CLAQR4 implements one level of recursion for CLAQR0.


    It is a complete implementation of the small bulge multi-shift


    QR algorithm.  It may be called by CLAQR0 and, for large enough


    deflation window size, it may be called by CLAQR3.  This


    subroutine is identical to CLAQR0 except that it calls CLAQR2


    instead of CLAQR3.


    CLAQR4 computes the eigenvalues of a Hessenberg matrix H


    and, optionally, the matrices T and Z from the Schur decomposition


    H = Z T Z**H, where T is an upper triangular matrix (the


    Schur form), and Z is the unitary matrix of Schur vectors.


    Optionally Z may be postmultiplied into an input unitary


    matrix Q so that this routine can give the Schur factorization


    of a matrix A which has been reduced to the Hessenberg form H


    by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.

WANTT

          WANTT is LOGICAL


          = .TRUE. : the full Schur form T is required;


          = .FALSE.: only eigenvalues are required.

WANTZ

          WANTZ is LOGICAL


          = .TRUE. : the matrix of Schur vectors Z is required;


          = .FALSE.: Schur vectors are not required.

N

          N is INTEGER


           The order of the matrix H.  N .GE. 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER


           It is assumed that H is already upper triangular in rows


           and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1,


           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a


           previous call to CGEBAL, and then passed to CGEHRD when the


           matrix output by CGEBAL is reduced to Hessenberg form.


           Otherwise, ILO and IHI should be set to 1 and N,


           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.


           If N = 0, then ILO = 1 and IHI = 0.

H

          H is COMPLEX array, dimension (LDH,N)


           On entry, the upper Hessenberg matrix H.


           On exit, if INFO = 0 and WANTT is .TRUE., then H


           contains the upper triangular matrix T from the Schur


           decomposition (the Schur form). If INFO = 0 and WANT is


           .FALSE., then the contents of H are unspecified on exit.


           (The output value of H when INFO.GT.0 is given under the


           description of INFO below.)


           This subroutine may explicitly set H(i,j) = 0 for i.GT.j and


           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

LDH

          LDH is INTEGER


           The leading dimension of the array H. LDH .GE. max(1,N).

W

          W is COMPLEX array, dimension (N)


           The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored


           in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are


           stored in the same order as on the diagonal of the Schur


           form returned in H, with W(i) = H(i,i).

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER


           Specify the rows of Z to which transformations must be


           applied if WANTZ is .TRUE..


           1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.

Z

          Z is COMPLEX array, dimension (LDZ,IHI)


           If WANTZ is .FALSE., then Z is not referenced.


           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is


           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the


           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).


           (The output value of Z when INFO.GT.0 is given under


           the description of INFO below.)

LDZ

          LDZ is INTEGER


           The leading dimension of the array Z.  if WANTZ is .TRUE.


           then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.

WORK

          WORK is COMPLEX array, dimension LWORK


           On exit, if LWORK = -1, WORK(1) returns an estimate of


           the optimal value for LWORK.

LWORK

          LWORK is INTEGER


           The dimension of the array WORK.  LWORK .GE. max(1,N)


           is sufficient, but LWORK typically as large as 6*N may


           be required for optimal performance.  A workspace query


           to determine the optimal workspace size is recommended.


           If LWORK = -1, then CLAQR4 does a workspace query.


           In this case, CLAQR4 checks the input parameters and


           estimates the optimal workspace size for the given


           values of N, ILO and IHI.  The estimate is returned


           in WORK(1).  No error message related to LWORK is


           issued by XERBLA.  Neither H nor Z are accessed.

INFO

          INFO is INTEGER


             =  0:  successful exit


           .GT. 0:  if INFO = i, CLAQR4 failed to compute all of


                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR


                and WI contain those eigenvalues which have been


                successfully computed.  (Failures are rare.)


                If INFO .GT. 0 and WANT is .FALSE., then on exit,


                the remaining unconverged eigenvalues are the eigen-


                values of the upper Hessenberg matrix rows and


                columns ILO through INFO of the final, output


                value of H.


                If INFO .GT. 0 and WANTT is .TRUE., then on exit


           (*)  (initial value of H)*U  = U*(final value of H)


                where U is a unitary matrix.  The final


                value of  H is upper Hessenberg and triangular in


                rows and columns INFO+1 through IHI.


                If INFO .GT. 0 and WANTZ is .TRUE., then on exit


                  (final value of Z(ILO:IHI,ILOZ:IHIZ)


                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U


                where U is the unitary matrix in (*) (regard-


                less of the value of WANTT.)


                If INFO .GT. 0 and WANTZ is .FALSE., then Z is not


                accessed.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

June 2017

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.
K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.