Lecture 25 Numerical Integration

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Lecture 25 - Numerical Integration

• CVEN 302
• October 24, 2001

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Lectures Goals

• Richardson Extrapolation
• Numerical Integration
• Trapezoid Rule
• Simpsons 1/3-Rule Quadratic
• Simpsons 3/8-Rule Cubic
• Booles Rule Fourth-order

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Richardson Extrapolation
This technique uses the concept of variable grid
sizes to reduce the error. The technique uses a
simple method for eliminating the error. Consider
a second order central difference technique.
Write the equation in the form
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Richardson Extrapolation
The central difference can be defined as
Write the equation with different grid sizes
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Richardson Extrapolation
Expand the terms
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Richardson Extrapolation
Multiply eqn 2 by 4 and subtract eqn 1 from
it.
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Richardson Extrapolation
The equation can be rewritten as
It can be rewritten in the form
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Richardson Extrapolation
The technique can be extrapolated to include the
higher order error elimination by using a finer
grid.
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Richardson Extrapolation Example
The function is given
Find the first derivative at x1.25 using a
central difference scheme and Dh 0.25.
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Richardson Extrapolation Example
The data points are
The derivatives using central difference
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Richardson Extrapolation Example
The results of the central difference scheme are
The Richardson Extrapolation uses these results
to find a better solution
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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
• 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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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
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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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Summary

• Partial Derivatives
• Richardson Extrapolation Technique

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Summary

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

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Homework

• Check the Homework webpage