Math Problem Statement

Fit the model F(t) = x1 + x2t^2 + x3e^t to the observed data from file 'lkn47.txt' using least squares method. Solve for the parameters x1, x2, and x3 to four decimal places.

Solution

To determine the model parameters x1,x2,x3x_1, x_2, x_3 for the function F(t)=x1+x2t2+x3etF(t) = x_1 + x_2 t^2 + x_3 e^t, we need to solve the least-squares fitting problem using the given data.

Here is the step-by-step outline for the solution:

1. Setup

The problem involves fitting the model F(t)=x1+x2t2+x3etF(t) = x_1 + x_2 t^2 + x_3 e^t to the observed data.

From the lkn47 matrix:

  • The first row contains the time values tt.
  • The second row contains the observed values ff.

We must construct the following system: Ax=fA x = f where:

  • AA is the design matrix containing basis functions [1,t2,et][1, t^2, e^t].
  • x=[x1,x2,x3]Tx = [x_1, x_2, x_3]^T are the unknown model parameters.
  • ff is the vector of observed values.

To solve for xx, we use the normal equation: ATAx=ATf.A^T A x = A^T f.

2. Define AA and ff

From the data in the lkn47 matrix:

  • tt values are in the first row: t=[1.0000,1.1429,…,3.0000]t = [1.0000, 1.1429, \dots, 3.0000]
  • ff values are in the second row: f=[2.4627,2.9042,…,23.1668]f = [2.4627, 2.9042, \dots, 23.1668]

The design matrix AA will be constructed as follows:

\begin{bmatrix} 1 & t_1^2 & e^{t_1} \\ 1 & t_2^2 & e^{t_2} \\ \vdots & \vdots & \vdots \\ 1 & t_n^2 & e^{t_n} \end{bmatrix}.$$ The observed values vector $$f$$ is: $$f = \begin{bmatrix} f_1 \\ f_2 \\ \vdots \\ f_n \end{bmatrix}.$$ --- ### **3. Solve the Normal Equation** To solve for $$x$$, follow these steps in MATLAB: 1. Construct the $$A$$ matrix using the time values: ```matlab t = lkn47(1, :); f = lkn47(2, :)'; A = [ones(size(t))', t'.^2, exp(t')]; ``` 2. Solve the least-squares problem using the normal equation: ```matlab x = (A' * A) \ (A' * f); ``` 3. Display the result: ```matlab fprintf('x1 = %.4f\n', x(1)); fprintf('x2 = %.4f\n', x(2)); fprintf('x3 = %.4f\n', x(3)); ``` --- ### **4. Expected Output** After running the MATLAB code, the values of $$x_1, x_2, x_3$$ will be displayed, rounded to four decimal places. --- ### **Do you want me to explain how this works further, or simulate this calculation for you?** --- ### **Related Questions** 1. How do you construct a design matrix for other types of models? 2. What is the geometric interpretation of least squares fitting? 3. Why are the normal equations used to solve least squares problems? 4. How does MATLAB's `\` operator differ from the normal equation approach? 5. Can this problem be solved using polynomial regression instead? --- ### **Tip** To avoid numerical instability, use MATLAB's `pinv` or `lsqr` for solving least squares problems instead of manually using the normal equation.

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Math Problem Analysis

Mathematical Concepts

Least Squares Method
Matrix Algebra
Model Fitting
Exponential and Polynomial Functions

Formulas

F(t) = x1 + x2t^2 + x3e^t
A^T A x = A^T f

Theorems

Normal Equations for Least Squares

Suitable Grade Level

Undergraduate

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