quartz/content/Obsidian Vault/Simpson's Rule.md

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#math/calculus

simpson's rule

Another way to Approximation, especially strong for approximating an especially curvy integral.

to get the integral with interval from -h to h, approximate by: #card

\frac{h}{3}(y_0+4y_1+y_2)

So that's basically like the midpoint or trapezoidal rule but curved in a primitive way. Simpson's rule is cut up into smaller pieces, like: $\int_{1}^{2} \frac{1}{x}dx$ h is split up into 10, $h=1/10$ intervals are 1,1.1,1.2, etc using simpsons looks like:

[1,1.1,1.2,1.3,1.4,1.5,1.6,1.7,1.8,1.9,2]

\frac{1}{3}* \frac{1}{10}*(1+4*1.1+1.2\ +\ 1.2+4*1.3+1.4\ +\ 1.4+4*1.5+1.6\ .....etc)

or more simply:

\frac{1}{3}* \frac{1}{10}*(1+4*1.1+2*1.2+4*1.3+2*1.4\ ... \ 4*1.9+2)

^1652282160437

error

#card

|E_S|=\frac{k(b-a)^5}{180n^4}

k is the maximum (absolute) value fourth derivative (f''''(x)=f^{(4)}) of the function $|f^{(4)}(x)|\le k$ ^1652381165392