Most equations worth solving have no tidy formula for the answer. There is no algebra that spits out the root of \cos x = x, no closed form for most integrals, no exact solution to the differential equations that describe weather or spacecraft. Numerical analysis is the science of computing the answer anyway — to as many decimal places as you like — with arithmetic a machine can actually perform.

It is the bridge between pure mathematics and the working computer. Every weather forecast, every engineering simulation, every rendered film frame and every trained machine-learning model runs on numerical methods. The craft is twofold: invent an algorithm that converges to the answer, and understand the error — because a computer that only knows finitely many digits will always be a little bit wrong, and a careless method can be catastrophically so.

The big idea: approximate, then control the error

One thread runs through everything here. Replace the impossible exact problem with a nearby one you can solve — a straight line for a curve, a finite sum for an integral, a few steps for a smooth flow — then ask two questions relentlessly: how fast does the approximation approach the truth as you work harder (convergence), and how much do the tiny rounding errors of finite arithmetic pile up along the way (stability)? Good numerical analysis is exactly the discipline of keeping both under control.

The shape of the journey

This course moves in four stages, each building on the last.

• Stage A — Foundations. How computers store numbers, where error comes from, and the iterative methods that hunt down a root.
• Stage B — Fitting and integrating. Passing curves through data and computing integrals when the antiderivative refuses to appear.
• Stage C — Linear systems. Solving many equations at once, exactly and iteratively, and knowing when the problem is treacherous.
• Stage D — Differential equations. Marching solutions of ODEs and PDEs forward through time and space.

Stage A — Foundations

1. Floating-Point and Error
2. Root Finding: Bisection and Newton
3. Fixed-Point Iteration

Stage B — Fitting and integrating

1. Polynomial Interpolation
2. Splines
3. Numerical Integration (Quadrature)

Stage C — Linear systems

1. Gaussian Elimination and LU Decomposition
2. Iterative Methods and Conditioning

Stage D — Differential equations

1. Numerical Solutions of ODEs
2. Numerical Methods for PDEs

Let's get started

We begin with the ground truth of all computation: how a machine actually stores a number, and why that means every answer carries a little error you must learn to measure and tame.

Let's get started → Floating-Point and Error