JDQZ

(This document in ps,pdf)

jdqz computes a partial generalized Schur decomposition (or QZ decomposition) of a pair of square matrices or operators.

Lambda = jdqz(A,B) and jdqz(A,B) return k eigenvalues of the matrix pair (A,B), where k = min(5,n) and n = size(A,1) if k has not been specified.

[X,Jordan] = jdqz(A,B) returns the eigenvectors Xx and the Jordan structure Jordan: AX = BXJordan. The diagonal of Jordan contains the eigenvalues: Lambda = diag(Jordan). Jordan is an k by k matrix with the eigenvalues on the diagonal and zero or one on the first upper diagonal elements. The other entries are zero.

[X,Jordan,history] = jdqz(A,B) returns also the convergence history.

[X,Jordan,Q,Z,S,T,history] = jdqz(A,B)
If between four and seven output arguments are required then Q and Z are n by k orthonormal, S and T are k by k upper triangular such that they form a partial generalized Schur decomposition: AQ = ZS and BQ = ZT. Then Lambda = diag(S)./diag(T) and X = QY with Y the eigenvectors of the pair (S,T): SY = TYJordan (see also Options.Schur).

jdqz(A,B)
jdqz('Afun','Bfun')
The first input argument is either a square matrix (which can be full or sparse, symmetric or nonsymmetric, real or complex), or a string containing the name of an M-file which applies a linear operator to the columns of a given matrix. In the latter case, the M-file, say Afun.m, must return the dimension N of the problem with n = Afun([ ],'dimension'). For example, jdqz('fft',...) is much faster than jdqz(F,...), where F is the explicit FFT matrix.
If another input argument is a square n by n matrix or the name of an M-file, then B is this argument (regardless whether A is an M-file or a matrix). If B has not been specified, then B is assumed to be the identity unless A is an M-file with two output vectors of dimension n with [Av,Bv] = Afun(v), or with Av = Afun(v,'A') and Bv = Afun(v,'B').

The remaining input arguments are optional and can be given in practically any order:

[X,Jordan,Q,Z,S,T,history] = jdqz(A,B,k,Sigma,Options)
[X,Jordan,Q,Z,S,T,history] = jdqz('Afun','Bfun',k,Sigma,Options),

where

	k	an integer, the number of desired eigenvalues.
	Sigma	a scalar shift or a two letter string.
	Options	a structure containing additional parameters.

If, in addition, there are other input arguments, then they are passed to 'Afun' and to 'Bfun' as input arguments.

If k is not specified, then k = min(n,5) eigenvalues are computed.

If Sigma is not specified, then the kth eigenvalues largest in magnitude are computed. If Sigma is a real or complex scalar, then the kth eigenvalues nearest Sigma are computed. If Sigma is column vector of size (m,1), then the jth eigenvalue nearest to Sigma(min(j,m)) is computed for j = 1:k. Sigma is the ``target'' for the desired eigenvalues. If Sigma is one of the following strings, then it specifies the desired eigenvalues.

	Sigma	Specified eigenvalues
	'LM'	Largest Magnitude
	'SM'	Smallest Magnitude (same as Sigma = 0)
	'LR'	Largest Real part
	'SR'	Smallest Real part
	'BE'	Both Ends. Computes k/2 eigenvalues from each end of the spectrum (one more from the high end if k is odd.)

If 'TestSpace' is 'Harmonic' (see Options), then Sigma = 0 is the default, otherwise Sigma = 'LM' is the default.

The Options structure specifies certain parameters in the algorithm.

	Field name	Parameter	Default
	Options.Tol	Convergence tolerance: norm(r) tol/sqrt(k)	1e-8
	Options.jmin	Minimum dimension search subspace V	k+5
	Options.jmax	Maximum dimension search subspace V	jmin+5
	Options.MaxIt	Maximum number of iterations.	100
	Options.v0	Starting space	ones+0.1rand
	Options.Schur	Gives schur decomposition If 'yes', then X and Jordan are not computed and [Q,Z,S,T,history] is the list of output arguments.	'no'
	Options.TestSpace	Defines the test subspace W 'Standard': W = sigmaAV+BV 'Harmonic': W = AV-sigmaBV 'SearchSpace': W = V W = V is justified if B is positive definite.	'Harmonic'
	Options.Disp	Shows size of intermediate residuals and the convergence history	'no'
	Options.LSolver	Linear solver	'GMRES'
	Options.LSTol	Residual reduction linear solver	1,0.7,0.7 2,...
	Options.LSMaxIt	Maximum number it. linear solver	5
	Options.LSell	ell for BiCGstab(ell)	4
	Options.Precond	Preconditioner	identity.

For instance,

    Options = struct('Tol',1.0e-8,'LSolver','BiCGstab','LSell',4,'Precond',M);

changes the convergence tolerance to 1.0e-8, takes BiCGstab as linear solver, and takes M as preconditioner (for ways of defining M, see below).

There are a few other Options that can be specified. They are listed here.

jdqz without input arguments returns the options and its defaults.

Subsections

• Preconditioning in jdqz
• Way of using a preconditioner
• Input arguments
• Additional Options
• Accuracy