.TH "SRC/shseqr.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
SRC/shseqr.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBshseqr\fP (job, compz, n, ilo, ihi, h, ldh, wr, wi, z, ldz, work, lwork, info)"
.br
.RI "\fBSHSEQR\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine shseqr (character job, character compz, integer n, integer ilo, integer ihi, real, dimension( ldh, * ) h, integer ldh, real, dimension( * ) wr, real, dimension( * ) wi, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer info)"

.PP
\fBSHSEQR\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!>    SHSEQR 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
\fIJOB\fP 
.PP
.nf
!>          JOB is CHARACTER*1
!>           = 'E':  compute eigenvalues only;
!>           = 'S':  compute eigenvalues and the Schur form T\&.
!> 
.fi
.PP
.br
\fICOMPZ\fP 
.PP
.nf
!>          COMPZ is CHARACTER*1
!>           = 'N':  no Schur vectors are computed;
!>           = 'I':  Z is initialized to the unit matrix and the matrix Z
!>                   of Schur vectors of H is returned;
!>           = 'V':  Z must contain an orthogonal matrix Q on entry, and
!>                   the product Q*Z is returned\&.
!> 
.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\&. ILO and IHI are normally
!>           set by a previous call to SGEBAL, and then passed to ZGEHRD
!>           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 JOB = 'S', 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 JOB = 'E', the
!>           contents of H are unspecified on exit\&.  (The output value of
!>           H when INFO > 0 is given under the description of INFO
!>           below\&.)
!>
!>           Unlike earlier versions of SHSEQR, this subroutine may
!>           explicitly 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 (N)
!> 
.fi
.PP
.br
\fIWI\fP 
.PP
.nf
!>          WI is REAL array, dimension (N)
!>
!>           The real and imaginary parts, respectively, of the computed
!>           eigenvalues\&. 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 JOB = 'S', 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
\fIZ\fP 
.PP
.nf
!>          Z is REAL array, dimension (LDZ,N)
!>           If COMPZ = 'N', Z is not referenced\&.
!>           If COMPZ = 'I', on entry Z need not be set and on exit,
!>           if INFO = 0, Z contains the orthogonal matrix Z of the Schur
!>           vectors of H\&.  If COMPZ = 'V', on entry Z must contain an
!>           N-by-N matrix Q, which is assumed to be equal to the unit
!>           matrix except for the submatrix Z(ILO:IHI,ILO:IHI)\&. On exit,
!>           if INFO = 0, Z contains Q*Z\&.
!>           Normally Q is the orthogonal matrix generated by SORGHR
!>           after the call to SGEHRD which formed the Hessenberg matrix
!>           H\&. (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 COMPZ = 'I' or
!>           COMPZ = 'V', then LDZ >= MAX(1,N)\&.  Otherwise, LDZ >= 1\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is REAL array, dimension (LWORK)
!>           On exit, if INFO = 0, 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 and delivers very good and sometimes
!>           optimal performance\&.  However, LWORK as large as 11*N
!>           may be required for optimal performance\&.  A workspace
!>           query is recommended to determine the optimal workspace
!>           size\&.
!>
!>           If LWORK = -1, then SHSEQR does a workspace query\&.
!>           In this case, SHSEQR 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, the i-th argument had an illegal
!>                    value
!>             > 0:  if INFO = i, SHSEQR 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 JOB = 'E', 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 JOB   = 'S', 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 COMPZ = 'V', then on exit
!>
!>                  (final value of Z)  =  (initial value of Z)*U
!>
!>                where U is the orthogonal matrix in (*) (regard-
!>                less of the value of JOB\&.)
!>
!>                If INFO > 0 and COMPZ = 'I', then on exit
!>                      (final value of Z)  = U
!>                where U is the orthogonal matrix in (*) (regard-
!>                less of the value of JOB\&.)
!>
!>                If INFO > 0 and COMPZ = 'N', 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
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>             Default values supplied by
!>             ILAENV(ISPEC,'SHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK)\&.
!>             It is suggested that these defaults be adjusted in order
!>             to attain best performance in each particular
!>             computational environment\&.
!>
!>            ISPEC=12: The SLAHQR vs SLAQR0 crossover point\&.
!>                      Default: 75\&. (Must be at least 11\&.)
!>
!>            ISPEC=13: Recommended deflation window size\&.
!>                      This depends on ILO, IHI and NS\&.  NS is the
!>                      number of simultaneous shifts returned
!>                      by ILAENV(ISPEC=15)\&.  (See ISPEC=15 below\&.)
!>                      The default for (IHI-ILO+1) <= 500 is NS\&.
!>                      The default for (IHI-ILO+1) >  500 is 3*NS/2\&.
!>
!>            ISPEC=14: Nibble crossover point\&. (See IPARMQ for
!>                      details\&.)  Default: 14% of deflation window
!>                      size\&.
!>
!>            ISPEC=15: Number of simultaneous shifts in a multishift
!>                      QR iteration\&.
!>
!>                      If IHI-ILO+1 is \&.\&.\&.
!>
!>                      greater than      \&.\&.\&.but less    \&.\&.\&. the
!>                      or equal to \&.\&.\&.      than        default is
!>
!>                           1               30          NS =   2(+)
!>                          30               60          NS =   4(+)
!>                          60              150          NS =  10(+)
!>                         150              590          NS =  **
!>                         590             3000          NS =  64
!>                        3000             6000          NS = 128
!>                        6000             infinity      NS = 256
!>
!>                  (+)  By default some or all matrices of this order
!>                       are passed to the implicit double shift routine
!>                       SLAHQR and this parameter is ignored\&.  See
!>                       ISPEC=12 above and comments in IPARMQ for
!>                       details\&.
!>
!>                 (**)  The asterisks (**) indicate an ad-hoc
!>                       function of N increasing from 10 to 64\&.
!>
!>            ISPEC=16: Select structured matrix multiply\&.
!>                      If the number of simultaneous shifts (specified
!>                      by ISPEC=15) is less than 14, then the default
!>                      for ISPEC=16 is 0\&.  Otherwise the default for
!>                      ISPEC=16 is 2\&.
!> 
.fi
.PP
 
.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 \fB314\fP of file \fBshseqr\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.