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VisualMath

Interactive, browser-based visualizations of numerical methods.

Author

Apurva Nakade

Published

June 25, 2026

VisualMath collects interactive, browser-based visualizations for exploring numerical methods and discrete mathematics. Each page pairs short exposition with a live app — adjust an input and watch the method respond immediately, rather than reading about its behavior in the abstract.

Polynomial Fits

Bias-variance tradeoff

Polynomial regression

Model complexity

Fitting a model to data always involves a choice: how complex should the model be? A model with more free parameters can match the training data more closely — in the…

Adaptive Methods for ODEs

Computational math

Differential equations

Runge-Kutta methods

Adaptive step size

Initial value problems

The classical RK4 method in Explicit Methods advances with a fixed step size \(h = b/n\) everywhere — spending the same effort even where the solution is nearly linear. RKF45…

Butcher Tableau Explorer

Computational math

Differential equations

Runge-Kutta methods

Butcher tableau

An \(s\)-stage single-step Runge-Kutta method is specified by its Butcher tableau: nodes \(c_i\), coefficients \(a_{ij}\) (\(1 \le i, j \le s\)), and weights \(b_i\)…

Explicit Methods for ODEs

Computational math

Differential equations

Runge-Kutta methods

Initial value problems

Given an initial value problem \(y' = f(t, y)\), \(y(0) = y_0\), an explicit method advances the solution step by step: each new value \(y_{i+1} \approx y(t_{i+1})\) is…

Implicit Methods for ODEs

Computational math

Differential equations

Implicit methods

Initial value problems

Given an initial value problem \(y' = f(t, y)\), \(y(0) = y_0\), Euler’s method (forward/explicit Euler) uses the slope at the start of the step:

Adaptive Integration

Computational math

Adaptive quadrature

Error estimation

A fixed grid of \(n\) equally spaced subintervals, as in Numerical Integration, spends the same effort everywhere — even on parts of \([a,b]\) where \(f\) is nearly linear…

Numerical Integration

Computational math

Trapezoidal rule

The definite integral \(\int_a^b f(x)\, dx\) can be approximated by replacing \(f\) with a simple piecewise interpolant on \(n\) equally spaced subintervals \([x_0, x_1]…

Linear vs. Cubic Spline Interpolation

Computational math

Linear interpolation

Given a function \(f\) on \([a, b]\), sample it at \(n+1\) equally spaced nodes \(x_0 < x_1 < \dots < x_n\) and connect the samples \((x_i, f(x_i))\) with simple pieces. A li…

Bisection Method

Computational math

Guaranteed convergence

The bisection method finds a root by repeatedly halving an interval \([a, b]\) where \(f\) changes sign, guaranteeing that each half still contains a root. At each step, the…

Fixed Point Iteration

Computational math

Linear convergence

Slow convergence

Fixed point iteration is a way to solve an equation after rewriting it in the form

Newton’s Method

Computational math

Fast convergence

Quadratic convergence

Newton’s method is one of the most common methods for solving equations of the form Newton’s method finds roots of \(f(x) = 0\) by repeatedly linearizing the function…

Secant Method

Computational math

Superlinear convergence

The secant method finds roots of \(f(x) = 0\) the way Newton’s method does — by repeatedly intersecting a line with the \(x\)-axis — but it replaces the tangent line with a…

Sperner’s Lemma - Geometric Proof

This is my new favorite proof of Sperner’s lemma. In the following app the vertices are slowly morphed to the “cardinal” R, G, B vertices with time while preserving the…

Sperner’s Lemma - Combinatorial Proof

combinatorial proof

This picture is a combinatorial proof of Sperner’s Lemma.

Sperner’s Lemma - Statement

Consider a triangulation of a triangle. (Below, we show a regular triangulation of an equilateral triangle, but the theorem applies to any triangulation of a triangle.) We…

Barycentric Coordinates

barycentric coordinates

Barycentric coordinates are a coordinate system for describing points inside a triangle.