Lecture 19 Numerical Integration

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Lecture 19 - Numerical Integration

• CVEN 302
• July 22, 2002

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Lectures Goals

• Trapezoidal Rule
• Simpsons Rule
• 1/3 Rule
• 3/8 Rule
• Midpoint
• Gaussian Quadrature

Basic Numerical Integration
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Basic Numerical Integration
We want to find integration of functions of
various forms of the equation known as the Newton
Cotes integration formulas.
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Basic Numerical Integration
Weighted sum of function values
f(x)
x
x0
x1
xn
xn-1
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Numerical Integration
Idea is to do integral in small parts, like the
way you first learned integration - a
summation Numerical methods just try to make it
faster and more accurate
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Numerical Integration

• Newton-Cotes Closed Formulae -- Use both end
  points
• Trapezoidal Rule Linear
• Simpsons 1/3-Rule Quadratic
• Simpsons 3/8-Rule Cubic
• Booles Rule Fourth-order

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Numerical Integration

• Newton-Cotes Open Formulae -- Use only interior
  points
• midpoint rule

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Trapezoid Rule

• Straight-line approximation

f(x)
L(x)
x
x0
x1
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Trapezoid Rule

• Lagrange interpolation

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Trapezoid Rule

• Integrate to obtain the rule

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ExampleTrapezoid Rule

• Evaluate the integral
• Exact solution
• Trapezoidal Rule

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Simpsons 1/3-Rule

• Approximate the function by a parabola

L(x)
f(x)
x
x0
x1
x2
h
h
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Simpsons 1/3-Rule
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Simpsons 1/3-Rule
Integrate the Lagrange interpolation
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Simpsons 3/8-Rule

• Approximate by a cubic polynomial

f(x)
L(x)
x
x0
x1
x2
h
h
x3
h
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Simpsons 3/8-Rule
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Example Simpsons Rules

• Evaluate the integral
• Simpsons 1/3-Rule
• Simpsons 3/8-Rule

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Midpoint Rule

• Newton-Cotes Open Formula

f(x)
x
a
b
xm
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Two-point Newton-Cotes Open Formula

• Approximate by a straight line

f(x)
x
x0
x1
x2
h
h
x3
h
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Three-Point Newton-Cotes Open Formula

• Approximate by a parabola

f(x)
x
x0
x1
x2
h
h
x3
h
h
x4
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Better Numerical Integration

• Composite integration
• Composite Trapezoidal Rule
• Composite Simpsons Rule
• Richardson Extrapolation
• Romberg integration

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Apply trapezoid rule to multiple segments over
integration limits
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Composite Trapezoid Rule
f(x)
x
x0
x1
x2
h
h
x3
h
h
x4
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Composite Trapezoid Rule

• Evaluate the integral

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Composite Trapezoid Example
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Composite Trapezoid Rule with Unequal Segments

• Evaluate the integral
• h1 2, h2 1, h3 0.5, h4 0.5

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Composite Simpsons Rule
Piecewise Quadratic approximations
f(x)
...
x
x0
x2
x4
h
h
xn-2
h
xn
h
x3
x1
xn-1
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Composite Simpsons Rule

• Multiple applications of Simpsons rule

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Composite Simpsons Rule

• Evaluate the integral
• n 2, h 2
• n 4, h 1

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Composite Simpsons Example
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Composite Simpsons Rule with Unequal Segments

• Evaluate the integral
• h1 1.5, h2 0.5

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Richardson Extrapolation

• Use trapezoidal rule as an example
• subintervals n 2j 1, 2, 4, 8, 16, .

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Richardson Extrapolation

• For trapezoidal rule
• kth level of extrapolation

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Romberg Integration

• Accelerated Trapezoid Rule

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Romberg Integration

• Accelerated Trapezoid Rule

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Romberg Integration Example
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Gaussian Quadratures

• Newton-Cotes Formulae
• use evenly-spaced functional values
• Gaussian Quadratures
• select functional values at non-uniformly
  distributed points to achieve higher accuracy
• change of variables so that the interval of
  integration is -1,1
• Gauss-Legendre formulae

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Gaussian Quadrature on -1, 1
x2
x1
-1
1

• Choose (c1, c2, x1, x2) such that the method
  yields exact integral for f(x) x0, x1, x2, x3

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Gaussian Quadrature on -1, 1

• Exact integral for f x0, x1, x2, x3
• Four equations for four unknowns

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Gaussian Quadrature on -1, 1

• Choose (c1, c2, c3, x1, x2, x3) such that the
  method yields exact integral for f(x) x0, x1,
  x2, x3,x4, x5

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Gaussian Quadrature on -1, 1
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Gaussian Quadrature on -1, 1

• Exact integral for f x0, x1, x2, x3, x4, x5

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Gaussian Quadrature on a, b

• Coordinate transformation from a,b to -1,1

t2
t1
a
b
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Example Gaussian Quadrature

• Evaluate
• Coordinate transformation
• Two-point formula

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Example Gaussian Quadrature

• Three-point formula
• Four-point formula

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Summary

• Integration Techniques
• Trapezoidal Rule Linear
• Simpsons 1/3-Rule Quadratic
• Simpsons 3/8-Rule Cubic
• Booles Rule Fourth-order
• Gaussian Quadrature

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Homework

• Check the Homework webpage