Problem 6 - A First Course in Mathematical Modeling

Question

Use the method of Lagrange multipliers to solve Problems 3-6. A Least Squares Plane-Given the four points \(\left(x_{k}, y_{k}, z_{k}\right)\) $$ (0,0,0),(0,1,1),(1,1,1),(1,0,-1) $$ find the values of \(A, B\), and \(C\) to minimize the sum of squared errors $$ \sum_{k=1}^{4}\left(A x_{k}+B y_{k}+C-z_{k}\right)^{2} $$ if the points must lie in the plane $$ z=A x+B y+C $$

Step-by-Step Solution

Answer

A = 0, B = 1, C = 0.

Step 1: Setting up the Objective Function

We want to minimize the sum of squared errors given by: \[ S = \sum_{k=1}^{4} (A x_k + B y_k + C - z_k)^2. \] For the four given points: \((0, 0, 0), (0, 1, 1), (1, 1, 1), (1, 0, -1)\), we substitute these into the formula. So the function becomes: \[ S = (A \cdot 0 + B \cdot 0 + C - 0)^2 + (A \cdot 0 + B \cdot 1 + C - 1)^2 + (A \cdot 1 + B \cdot 1 + C - 1)^2 + (A \cdot 1 + B \cdot 0 + C - (-1))^2. \] Simplifying, \[ S = C^2 + (B + C - 1)^2 + (A + B + C - 1)^2 + (A + C + 1)^2. \]

Step 2: Differentiate with Respect to A, B, and C

To find the minimum, we differentiate the sum of squares \(S\) with respect to \(A\), \(B\), and \(C\).For \(A\), take the partial derivative: \[ \frac{\partial S}{\partial A} = 2(A + B + C - 1) + 2(A + C + 1). \] For \(B\), take the partial derivative: \[ \frac{\partial S}{\partial B} = 2(B + C - 1) + 2(A + B + C - 1). \] For \(C\), take the partial derivative: \[ \frac{\partial S}{\partial C} = 2C + 2(B + C - 1) + 2(A + B + C - 1) + 2(A + C + 1). \]

Step 3: Set Partial Derivatives to Zero

After differentiating, set each equation to zero to find the critical points:1. From \( \frac{\partial S}{\partial A} = 0 \): \[ 2A + 2B + 2C - 2 + 2A + 2C + 2 = 0 \] ⟹ \( 4A + 2B + 4C = 0 \).2. From \( \frac{\partial S}{\partial B} = 0 \): \[ 2B + 2C - 2 + 2A + 2B + 2C - 2 = 0 \] ⟹ \( 2A + 4B + 4C - 4 = 0 \).3. From \( \frac{\partial S}{\partial C} = 0 \): \[ 2C + 2B + 2C - 2 + 2A + 2B + 2C - 2 + 2A + 2C + 2 = 0 \] ⟹ \( 4A + 4B + 6C - 2 = 0 \).

Step 4: Solve the System of Equations

Now, we have a system of linear equations:1. \( 4A + 2B + 4C = 0 \).2. \( 2A + 4B + 4C = 4 \).3. \( 4A + 4B + 6C = 2 \).Solve these equations simultaneously using substitution or elimination methods:- Subtract the first equation from the third to eliminate \(4A\), yielding \( 2B + 2C = 2 \) ⟹ \( B + C = 1 \).- Substitute \( B + C = 1 \) into the second equation: \( 2A + 4(1 - C) + 4C = 4 \) ⟹ \( 2A + 4 = 4 \) ⟹ \( A = 0 \).- Substitute \( A = 0 \) into \( B + C = 1 \): Choose \(B = 1, C = 0\) as a solution.

Key Concepts

Lagrange Multipliers

Lagrange multipliers are a powerful mathematical tool used to find the extrema of functions subject to constraints. They allow us to optimize a function within a defined feasible region. This technique is particularly useful in various fields such as economics and engineering. The core idea is to introduce an extra variable, known as the Lagrange multiplier, into our optimization problem. This method transforms a constrained problem into a format that is easier to solve.

In the context of this exercise, we use Lagrange multipliers when the points must lie on a plane, thus imposing the condition \( z = Ax + By + C \). This constraint can be incorporated into the objective function \( S \) that is aimed to be minimized. Although the solution steps provided do not explicitly involve Lagrange multipliers, their theoretical foundation certainly aids deeper understanding of multilayer constraints and optimization. The technique relies on equating the gradient of the objective function and the gradient of the constraints to zero, signifying no further optimization is possible at these extrema.

Least Squares Method

The least squares method is a standard approach to find the best-fit line or plane, minimizing the discrepancies between observed data points and estimation. It's widely used in regression analysis to approximate solutions for overdetermined systems.

In mathematical modeling, the least squares method aims to minimize the sum of squared errors. This can be visualized as finding a plane that closely fits given points. The sum of squared errors can be expressed as:\[ S = \sum_{k=1}^{4} (Ax_k + By_k + C - z_k)^2\]For the points provided: (0,0,0), (0,1,1), (1,1,1), (1,0,-1), the method involves substituting these coordinates into the equation to analyze the fit. Subsequently, the process involves solving the equations derived from the partial derivatives for critical values. The simplicity and efficiency of this method make it an invaluable tool in data fitting exercises.

Linear Equations

Linear equations form the backbone of mathematical modeling. They express a linear relationship between variables without involving powers or products of the variables.

In the context of least squares, after differentiating the sum of squared errors with respect to variables \(A\), \(B\), and \(C\), we derive a system of linear equations. The linear equations are then solved to find the values that minimize the sum of squared errors. The process is illustrated in the equation set:

• \( 4A + 2B + 4C = 0 \)
• \( 2A + 4B + 4C = 4 \)
• \( 4A + 4B + 6C = 2 \)

Finding solutions to these simultaneous equations yields the optimal values of \(A\), \(B\), and \(C\) needed to describe the plane in three-dimensional space accurately. Understanding and solving these linear equations is crucial for effective application of mathematical modeling principles.