The Padé iterations for the matrix sign function and their reciprocals are optimal

What is it about?

It is possible to define extensions of several functions to a matrix argument; for instance, the square root, or the sign function. One of the strategies used to compute them is approximating them with ratios of polynomials. We show that, among all possible approximations of this kind for the sign function, the best convergence speed is attained by two families of rational functions: one, the so-called Padé family of iterations, is in common use; we introduce and identify the second one, which is a variant that we call reciprocal Padé family.

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Why is it important?

The matrix sign function appears in several applications, ranging from quantum physics to control engineering and probability. It is strictly related to the matrix square root A^{1/2}. Identifying the best methods to compute it ensures that these problems can be computed with the maximum accuracy. This work not only confirms that the approximations in common use were already the best ones, but, more importantly, it identifies new ones with the same optimal features, opening new possibilities for numerical computations. Later, K. Ziȩtak has found a similar family of approximations for the p-sector, a matrix function that generalizes the sign.

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