,
C if performed. If the eigenvectors of the Hessenberg matrix
C are desired, these elements may be arbitrary. HR and HI
C are two-dimensional REAL arrays, dimensioned HR(NM,N) and
C HI(NM,N).
C
C On OUTPUT
C
C ORTR, ORTI, and the upper Hessenberg portions of HR and HI
C have been destroyed.
C
C WR and WI contain the real and imaginary parts, respectively,
C of the eigenvalues of the upper Hessenberg matrix. If an
C error exit is made, the eigenvalues should be correct for
C indices IERR+1, IERR+2, ..., N. WR and WI are one-
C dimensional REAL arrays, dimensioned WR(N) and WI(N).
C
C ZR and ZI contain the real and imaginary parts, respectively,
C of the eigenvectors. The eigenvectors are unnormalized.
C If an error exit is made, none of the eigenvectors has been
C found. ZR and ZI are two-dimensional REAL arrays,
C dimensioned ZR(NM,N) and ZI(NM,N).
C
C IERR is an INTEGER flag set to
C Zero for normal return,
C J if the J-th eigenvalue has not been
C determined after a total of 30*N iterations.
C The eigenvalues should be correct for indices
C IERR+1, IERR+2, ..., N, but no eigenvectors are
C computed.
C
C Calls CSROOT for complex square root.
C Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
C Calls CDIV for complex division.
C
C Questions and comments should be directed to B. S. Garbow,
C APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
C ------------------------------------------------------------------
C
C***REFERENCES B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
C Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
C system Routines - EISPACK Guide, Springer-Verlag,
C 1976.
C***ROUTINES CALLED CDIV, CSROOT, PYTHAG
C***REVISION HISTORY (YYMMDD)
C 760101 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890831 Modified array declarations. (WRB)
C 890831 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE COMQR2