.TH "slahqr.f" 3 "Tue Nov 14 2017" "Version 3.8.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
slahqr.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBslahqr\fP (WANTT, WANTZ, \fBN\fP, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, INFO)"
.br
.RI "\fBSLAHQR\fP computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm\&. "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine slahqr (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, integer INFO)"

.PP
\fBSLAHQR\fP computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm\&.  
.PP
\fBPurpose: \fP
.RS 4

.PP
.nf
    SLAHQR is an auxiliary routine called by SHSEQR to update the
    eigenvalues and Schur decomposition already computed by SHSEQR, by
    dealing with the Hessenberg submatrix in rows and columns ILO to
    IHI.
.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 quasi-triangular in
          rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
          ILO = 1). SLAHQR works primarily with the Hessenberg
          submatrix in rows and columns ILO to IHI, but applies
          transformations to all of H if WANTT is .TRUE..
          1 <= ILO <= max(1,IHI); IHI <= N.
.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 is zero and if WANTT is .TRUE., H is upper
          quasi-triangular in rows and columns ILO:IHI, with any
          2-by-2 diagonal blocks in standard form. If INFO is zero
          and WANTT is .FALSE., the contents of H are unspecified on
          exit.  The output state of H if INFO is nonzero is given
          below under the description of INFO.
.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 (N)
.fi
.PP
.br
\fIWI\fP 
.PP
.nf
          WI is REAL array, dimension (N)
          The real and imaginary parts, respectively, of the computed
          eigenvalues ILO to IHI are stored in the corresponding
          elements of WR and WI. 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., 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,N)
          If WANTZ is .TRUE., on entry Z must contain the current
          matrix Z of transformations accumulated by SHSEQR, and on
          exit Z has been updated; transformations are applied only to
          the submatrix Z(ILOZ:IHIZ,ILO:IHI).
          If WANTZ is .FALSE., Z is not referenced.
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
          LDZ is INTEGER
          The leading dimension of the array Z. LDZ >= max(1,N).
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
           =   0: successful exit
          .GT. 0: If INFO = i, SLAHQR failed to compute all the
                  eigenvalues ILO to IHI in a total of 30 iterations
                  per eigenvalue; elements i+1:ihi of WR and WI
                  contain those eigenvalues which have been
                  successfully computed.

                  If INFO .GT. 0 and WANTT is .FALSE., then on exit,
                  the remaining unconverged eigenvalues are the
                  eigenvalues of the upper Hessenberg matrix rows
                  and columns ILO thorugh 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 an orthognal 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)  = (initial value of Z)*U
                  where U is the orthogonal matrix in (*)
                  (regardless of the value of WANTT.)
.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
\fBDate:\fP
.RS 4
December 2016 
.RE
.PP
\fBFurther Details: \fP
.RS 4

.PP
.nf
     02-96 Based on modifications by
     David Day, Sandia National Laboratory, USA

     12-04 Further modifications by
     Ralph Byers, University of Kansas, USA
     This is a modified version of SLAHQR from LAPACK version 3.0.
     It is (1) more robust against overflow and underflow and
     (2) adopts the more conservative Ahues & Tisseur stopping
     criterion (LAWN 122, 1997).
.fi
.PP
 
.RE
.PP

.PP
Definition at line 209 of file slahqr\&.f\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.
