laqr4(3)

NAME
laqr4 - laqr4: eig of Hessenberg, step in hseqr

SYNOPSIS

  Functions

subroutine claqr4 (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, work, lwork, info)
CLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
subroutine dlaqr4 (wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, work, lwork, info)
DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
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.
subroutine zlaqr4 (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, work, lwork, info)
ZLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Detailed Description

Function Documentation

  subroutine claqr4 (logical wantt, logical wantz, integer n, integer ilo, integer ihi, complex, dimension( ldh, * ) h, integer ldh, complex, dimension( * ) w, integer iloz, integer ihiz, complex, dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work, integer lwork, integer info)

CLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Purpose:

    !>
    !>    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.
    !>

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 >= 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 > 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 > 0, then 1 <= ILO <= IHI <= 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 > 0 is given under the
    !>           description of INFO below.)
    !>
    !>           This subroutine may explicitly set H(i,j) = 0 for i > 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 >= 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 <= ILOZ <= ILO; IHI <= IHIZ <= 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 > 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 >= MAX(1,IHIZ).  Otherwise, LDZ >= 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 >= 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
    !>             > 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 > 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 > 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 > 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 > 0 and WANTZ is .FALSE., then Z is not
    !>                accessed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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.

  subroutine dlaqr4 (logical wantt, logical wantz, integer n, integer ilo, integer ihi, double precision, dimension( ldh, * ) h, integer ldh, double precision, dimension( * ) wr, double precision, dimension( * ) wi, integer iloz, integer ihiz, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer lwork, integer info)

DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Purpose:

    !>
    !>    DLAQR4 implements one level of recursion for DLAQR0.
    !>    It is a complete implementation of the small bulge multi-shift
    !>    QR algorithm.  It may be called by DLAQR0 and, for large enough
    !>    deflation window size, it may be called by DLAQR3.  This
    !>    subroutine is identical to DLAQR0 except that it calls DLAQR2
    !>    instead of DLAQR3.
    !>
    !>    DLAQR4 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 >= 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 > 1,
    !>           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
    !>           previous call to DGEBAL, and then passed to DGEHRD when the
    !>           matrix output by DGEBAL is reduced to Hessenberg form.
    !>           Otherwise, ILO and IHI should be set to 1 and N,
    !>           respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
    !>           If N = 0, then ILO = 1 and IHI = 0.
    !>

    H

    !>          H is DOUBLE PRECISION 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) < 0. If INFO = 0 and WANTT is
    !>           .FALSE., then the contents of H are unspecified on exit.
    !>           (The output value of H when INFO > 0 is given under the
    !>           description of INFO below.)
    !>
    !>           This subroutine may explicitly set H(i,j) = 0 for i > 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 >= max(1,N).
    !>

    WR

    !>          WR is DOUBLE PRECISION array, dimension (IHI)
    !>

    WI

    !>          WI is DOUBLE PRECISION 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) > 0 and WI(i+1) < 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 <= ILOZ <= ILO; IHI <= IHIZ <= N.
    !>

    Z

    !>          Z is DOUBLE PRECISION 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 > 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 >= MAX(1,IHIZ).  Otherwise, LDZ >= 1.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION 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 >= 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 DLAQR4 does a workspace query.
    !>           In this case, DLAQR4 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
    !>             > 0:  if INFO = i, DLAQR4 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 > 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 > 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 > 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 > 0 and WANTZ is .FALSE., then Z is not
    !>                accessed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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.

  subroutine slaqr4 (logical wantt, logical wantz, integer n, integer ilo, integer ihi, real, dimension( ldh, * ) h, integer ldh, real, dimension( * ) wr, real, dimension( * ) wi, integer iloz, integer ihiz, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer info)

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 >= 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 > 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 > 0, then 1 <= ILO <= IHI <= 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) < 0. If INFO = 0 and WANTT is
    !>           .FALSE., then the contents of H are unspecified on exit.
    !>           (The output value of H when INFO > 0 is given under the
    !>           description of INFO below.)
    !>
    !>           This subroutine may explicitly set H(i,j) = 0 for i > 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 >= 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) > 0 and WI(i+1) < 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 <= ILOZ <= ILO; IHI <= IHIZ <= 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 > 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 >= MAX(1,IHIZ).  Otherwise, LDZ >= 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 >= 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
    !> \verbatim
    !>          INFO is INTEGER
    !>             = 0:  successful exit
    !>             > 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 > 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 > 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 > 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 > 0 and WANTZ is .FALSE., then Z is not
    !>                accessed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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.

  subroutine zlaqr4 (logical wantt, logical wantz, integer n, integer ilo, integer ihi, complex*16, dimension( ldh, * ) h, integer ldh, complex*16, dimension( * ) w, integer iloz, integer ihiz, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork, integer info)

ZLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Purpose:

    !>
    !>    ZLAQR4 implements one level of recursion for ZLAQR0.
    !>    It is a complete implementation of the small bulge multi-shift
    !>    QR algorithm.  It may be called by ZLAQR0 and, for large enough
    !>    deflation window size, it may be called by ZLAQR3.  This
    !>    subroutine is identical to ZLAQR0 except that it calls ZLAQR2
    !>    instead of ZLAQR3.
    !>
    !>    ZLAQR4 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.
    !>

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 >= 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 > 1,
    !>           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
    !>           previous call to ZGEBAL, and then passed to ZGEHRD when the
    !>           matrix output by ZGEBAL is reduced to Hessenberg form.
    !>           Otherwise, ILO and IHI should be set to 1 and N,
    !>           respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
    !>           If N = 0, then ILO = 1 and IHI = 0.
    !>

    H

    !>          H is COMPLEX*16 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 > 0 is given under the
    !>           description of INFO below.)
    !>
    !>           This subroutine may explicitly set H(i,j) = 0 for i > 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 >= max(1,N).
    !>

    W

    !>          W is COMPLEX*16 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 <= ILOZ <= ILO; IHI <= IHIZ <= N.
    !>

    Z

    !>          Z is COMPLEX*16 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 > 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 >= MAX(1,IHIZ).  Otherwise, LDZ >= 1.
    !>

    WORK

    !>          WORK is COMPLEX*16 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 >= 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 ZLAQR4 does a workspace query.
    !>           In this case, ZLAQR4 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
    !>             > 0:  if INFO = i, ZLAQR4 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 > 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 > 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 > 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 > 0 and WANTZ is .FALSE., then Z is not
    !>                accessed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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.

Author

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