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Numerical Methods Library

Clear explanations, formulas, worked examples, and an interactive solver for every major numerical analysis algorithm. Bookmark this for your coursework.

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Root Finding

Newton-Raphson Method

The fastest root-finding method for smooth functions — converges quadratically when it works.

Convergence: Quadratic (order 2)·Derivative: Required

Root Finding

Bisection Method

The simplest, most reliable root-finding algorithm — slow but guaranteed to converge.

Convergence: Linear (order 1, ~0.3 digits/iteration)·Derivative: Not needed

Root Finding

Secant Method

Newton-Raphson without the derivative — superlinear convergence using two starting points.

Convergence: Superlinear (order ≈ 1.618)·Derivative: Not needed

Root Finding

Fixed-Point Iteration

Solve f(x) = 0 by rearranging to x = g(x) — elegant but convergence depends on the rearrangement.

Convergence: Linear (rate = |g′(x*)|)·Derivative: Not needed

Linear Systems

Gaussian Elimination

The standard direct method for solving systems of linear equations — the backbone of scientific computing.

Convergence: Direct (exact in O(n³) operations)·Derivative: Not needed

Approximation

Lagrange Interpolation

Construct the unique polynomial passing exactly through any set of data points.

Convergence: Exact through all n points·Derivative: Not needed

Approximation

Simpson's Rule

Approximate definite integrals by fitting parabolas — far more accurate than the Trapezoidal Rule.

Convergence: O(h⁴) — exact for degree ≤ 3 polynomials·Derivative: Not needed

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