nag_zgeesx (f08ppc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zgeesx (f08ppc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zgeesx (f08ppc) 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_zgeesx (Nag_OrderType order, Nag_JobType jobvs, Nag_SortEigValsType sort,
Nag_Boolean (*select)(Complex w),
Nag_RCondType sense, Integer n, Complex a[], Integer pda, Integer *sdim, Complex w[], Complex vs[], Integer pdvs, double *rconde, double *rcondv, 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_zgeesx (f08ppc) also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left; computes a reciprocal condition number for the average of the selected eigenvalues (rconde); and computes a reciprocal condition number for the right invariant subspace corresponding to the selected eigenvalues (rcondv). The leading columns of Z form an orthonormal basis for this invariant subspace.
For further explanation of the reciprocal condition numbers rconde and rcondv, see Section 4.8 of Anderson et al. (1999) (where these quantities are called s and sep respectively).

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:     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.
sort=Nag_NoSortEigVals
Eigenvalues are not ordered.
sort=Nag_SortEigVals
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_zgeesx (f08ppc) 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:     senseNag_RCondTypeInput
On entry: determines which reciprocal condition numbers are computed.
sense=Nag_NotRCond
None are computed.
sense=Nag_RCondEigVals
Computed for average of selected eigenvalues only.
sense=Nag_RCondEigVecs
Computed for selected right invariant subspace only.
sense=Nag_RCondBoth
Computed for both.
If sense=Nag_RCondEigVals, Nag_RCondEigVecs or Nag_RCondBoth, sort=Nag_SortEigVals.
Constraint: sense=Nag_NotRCond, Nag_RCondEigVals, Nag_RCondEigVecs or Nag_RCondBoth.
6:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
7:     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.
8:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: pdamax1,n.
9:     sdimInteger *Output
On exit: if sort=Nag_NoSortEigVals, sdim=0.
If sort=Nag_SortEigVals, sdim= number of eigenvalues for which select is Nag_TRUE.
10:   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.
11:   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.
12:   pdvsIntegerInput
On entry: the stride used in the array vs.
Constraints:
  • if jobvs=Nag_Schur, pdvs max1,n ;
  • otherwise pdvs1.
13:   rcondedouble *Output
On exit: if sense=Nag_RCondEigVals or Nag_RCondBoth, contains the reciprocal condition number for the average of the selected eigenvalues.
If sense=Nag_NotRCond or Nag_RCondEigVecs, rconde is not referenced.
14:   rcondvdouble *Output
On exit: if sense=Nag_RCondEigVecs or Nag_RCondBoth, rcondv contains the reciprocal condition number for the selected right invariant subspace.
If sense=Nag_NotRCond or Nag_RCondEigVals, rcondv is not referenced.
15:   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.
NE_ENUM_INT_2
On entry, jobvs=value, pdvs=value and n=value.
Constraint: if jobvs=Nag_Schur, pdvs max1,n ;
otherwise pdvs1.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdvs=value.
Constraint: pdvs>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.
NE_SCHUR_REORDER
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
NE_SCHUR_REORDER_SELECT
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
A+E = ZTZH ,
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

The total number of floating point operations is proportional to n3.
The real analogue of this function is nag_dgeesx (f08pbc).

9  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 ,
such that the eigenvalues of A with positive real part of are the top left diagonal elements of the Schur form, T. Estimates of the condition numbers for the selected eigenvalue cluster and corresponding invariant subspace are also returned.

9.1  Program Text

Program Text (f08ppce.c)

9.2  Program Data

Program Data (f08ppce.d)

9.3  Program Results

Program Results (f08ppce.r)


nag_zgeesx (f08ppc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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