\brief \b ESMF_DHSEQR \htmlonly Download ESMF_DHSEQR + dependencies [TGZ] [ZIP] [TXT] \endhtmlonly \par Purpose:
\verbatim
ESMF_DHSEQR 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 = QHQT = (QZ)T(QZ)T. \endverbatim \param[in] JOB \verbatim JOB is CHARACTER*1 = ‘E’: compute eigenvalues only; = ‘S’: compute eigenvalues and the Schur form T. \endverbatim
\param[in] COMPZ \verbatim COMPZ is CHARACTER1 = ‘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 QZ is returned. \endverbatim
\param[in] N \verbatim N is INTEGER The order of the matrix H. N .GE. 0. \endverbatim
\param[in] ILO \verbatim ILO is INTEGER \endverbatim
\param[in] IHI \verbatim 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 ESMF_DGEBAL, and then passed to ZGEHRD
when the matrix output by ESMF_DGEBAL 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.
\endverbatim
\param[in,out] H \verbatim H is DOUBLE PRECISION 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).LT.0. If INFO = 0 and JOB = ‘E’, 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.)
Unlike earlier versions of ESMF_DHSEQR, this subroutine may
explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1
or j = IHI+1, IHI+2, ... N.
\endverbatim
\param[in] LDH \verbatim LDH is INTEGER The leading dimension of the array H. LDH .GE. max(1,N). \endverbatim
\param[out] WR \verbatim WR is DOUBLE PRECISION array, dimension (N) \endverbatim
\param[out] WI \verbatim WI is DOUBLE PRECISION 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) .GT. 0 and
WI(i+1) .LT. 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).
\endverbatim
\param[in,out] Z \verbatim Z is DOUBLE PRECISION 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 ESMF_DORGHR after the call to ESMF_DGEHRD which formed the Hessenberg matrix H. (The output value of Z when INFO.GT.0 is given under the description of INFO below.) \endverbatim
\param[in] LDZ \verbatim LDZ is INTEGER The leading dimension of the array Z. if COMPZ = ‘I’ or COMPZ = ‘V’, then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1. \endverbatim
\param[out] WORK \verbatim WORK is DOUBLE PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns an estimate of the optimal value for LWORK. \endverbatim
\param[in] LWORK \verbatim LWORK is INTEGER The dimension of the array WORK. LWORK .GE. 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 ESMF_DHSEQR does a workspace query.
In this case, ESMF_DHSEQR 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 ESMF_XERBLA. Neither H nor Z are accessed.
\endverbatim
\param[out] INFO \verbatim INFO is INTEGER = 0: successful exit .LT. 0: if INFO = -i, the i-th argument had an illegal value .GT. 0: if INFO = i, ESMF_DHSEQR 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 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 .GT. 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 .GT. 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 .GT. 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 .GT. 0 and COMPZ = 'N', then Z is not
accessed.
\endverbatim \author Univ. of Tennessee \author Univ. of California Berkeley \author Univ. of Colorado Denver \author NAG Ltd. \date December 2016 \ingroup doubleOTHERcomputational \par Contributors:
Karen Braman and Ralph Byers, Department of Mathematics,
University of Kansas, USA
\par Further Details:
\verbatim
Default values supplied by
ESMF_ILAENV(ISPEC,'DHSEQR',JOB(:1)
It is suggested that these defaults be adjusted in order
to attain best performance in each particular
computational environment.
ISPEC=12: The ESMF_DLAHQR vs ESMF_DLAQR0 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 ESMF_ILAENV(ISPEC=15). (See ISPEC=15 below.)
The default for (IHI-ILO+1).LE.500 is NS.
The default for (IHI-ILO+1).GT.500 is 3*NS/2.
ISPEC=14: Nibble crossover point. (See ESMF_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
ESMF_DLAHQR and this parameter is ignored. See
ISPEC=12 above and comments in ESMF_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.
\endverbatim \par 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.
\n 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.
| Type | Intent | Optional | Attributes | Name | ||
|---|---|---|---|---|---|---|
| character(len=1) | :: | JOB | ||||
| character(len=1) | :: | COMPZ | ||||
| integer | :: | N | ||||
| integer | :: | ILO | ||||
| integer | :: | IHI | ||||
| double precision | :: | H(LDH,*) | ||||
| integer | :: | LDH | ||||
| double precision | :: | WR(*) | ||||
| double precision | :: | WI(*) | ||||
| double precision | :: | Z(LDZ,*) | ||||
| integer | :: | LDZ | ||||
| double precision | :: | WORK(*) | ||||
| integer | :: | LWORK | ||||
| integer | :: | INFO |