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Numerical Methods
Root finding, approximation, numerical integration and error.
经验教训
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.
更多经验教训
Finding roots
Integration and interpolation
Linear systems and ODEs
Error
其中一些经验教训从开放文本中改编: Wikipedia. 每页课目都列出其来源和执照。
此处所用符号
输入完整定义、图片和其中每个字母的含义的符号。
人们对人们提出的问题
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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