[SciPy-user] Matrix sign function

[Nils Wagner]

Pearu Peterson schrieb:
> 
> On Wed, 24 Sep 2003, Nils Wagner wrote:
> 
> > It would be nice to have the matrix sign function in scipy.
> 
> Try the following:
> 
>   signm=lambda a:linalg.funm(a,lambda v:sign(v)+0j)
> 
> This assumes that `a` has real eigenvalues. Otherwise one most use some
> other sign function that is properly defined for complex values, the
> current sign is acctually
>   v |-> sign(real(v))
> 
> Pearu
>

To account for complex values, a generalized sign matrix can be defined
as follows

sign(v) = \left\{ \begin{array}{c} 1 \Re v > 0 \\ 0 \Re v = 0 \\ -1 \Re
v < 0 \end{array}\right.

Details are given in the reference (previous e-mail). Anyway, I think
scipy should have a generalized
sign function as well.

A suitable test case is the following matrix

array([[ 29.2, -24.2,  69.5,  49.8,   7. ],
       [ -9.2,   5.2, -18. , -16.8,  -2. ],
       [-10. ,   6. , -20. , -18. ,  -2. ],
       [ -9.6,   9.6, -25.5, -15.4,  -2. ],
       [  9.8,  -4.8,  18. ,  18.2,   2. ]])

Any suggestion for the implementation of a generalized matrix sign
function ?

BTW, Davies and Higham have published an interesting paper on matrix
functions. They have improved Matlab's function funm.


Davies, P.~I., Higham, N.~J.
A Schur-Parlett algorithm for computing matrix functions,
SIAM Journal Matrix Analysis and Applications, Vol. 25 No.2 pp. 464-485


Nils
  
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