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LightKrylov

Targeting large-scale linear algebra applications where the matrix $\mathbf{A}$ is only defined implicitly (e.g. through a call to a matvec subroutine), this package provides lightweight Fortran implementations of the most useful Krylov methods to solve a variety of problems, among which:

1. Eigenvalue Decomposition $$\mathbf{A} \mathbf{x} = \lambda \mathbf{x}$$

2. Singular Value Decomposition $$\mathbf{A} = \mathbf{U} \boldsymbol{\Sigma} \mathbf{V}^T$$

3. Linear system of equations $$\mathbf{Ax} = \mathbf{b}$$

Krylov methods are particularly appropriate in situations where such problems must be solved but factorizing the matrix $\mathbf{A}$ is not possible because:

• $\mathbf{A}$ is not available explicitly but only implicitly through a matvec subroutine computing the matrix-vector product $\mathbf{Ax}$.
• $\mathbf{A}$ or its factors (e.g. LU or Cholesky) are dense and would consume an excessive amount of memory.

Krylov methods are iterative methods, i.e. they iteratively refine the solution of the problem until a desired accuracy is reached. While they are not recommended when a machine-precision solution is needed, they can nonetheless provide highly accurate approximations of the solution after a relatively small number of iterations. Krylov methods form the workhorses of large-scale numerical linear algebra.

Capabilities

LightKrylov leverages Fortran's abstract type feature to provide generic implementations of the various Krylov methods. The only requirement from the user to benefit from the capabilities of LightKrylov is to extend the abstract_vector and abstract_linop types to define their notion of vectors and linear operators. LightKrylov then provides the following functionalities:

• Krylov factorizations : arnoldi, lanczos, bidiagonalization.
• Spectral analysis : eigs, eighs, svds.
• Linear systems : gmres, cg.
• Nonlinear system: newton.

To date, LightKrylov can handle real and complex-valued vectors and linear operators, using both single and double precision arithmetic. This was made possible thanks to fypp, a python powered Fortran meta programming utility.

Examples

Some examples can be found in the example folder. These include:

• Ginzburg-Landau : Serial computation of the leading eigenpairs of a complex-valued linear operator via time-stepping.
• Laplace operator : Parallel computation of the leading eigenpairs of the Laplace operator defined on the unit-square.
• Roessler system : Computation of an unstable periodic orbit embedded in the strange attractor of the system along with an OTD analysis of this orbit.

Alternatively, you can also look at neklab, a bifurcation and stability analysis toolbox based on LightKrylov and designed to augment the functionalities of the massively parallel spectral element solver Nek5000.

Ginzburg-Landau	Laplace operator	Roesler system
ADD FIGURE	ADD FIGURE	ADD FIGURE

Installation

Provided you have git installed, getting the code is as simple as:

git clone https://github.com/nekStab/LightKrylov

Alternatively, using gh-cli, you can type

gh repo clone nekStab/LightKrylov

Dependencies

LightKrylov has a very minimal set of dependencies. These only include:

• a Fortran compiler,
• fpm for building the code.

All other dependencies are directly handled by the Fortran Package Manage fpm. To date, the tested compilers include:

• gfortran 12 (Linux)
• gfortran 13 (Linux, Windows, MacOS)
• ifort (Linux)
• ifx (Linux)

Building with fpm

Provided you have cloned the repo, installing LightKrylov with fpm is as simple as

fpm build --profile release

To install it and make it accessible for other non-fpm related programs, simply run

fpm install --profile release

Both of these will make use of the standard compilation options set by the fpm team. Please refer to their documentation (here) for more details.

Running the tests

To see if the library has been compiled correctly, a set of unit tests are provided in the test folder. Run the following command.

fpm test

If everything went fine, you should see

All tests successfully passed!

If not, please feel free to open an Issue.

Running the examples

To run the examples:

fpm run --example

This command will run all of the examples sequentially. You can alternatively run a specific example using e.g.

fpm run --example Ginzburg-Landau

For more details, please refer to each of the examples.

Contributing

Current developers

LightKrylov is currently developed and maintained by a team of three:

• Jean-Christophe Loiseau : Assistant Professor of Applied maths and Fluid dynamics at DynFluid, Arts et Métiers Institute of Technology, Paris, France.
• Ricardo Frantz : PhD in Fluid dynamics (Arts et Métiers, France, 2022) and currently postdoctoral researcher at DynFluid.
• Simon Kern : PhD in Fluid dynamics (KTH, Sweden, 2023) and currently postdoctoral researcher at DynFluid.

Anyone else interested in contributing is obviously most welcomed!

Acknowledgment

The development of LightKrylov is part of an on-going research project funded by Agence Nationale pour la Recherche (ANR) under the grant agreement ANR-22-CE46-0008. The project started in January 2023 and will run until December 2026. We are also very grateful to the fortran-lang community and the maintainers of stdlib, in particular to @perazz, @jalvesz and @jvdp1 for their awesome work on the stdlib_linalg module which greatly simplified the developlement of LightKrylov.

Related projects

LightKrylov is the base package of our ecosystem. If you like it, you may also be interested in :

• LightROM : a lightweight Fortran package providing a set of functions for reduced-order modeling, control and estimation of large-scale linear time invariant dynamical systems.
• neklab : a bifurcation and stability analysis toolbox based on LightKrylov for the massively parallel spectral element solver Nek5000.