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Numerical Methods

Root finding, approximation, numerical integration and error.

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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.

மேலும் பாடங்கள்

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

இந்த பாடங்களில் சில திறந்த உரையிலிருந்து மாற்றப்பட்டுள்ளன: Wikipedia. ஒவ்வொரு பாடப் பக்கமும் அதன் மூலத்தையும் உரிமத்தையும் குறிப்பிடுகிறது.

இங்கே பயன்படுத்தப்பட்ட குறியீடுகள்

முழுமையான வரையறை, ஒரு படத்தை, மற்றும் ஒவ்வொரு எழுத்தும் என்ன அர்த்தம் என்று எந்த சின்னத்தையும் அழுத்து.

கேள்விகள்

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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தீர்ப்பாளர் பக்கத்தில் அதை எழுதவும் அல்லது வரையவும்: ஒவ்வொரு படியும், ஒரு படமாக, நீங்கள் கேட்கும் வரை பதில் மறைக்கப்படும். தீர்வு →

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