Implement parallel GMRES solver with MPI support

[Copilot]

Adds a standalone GMRES(m) iterative solver for sparse linear systems with MPI parallelization, as requested in the issue referencing the GITM Fortran implementation.

Implementation

Core solver (src/solver/)

• Full GMRES(m) with configurable restart parameter
• Arnoldi iteration with Modified Gram-Schmidt orthogonalization
• Givens rotations for QR factorization
• Matrix-free interface via function pointer callback
• MPI-parallel vector operations (dot products, norms via MPI_Allreduce)

API design

// Configure solver
gmres_config_t config;
gmres_config_init(&config, MPI_COMM_WORLD);
config.max_iter = 1000;
config.restart = 30;
config.tol = 1e-6;

// Solve Ax = b with user-defined matvec
gmres_result_t result;
gmres_solve(&A, user_matvec_fn, b, x, n_local, &config, &result);

Test program

• 1D Laplacian with Dirichlet BCs validates correctness
• Parallel tests with 1, 2, 4 processes confirm MPI implementation
• Discretization error ~6e-5 matches analytical solution

Build integration

• Optional CMake component (requires MPI)
• Standalone Makefile for independent use
• Includes comprehensive documentation and security analysis

Memory safety

• NULL-initialized pointers with goto cleanup pattern
• Allocation failure handling on all paths
• No buffer overflows or double-free vulnerabilities

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Original prompt

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This section details on the original issue you should resolve

<issue_title>Implement a parallel GMRES solver in C</issue_title>
<issue_description>Please consult the following code for implementing a similar standalone GMRES solver in C with MPI support

https://github.com/aaronjridley/GITM/blob/476f28d7cf72a39bfc356ce1dce70f8ed9612533/share/Library/src/ModLinearAdvection.f90
</issue_description>

Comments on the Issue (you are @copilot in this section)

• Fixes Implement a parallel GMRES solver in C #55

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