nag_zgees (f08pnc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_zgees (f08pnc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zgees (f08pnc) computes the eigenvalues, the Schur form T, and, optionally, the matrix of Schur vectors Z for an n by n complex nonsymmetric matrix A.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_zgees (Nag_OrderType order, Nag_JobType jobvs, Nag_SortEigValsType sort,
Nag_Boolean (*select)(Complex w),
Integer n, Complex a[], Integer pda, Integer *sdim, Complex w[], Complex vs[], Integer pdvs, NagError *fail)

3  Description

The Schur factorization of A is given by
A = Z T ZH ,
where Z, the matrix of Schur vectors, is unitary and T is the Schur form. A complex matrix is in Schur form if it is upper triangular.
Optionally, nag_zgees (f08pnc) also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left. The leading columns of Z form an orthonormal basis for the invariant subspace corresponding to the selected eigenvalues.

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
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 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     jobvsNag_JobTypeInput
On entry: if jobvs=Nag_DoNothing, Schur vectors are not computed.
If jobvs=Nag_Schur, Schur vectors are computed.
Constraint: jobvs=Nag_DoNothing or Nag_Schur.
3:     sortNag_SortEigValsTypeInput
On entry: specifies whether or not to order the eigenvalues on the diagonal of the Schur form.
Eigenvalues are not ordered.
Eigenvalues are ordered (see select).
Constraint: sort=Nag_NoSortEigVals or Nag_SortEigVals.
4:     selectfunction, supplied by the userExternal Function
If sort=Nag_SortEigVals, select is used to select eigenvalues to sort to the top left of the Schur form.
If sort=Nag_NoSortEigVals, select is not referenced and nag_zgees (f08pnc) may be specified as NULLFN.
An eigenvalue w[j-1] is selected if selectw[j-1] is Nag_TRUE.
The specification of select is:
Nag_Boolean  select (Complex w)
1:     wComplexInput
On entry: the real and imaginary parts of the eigenvalue.
5:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
6:     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 is overwritten by its Schur form T.
7:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,n.
8:     sdimInteger *Output
On exit: if sort=Nag_NoSortEigVals, sdim=0.
If sort=Nag_SortEigVals, sdim= number of eigenvalues for which select is Nag_TRUE.
9:     w[dim]ComplexOutput
Note: the dimension, dim, of the array w must be at least max1,n.
On exit: contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form T.
10:   vs[dim]ComplexOutput
Note: the dimension, dim, of the array vs must be at least
  • max1,pdvs×n when jobvs=Nag_Schur;
  • 1 otherwise.
The ith element of the jth vector is stored in
  • vs[j-1×pdvs+i-1] when order=Nag_ColMajor;
  • vs[i-1×pdvs+j-1] when order=Nag_RowMajor.
On exit: if jobvs=Nag_Schur, vs contains the unitary matrix Z of Schur vectors.
If jobvs=Nag_DoNothing, vs is not referenced.
11:   pdvsIntegerInput
On entry: the stride used in the array vs.
  • if jobvs=Nag_Schur, pdvs max1,n ;
  • otherwise pdvs1.
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
The QR algorithm failed to compute all the eigenvalues.
On entry, jobvs=value, pdvs=value and n=value.
Constraint: if jobvs=Nag_Schur, pdvs max1,n ;
otherwise pdvs1.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdvs=value.
Constraint: pdvs>0.
On entry, pda=value and n=value.
Constraint: pdamax1,n.
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.
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy select=Nag_TRUE. This could also be caused by underflow due to scaling.

7  Accuracy

The computed Schur factorization satisfies
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_zgees (f08pnc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zgees (f08pnc) 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 Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The total number of floating-point operations is proportional to n3.
The real analogue of this function is nag_dgees (f08pac).

10  Example

This example finds the Schur factorization 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 (f08pnce.c)

10.2  Program Data

Program Data (f08pnce.d)

10.3  Program Results

Program Results (f08pnce.r)

nag_zgees (f08pnc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014