Lecture 17 - Approximation Methods CVEN 302 July 17, 2002

Lecture’s Goals Discrete Least Square Approximation (cont.) –Nonlinear Continuous Least Square –Orthogonal Polynomials –Gram Schmidt -Legendre Polynomial –Tchebyshev Polynomial –Fourier Series

Nonlinear Least Squared Approximation Method How would you handle a problem, which is modeled as:

Nonlinear Least Squared Approximation Method Take the natural log of the equations and

Least Square Fit Approximations Suppose we want to fit the data set.

Linear Least Square Approximations Use:

Least Square Fit Approximations We would like to find the best straight line to fit the data? y = x

Nonlinear Least Square Approximations Use:

Nonlinear Least Square Example The equation is:

Nonlinear Least Square Approximations Use:

Nonlinear Least Square Approximations The exponential approximation fits the data. The power approximation does not fit the data.

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.

Continuous Least Square Functions The technique minimizes the error of the function uses an integral. where

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.

Continuous Least Square Function Example Given the following function: Model the function with a quadratic polynomial. from 0 to 1

Continuous Least Square Function Example The integral for the error is:

Continuous Least Square Function Example The integral for the components are:

Continuous Least Square Function Example The coefficient matrix becomes:

Continuous Least Square Function Example The right-hand side of the equation becomes:

Continuous Least Square Function Example The function becomes : f(x) = x x 2

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.

Legendre Polynomial The Legendre polynomials are a set of orthogonal functions, which can be used to represent a function as components of a function.

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:

Legendre Polynomial The Legendre functions are:

Legendre Polynomial How would you work with a least square fit of a function.

Legendre Polynomial How would you work with a least square fit of a function.

Legendre Polynomial The coefficient a 0 is determined by the orthogonality of the Legendre polynomials:

Legendre Polynomial Example Given a simple polynomial: We want to throw a loop, let’s model it from 0 to 4 with f(x):

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

Legendre Polynomial Example The coefficients are

Legendre Polynomial Example The Legendre functions must be adjusted to handle the scaling:

Tchebyshev Polynomial The Tchebyshev polynomials are another set of orthogonal functions, which can be used to represent a function as components of a function.

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:

Tchebyshev Polynomial The Tchebyschev functions are:

Tchebyshev Polynomial How would you work with a least square fit of a function.

Tchebyshev Polynomial How would you work with a least square fit of a function. Rearrange

Tchebyshev Polynomial The coefficients are determined as:

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.

Fourier Series The Fourier series takes advantage of the orthogonality of sines and cosines.

Fourier Series The time series or spatial series is generally clipped and the resulting coefficients are determined using Least Squared technique.

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 60’s. To take advantage of the computer. It is a method to analysis the series.

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.

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

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

Homework Check the homework webpage