maths.free › Numerical Methods

Numerical Methods

Root finding, approximation, numerical integration and error.

Lärdomar

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

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.

Fler lärdomar

Finding roots

Numerical analysis Newton's method Bisection method Secant method Fixed-point iteration Rate of convergence Root-finding algorithm

Integration and interpolation

Simpson's rule Trapezoidal rule Gaussian quadrature Interpolation Polynomial interpolation Spline interpolation Least squares Curve fitting

Linear systems and ODEs

Gaussian elimination LU decomposition QR decomposition Condition number Iterative method Euler method Runge–Kutta methods Finite difference method Finite element method

Error

Floating-point arithmetic Round-off error Truncation error Numerical stability Machine epsilon Catastrophic cancellation

Några av dessa lärdomar är anpassade till öppna texter: Wikipedia. Varje lektion sida namnger dess källa och licens.

Symboler som används här

Knacka på alla symboler för den fullständiga definitionen, en bild, och vad varje bokstav i den betyder.

Frågor folk frågar

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.

Har du några egna problem?
Skriv in det eller rita det på lösaren sida: varje steg, en bild, svaret dolt tills du frågar. Lös →

Övriga filialer

＋ Arithmetic 𝑥 Algebra △ Geometry ∿ Trigonometry ƒ Precalculus σ Statistics & Probability ⊢ Discrete Math & Logic ∫ Calculus ⊞ Linear Algebra ℕ Number Theory ẏ Differential Equations ∇ Multivariable Calculus ε Real Analysis ⋔ Combinatorics & Graph Theory ∈ Set Theory & Logic 𝔾 Abstract Algebra 𝑃 Probability Theory ℂ Complex Analysis ◎ Topology μ Measure Theory ‖·‖ Functional Analysis κ Differential Geometry π₁ Algebraic Topology → Category Theory ∂ Partial Differential Equations ≋ Fluid Dynamics ∞ Frontiers