Lecture 17 Approximation Methods

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Lecture 17 - Approximation Methods

• CVEN 302
• July 17, 2002

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Lectures Goals

• Discrete Least Square Approximation(cont.)
• Nonlinear
• Continuous Least Square
• Orthogonal Polynomials
• Gram Schmidt -Legendre Polynomial
• Tchebyshev Polynomial
• Fourier Series

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Nonlinear Least Squared Approximation Method
How would you handle a problem, which is modeled
as
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Nonlinear Least Squared Approximation Method
Take the natural log of the equations
and
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Least Square Fit Approximations
Suppose we want to fit the data set.
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Linear Least Square Approximations
Use
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Least Square Fit Approximations
We would like to find the best straight line to
fit the data?
y -1.11733x 8.66608
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Nonlinear Least Square Approximations
Use
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Nonlinear Least Square Example
The equation is
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Nonlinear Least Square Approximations
Use
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Nonlinear Least Square Approximations
The exponential approximation fits the data. The
power approximation does not fit the data.
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Continuous Least Square Functions
Instead of modeling a known complex function over
a region, we would like to model the values with
a simple polynomial. This technique uses a
least squares over a continuous region. The
coefficients of the polynomial can be determined
using same technique that was used in discrete
method.
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Continuous Least Square Functions
The technique minimizes the error of the function
uses an integral.
where
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Continuous Least Square Functions
Take the derivative of the error with respect to
the coefficients and set it equal to zero.
And compute the components of the coefficient
matrix. The right hand side of the matrix will
be the function we are modeling times a x value.
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Continuous Least Square Function Example
Given the following function
from 0 to 1
Model the function with a quadratic polynomial.
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Continuous Least Square Function Example
The integral for the error is
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Continuous Least Square Function Example
The integral for the components are
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Continuous Least Square Function Example
The coefficient matrix becomes
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Continuous Least Square Function Example
The right-hand side of the equation becomes
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Continuous Least Square Function Example
The function becomes f(x) 0.3328 -2.5806x
9.1642x2
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Continuous Least Square Function

• There are other forms of equations, which can be
  used to represent continuous functions. Examples
  of these functions are
• Legrendre Polynomials
• Tchebyshev Polynomials
• Cosines and sines.

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Legendre Polynomial
The Legendre polynomials are a set of orthogonal
functions, which can be used to represent a
function as components of a function.
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Legendre Polynomial
These function are orthogonal over a range -1,
1 . This range can be scaled to fit the
function. The orthogonal functions are defined
as
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Legendre Polynomial
The Legendre functions are
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Legendre Polynomial
How would you work with a least square fit of a
function.
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Legendre Polynomial
How would you work with a least square fit of a
function.
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Legendre Polynomial
The coefficient a0 is determined by the
orthogonality of the Legendre polynomials
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Legendre Polynomial Example
Given a simple polynomial
We want to throw a loop, lets model it from 0 to
4 with f(x)
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Legendre Polynomial Example
The first step will be to scale the function
We know that at the ends are 0 and 4 for x and -1
to 1 for u so
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Legendre Polynomial Example
The coefficients are
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Legendre Polynomial Example
The Legendre functions must be adjusted to handle
the scaling
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Tchebyshev Polynomial
The Tchebyshev polynomials are another set of
orthogonal functions, which can be used to
represent a function as components of a function.
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Tchebyshev Polynomial
These function are orthogonal over a range -1,
1 . This range can be scaled to fit the
function. The orthogonal functions are defined
as
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Tchebyshev Polynomial
The Tchebyschev functions are
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Tchebyshev Polynomial
How would you work with a least square fit of a
function.
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Tchebyshev Polynomial
How would you work with a least square fit of a
function.
Rearrange
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Tchebyshev Polynomial
The coefficients are determined as
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Continuous Functions
Other forms of orthogonal functions are sines and
cosines, which are used in Fourier approximation.
The advantages for the sines and cosines are
that they can model large time scales. You will
need to clip the ends of the series so that it
will have zeros at the ends.
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Fourier Series
The Fourier series takes advantage of the
orthogonality of sines and cosines.
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Fourier Series
The time series or spatial series is generally
clipped and the resulting coefficients are
determined using Least Squared technique.
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Fast Fourier Transforms
The Fast Fourier Transforms (FFT) are discrete
form of the equations. It takes advantage of the
power of 2 to find the coefficients in the
analysis of data. So when you hear FFT, it is
technique developed by Black and Tukey in the
early 60s. To take advantage of the computer.
It is a method to analysis the series.
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Other Continuous Functions
Wavelets are another form of orthogonal
functions, which maintain the amplitude and phase
information of the series. The techniques are
used in data compression, earthquake modeling,
wave modeling, and other forms of environmental
loading. Etc.
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Summary

• Developed a technique using Least Squared
  applications to nonlinear functions.

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Summary

• Modeled the equations with continuous functions
  to describe the functions.
• Polynomials
• Legrendre Polynomials
• Tchebyshev Polynomials
• Cosines and Sines

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Homework

• Check the homework webpage