NAG Library Function Document
nag_zhseqr (f08psc)
1 Purpose
nag_zhseqr (f08psc) computes all the eigenvalues and, optionally, the Schur factorization of a complex Hessenberg matrix or a complex general matrix which has been reduced to Hessenberg form.
2 Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_zhseqr (Nag_OrderType order,
Nag_JobType job,
Nag_ComputeZType compz,
Integer n,
Integer ilo,
Integer ihi,
Complex h[],
Integer pdh,
Complex w[],
Complex z[],
Integer pdz,
NagError *fail) |
|
3 Description
nag_zhseqr (f08psc) computes all the eigenvalues and, optionally, the Schur factorization of a complex upper Hessenberg matrix
:
where
is an upper triangular matrix (the Schur form of
), and
is the unitary matrix whose columns are the Schur vectors
. The diagonal elements of
are the eigenvalues of
.
The function may also be used to compute the Schur factorization of a complex general matrix
which has been reduced to upper Hessenberg form
:
In this case, after
nag_zgehrd (f08nsc) has been called to reduce
to Hessenberg form,
nag_zunghr (f08ntc) must be called to form
explicitly;
is then passed to nag_zhseqr (f08psc), which must be called with
.
The function can also take advantage of a previous call to
nag_zgebal (f08nvc) which may have balanced the original matrix before reducing it to Hessenberg form, so that the Hessenberg matrix
has the structure:
where
and
are upper triangular. If so, only the central diagonal block
(in rows and columns
to
) needs to be further reduced to Schur form (the blocks
and
are also affected). Therefore the values of
and
can be supplied to nag_zhseqr (f08psc) directly. Also,
nag_zgebak (f08nwc) must be called after this function to permute the Schur vectors of the balanced matrix to those of the original matrix. If
nag_zgebal (f08nvc) has not been called however, then
must be set to
and
to
. Note that if the Schur factorization of
is required,
nag_zgebal (f08nvc) must
not be called with
or
, because the balancing transformation is not unitary.
nag_zhseqr (f08psc) uses a multishift form of the upper Hessenberg
algorithm, due to
Bai and Demmel (1989). The Schur vectors are normalized so that
, but are determined only to within a complex factor of absolute value
.
4 References
Bai Z and Demmel J W (1989) On a block implementation of Hessenberg multishift iteration Internat. J. High Speed Comput. 1 97–112
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Arguments
- 1:
order – Nag_OrderTypeInput
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
job – Nag_JobTypeInput
On entry: indicates whether eigenvalues only or the Schur form
is required.
- Eigenvalues only are required.
- The Schur form is required.
Constraint:
or .
- 3:
compz – Nag_ComputeZTypeInput
On entry: indicates whether the Schur vectors are to be computed.
- No Schur vectors are computed (and the array z is not referenced).
- The Schur vectors of are computed (and the array z is initialized by the function).
- The Schur vectors of are computed (and the array z must contain the matrix on entry).
Constraint:
, or .
- 4:
n – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
- 5:
ilo – IntegerInput
- 6:
ihi – IntegerInput
On entry: if the matrix
has been balanced by
nag_zgebal (f08nvc), then
ilo and
ihi must contain the values returned by that function. Otherwise,
ilo must be set to
and
ihi to
n.
Constraint:
and .
- 7:
h[] – ComplexInput/Output
-
Note: the dimension,
dim, of the array
h
must be at least
.
Where
appears in this document, it refers to the array element
- when ;
- when .
On entry: the
by
upper Hessenberg matrix
, as returned by
nag_zgehrd (f08nsc).
On exit: if
, the array contains no useful information.
If
,
h is overwritten by the upper triangular matrix
from the Schur decomposition (the Schur form) unless
NE_CONVERGENCE.
- 8:
pdh – IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
h.
Constraint:
.
- 9:
w[] – ComplexOutput
-
Note: the dimension,
dim, of the array
w
must be at least
.
On exit: the computed eigenvalues, unless
NE_CONVERGENCE (in which case see
Section 6). The eigenvalues are stored in the same order as on the diagonal of the Schur form
(if computed).
- 10:
z[] – ComplexInput/Output
-
Note: the dimension,
dim, of the array
z
must be at least
- when
or and
;
- when
or and
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: if
,
z must contain the unitary matrix
from the reduction to Hessenberg form.
If
,
z need not be set.
On exit: if
or
,
z contains the unitary matrix of the required Schur vectors, unless
NE_CONVERGENCE.
If
,
z is not referenced.
- 11:
pdz – IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
z.
Constraints:
- if ,
- if or , ;
- if , ;
- if ,
- if or ,
;
- if ,
.
- 12:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_CONVERGENCE
-
The algorithm has failed to find all the eigenvalues after a total of iterations.
- NE_ENUM_INT_2
-
On entry, , and .
Constraint: if or , ;
if , .
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
- NE_INT_3
-
On entry, , and .
Constraint: and .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
7 Accuracy
The computed Schur factorization is the exact factorization of a nearby matrix
, where
and
is the
machine precision.
If
is an exact eigenvalue, and
is the corresponding computed value, then
where
is a modestly increasing function of
, and
is the reciprocal condition number of
. The condition numbers
may be computed by calling
nag_ztrsna (f08qyc).
8 Parallelism and Performance
nag_zhseqr (f08psc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zhseqr (f08psc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
Users' Note for your implementation for any additional implementation-specific information.
The total number of real floating-point operations depends on how rapidly the algorithm converges, but is typically about:
- if only eigenvalues are computed;
- if the Schur form is computed;
- if the full Schur factorization is computed.
The real analogue of this function is
nag_dhseqr (f08pec).
10 Example
This example computes all the eigenvalues and the Schur factorization of the upper Hessenberg matrix
, where
See also
Section 10 in nag_zunghr (f08ntc), which illustrates the use of this function to compute the Schur factorization of a general matrix.
10.1 Program Text
Program Text (f08psce.c)
10.2 Program Data
Program Data (f08psce.d)
10.3 Program Results
Program Results (f08psce.r)