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intro numerical integration

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Numerical integration: A short introduction

outline

intro

Suppose that there is $f(x)$ with $x \in [a, b]$, where

$$\tag{1} \int_a^b f(x) dx. $$

is to be calculated.

discretization

Value of $x$ can be disretized as follow

$$\tag{2} x_i = a + \Delta x (i - 1), \ \ \ \ i = 0, 1, .., n-1, n, $$

where

$$\tag{3} \Delta x = \frac{b-a}{n} $$

is increment step.

Assignment 1. Proove that $x_0 = a$ and $x_n = b$ using (2) and (3).

formula

In general (4) can be approximated

$$\tag{4} \int_a^b f(x) dx \approx \sum_{i = 1}^n g_e(x_{i-1}, x_i) \Delta x, $$

where $g_e(x_i, x_j)$ is different for various types of Riemann sums and index $e$ stands for certain type of Rieman sum.

lhr & rhl

Left hand rule (LHR) is using

$$\tag{5} g_L(x_i, x_j) = f(x_i), $$

while

$$\tag{6} g_R(x_i, x_j) = f(x_j), $$

is used for right hand rule (RHR).

mpr & trapezoid rule

Mid point rule (MPR) is using

$$\tag{7} g_M(x_i, x_j) = f( \tfrac12 (x_i + x_j) ), $$

which is different than trapezoid rule

$$\tag{8} g_T(x_i, x_j) = \tfrac12 ( f(x_i) + f(x_j) ). $$

Notice the difference between (6) and (7).

simpson’s rule

Other approach in Riemann sum is

$$\tag{9} g_S(x_i, x_j) = \left\{ \begin{array}{cc} \tfrac13 ( f(x_i) + 2 f(x_j)), & \lfloor i / 2 \rfloor = i/2, \newline \newline \tfrac13 ( 2 f(x_i) + f(x_j)), & \lfloor i / 2 \rfloor \ne i/2, \end{array} \right. $$

which is known as Simpson’s rule. Note that $n$ or number of partitions must be even or there should be at least three points $x_i$, $x_{i+1}$, and $x_{i+2}$ for (8) to work.

refs

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