MatrixFunctions.jl

A Julia package for computing scalar and matrix-valued functions of matrix variables and their Fréchet derivatives.
Released under the MIT license, see LICENSE file for details.
Copyright (C) 2023 Aravindh Krishnamoorthy and contributors.

Note: This package is still under development. The features with check marks below are available. Those without are to be developed.

Table of Contents

• Preparation
• Implementation
• Usage
• References

Preparation

The Book [1]: Reading progress

• Chapter 1: Theory of Matrix Functions
• Chapter 4: Techniques for General Functions
• Chapter 9: Schur-Parlett Algorithm
• Chapter 13: Function of Matrix Times Vector: $f(\boldsymbol{A})\boldsymbol{b}$
• Chapter 2: Applications
• Chapter 3: Conditioning
• Others

Implementation

Matrix-valued functions of matrix variables

• Matrix function fm(f, X)
  • fm_schur_parlett_recurrence(f, X) File: schur_parlett.jl
  • ⚡ fm_schur_parlett_block(f, X) File: schur_parlett.jl
  • ...
  • Tests
  • Code optimisation

Fréchet derivatives of matrix functions

• Fréchet derivative of a matrix function fm_frechet(f, X, E)
  • fm_frechet_[...](f, X, E)
  • ...
  • Tests
  • Code optimisation

Function of matrix times vector

• Matrix times vector $f(\boldsymbol{X})\boldsymbol{y}$ fmv(f, X, y)
  • fmv_[..](f, X, y)
  • ...
  • Tests
  • Code optimisation

Matrix polynomials

• Matrix polynomial pm(p, X)
  • pm_[...](f, X, E)
  • ...
  • Tests
  • Code optimisation
• Fréchet derivative of a matrix polynomial pm_frechet(f, X, E)
  • pm_frechet_[...](f, X, E)
  • ...
  • Tests
  • Code optimisation

Usage

A Julia Pkg package is not setup yet and will be done once implementation is complete. Until then, this package can be loaded as follows.

julia> include("/<path>/MatrixFunctions.jl/src/MatrixFunctions.jl")

References

[1]	Higham, Nicholas J. Functions of matrices: theory and computation. Society for Industrial and Applied Mathematics, 2008.