Numerical methods

Small numerical-method examples that demonstrate how iterative and sampled-grid algorithms look in matten. They use only the default numeric Tensor API (plus the RFC-038 comfort APIs), small hard-coded inputs, and deterministic output.

These are teaching examples, not a SciPy replacement.

Examples

35_linear_regression_gradient_descent.rs

Difficulty: Advanced-small. Fits y = w·x + b by batch gradient descent on mean-squared error. The data is stacked into a design matrix with a bias column, so predictions are X · θ and the gradient is (2/n)·Xᵀ·(ŷ - y) — one matmul for each, with transpose forming Xᵀ once. Converges to the true line y = 2x + 1.

cargo run --example 35_linear_regression_gradient_descent

Source: 35_linear_regression_gradient_descent.rs

36_heat_equation_1d.rs

Difficulty: Advanced-small. Evolves the 1D heat equation on a rod with fixed-end temperatures using the explicit (forward-Euler) finite-difference update. The stencil is encoded as a tridiagonal matrix A (with identity rows at the boundaries), so each time step is u_next = A · u. The profile converges to the steady-state straight line between the boundary temperatures.

cargo run --example 36_heat_equation_1d

Source: 36_heat_equation_1d.rs

39_finite_difference_derivative.rs

Difficulty: Intermediate. Approximates the derivative of f(x) = x³ sampled on a linspace grid using the central difference (f(x+h) − f(x−h)) / (2h). The grid and the function values are Tensors (the latter via elementwise &x * &x). For a cubic the central-difference error is exactly h², so the example shows the approximation quality directly. It is a numerical approximation, not symbolic differentiation.

cargo run --example 39_finite_difference_derivative

Source: 39_finite_difference_derivative.rs

40_trapezoidal_integration.rs

Difficulty: Intermediate. Approximates ∫₀¹ x² dx with the composite trapezoidal rule and compares against the known exact value 1/3. The grid comes from linspace, the values from elementwise squaring, and the running total from a Tensor::sum reduction. It is a numerical approximation, not an integration library.

cargo run --example 40_trapezoidal_integration

Source: 40_trapezoidal_integration.rs

What this is not

These are single-file demonstrations of accepted APIs. They do not imply that matten is an optimization library, a PDE/finite-element framework, or a SciPy replacement.