nag_dgeesx (f08pbc) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_dgeesx (f08pbc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgeesx (f08pbc) computes the eigenvalues, the real Schur form T, and, optionally, the matrix of Schur vectors Z for an n by n real nonsymmetric matrix A.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dgeesx (Nag_OrderType order, Nag_JobType jobvs, Nag_SortEigValsType sort,
Nag_Boolean (*select)(double wr, double wi),
Nag_RCondType sense, Integer n, double a[], Integer pda, Integer *sdim, double wr[], double wi[], double vs[], Integer pdvs, double *rconde, double *rcondv, NagError *fail)

3  Description

The real Schur factorization of A is given by
A = Z T ZT ,
where Z, the matrix of Schur vectors, is orthogonal and T is the real Schur form. A matrix is in real Schur form if it is upper quasi-triangular with 1 by 1 and 2 by 2 blocks. 2 by 2 blocks will be standardized in the form
a b c a
where bc<0. The eigenvalues of such a block are a±bc.
Optionally, nag_dgeesx (f08pbc) also orders the eigenvalues on the diagonal of the real 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
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_dgeesx (f08pbc) may be specified as NULLFN.
An eigenvalue wr[j-1]+-1×wi[j-1] is selected if selectwr[j-1],wi[j-1] is Nag_TRUE. If either one of a complex conjugate pair of eigenvalues is selected, then both are. Note that a selected complex eigenvalue may no longer satisfy selectwr[j-1],wi[j-1]=Nag_TRUE after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned); in this case fail.errnum is set to n+2.
The specification of select is:
Nag_Boolean  select (double wr, double wi)
1:     wrdoubleInput
2:     widoubleInput
On entry: the real and imaginary parts of the eigenvalue.
5:     senseNag_RCondTypeInput
On entry: determines which reciprocal condition numbers are computed.
None are computed.
Computed for average of selected eigenvalues only.
Computed for selected right invariant subspace only.
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]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 is overwritten by its real 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 (after sorting) for which select is Nag_TRUE. (Complex conjugate pairs for which select is Nag_TRUE for either eigenvalue count as 2.)
10:   wr[dim]doubleOutput
Note: the dimension, dim, of the array wr must be at least max1,n.
On exit: see the description of wi.
11:   wi[dim]doubleOutput
Note: the dimension, dim, of the array wi must be at least max1,n.
On exit: wr and wi contain the real and imaginary parts, respectively, of the computed eigenvalues in the same order that they appear on the diagonal of the output Schur form T. Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.
12:   vs[dim]doubleOutput
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 orthogonal matrix Z of Schur vectors.
If jobvs=Nag_DoNothing, vs is not referenced.
13:   pdvsIntegerInput
On entry: the stride used in the array vs.
  • if jobvs=Nag_Schur, pdvs max1,n ;
  • otherwise pdvs1.
14:   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.
15:   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.
16:   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
A+E = ZTZT ,
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_dgeesx (f08pbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dgeesx (f08pbc) 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 complex analogue of this function is nag_zgeesx (f08ppc).

10  Example

This example finds the Schur factorization 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 ,
such that the real eigenvalues of A 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.

10.1  Program Text

Program Text (f08pbce.c)

10.2  Program Data

Program Data (f08pbce.d)

10.3  Program Results

Program Results (f08pbce.r)

nag_dgeesx (f08pbc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

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