# Internals

Documentation for `ExponentialAction.jl`

's internal functions.

See the Public Documentation section for documentation of the public interface.

## Index

## Internal Interface

`ExponentialAction.expv_taylor`

— Method`expv_taylor(t, A, B, degree_max; tol)`

Compute $\exp(tA)B$ using the truncated Taylor series with degree $m=$ `degree_max`

.

Instead of computing the Taylor series $T_m(tA)$ of the matrix exponential directly, its action on $B$ is computed instead.

The series is truncated early if

\[\frac{\lVert \exp(t A) B - T_m(tA) B \rVert_1}{\lVert T_m(tA) B \rVert_1} \le \mathrm{tol},\]

where $\lVert X \rVert_1$ is the operator 1-norm of the matrix $X$. This condition is only approximately checked.

`ExponentialAction.expv_taylor_cache`

— Method`expv_taylor_cache(t, A, B, degree_max, k, Z; tol)`

Compute $\exp(tkA)B$ using the truncated Taylor series with degree $m=$ `degree_max`

.

This method stores all matrix products in a cache `Z`

, where $Z_p = \frac{1}{(p-1)!} (t A)^{p-1} B$. This cache can be reused if $k$ changes but $t$, $A$, and $B$ are unchanged.

`Z`

is a vector of arrays of the same shape as `B`

and is not mutated; instead the (possibly updated) cache is returned.

**Returns**

`F::AbstractMatrix`

: The action of the truncated Taylor series`Z::AbstractVector`

: The cache of matrix products of the same shape as`F`

. If the cache is updated, then this is a different object than the input`Z`

.

See `expv_taylor`

.

`ExponentialAction.parameters`

— Method`parameters(t, A, ncols_B; kwargs...) -> (degree_opt, scale)`

Compute Taylor series parameters needed for $\exp(tA)B$.

This is Code Fragment 3.1 from ^{[AlMohyHigham2011]}.

**Keywords**

`tol`

: the desired relative tolerance`degree_max=55`

: the maximum degree of the truncated Taylor series that will be used. This is $m_{\mathrm{max}}$ in^{[AlMohyHigham2011]}, where they recommend a value of 55 in §3.`ℓ=2`

: the number of columns in the matrix that is multiplied for norm estimation (note: currently only used for control flow.). Recommended values are 1 or 2.

**Returns**

`degree_opt`

: the degree of the truncated Taylor series that will be used. This is $m^*$ in^{[AlMohyHigham2011]},`scale`

: the amount of scaling $s$ that will be applied to $A$. The truncated Taylor series of $\exp(t A / s)$ will be applied $s$ times to $B$.

- AlMohyHigham2011Al-Mohy, Awad H. and Higham, Nicholas J. (2011) Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators. SIAM Journal on Scientific Computing, 33 (2). pp. 488-511. ISSN 1064-8275 doi: 10.1137/100788860 eprint: eprints.maths.manchester.ac.uk/id/eprint/1591