scipy.linalg.matrix_balance

scipy.linalg.matrix_balance(A, permute=True, scale=True, separate=False, overwrite_a=False)

A wrapper around LAPACK’s xGEBAL routine family for matrix balancing.

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

Moreover, if enabled, the matrix is first permuted to isolate the upper triangular 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

\[B = T^{-1} A T\]

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

Parameters:

A : (n, n) array_like

Square data matrix for the balancing.

permute : bool, optional

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

scale : bool, optional

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

separate : bool, optional

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

overwrite_a : bool, optional

This is passed to xGEBAL directly. Essentially, overwrites the result to the data. It might increase the space efficiency. See LAPACK manual for 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 are integer powers of 2 to avoid numerical truncation errors.

scale, perm : (n,) ndarray

If separate keyword is set to True then instead of the array T above, the scaling and the permutation vector is given separately without allocating the full array T.

New in version 0.19.0.

Notes

This algorithm is particularly useful for eigenvalue and matrix decompositions and in many cases it is already called by various LAPACK routines.

The algorithm is based on the well-known technique of [R108] and has been modified to account for special cases. See [R109] for details which have been implemented since LAPACK v3.5.0. Before this version there are corner cases where balancing can actually worsen the conditioning. See [R110] for such examples.

References

[R108]

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

[R109]

(1, 2) : R. James, J. Langou, B.R. Lowery, “On matrix balancing and eigenvector computation”, 2014, Available online: http://arxiv.org/abs/1401.5766

[R110]

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

Examples

>>> from scipy import linalg
>>> x = np.array([[1,2,0], [9,1,0.01], [1,2,10*np.pi]])

>>> y, permscale = linalg.matrix_balance(x)
>>> np.abs(x).sum(axis=0) / np.abs(x).sum(axis=1)
array([ 3.66666667,  0.4995005 ,  0.91312162])

>>> np.abs(y).sum(axis=0) / np.abs(y).sum(axis=1) # 1-norms approx. equal
array([ 1.10625   ,  0.90547703,  1.00011878])

>>> permscale  # only powers of 2 (0.5 == 2^(-1))
array([[  0.5,   0. ,   0. ],
       [  0. ,   1. ,   0. ],
       [  0. ,   0. ,  16. ]])