nag_zgeev (f08nnc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zgeev (f08nnc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zgeev (f08nnc) computes the eigenvalues and, optionally, the left and/or right eigenvectors for an n by n complex nonsymmetric matrix A.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_zgeev (Nag_OrderType order, Nag_LeftVecsType jobvl, Nag_RightVecsType jobvr, Integer n, Complex a[], Integer pda, Complex w[], Complex vl[], Integer pdvl, Complex 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 unitary similarity transformations, and the QR algorithm is then used to further reduce the matrix to upper triangular Schur form, T, from which the eigenvalues are computed. Optionally, the eigenvectors of T are also computed and backtransformed to those 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:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     jobvlNag_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:     jobvrNag_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:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
5:     a[dim]ComplexInput/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:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,n.
7:     w[dim]ComplexOutput
Note: the dimension, dim, of the array w must be at least max1,n.
On exit: contains the computed eigenvalues.
8:     vl[dim]ComplexOutput
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; that is uj=VLi,j, for i=1,2,,n.
If jobvl=Nag_NotLeftVecs, vl is not referenced.
9:     pdvlIntegerInput
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.
10:   vr[dim]ComplexOutput
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; that is vj=VRi,j, for i=1,2,,n.
If jobvr=Nag_NotRightVecs, vr is not referenced.
11:   pdvrIntegerInput
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.
12:   failNagError *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 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 w 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.

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  Further Comments

Each eigenvector is normalized to have Euclidean norm equal to unity and the element of largest absolute value real and positive.
The total number of floating point operations is proportional to n3.
The real analogue of this function is nag_dgeev (f08nac).

9  Example

This example finds all the eigenvalues and right eigenvectors of the matrix
A = -3.97-5.04i -4.11+3.70i -0.34+1.01i 1.29-0.86i 0.34-1.50i 1.52-0.43i 1.88-5.38i 3.36+0.65i 3.31-3.85i 2.50+3.45i 0.88-1.08i 0.64-1.48i -1.10+0.82i 1.81-1.59i 3.25+1.33i 1.57-3.44i .

9.1  Program Text

Program Text (f08nnce.c)

9.2  Program Data

Program Data (f08nnce.d)

9.3  Program Results

Program Results (f08nnce.r)


nag_zgeev (f08nnc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012