Implementing the Thomas Algorithm for Diffusion Equation Steps

Summary

The document explains a C++ implementation of the Thomas algorithm, a specialized form of Gaussian elimination for solving tridiagonal linear systems. It describes the diagonal and off-diagonal coefficient vectors, the forward sweep that computes modified coefficients, and the reverse sweep that recovers the solution vector. Temporary vectors hold intermediate values, while the solution is written into a caller-provided vector.

A sample program applies the solver to one Crank–Nicolson time step for a discretized one-dimensional heat equation. Its output shows an initial localized profile spreading across the grid after the step. The example illustrates numerical computing that can also support later financial equation work, such as Black–Scholes models, but it is not itself a trading strategy or market study. The implementation notes that temporary storage is allocated on every call, which is inefficient for repeated time steps, and flags division-by-zero checks as an omitted safeguard. It also assumes valid coefficient dimensions and a nonsingular system.

Key ideas

• The Thomas algorithm solves tridiagonal systems using forward elimination followed by backward substitution.
• Separate vectors represent the lower diagonal, main diagonal, upper diagonal, and right-hand side.
• The example applies the solver to a Crank–Nicolson discretization of the one-dimensional heat equation.
• Repeated allocation of temporary vectors can be avoided by reusing external workspace.
• A robust implementation should check divisors and validate system dimensions.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.