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Visual Math Lab

Interactive, browser-based visualizations of numerical methods.

Author

Apurva Nakade

Published

June 25, 2026

Visual Math Lab collects interactive, browser-based visualizations for exploring computational math, probability theory, data science, and discrete mathematics.

Positive Predictive Value

Conditional Probability

Medical Testing

A standard mammogram has 80% sensitivity and 90% specificity: if you have cancer, it comes back positive 80% of the time; if you don’t, it comes back negative 90% of the…

L1 vs. L2 Regression

Robust Regression

Linear Regression

Drag the outlier severity slider (in either direction) to see the two fits diverge. L2 minimizes \(\sum_i (y_i - \hat y_i)^2\); L1 minimizes \(\sum_i |y_i - \hat y_i|\).…

Polynomial Fits

Bias-Variance Tradeoff

Polynomial Regression

Model Complexity

A low-degree polynomial (e.g. a line) can’t bend to match every point — high bias, low variance, underfitting. A high-degree polynomial can fit the training data almost…

Adaptive Methods for ODEs

Computational Math

Differential Equations

Runge-Kutta Methods

Adaptive Step Size

Initial Value Problems

The chart below shows each accepted step’s increment \(\Delta y_i = y_{i+1}^{(5)} - y_i\) across its time interval \([t_i, t_{i+1}]\), with a semi-transparent overlay of the…

Butcher Tableau Explorer

Computational Math

Differential Equations

Runge-Kutta Methods

Butcher Tableau

An \(s\)-stage Runge-Kutta method is specified by a Butcher tableau of nodes \(c_i\), coefficients \(a_{ij}\), and weights \(b_i\). Enter coefficients as integers…

Explicit Methods for ODEs

Computational Math

Differential Equations

Runge-Kutta Methods

Initial Value Problems

Given \(y' = f(t, y)\), \(y(0) = y_0\), an explicit method computes each \(y_{i+1} \approx y(t_{i+1})\) directly from already-known values. Compare Euler’s method (one slope…

Implicit Methods for ODEs

Computational Math

Differential Equations

Implicit Methods

Initial Value Problems

Implicit Euler (backward Euler) uses the slope at the end of the step, \(y_{i+1} = y_i + h\, f(t_{i+1}, y_{i+1})\), which requires solving for \(y_{i+1}\) at each step (this…

Adaptive Integration

Computational Math

Adaptive Quadrature

Error Estimation

Adaptive quadrature refines the partition of \([a,b]\) only where it’s needed, instead of the fixed grid used in Numerical Integration: it compares Simpson’s rule on \([a,b]\)…

Numerical Integration

Computational Math

Trapezoidal Rule

Approximates \(\int_a^b f(x)\, dx\) by replacing \(f\) with a simple piecewise interpolant on \(n\) equally spaced subintervals and integrating that interpolant exactly — a…

Polynomial Interpolation

Computational Math

Linear Interpolation

Connects \(n+1\) samples of \(f\) on \([a,b]\) with straight segments (linear) or with cubics matching value and first two derivatives at each node (cubic spline), then…

Bisection Method

Computational Math

Guaranteed Convergence

The bisection method finds a root of \(f(x)=0\) by repeatedly halving an interval \([a,b]\) where \(f\) changes sign, keeping whichever half still brackets the root.

Fixed Point Iteration

Computational Math

Linear Convergence

Slow Convergence

Fixed point iteration solves \(x=g(x)\) by repeatedly applying \(g\): \(x_{n+1}=g(x_n)\). If the sequence converges to a value \(\alpha\), then \(\alpha=g(\alpha)\) — a…

Newton’s Method

Computational Math

Fast Convergence

Quadratic Convergence

Newton’s method finds roots of \(f(x)=0\) by repeatedly linearizing: from \(x_n\), it follows the tangent line at \((x_n, f(x_n))\) to its \(x\)-intercept,

Secant Method

Computational Math

Superlinear Convergence

The secant method finds roots of \(f(x)=0\) like Newton’s method, but replaces the tangent line with the secant line through the two most recent approximations, so no…

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.