nag_dgges (f08xac) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_dgges (f08xac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgges (f08xac) computes the generalized eigenvalues, the generalized real Schur form S,T  and, optionally, the left and/or right generalized Schur vectors for a pair of n by n real nonsymmetric matrices A,B .

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dgges (Nag_OrderType order, Nag_LeftVecsType jobvsl, Nag_RightVecsType jobvsr, Nag_SortEigValsType sort,
Nag_Boolean (*selctg)(double ar, double ai, double b),
Integer n, double a[], Integer pda, double b[], Integer pdb, Integer *sdim, double alphar[], double alphai[], double beta[], double vsl[], Integer pdvsl, double vsr[], Integer pdvsr, NagError *fail)

3  Description

The generalized Schur factorization for a pair of real matrices A,B  is given by
A = QSZT ,   B = QTZT ,
where Q and Z are orthogonal, T is upper triangular and S is upper quasi-triangular with 1 by 1 and 2 by 2 diagonal blocks. The generalized eigenvalues, λ , of A,B  are computed from the diagonals of S and T and satisfy
Az = λBz ,
where z is the corresponding generalized eigenvector. λ  is actually returned as the pair α,β  such that
λ = α/β
since β , or even both α  and β  can be zero. The columns of Q and Z are the left and right generalized Schur vectors of A,B .
Optionally, nag_dgges (f08xac) can order the generalized eigenvalues on the diagonals of S,T  so that selected eigenvalues are at the top left. The leading columns of Q and Z then form an orthonormal basis for the corresponding eigenspaces, the deflating subspaces.
nag_dgges (f08xac) computes T to have non-negative diagonal elements, and the 2 by 2 blocks of S correspond to complex conjugate pairs of generalized eigenvalues. The generalized Schur factorization, before reordering, is computed by the QZ algorithm.

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:     jobvslNag_LeftVecsTypeInput
On entry: if jobvsl=Nag_NotLeftVecs, do not compute the left Schur vectors.
If jobvsl=Nag_LeftVecs, compute the left Schur vectors.
Constraint: jobvsl=Nag_NotLeftVecs or Nag_LeftVecs.
3:     jobvsrNag_RightVecsTypeInput
On entry: if jobvsr=Nag_NotRightVecs, do not compute the right Schur vectors.
If jobvsr=Nag_RightVecs, compute the right Schur vectors.
Constraint: jobvsr=Nag_NotRightVecs or Nag_RightVecs.
4:     sortNag_SortEigValsTypeInput
On entry: specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.
sort=Nag_NoSortEigVals
Eigenvalues are not ordered.
sort=Nag_SortEigVals
Eigenvalues are ordered (see selctg).
Constraint: sort=Nag_NoSortEigVals or Nag_SortEigVals.
5:     selctgfunction, supplied by the userExternal Function
If sort=Nag_SortEigVals, selctg is used to select generalized eigenvalues to the top left of the generalized Schur form.
If sort=Nag_NoSortEigVals, selctg is not referenced by nag_dgges (f08xac), and may be specified as NULLFN.
The specification of selctg is:
Nag_Boolean  selctg (double ar, double ai, double b)
1:     ardoubleInput
2:     aidoubleInput
3:     bdoubleInput
On entry: an eigenvalue ar[j-1] + -1 × ai[j-1] / b[j-1]  is selected if selctg ar[j-1],ai[j-1],b[j-1]=Nag_TRUE . If either one of a complex conjugate pair is selected, then both complex generalized eigenvalues are selected.
Note that in the ill-conditioned case, a selected complex generalized eigenvalue may no longer satisfy selctg ar[j-1],ai[j-1],b[j-1]=Nag_TRUE  after ordering. fail.code= NE_SCHUR_REORDER_SELECT in this case.
6:     nIntegerInput
On entry: n, the order of the matrices A and B.
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 first of the pair of matrices, A.
On exit: a has been overwritten by its generalized Schur form S.
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:     b[dim]doubleInput/Output
Note: the dimension, dim, of the array b must be at least max1,pdb×n.
The i,jth element of the matrix B is stored in
  • b[j-1×pdb+i-1] when order=Nag_ColMajor;
  • b[i-1×pdb+j-1] when order=Nag_RowMajor.
On entry: the second of the pair of matrices, B.
On exit: b has been overwritten by its generalized Schur form T.
10:   pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: pdbmax1,n.
11:   sdimInteger *Output
On exit: if sort=Nag_NoSortEigVals, sdim=0.
If sort=Nag_SortEigVals, sdim= number of eigenvalues (after sorting) for which selctg is Nag_TRUE. (Complex conjugate pairs for which selctg is Nag_TRUE for either eigenvalue count as 2.)
12:   alphar[n]doubleOutput
On exit: see the description of beta.
13:   alphai[n]doubleOutput
On exit: see the description of beta.
14:   beta[n]doubleOutput
On exit: alphar[j-1]+alphai[j-1]×i/beta[j-1], for j=1,2,,n, will be the generalized eigenvalues. alphar[j-1]+alphai[j-1]×i, and beta[j-1], for j=1,2,,n, are the diagonals of the complex Schur form S,T that would result if the 2 by 2 diagonal blocks of the real Schur form of A,B were further reduced to triangular form using 2 by 2 complex unitary transformations.
If alphai[j-1] is zero, then the jth eigenvalue is real; if positive, then the jth and j+1st eigenvalues are a complex conjugate pair, with alphai[j] negative.
Note:  the quotients alphar[j-1]/beta[j-1] and alphai[j-1]/beta[j-1] may easily overflow or underflow, and beta[j-1] may even be zero. Thus, you should avoid naively computing the ratio α/β. However, alphar and alphai will always be less than and usually comparable with a2 in magnitude, and beta will always be less than and usually comparable with b2.
15:   vsl[dim]doubleOutput
Note: the dimension, dim, of the array vsl must be at least
  • max1,pdvsl×n when jobvsl=Nag_LeftVecs;
  • 1 otherwise.
The i,jth element of the matrix is stored in
  • vsl[j-1×pdvsl+i-1] when order=Nag_ColMajor;
  • vsl[i-1×pdvsl+j-1] when order=Nag_RowMajor.
On exit: if jobvsl=Nag_LeftVecs, vsl will contain the left Schur vectors, Q.
If jobvsl=Nag_NotLeftVecs, vsl is not referenced.
16:   pdvslIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array vsl.
Constraints:
  • if jobvsl=Nag_LeftVecs, pdvsl max1,n ;
  • otherwise pdvsl1.
17:   vsr[dim]doubleOutput
Note: the dimension, dim, of the array vsr must be at least
  • max1,pdvsr×n when jobvsr=Nag_RightVecs;
  • 1 otherwise.
The i,jth element of the matrix is stored in
  • vsr[j-1×pdvsr+i-1] when order=Nag_ColMajor;
  • vsr[i-1×pdvsr+j-1] when order=Nag_RowMajor.
On exit: if jobvsr=Nag_RightVecs, vsr will contain the right Schur vectors, Z.
If jobvsr=Nag_NotRightVecs, vsr is not referenced.
18:   pdvsrIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array vsr.
Constraints:
  • if jobvsr=Nag_RightVecs, pdvsr max1,n ;
  • otherwise pdvsr1.
19:   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_ENUM_INT_2
On entry, jobvsl=value, pdvsl=value and n=value.
Constraint: if jobvsl=Nag_LeftVecs, pdvsl max1,n ;
otherwise pdvsl1.
On entry, jobvsr=value, pdvsr=value and n=value.
Constraint: if jobvsr=Nag_RightVecs, pdvsr max1,n ;
otherwise pdvsr1.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdvsl=value.
Constraint: pdvsl>0.
On entry, pdvsr=value.
Constraint: pdvsr>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax1,n.
On entry, pdb=value and n=value.
Constraint: pdbmax1,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_ITERATION_QZ
The QZ iteration failed. No eigenvectors have been calculated but alphar[j], alphai[j] and beta[j] should be correct from element value.
The QZ iteration failed with an unexpected error, please contact NAG.
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 generalized Schur form no longer satisfy selctg=Nag_TRUE. This could also be caused by underflow due to scaling.

7  Accuracy

The computed generalized Schur factorization satisfies
A+E = QS ZT ,   B+F = QT ZT ,
where
E,F F = Oε A,B F
and ε is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details.

8  Parallelism and Performance

nag_dgges (f08xac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dgges (f08xac) 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_zgges (f08xnc).

10  Example

This example finds the generalized Schur factorization of the matrix pair A,B, where
A = 3.9 12.5 -34.5 -0.5 4.3 21.5 -47.5 7.5 4.3 21.5 -43.5 3.5 4.4 26.0 -46.0 6.0   and   B= 1.0 2.0 -3.0 1.0 1.0 3.0 -5.0 4.0 1.0 3.0 -4.0 3.0 1.0 3.0 -4.0 4.0 ,
such that the real eigenvalues of A,B correspond to the top left diagonal elements of the generalized Schur form, S,T.

10.1  Program Text

Program Text (f08xace.c)

10.2  Program Data

Program Data (f08xace.d)

10.3  Program Results

Program Results (f08xace.r)


nag_dgges (f08xac) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

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