NAG CL Interface
f08nnc (zgeev)

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1 Purpose

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

2 Specification

#include <nag.h>
void  f08nnc (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)
The function may be called by the names: f08nnc, nag_lapackeig_zgeev or nag_zgeev.

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 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: order 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 order=Nag_RowMajor. 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: order=Nag_RowMajor or Nag_ColMajor.
2: jobvl Nag_LeftVecsType Input
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_RightVecsType Input
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 Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
5: a[dim] Complex Input/Output
Note: the dimension, dim, of the array a must be at least max(1,pda×n).
The (i,j)th 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×n matrix A.
On exit: a has been overwritten.
6: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax(1,n).
7: w[dim] Complex Output
Note: the dimension, dim, of the array w must be at least max(1,n).
On exit: contains the computed eigenvalues.
8: vl[dim] Complex Output
Note: the dimension, dim, of the array vl must be at least
  • max(1,pdvl×n) when jobvl=Nag_LeftVecs;
  • 1 otherwise.
where VL(i,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=VL(i,j), for i=1,2,,n.
If jobvl=Nag_NotLeftVecs, vl is not referenced.
9: pdvl Integer Input
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 max(1,n) ;
  • otherwise pdvl1.
10: vr[dim] Complex Output
Note: the dimension, dim, of the array vr must be at least
  • max(1,pdvr×n) when jobvr=Nag_RightVecs;
  • 1 otherwise.
where VR(i,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=VR(i,j), for i=1,2,,n.
If jobvr=Nag_NotRightVecs, vr is not referenced.
11: pdvr Integer Input
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 max(1,n) ;
  • otherwise pdvr1.
12: fail 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 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 max(1,n) ;
otherwise pdvl1.
On entry, jobvr=value, pdvr=value and n=value.
Constraint: if jobvr=Nag_RightVecs, pdvr max(1,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: pdamax(1,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 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 (A+E), where
E2 = O(ε) A2 ,  
and ε is the machine precision. See Section 4.8 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

f08nnc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08nnc 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 real analogue of this function is f08nac.

10 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 ) .  

10.1 Program Text

Program Text (f08nnce.c)

10.2 Program Data

Program Data (f08nnce.d)

10.3 Program Results

Program Results (f08nnce.r)