Question

The following are the values and standard errors of a physical quantity \(f(\theta)\) measured at various values of \(\theta\) (in which there is negligible error): \(\begin{array}{lcccc}\theta & 0 & \pi / 6 & \pi / 4 & \pi / 3 \\ f(\theta) & 3.72 \pm 0.2 & 1.98 \pm 0.1 & -0.06 \pm 0.1 & -2.05 \pm 0.1 \\ \theta & \pi / 2 & 2 \pi / 3 & 3 \pi / 4 & \pi \\ f(\theta) & -2.83 \pm 0.2 & 1.15 \pm 0.1 & 3.99 \pm 0.2 & 9.71 \pm 0.4\end{array}\) Theory suggests that \(f\) should be of the form \(a_{1}+a_{2} \cos \theta+a_{3} \cos 2 \theta\). Show that the normal equations for the coefficients \(a_{i}\) are $$ \begin{aligned} 481.3 a_{1}+158.4 a_{2}-43.8 a_{3} &=284.7 \\ 158.4 a_{1}+218.8 a_{2}+62.1 a_{3} &=-31.1 \\ -43.8 a_{1}+62.1 a_{2}+131.3 a_{3} &=368.4 \end{aligned} $$ (a) If you have matrix inversion routines available on a computer, determine the best values and variances for the coefficients \(a_{i}\) and the correlation between the coefficients \(a_{1}\) and \(a_{2}\). (b) If you have only a calculator available, solve for the values using GaussSeidel iteration starting from the approximate solution \(a_{1}=2, a_{2}=-2, a_{3}=\) 4.

Short answer

Use matrix inversion to find \( a_1, a_2, a_3 \). Alternatively, apply Gauss-Seidel starting from \( a_1 = 2, a_2 = -2, a_3 = 4 \).

Step-by-step solution

Write Down the Given Values

The values of \( f(\theta) \) and their errors are given for various \( \theta \) values. Write these values in a tabular format for easy reference.

Formulate the Function Model

According to the theory, \(f(\theta) = a_1 + a_2 \cos \theta + a_3 \cos 2\theta \). Substitute the given \( \theta \) values into this function.

Set Up the Normal Equations

Use the method of least squares to derive the normal equations. The given normal equations are already provided: \[481.3 a_{1}+158.4 a_{2}-43.8 a_{3} =284.7 \ 158.4 a_{1}+218.8 a_{2}+62.1 a_{3} =-31.1 \ -43.8 a_{1}+62.1 a_{2}+131.3 a_{3}=368.4\]

Matrix Form Representation

Represent the normal equations in matrix form: \[ \begin{pmatrix} 481.3 & 158.4 & -43.8 \ 158.4 & 218.8 & 62.1 \ -43.8 & 62.1 & 131.3 \end{pmatrix} \begin{pmatrix} a_1 \ a_2 \ a_3 \end{pmatrix} = \begin{pmatrix} 284.7 \ -31.1 \ 368.4 \end{pmatrix} \]

Solve Using Matrix Inversion (Part a)

Use a computer with matrix inversion routines to solve the matrix equation: \( \mathbf{A} \mathbf{x} = \mathbf{b} \). Calculate \( \mathbf{x} = \mathbf{A}^{-1} \mathbf{b} \) to determine the best values for \( a_1, a_2, \text{ and } a_3 \). Find the variances and correlation between the coefficients.

Gauss-Seidel Iteration (Part b)

Start with the approximate solution: \(a_1 = 2, a_2 = -2, a_3 = 4\). Use Gauss-Seidel iteration to update the values iteratively. Update each coefficient using the others' current values until the values converge.

Key concepts

Matrix Inversion

Matrix inversion is a fundamental concept in linear algebra. It involves finding a matrix, known as the inverse matrix, that, when multiplied by the original matrix, yields the identity matrix. This process is essential in solving systems of linear equations, particularly in the form \(\mathbf{A} \mathbf{x} = \mathbf{b}\).
To find the inverse of a matrix, several methods can be employed, including

• Gaussian elimination
• LU decomposition
• Adjugate method.

These methods are particularly useful when dealing with complex systems that require precise solutions.
In the context of our exercise, matrix inversion helps determine the coefficients \(a_1, a_2, a_3\) by solving the matrix equation:
\ \mathbf{A} \mathbf{x} = \mathbf{b} \
Once the inverse matrix \ \mathbf{A}^{-1} \ is found, the coefficients can be computed using:
\ \mathbf{x} = \mathbf{A}^{-1} \mathbf{b} \
This method is efficient and accurate, especially when implemented through computational software.

Gauss-Seidel Iteration

The Gauss-Seidel iteration is an iterative method for solving a system of linear equations. This method is particularly useful when a quick approximation is needed, or computational resources are limited.
Here's how it works:

• Start with an initial guess for the solution vector.
• Iteratively update each variable using the most recent values of other variables.
• Continue the process until the changes become negligibly small.

This approach is beneficial when solving our system of normal equations, especially when matrix inversion routines are unavailable. Starting with initial guesses (\(a_1 = 2, a_2 = -2, a_3 = 4\)), the Gauss-Seidel iteration can gradually refine the values until they converge to accurate solutions.
This method relies on updating each coefficient sequentially, improving the solution with each iteration.

Least Squares Method

The least squares method is a powerful statistical tool used to find the best-fitting curve or line through a set of data points. It minimizes the sum of the squares of the differences between observed values and the values provided by the model.
In our exercise, the least squares method is used to derive the normal equations for the coefficients \(a_1, a_2, a_3\). Given the theoretical model \(f(\theta) = a_1 + a_2 \cos\theta + a_3 \cos 2\theta\), we fit the model to the data points by minimizing the error.
The process involves:

• Formulating the model equation.
• Setting up the normal equations by equating partial derivatives of the error function to zero.
• Solving the normal equations to find the optimal coefficients.

This method ensures that the model best represents the given data set with minimal discrepancies.

Correlation Coefficients

Correlation coefficients measure the strength and direction of the linear relationship between two variables. They range from -1 to 1, where:

• -1 indicates a perfect negative correlation.
• 0 indicates no correlation.
• 1 indicates a perfect positive correlation.

In the context of our exercise, the correlation coefficient between coefficients (\(a_1\) and \(a_2\)) helps us understand the interdependence between these variables.
Using the coefficients found from the normal equations, calculating the correlation coefficient offers insights into how changes in one coefficient might influence the other. This is crucial for understanding the dynamics of our model and ensuring its accuracy and reliability.