scalingandsquaring

Scaling and squaring, sometimes written scaling-and-squaring, is a numerical method for computing the matrix exponential exp(A). The central idea is to reduce the magnitude of A by a scale factor 2^s, compute exp(A/2^s) with a stable small-norm approximation, and then square the result s times to recover exp(A). This approach improves numerical accuracy by working with a matrix whose norm is within a range where reliable polynomial or rational approximations are available.

The standard implementation uses a Padé approximant of exp(A/2^s), typically of symmetric order (m,m), or an equivalent polynomial form. After choosing s so that A/2^s lies in a suitable norm bound, the Padé approximation is evaluated in a stable manner, often via Horner-type recurrences. The final step is to perform s repeated squarings of the computed exp(A/2^s) to obtain exp(A). The choice of s and the Padé order m is driven by a balance between accuracy and computational cost; common practice features orders such as m = 6 or m = 13 in many libraries. The method also adapts to sparse or structured matrices to improve performance.

Applications and variants: scaling and squaring is widely used to compute the matrix exponential in scientific computing, including solving linear differential equations, control theory, and physics simulations. It is a foundational technique in libraries for matrix functions, often paired with refinements for numerical stability and backward error analysis. While primarily associated with exp(A), the scaling-and-squaring paradigm has influenced related approaches for other matrix functions and larger, structured problems.