Computational Physics

The first four programs solve linear systems by Gaussian elimination, successive over-relaxation, and the Thomas algorithm, or compute a small matrix inverse by repeated solves. The remaining programs find one dominant eigenvalue, one eigenvalue near a chosen shift, or all eigenvalues of a real symmetric matrix, together with their eigenvectors. Each program is available in MATLAB, Python, C++, and Fortran. Start with the matrix structure and the quantity the physics problem requires: a solution, a selected eigenmode, or the full spectrum. These seven programs are a focused selection from the chapter. The matrices in other printed examples can also be studied with the library routines listed on the resources page, although their iteration histories may differ.

The linear-system programs report a residual or an inverse check against the original matrix. A small residual does not by itself guarantee an accurate solution when the matrix is ill-conditioned. The eigenvalue programs check Av − λv; Jacobi also checks eigenvector orthogonality. The Thomas and matrix inverse examples do not pivot, so their chosen matrices are important. The Chapter 8 resources give further guidance on choosing a solver and checking results. Several default matrices, including those for SOR, inversion, and eigenvalues, differ from the book’s printed examples. Compare each program with its own documented input and reference result.

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