.TH "laqr0" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
laqr0 \- laqr0: eig of Hessenberg, step in hseqr
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBclaqr0\fP (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, work, lwork, info)"
.br
.RI "\fBCLAQR0\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&. "
.ti -1c
.RI "subroutine \fBdlaqr0\fP (wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, work, lwork, info)"
.br
.RI "\fBDLAQR0\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&. "
.ti -1c
.RI "subroutine \fBslaqr0\fP (wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, work, lwork, info)"
.br
.RI "\fBSLAQR0\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&. "
.ti -1c
.RI "subroutine \fBzlaqr0\fP (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, work, lwork, info)"
.br
.RI "\fBZLAQR0\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&. "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine claqr0 (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)"

.PP
\fBCLAQR0\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!>    CLAQR0 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIWANTT\fP 
.PP
.nf
!>          WANTT is LOGICAL
!>          = \&.TRUE\&. : the full Schur form T is required;
!>          = \&.FALSE\&.: only eigenvalues are required\&.
!> 
.fi
.PP
.br
\fIWANTZ\fP 
.PP
.nf
!>          WANTZ is LOGICAL
!>          = \&.TRUE\&. : the matrix of Schur vectors Z is required;
!>          = \&.FALSE\&.: Schur vectors are not required\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>           The order of the matrix H\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
!>          ILO is INTEGER
!> 
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIH\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDH\fP 
.PP
.nf
!>          LDH is INTEGER
!>           The leading dimension of the array H\&. LDH >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIW\fP 
.PP
.nf
!>          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)\&.
!> 
.fi
.PP
.br
\fIILOZ\fP 
.PP
.nf
!>          ILOZ is INTEGER
!> 
.fi
.PP
.br
\fIIHIZ\fP 
.PP
.nf
!>          IHIZ is INTEGER
!>           Specify the rows of Z to which transformations must be
!>           applied if WANTZ is \&.TRUE\&.\&.
!>           1 <= ILOZ <= ILO; IHI <= IHIZ <= N\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          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\&.)
!> 
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
!>          LDZ is INTEGER
!>           The leading dimension of the array Z\&.  if WANTZ is \&.TRUE\&.
!>           then LDZ >= MAX(1,IHIZ)\&.  Otherwise, LDZ >= 1\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is COMPLEX array, dimension LWORK
!>           On exit, if LWORK = -1, WORK(1) returns an estimate of
!>           the optimal value for LWORK\&.
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          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 CLAQR0 does a workspace query\&.
!>           In this case, CLAQR0 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\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>             = 0:  successful exit
!>             > 0:  if INFO = i, CLAQR0 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA 
.RE
.PP
\fBReferences:\fP
.RS 4

.PP
.nf
 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\&.

.fi
.PP
 
.br
 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\&. 
.RE
.PP

.PP
Definition at line \fB238\fP of file \fBclaqr0\&.f\fP\&.
.SS "subroutine dlaqr0 (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)"

.PP
\fBDLAQR0\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!>    DLAQR0 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIWANTT\fP 
.PP
.nf
!>          WANTT is LOGICAL
!>          = \&.TRUE\&. : the full Schur form T is required;
!>          = \&.FALSE\&.: only eigenvalues are required\&.
!> 
.fi
.PP
.br
\fIWANTZ\fP 
.PP
.nf
!>          WANTZ is LOGICAL
!>          = \&.TRUE\&. : the matrix of Schur vectors Z is required;
!>          = \&.FALSE\&.: Schur vectors are not required\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>           The order of the matrix H\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
!>          ILO is INTEGER
!> 
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIH\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDH\fP 
.PP
.nf
!>          LDH is INTEGER
!>           The leading dimension of the array H\&. LDH >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIWR\fP 
.PP
.nf
!>          WR is DOUBLE PRECISION array, dimension (IHI)
!> 
.fi
.PP
.br
\fIWI\fP 
.PP
.nf
!>          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)\&.
!> 
.fi
.PP
.br
\fIILOZ\fP 
.PP
.nf
!>          ILOZ is INTEGER
!> 
.fi
.PP
.br
\fIIHIZ\fP 
.PP
.nf
!>          IHIZ is INTEGER
!>           Specify the rows of Z to which transformations must be
!>           applied if WANTZ is \&.TRUE\&.\&.
!>           1 <= ILOZ <= ILO; IHI <= IHIZ <= N\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          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\&.)
!> 
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
!>          LDZ is INTEGER
!>           The leading dimension of the array Z\&.  if WANTZ is \&.TRUE\&.
!>           then LDZ >= MAX(1,IHIZ)\&.  Otherwise, LDZ >= 1\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is DOUBLE PRECISION array, dimension LWORK
!>           On exit, if LWORK = -1, WORK(1) returns an estimate of
!>           the optimal value for LWORK\&.
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          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 DLAQR0 does a workspace query\&.
!>           In this case, DLAQR0 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\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>             = 0:  successful exit
!>             > 0:  if INFO = i, DLAQR0 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 an orthogonal matrix\&.  The final
!>                value of H is upper Hessenberg and quasi-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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBContributors:\fP
.RS 4
Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA 
.RE
.PP
\fBReferences:\fP
.RS 4

.PP
.nf
 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\&.

.fi
.PP
 
.br
 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\&. 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP

.PP
Definition at line \fB254\fP of file \fBdlaqr0\&.f\fP\&.
.SS "subroutine slaqr0 (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)"

.PP
\fBSLAQR0\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!>    SLAQR0 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIWANTT\fP 
.PP
.nf
!>          WANTT is LOGICAL
!>          = \&.TRUE\&. : the full Schur form T is required;
!>          = \&.FALSE\&.: only eigenvalues are required\&.
!> 
.fi
.PP
.br
\fIWANTZ\fP 
.PP
.nf
!>          WANTZ is LOGICAL
!>          = \&.TRUE\&. : the matrix of Schur vectors Z is required;
!>          = \&.FALSE\&.: Schur vectors are not required\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>           The order of the matrix H\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
!>          ILO is INTEGER
!> 
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIH\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDH\fP 
.PP
.nf
!>          LDH is INTEGER
!>           The leading dimension of the array H\&. LDH >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIWR\fP 
.PP
.nf
!>          WR is REAL array, dimension (IHI)
!> 
.fi
.PP
.br
\fIWI\fP 
.PP
.nf
!>          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)\&.
!> 
.fi
.PP
.br
\fIILOZ\fP 
.PP
.nf
!>          ILOZ is INTEGER
!> 
.fi
.PP
.br
\fIIHIZ\fP 
.PP
.nf
!>          IHIZ is INTEGER
!>           Specify the rows of Z to which transformations must be
!>           applied if WANTZ is \&.TRUE\&.\&.
!>           1 <= ILOZ <= ILO; IHI <= IHIZ <= N\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          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\&.)
!> 
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
!>          LDZ is INTEGER
!>           The leading dimension of the array Z\&.  if WANTZ is \&.TRUE\&.
!>           then LDZ >= MAX(1,IHIZ)\&.  Otherwise, LDZ >= 1\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is REAL array, dimension LWORK
!>           On exit, if LWORK = -1, WORK(1) returns an estimate of
!>           the optimal value for LWORK\&.
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          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 SLAQR0 does a workspace query\&.
!>           In this case, SLAQR0 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\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>             = 0:  successful exit
!>             > 0:  if INFO = i, SLAQR0 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 an orthogonal matrix\&.  The final
!>                value of H is upper Hessenberg and quasi-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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA 
.RE
.PP
\fBReferences:\fP
.RS 4

.PP
.nf
 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\&.

.fi
.PP
 
.br
 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\&. 
.RE
.PP

.PP
Definition at line \fB254\fP of file \fBslaqr0\&.f\fP\&.
.SS "subroutine zlaqr0 (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)"

.PP
\fBZLAQR0\fP computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!>    ZLAQR0 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIWANTT\fP 
.PP
.nf
!>          WANTT is LOGICAL
!>          = \&.TRUE\&. : the full Schur form T is required;
!>          = \&.FALSE\&.: only eigenvalues are required\&.
!> 
.fi
.PP
.br
\fIWANTZ\fP 
.PP
.nf
!>          WANTZ is LOGICAL
!>          = \&.TRUE\&. : the matrix of Schur vectors Z is required;
!>          = \&.FALSE\&.: Schur vectors are not required\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>           The order of the matrix H\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
!>          ILO is INTEGER
!> 
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIH\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDH\fP 
.PP
.nf
!>          LDH is INTEGER
!>           The leading dimension of the array H\&. LDH >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIW\fP 
.PP
.nf
!>          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)\&.
!> 
.fi
.PP
.br
\fIILOZ\fP 
.PP
.nf
!>          ILOZ is INTEGER
!> 
.fi
.PP
.br
\fIIHIZ\fP 
.PP
.nf
!>          IHIZ is INTEGER
!>           Specify the rows of Z to which transformations must be
!>           applied if WANTZ is \&.TRUE\&.\&.
!>           1 <= ILOZ <= ILO; IHI <= IHIZ <= N\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          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\&.)
!> 
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
!>          LDZ is INTEGER
!>           The leading dimension of the array Z\&.  if WANTZ is \&.TRUE\&.
!>           then LDZ >= MAX(1,IHIZ)\&.  Otherwise, LDZ >= 1\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is COMPLEX*16 array, dimension LWORK
!>           On exit, if LWORK = -1, WORK(1) returns an estimate of
!>           the optimal value for LWORK\&.
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          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 ZLAQR0 does a workspace query\&.
!>           In this case, ZLAQR0 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\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>             = 0:  successful exit
!>             > 0:  if INFO = i, ZLAQR0 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBContributors:\fP
.RS 4
Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA 
.RE
.PP
\fBReferences:\fP
.RS 4

.PP
.nf
 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\&.

.fi
.PP
 
.br
 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\&. 
.RE
.PP

.PP
Definition at line \fB239\fP of file \fBzlaqr0\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.