NAG CL Interface
f08nac (dgeev)
1
Purpose
f08nac computes the eigenvalues and, optionally, the left and/or right eigenvectors for an by real nonsymmetric matrix .
2
Specification
void |
f08nac (Nag_OrderType order,
Nag_LeftVecsType jobvl,
Nag_RightVecsType jobvr,
Integer n,
double a[],
Integer pda,
double wr[],
double wi[],
double vl[],
Integer pdvl,
double vr[],
Integer pdvr,
NagError *fail) |
|
The function may be called by the names: f08nac, nag_lapackeig_dgeev or nag_dgeev.
3
Description
The right eigenvector
of
satisfies
where
is the
th eigenvalue of
. The left eigenvector
of
satisfies
where
denotes the conjugate transpose of
.
The matrix is first reduced to upper Hessenberg form by means of orthogonal similarity transformations, and the algorithm is then used to further reduce the matrix to upper quasi-triangular Schur form, , with by and by blocks on the main diagonal. The eigenvalues are computed from , the by blocks corresponding to complex conjugate pairs and, optionally, the eigenvectors of are computed and backtransformed to the eigenvectors of .
4
References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments
-
1:
– Nag_OrderType
Input
-
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
-
2:
– Nag_LeftVecsType
Input
-
On entry: if
, the left eigenvectors of
are not computed.
If , the left eigenvectors of are computed.
Constraint:
or .
-
3:
– Nag_RightVecsType
Input
-
On entry: if
, the right eigenvectors of
are not computed.
If , the right eigenvectors of are computed.
Constraint:
or .
-
4:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
5:
– double
Input/Output
-
Note: the dimension,
dim, of the array
a
must be at least
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by matrix .
On exit:
a has been overwritten.
-
6:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
.
-
7:
– double
Output
-
8:
– double
Output
-
Note: the dimension,
dim, of the arrays
wr and
wi
must be at least
.
On exit:
wr and
wi contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.
-
9:
– double
Output
-
Note: the dimension,
dim, of the array
vl
must be at least
- when
;
- otherwise.
where
appears in this document, it refers to the array element
- when ;
- when .
On exit: if
, the left eigenvectors
are stored one after another in
vl, in the same order as their corresponding eigenvalues. If the
th eigenvalue is real, then
, for
. If the
th and
st eigenvalues form a complex conjugate pair, then
and
, for
.
If
,
vl is not referenced.
-
10:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
vl.
Constraints:
- if , ;
- otherwise .
-
11:
– double
Output
-
Note: the dimension,
dim, of the array
vr
must be at least
- when
;
- otherwise.
where
appears in this document, it refers to the array element
- when ;
- when .
On exit: if
, the right eigenvectors
are stored one after another in
vr, in the same order as their corresponding eigenvalues. If the
th eigenvalue is real, then
, for
. If the
th and
st eigenvalues form a complex conjugate pair, then
and
, for
.
If
,
vr is not referenced.
-
12:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
vr.
Constraints:
- if , ;
- otherwise .
-
13:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_CONVERGENCE
-
The
algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements
to
n of
wr and
wi contain eigenvalues which have converged.
- NE_ENUM_INT_2
-
On entry, , and .
Constraint: if , ;
otherwise .
On entry, , and .
Constraint: if , ;
otherwise .
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
- 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.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
7
Accuracy
The computed eigenvalues and eigenvectors are exact for a nearby matrix
, where
and
is the
machine precision. See Section 4.8 of
Anderson et al. (1999) for further details.
8
Parallelism and Performance
f08nac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08nac 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
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
Each eigenvector is normalized to have Euclidean norm equal to unity and the element of largest absolute value real.
The total number of floating-point operations is proportional to .
The complex analogue of this function is
f08nnc.
10
Example
This example finds all the eigenvalues and right eigenvectors of the matrix
10.1
Program Text
10.2
Program Data
10.3
Program Results