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NUMERICAL INTEGRATION

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NUMERICAL INTEGRATION Motivation: Most such integrals cannot be evaluated explicitly. Many others it is often faster to integrate them numerically rather than ... – PowerPoint PPT presentation

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Title: NUMERICAL INTEGRATION


1
NUMERICAL INTEGRATION
  • Motivation
  • Most such integrals cannot be evaluated
    explicitly. Many others it is often faster to
    integrate them numerically rather than evaluating
    them exactly using a complicated antiderivative
    of f(x)
  • Example
  • The solution of this integral equation with
    Matlab is 1/22(1/2)pi(1/2)FresnelS(2(1/2)/p
    i(1/2)x)
  • we cannot find this solution analytically by
    techniques in calculus.

2
Course content
  • Methods of Numerical Integration
  • Trapezoidal Rules
  • 1/3 Simpsons method
  • 3/8 Simpsons method
  • Applied in two dimensional domain

3
Trapezoidal Rules
f
fp
4
  • Function f approximately by function fp. Then,
  • where fp is a linear polynomial interpolation,
    that is
  • By substitution ux-x0 we have
  • where

5
Trapezoidal Rules
f
fp
6
  • For two interval, we can use summation operation
    to derive the formula of two interval trapezoidal
    that is
  • where

7
Trapezoidal Rules
f
fp
8
  • Similar to two interval trapezoidal, we can
    derive three interval trapezoidal formula that
    is
  • where
  • Thus, for n interval we have
  • where and
  • for

9
1/3 Simpsons
f
fp
10
  • Function f approximately by function fp. Then,
  • where fp is a quadratic polynomial
    interpolation, that is
  • By substitution ux-x0 we have
  • where

11
1/3 Simpsons
f
fp
12
  • For 4 subinterval we have
  • where
  • Thus, for n subinterval we have
  • where and

13
3/8 Simpsons
fp
f
14
  • Similar to 1/3 Simpsons method, f approximately
    by function fp where fp is a cubic polynomial
    interpolation, that is
  • By substitution ux-x0 we have
  • where and

15
Numerical Integration in a Two Dimensional Domain
c(x)
d(x)
b
a
16
  • A double integration in the domain is written as
  • The numerical integration of above equation is to
    reduce to a combination of one-dimensional
    problems

17
  • Procedure
  • Step 1 Define
  • So, the solution is
  • Step 2 Divided the range of integration a,b
    into
  • N equispaced intervals with the interval size
  • So, the grid points will be denoted by
  • and then we have

18
  • Step 3 Divided the domain of integration
  • into N equispaced intervals with the interval
    size
  • So, the grid points denoted by
  • Step 4 By Applying numerical integration for
    one-dimensional (for example the trapezoidal
    rule) we have
  • for

19
  • Step 5 By applying numerical integration (for
    example trapezoidal rule) in one-dimensional
    domain we have the solution of double integration
    is
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