## The Paterson–Stockmeyer method for evaluating polynomials and rational functions of matrices

According to Moler and Van Loan, truncating the Taylor series expansion to the exponential at 0 is the least effective of their nineteen dubious ways to compute the exponential of a matrix. Such an undoubtedly questionable approach can, however, become a powerful tool for evaluating matrix functions when used in conjunction with other strategies, such as, for example, the scaling and squaring technique. In fact, truncated Taylor series are just a special case of a much larger class of rational approximants, known as the Padé family, on which many state-of-the-art algorithms rely in order to compute matrix functions.

A customary choice for evaluating these approximants at a matrix argument is the Paterson–Stockmeyer method, an evaluation scheme that was originally proposed as an asymptotically optimal algorithm for evaluating polynomials of matrices, but generalizes quite naturally to rational functions, which are nothing but the solutions of a linear system whose coefficients and right-hand side are polynomials of the same matrix. This technique exploits a clever rewriting of a polynomial in $A$ as a polynomials in $A^s$, for some positive integer $s$, and overall requires about $2\sqrt{k}$ matrix products to evaluate a polynomial of degree $k$. Number of matrix multiplications required to evaluate a polynomial of degree between 0 and 50.

As shown in the figure, when the Paterson–Stockmeyer scheme is used the number of matrix multiplications required to evaluate a polynomial of degree $k$ grows slower than $k$ itself, with the result that evaluating polynomials of different degree will asymptotically have the same cost. For example, evaluating a polynomial of any degree between 43 and 49 requires 12 matrix multiplications, thus there is little point in considering an approximant of degree 43 when evaluating that of degree 49 has roughly the same cost but will in all likelihood deliver a more accurate result.

When designing algorithms to evaluate functions of matrices, one is interested in finding the optimal degrees, those marked with a red circle in the figure above, since they guarantee maximal accuracy for a given computational cost. When fixed precision is considered, finding all such degrees is not a problem: a backward error analysis can be used to determine the maximum degree that will ever be needed, $m_{max}$ say, and then looking at the plot is enough to find all the optimal degrees smaller than $m_{max}$. In order to deal with arbitrary precision arithmetic, however, a different strategy is needed, as depending on the working precision and the required tolerance, approximants of arbitrarily high degree may be needed. The new Eprint Optimality of the Paterson–Stockmeyer Method for Evaluating Matrix Polynomials and Rational Matrix Functions studies the cost of the Paterson–Stockmeyer method for polynomial evaluation and shows that a degree is optimal if and only if it is a quarter-square, that is, a number of the form $\lfloor n^2/4 \rfloor$ for some nonnegative integer $n$, where $\lfloor \cdot \rfloor$ is the floor function.

Similar results can be obtained for Paterson–Stockmeyer-like algorithms for evaluating diagonal Padé approximants, rational functions whose numerator and denominator have same degree. In that case, one can show that an approximant is optimal if and only if the degree of numerator and denominator is an eight-square, an integer of the form $\lfloor n^2/8 \rfloor$ for some $n \in \mathbb{N}$. In particular, for diagonal Padé approximants to the exponential, due to a symmetry of the coefficients of numerator and denominator, faster algorithms can be developed, and an explicit formula—not as nice as that in the two previous cases—can be derived for the optimal orders of these approximants.