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Numerical Methods

Most equations have no closed form; numerical methods find answers to as many decimals as you like and, crucially, tell you how wrong they might be. Every iteration here is shown as a step.

درسها

Core Root finding: bisection and Newton's method Finding where f(x) = 0 to any accuracy, and how fast each method gets there. newton's method on x^2 - 2 from 1 Core Numerical integration: trapezoid and Simpson Approximating an integral from sampled values, with error bounds. integrate e^(-x^2) dx from 0 to 1 Core Interpolation and Taylor approximation Fitting polynomials through points, and approximating functions near a centre. taylor series of e^x Core Floating point and error Why 0.1 + 0.2 ≠ 0.3 on a computer, and how error propagates. 0.1 + 0.2

Symbols used here

\approx
approximately equal
Equal to the precision shown, not exactly.
f'(x),\ \frac{dy}{dx}
derivative
Instantaneous rate of change; slope of the graph.
\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
O(n^2),\ \Theta,\ \Omega
big-O notation
Grows no faster than n² (up to a constant), for large n.
x_{n+1} = x_n - \frac{f(x_n)}{f\'(x_n)}
Newton iteration
The next approximation follows the tangent to the axis.

Questions people ask

Why not just solve exactly?

Most equations have no closed-form solution at all, and many that do are unusable in practice. A numerical method delivers as many correct digits as you need, and a good one tells you how many that is.

Why can Newton's method fail?

If it starts where the tangent is nearly flat it shoots far away; near a repeated root it slows to a crawl; and with several roots it may land on the wrong one. A bracketing method like bisection is slower but cannot fail.

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