Permute 3 V3 1

  1. Permute 3 V3 1000
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Compute a diagonal similarity transformation for row/column balancing.

The balancing tries to equalize the row and column 1-norms by applyinga similarity transformation such that the magnitude variation of thematrix entries is reflected to the scaling matrices.

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Moreover, if enabled, the matrix is first permuted to isolate the uppertriangular parts of the matrix and, again if scaling is also enabled,only the remaining subblocks are subjected to scaling.

The balanced matrix satisfies the following equality

4 permute 3

The scaling coefficients are approximated to the nearest power of 2to avoid round-off errors.

Parameters
A(n, n) array_like

Square data matrix for the balancing.

permutebool, optional

The selector to define whether permutation of A is also performedprior to scaling.

scalebool, optional

The selector to turn on and off the scaling. If False, the matrixwill not be scaled.

separatebool, optional

This switches from returning a full matrix of the transformationto a tuple of two separate 1D permutation and scaling arrays.

overwrite_abool, optional

This is passed to xGEBAL directly. Essentially, overwrites the resultto the data. It might increase the space efficiency. See LAPACK manualfor details. This is False by default.

Returns
B(n, n) ndarray

Balanced matrix

T(n, n) ndarray

A possibly permuted diagonal matrix whose nonzero entries areinteger powers of 2 to avoid numerical truncation errors.

scale, perm(n,) ndarray

If separate keyword is set to True then instead of the arrayT above, the scaling and the permutation vectors are givenseparately as a tuple without allocating the full array T.

Notes

This algorithm is particularly useful for eigenvalue and matrixdecompositions and in many cases it is already called by variousLAPACK routines.

Permute 3 V3 1000

The algorithm is based on the well-known technique of [1] and hasbeen modified to account for special cases. See [2] for detailswhich have been implemented since LAPACK v3.5.0. Before this versionthere are corner cases where balancing can actually worsen theconditioning. See [3] for such examples.

The code is a wrapper around LAPACK’s xGEBAL routine family for matrixbalancing.

New in version 0.19.0.

References

1(1,2)

: B.N. Parlett and C. Reinsch, “Balancing a Matrix forCalculation of Eigenvalues and Eigenvectors”, Numerische Mathematik,Vol.13(4), 1969, DOI:10.1007/BF02165404

2(1,2)

: R. James, J. Langou, B.R. Lowery, “On matrix balancing andeigenvector computation”, 2014, Available online:https://arxiv.org/abs/1401.5766

3(1,2)

: D.S. Watkins. A case where balancing is harmful.Electron. Trans. Numer. Anal, Vol.23, 2006.

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Examples