The scaling and squaring method is an algorithm for computing the matrix exponential by approximating
it at a scaled matrix and recovering the result by repeated
squaring. For a square matrix 
and a nonnegative integer 
,
the exact identity is
 |
(1)
|
Scaling makes a low-degree Padé approximant or Taylor polynomial effective near zero (Higham 2005, Fasi and Higham 2019).
For a rational function approximation 
to the exponential
function, put
 |
(2)
|
where 
,
provided 
is invertible. The matrix
inverse is usually avoided by solving a matrix
equation. Repeated squaring then gives
 |
(3)
|
for 
,
so 
.
The approximation degree and the scaling must
balance computational cost and accuracy. Excessive scaling can increase both cost
and roundoff error (Al-Mohy and Higham 2009).
The Fasi-Higham arbitrary-precision matrix exponential algorithm takes the unit roundoff of the arithmetic as input. It adaptively chooses between increasing the approximation degree and scaling again, using bounds on the error of the underlying approximation. It supports diagonal Padé approximants or Taylor polynomials, with or without a Schur decomposition (Fasi and Higham 2019). The Fasi-Higham positivity conjecture arose in its error analysis. Counterexamples to the unrestricted conjecture do not establish failure of the algorithm (Scardino 2026).
