The function may be called by the names: f08nac, nag_lapackeig_dgeev or nag_dgeev.
3Description
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 and blocks on the main diagonal. The eigenvalues are computed from , the blocks corresponding to complex conjugate pairs and, optionally, the eigenvectors of are computed and backtransformed to the eigenvectors of .
4References
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
5Arguments
1: – 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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
2: – Nag_LeftVecsTypeInput
On entry: if , the left eigenvectors of are not computed.
If , the left eigenvectors of are computed.
Constraint:
or .
3: – Nag_RightVecsTypeInput
On entry: if , the right eigenvectors of are not computed.
If , the right eigenvectors of are computed.
Constraint:
or .
4: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
5: – doubleInput/Output
Note: the dimension, dim, of the array a
must be at least
.
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint:
.
7: – doubleOutput
8: – doubleOutput
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: – doubleOutput
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 .
On entry: the stride separating row or column elements (depending on the value of order) in the array vl.
Constraints:
if , ;
otherwise .
11: – doubleOutput
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 .
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).
6Error 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.
7Accuracy
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.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
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.
9Further Comments
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 .