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qz

QZ factorization for generalized eigenvalues

Syntax

Description

The qz function gives access to intermediate results in the computation of generalized eigenvalues.

[AA,BB,Q,Z] = qz(A,B) for square matrices A and B, produces upper quasitriangular matrices AA and BB, and unitary matrices Q and Z such that Q*A*Z = AA, and Q*B*Z = BB. For complex matrices, AA and BB are triangular.

[AA,BB,Q,Z,V,W] = qz(A,B) also produces matrices V and W whose columns are generalized eigenvectors.

qz(A,B,flag) for real matrices A and B, produces one of two decompositions depending on the value of flag:

'complex'
Produces a possibly complex decomposition with a triangular AA. For compatibility with earlier versions, 'complex' is the default.
'real'
Produces a real decomposition with a quasitriangular AA, containing 1-by-1 and 2-by-2 blocks on its diagonal.

If AA is triangular, the diagonal elements of AA and BB,

are the generalized eigenvalues that satisfy

The eigenvalues produced by

are the element-wise ratios of alpha and beta.

If AA is not triangular, it is necessary to further reduce the 2-by-2 blocks to obtain the eigenvalues of the full system.

Algorithm

For full matrices A and B, qz uses the LAPACK routines listed in the following table.


A and B Real
A or B Complex
A and B double
DGGES, DTGEVC (if you request the fifth output V)
ZGGES, ZTGEVC (if you request the fifth output V)
A or B single
SGGES, STGEVC (if you request the fifth output V)
CGGES, CTGEVC (if you request the fifth output V)

See Also

eig

References

[1]  Anderson, E., Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK User's Guide (http://www.netlib.org/lapack/lug/ lapack_lug.html), Third Edition, SIAM, Philadelphia, 1999.


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