# GraphMatFun.jl

GraphMatFun.jl is a Julia package for working with computational graphs representing functions of matrices.

## Installation

The package is registered with the Julia central registry and can be installed with the command

```
julia> ]
(v1.7) pkg> add GraphMatFun
```

When installed, the package can be loaded into the current session by writing

`julia> using GraphMatFun`

## Building a first graph

Consider the following polynomial,

\[p(x) = (1+2x+x^2)^2.\]

A graph representing $p(x)$ can be manually constructed by defining the operations.

```
julia> graph = Compgraph(Float64);
julia> add_mult!(graph,:A2,:A,:A);
julia> add_lincomb!(graph,:P1,1.0,:I,2.0,:A);
julia> add_lincomb!(graph,:P2,1.0,:A2,1.0,:P1);
julia> add_mult!(graph,:P,:P2,:P2);
julia> add_output!(graph,:P);
```

The graph can be evaluated in both scalars, vectors, and matrices. For vectors the evaluation is elementwise.

```
julia> x = [1, 2, 3.5];
julia> eval_graph(graph, x)
3-element Vector{Float64}:
16.0
81.0
410.0625
julia> (1 .+ 2*x + x.^2).^2
3-element Vector{Float64}:
16.0
81.0
410.0625
julia> A = [1 3; 7 2.0];
julia> eval_graph(graph, A)
2×2 Matrix{Float64}:
1150.0 825.0
1925.0 1425.0
julia> (I + 2*A + A^2)^2
2×2 Matrix{Float64}:
1150.0 825.0
1925.0 1425.0
```

Since the graph is a polynomial the coefficients can be directly extracted.

```
julia> get_polynomial_coefficients(graph)
5-element Vector{Float64}:
1.0
4.0
6.0
4.0
1.0
```

## How do I cite it?

If you find this software useful, please cite the open access article:

```
@Article{jfr22,
author = "Jarlebring, Elias and Fasi, Massimiliano and Ringh, Emil",
title = "Computational Graphs for Matrix Functions",
journal = "ACM Trans. Math. Software",
year = 2023,
volume = 48,
number = 4,
pages = "1-35",
month = mar,
doi = "10.1145/3568991"
}
```

Previous versions of the manuscript are available as arXiv:2107.12198 [math.NA] and MIMS Eprint 2021.12.