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NAME

       slaqr4.f -

SYNOPSIS

   Functions/Subroutines
       subroutine slaqr4 (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK,
           LWORK, INFO)
           SLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices
           from the Schur decomposition.

Function/Subroutine Documentation

   subroutine slaqr4 (logicalWANTT, logicalWANTZ, integerN, integerILO, integerIHI, real,
       dimension( ldh, * )H, integerLDH, real, dimension( * )WR, real, dimension( * )WI,
       integerILOZ, integerIHIZ, real, dimension( ldz, * )Z, integerLDZ, real, dimension( *
       )WORK, integerLWORK, integerINFO)
       SLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from
       the Schur decomposition.

       Purpose:

               SLAQR4 implements one level of recursion for SLAQR0.
               It is a complete implementation of the small bulge multi-shift
               QR algorithm.  It may be called by SLAQR0 and, for large enough
               deflation window size, it may be called by SLAQR3.  This
               subroutine is identical to SLAQR0 except that it calls SLAQR2
               instead of SLAQR3.

               SLAQR4 computes the eigenvalues of a Hessenberg matrix H
               and, optionally, the matrices T and Z from the Schur decomposition
               H = Z T Z**T, where T is an upper quasi-triangular matrix (the
               Schur form), and Z is the orthogonal matrix of Schur vectors.

               Optionally Z may be postmultiplied into an input orthogonal
               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 orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

       Parameters:
           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 SGEBAL, and then passed to SGEHRD when the
                      matrix output by SGEBAL 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 REAL 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 quasi-triangular matrix T from the Schur
                      decomposition (the Schur form); 2-by-2 diagonal blocks
                      (corresponding to complex conjugate pairs of eigenvalues)
                      are returned in standard form, with H(i,i) = H(i+1,i+1)
                      and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT 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).

           WR

                     WR is REAL array, dimension (IHI)

           WI

                     WI is REAL array, dimension (IHI)
                      The real and imaginary parts, respectively, of the computed
                      eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
                      and WI(ILO:IHI). If two eigenvalues are computed as a
                      complex conjugate pair, they are stored in consecutive
                      elements of WR and WI, say the i-th and (i+1)th, with
                      WI(i) .GT. 0 and WI(i+1) .LT. 0. 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
                      WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
                      block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
                      WI(i+1) = -WI(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 REAL 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 REAL 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 SLAQR4 does a workspace query.
                      In this case, SLAQR4 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
            batim
                     INFO is INTEGER
                        =  0:  successful exit
                      .GT. 0:  if INFO = i, SLAQR4 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 orthogonal 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 orthogonal matrix in (*) (regard-
                           less of the value of WANTT.)

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

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           September 2012

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

       References:
           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.

       Definition at line 265 of file slaqr4.f.

Author

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