Chapter 6 Linearized Nonlinear Equations 100309

Chapter 6 Linearized Nonlinear Equations 100309 - Chapter 6...

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1 Chapter 6 Least Squares Curve Fitting of Linearized Non-Linear Models Contents Introduction . ........................................................................................................................ 1 6.1 The Importance of Non-Polynomial Models . ............................................................... 2 6.2 Converting Nonlinear Models to Linear Least Squares Problems . ............................... 3 6.3 Linear Least Squares for the Exponential Model . ........................................................ 3 6.3.1 Linearizing the Relationship between the Variables . .......................................... 3 Example 6.1 Fitting Data with an Exponential Model . ................................................ 4 6.3.2 The Coefficient of Determination for Non-Linear Models . ................................. 6 6.4 Linearizing the Power Law Model . .............................................................................. 8 Example 6.2 Fitting Data with a Power Law Model . ................................................... 8 6.5 Linearizing the Saturation-Growth Model . ................................................................. 10 Example 6.3 Fitting Data with a Saturation-Growth Model . ...................................... 10 6.6 Other Non-Linear Models . .......................................................................................... 12 Example 6.4 Other Nonlinear Models . ....................................................................... 12 Chapter 6 Summary . ......................................................................................................... 15 Exercises . .......................................................................................................................... 16 Introduction Some phenomena obey a nonlinear relationship that cannot be captured by any order polynomial. As we saw in Chapter 5, polynomials are powerful, flexible functions capable of representing many different relationships between dependent and independent variables as continuous functions. The Weierstrass approximation theorem also tells us that for any analytic function on a fixed interval a polynomial exists that will represent the function to any desired degree of accuracy. This does not mean, however, that a polynomial is always the best representation of the data. When the data obey an exponential or power law relationship, for example, then polynomial models will never represent the data as efficiently. Another advantage of polynomial models is that when we perform a least squares curve fit for a polynomial model, the set of equations we must solve is a set of linear equations. For exponential, power law and other types of non-polynomial models, the least squares equations are no longer linear and we find ourselves looking at a system of nonlinear equations that must be solved simultaneously. Many nonlinear relationships, however, including the important exponential and power law relationships, are easily converted
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This note was uploaded on 01/26/2011 for the course CH E 2112 taught by Professor Dr.harwell during the Spring '10 term at The University of Oklahoma.

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Chapter 6 Linearized Nonlinear Equations 100309 - Chapter 6...

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