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bdqz

Block-diagonal QZ decomposition

Since R2023b

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    Syntax

    [AA,BB,TL,TR,blks] = bdqz(A,B,condmax)
    [AA,BB,TL,TR] = bdqz(A,B,[],blks)

    Description

    example

    [AA,BB,TL,TR,blks] = bdqz(A,B,condmax) transforms the pair (A,B) to block-diagonal form.

    The output matrices AA, BB, TL, and TR are such that AA=TLTATR and BB=TLTBTR. AA and BB are block diagonal and each diagonal pair (AAj,BBj) is in QZ form:

    • AAj is quasi upper-triangular (real case) or upper-triangular (complex case).

    • BBj is upper-triangular with real diagonal.

    blks is a vector containing block sizes down the diagonal. Use condmax to specify an upper bound on the condition number of TL and TR. If you omit condmax, bdqz uses 1e4 as the upper bound.

    [AA,BB,TL,TR] = bdqz(A,B,[],blks) specifies the desired block sizes. For this syntax, the input pair A and B must be in real or complex QZ form (see qz).

    Examples

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    Open Live Script

    This example shows how to perform the block-diagonal QZ decomposition.

    Generate random 10-by-10 matrices with real values.

    rng(0)
A = randn(10);
B = randn(10);

    Perform the block-diagonal QZ decomposition.

    [AA,BB,TL,TR,blks] = bdqz(A,B);
AA
    AA = 10×10

    2.2815         0         0         0         0         0         0         0         0         0
         0   -1.4630         0         0         0         0         0         0         0         0
         0         0    1.0999         0         0         0         0         0         0         0
         0         0         0   -1.1764         0         0         0         0         0         0
         0         0         0         0    0.3870    0.8050         0         0         0         0
         0         0         0         0   -0.5243    0.1540         0         0         0         0
         0         0         0         0         0         0   -0.4186    0.6583         0         0
         0         0         0         0         0         0   -1.5091   -1.1076         0         0
         0         0         0         0         0         0         0         0    0.4881    0.5093
         0         0         0         0         0         0         0         0   -0.0768    0.4697


    BB
    BB = 10×10

    0.6842         0         0         0         0         0         0         0         0         0
         0    0.4721         0         0         0         0         0         0         0         0
         0         0    0.6384         0         0         0         0         0         0         0
         0         0         0    0.8556         0         0         0         0         0         0
         0         0         0         0    0.6615         0         0         0         0         0
         0         0         0         0         0    0.5652         0         0         0         0
         0         0         0         0         0         0    2.9601         0         0         0
         0         0         0         0         0         0         0    1.4418         0         0
         0         0         0         0         0         0         0         0    0.8740         0
         0         0         0         0         0         0         0         0         0    0.8314


    blks
    blks = 7×1

     1
     1
     1
     1
     2
     2
     2

    bdqz returns diagonal AA and BB. The size of the blocks in this decomposition is either 1-by-1 or 2-by-2.

    Input Arguments

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    Input matrices, specified as matrices with real or complex values.

    Complex Number Support: Yes

    Upper bound on condition number, of TL and TR matrices, specified as a positive scalar greater than 1. If you omit this argument, bdqz uses 1e4 as the upper bound.

    Use condmax to control the trade off between block size and accuracy of the decomposition. The larger condmax is, the smaller the blocks are, but TL and TR can become more ill-conditioned.

    Output Arguments

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    Block diagonal matrices, returned as matrices with real or complex values.

    For real matrices A and B, all the diagonal blocks of AA and BB pair are real. For complex matrices A and B, each block of AA and BB is associated with a cluster of nearby eigenvalues.

    For diagonalizable A and B, AA and BB are diagonal.

    Block-diagonalizing transformation, returned as matrices.

    bdqz transforms A,B to AA,BB such that AA=TLTATR and BB=TLTBTR.

    Block sizes in the block-diagonal decomposition, returned as a vector.

    Version History

    Introduced in R2023b

    See Also

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