NAG C Library Function Document

nag_dgeev (f08nac)

1
Purpose

nag_dgeev (f08nac) computes the eigenvalues and, optionally, the left and/or right eigenvectors for an n by n real nonsymmetric matrix A.

2
Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dgeev (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)

3
Description

The right eigenvector vj of A satisfies
A vj = λj vj  
where λj is the jth eigenvalue of A. The left eigenvector uj of A satisfies
ujH A = λj ujH  
where ujH denotes the conjugate transpose of uj.
The matrix A is first reduced to upper Hessenberg form by means of orthogonal similarity transformations, and the QR algorithm is then used to further reduce the matrix to upper quasi-triangular Schur form, T, with 1 by 1 and 2 by 2 blocks on the main diagonal. The eigenvalues are computed from T, the 2 by 2 blocks corresponding to complex conjugate pairs and, optionally, the eigenvectors of T are computed and backtransformed to the eigenvectors of A.

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 http://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:     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 order=Nag_RowMajor. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     jobvl Nag_LeftVecsTypeInput
On entry: if jobvl=Nag_NotLeftVecs, the left eigenvectors of A are not computed.
If jobvl=Nag_LeftVecs, the left eigenvectors of A are computed.
Constraint: jobvl=Nag_NotLeftVecs or Nag_LeftVecs.
3:     jobvr Nag_RightVecsTypeInput
On entry: if jobvr=Nag_NotRightVecs, the right eigenvectors of A are not computed.
If jobvr=Nag_RightVecs, the right eigenvectors of A are computed.
Constraint: jobvr=Nag_NotRightVecs or Nag_RightVecs.
4:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
5:     a[dim] doubleInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
The i,jth element of the matrix A is stored in
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
On entry: the n by n matrix A.
On exit: a has been overwritten.
6:     pda IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,n.
7:     wr[dim] doubleOutput
8:     wi[dim] doubleOutput
Note: the dimension, dim, of the arrays wr and wi must be at least max1,n.
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:     vl[dim] doubleOutput
Note: the dimension, dim, of the array vl must be at least
  • max1,pdvl×n when jobvl=Nag_LeftVecs;
  • 1 otherwise.
Where VLi,j appears in this document, it refers to the array element
  • vl[j-1×pdvl+i-1] when order=Nag_ColMajor;
  • vl[i-1×pdvl+j-1] when order=Nag_RowMajor.
On exit: if jobvl=Nag_LeftVecs, the left eigenvectors uj are stored one after another in vl, in the same order as their corresponding eigenvalues. If the jth eigenvalue is real, then uj=VLi,j, for i=1,2,,n. If the jth and j+1st eigenvalues form a complex conjugate pair, then uj=VLi,j+i×VLi,j+1 and uj+1=VLi,j-i×VLi,j+1, for i=1,2,,n.
If jobvl=Nag_NotLeftVecs, vl is not referenced.
10:   pdvl IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array vl.
Constraints:
  • if jobvl=Nag_LeftVecs, pdvl max1,n ;
  • otherwise pdvl1.
11:   vr[dim] doubleOutput
Note: the dimension, dim, of the array vr must be at least
  • max1,pdvr×n when jobvr=Nag_RightVecs;
  • 1 otherwise.
Where VRi,j appears in this document, it refers to the array element
  • vr[j-1×pdvr+i-1] when order=Nag_ColMajor;
  • vr[i-1×pdvr+j-1] when order=Nag_RowMajor.
On exit: if jobvr=Nag_RightVecs, the right eigenvectors vj are stored one after another in vr, in the same order as their corresponding eigenvalues. If the jth eigenvalue is real, then vj=VRi,j, for i=1,2,,n. If the jth and j+1st eigenvalues form a complex conjugate pair, then vj=VRi,j+i×VRi,j+1 and vj+1=VRi,j-i×VRi,j+1, for i=1,2,,n.
If jobvr=Nag_NotRightVecs, vr is not referenced.
12:   pdvr IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array vr.
Constraints:
  • if jobvr=Nag_RightVecs, pdvr max1,n ;
  • otherwise pdvr1.
13:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_CONVERGENCE
The QR algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements value to n of wr and wi contain eigenvalues which have converged.
NE_ENUM_INT_2
On entry, jobvl=value, pdvl=value and n=value.
Constraint: if jobvl=Nag_LeftVecs, pdvl max1,n ;
otherwise pdvl1.
On entry, jobvr=value, pdvr=value and n=value.
Constraint: if jobvr=Nag_RightVecs, pdvr max1,n ;
otherwise pdvr1.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdvl=value.
Constraint: pdvl>0.
On entry, pdvr=value.
Constraint: pdvr>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax1,n.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7
Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix A+E, where
E2 = Oε A2 ,  
and ε is the machine precision. See Section 4.8 of Anderson et al. (1999) for further details.

8
Parallelism and Performance

nag_dgeev (f08nac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dgeev (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.

9
Further 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 n3.
The complex analogue of this function is nag_zgeev (f08nnc).

10
Example

This example finds all the eigenvalues and right eigenvectors of the matrix
A = 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 .  

10.1
Program Text

Program Text (f08nace.c)

10.2
Program Data

Program Data (f08nace.d)

10.3
Program Results

Program Results (f08nace.r)