f08nbc computes the eigenvalues and, optionally, the left and/or right eigenvectors for an real nonsymmetric matrix .
Optionally, it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors, reciprocal condition numbers for the eigenvalues, and reciprocal condition numbers for the right eigenvectors.
The function may be called by the names: f08nbc, nag_lapackeig_dgeevx or nag_dgeevx.
3Description
The right eigenvector of satisfies
where is the th eigenvalue of . The left eigenvector of satisfies
where denotes the conjugate transpose of .
Balancing a matrix means permuting the rows and columns to make it more nearly upper triangular, and applying a diagonal similarity transformation , where is a diagonal matrix, with the aim of making its rows and columns closer in norm and the condition numbers of its eigenvalues and eigenvectors smaller. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see Section 4.8.1.2 of Anderson et al. (1999).
Following the optional balancing, the matrix is first reduced to upper Hessenberg form by means of unitary similarity transformations, and the algorithm is then used to further reduce the matrix to upper triangular Schur form, , from which the eigenvalues are computed. Optionally, the eigenvectors of are also computed and backtransformed to those 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_BalanceTypeInput
On entry: indicates how the input matrix should be diagonally scaled and/or permuted to improve the conditioning of its eigenvalues.
Do not diagonally scale or permute.
Perform permutations to make the matrix more nearly upper triangular. Do not diagonally scale.
Diagonally scale the matrix, i.e., replace , where is a diagonal matrix chosen to make the rows and columns of more equal in norm. Do not permute.
Both diagonally scale and permute .
Computed reciprocal condition numbers will be for the matrix after balancing and/or permuting. Permuting does not change condition numbers (in exact arithmetic), but balancing does.
Constraint:
, , or .
3: – Nag_LeftVecsTypeInput
On entry: if , the left eigenvectors of are not computed.
On entry: determines which reciprocal condition numbers are computed.
None are computed.
Computed for eigenvalues only.
Computed for right eigenvectors only.
Computed for eigenvalues and right eigenvectors.
If or , both left and right eigenvectors must also be computed ( and ).
Constraint:
, , or .
6: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
7: – doubleInput/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 matrix .
On exit: a has been overwritten. If or , contains the real Schur form of the balanced version of the input matrix .
8: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint:
.
9: – doubleOutput
10: – 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.
11: – 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 .
13: – 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 .
15: – Integer *Output
16: – Integer *Output
On exit: ilo and ihi are integer values determined when was balanced. The balanced has if and or .
17: – doubleOutput
Note: the dimension, dim, of the array scale
must be at least
.
On exit: details of the permutations and scaling factors applied when balancing .
If is the index of the row and column interchanged with row and column , and is the scaling factor applied to row and column , then
, for ;
, for ;
, for .
The order in which the interchanges are made is n to , then to .
18: – double *Output
On exit: the -norm of the balanced matrix (the maximum of the sum of absolute values of elements of any column).
19: – doubleOutput
Note: the dimension, dim, of the array rconde
must be at least
.
On exit: is the reciprocal condition number of the th eigenvalue.
20: – doubleOutput
Note: the dimension, dim, of the array rcondv
must be at least
.
On exit: is the reciprocal condition number of the th right eigenvector.
21: – 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 or condition numbers have been computed; elements to and 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.
f08nbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08nbc 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 .
This example finds all the eigenvalues and right eigenvectors of the matrix
together with estimates of the condition number and forward error bounds for each eigenvalue and eigenvector. The option to balance the matrix is used. In order to compute the condition numbers of the eigenvalues, the left eigenvectors also have to be computed, but they are not printed out in this example.