Question

Find a least squares approximating function of the form \(r_{0} x+r_{1} x^{2}+r_{2} 2^{x}\) for each of the following sets of data pairs. $$ \begin{array}{l} \text { a. }(-1,1),(0,3),(1,1),(2,0) \\ \text { b. }(0,1),(1,1),(2,5),(3,10) \end{array} $$

Short answer

The least squares approximating functions are derived by solving the constructed linear systems for each data set.

Step-by-step solution

Formulate the Linear System

To find the least squares approximation, set up the equation \(Ax = b\). Here, \(x\) is the vector of coefficients \([r_0, r_1, r_2]^T\), and \(b\) is the vector of output values from the data pairs. The matrix \(A\) consists of rows of the form \([x, x^2, 2^x]\) for each data pair.

Construct Matrix A and Vector b for Set a

For the data set \((-1,1),(0,3),(1,1),(2,0)\), calculate each row of matrix \(A\) using the respective \(x\) values: \([x, x^2, 2^x]\). Hence, \(A\) becomes \[\begin{bmatrix}-1 & 1 & \frac{1}{2} \0 & 0 & 1 \1 & 1 & 2 \2 & 4 & 4\end{bmatrix}\]and \(b\) becomes \([1, 3, 1, 0]^T\).

Solve Normal Equations for Set a

Form the normal equations \((A^TA)x = A^Tb\) and solve for \(x\) which is the vector \([r_0, r_1, r_2]^T\). Use methods like matrix inversion or numerical solvers to find \(x\).

Repeat Steps 2 and 3 for Set b

For the data set \((0,1),(1,1),(2,5),(3,10)\), construct the matrix \(A\) and vector \(b\):\[\begin{bmatrix}0 & 0 & 1 \1 & 1 & 2 \2 & 4 & 4 \3 & 9 & 8\end{bmatrix}\]and \(b = [1, 1, 5, 10]^T\). Then solve the normal equations for the least squares solution \([r_0, r_1, r_2]^T\).

Interpret the Results

After finding the least squares coefficients for set a and b, substitute \(r_0, r_1, r_2\) into the approximating function \(r_0x + r_1x^2 + r_2 2^x\). These functions represent the best fit for the given data sets.

Key concepts

Matrix Algebra

Matrix algebra is essential when dealing with the least squares approximation. It involves creating matrices and vectors that represent the data and unknowns in a problem. Let's simplify it step-by-step:

To begin, we set up a system of linear equations. For least squares approximation, we express the system in the form \(Ax = b\), where:

• \(A\) is the matrix containing terms generated from the input data, such as powers of \(x\) and computed values like \(2^x\).
• \(x\) is the vector that contains the coefficients we want to determine, represented by \([r_0, r_1, r_2]^T\).
• \(b\) is the vector containing the output or dependent variable values from the data pairs.

By organizing the data in matrix form, we simplify computations and make it easier to apply further algebraic operations, like calculating transposes or inverses.

Normal Equations

Normal equations are derived from the matrix algebra setup to solve for the least squares approximation. They create a straightforward pathway to find the desired coefficients. Here's how:

The normal equations stem from rewriting the equation \(Ax = b\). We multiply both sides by the transpose of the matrix \(A\), resulting in the system \((A^TA)x = A^Tb\). This transformation preserves the solution but allows us to deal with a symmetric, usually smaller system.

Solving the normal equations gives us the best approximation coefficients. These coefficients minimize the sum of squared differences between the observed values (from the data) and the values predicted by our function.

Numerical Solvers

Numerical solvers play a critical role when the equations become complex or the matrices become large, making analytical solutions impractical.

In the context of least squares approximation, after setting up the normal equations, we can use numerical solvers to find the solution vector \(x\). Techniques include:

• Matrix Inversion: This involves computing the inverse of \(A^TA\), although it's not always the best due to computational intensity and potential instability.
• LU Decomposition: A more stable alternative that breaks the matrix into lower and upper triangular matrices.
• Iterative Methods: Useful for very large systems, requiring fewer resources and often achieving satisfactory precision.

Depending on the context, different solvers can be more suitable, balancing the trade-off between accuracy and resource constraints.

Polynomial Functions

Polynomial functions are fundamentally significant in least squares approximation, particularly because they act as the models we're fitting to our data.

In the given exercise, the polynomial function is not a simple polynomial but a combination of terms: a linear term \(r_0x\), a quadratic term \(r_1x^2\), and an exponential term \(r_2\cdot2^x\). This particular form accommodates non-linear growth well.

Polynomial functions are advantageous for several reasons:

• They're versatile, providing a close fit to various data behaviors by adjusting degree and coefficients.
• They simplify computation, as derivatives and integrals are straightforward.
• Combination of polynomial and exponential terms in our model captures complex trends in data.

Fitting the right combination improves predictions and ensures we gain meaningful insights from our data.