Question

Use Jacobi's method to solve $$ \begin{aligned} 4 x-y &=-9.4 \\ 3 x+5 y &=7.9 \end{aligned} $$ Take \(x_{0}=-1, y_{0}=1.5\) and perform five iterations.

Short answer

Answer: After using Jacobi's method, the approximate solutions for x and y after five iterations are x ≈ -2.075 and y ≈ 1.

Step-by-step solution

Rewrite the equations to isolate x and y

We will rewrite the given equations to isolate x and y: Equation 1: \(4x - y = -9.4\) Add y to both sides and divide by 4: \(x = \frac{1}{4}(y - 9.4)\) Equation 2: \(3x + 5y = 7.9\) Subtract 5y from both sides and divide by 3: \(x = \frac{1}{3}(7.9 - 5y)\) Now we have the equations in terms of x and y: \(x_{k+1} = \frac{1}{4}(y_{k} - 9.4)\) \(x_{k+1} = \frac{1}{3}(7.9 - 5y_{k})\)

Initialize x and y values

The initial values for x₀ and y₀ are given as: \(x_{0} = -1\) \(y_{0} = 1.5\)

Perform five iterations

Now, we will perform five iterations of Jacobi's method, updating x and y at each iteration. Iteration 1: \(x_{1} = \frac{1}{4}(y_{0} - 9.4) = \frac{1}{4}(1.5 - 9.4) = -1.975\) \(y_{1} = \frac{1}{3}(7.9 - 5y_{0}) = \frac{1}{3}(7.9 - 5(1.5)) = 0.3\) Iteration 2: \(x_{2} = \frac{1}{4}(y_{1} - 9.4) = \frac{1}{4}(0.3 - 9.4) = -2.275\) \(y_{2} = \frac{1}{3}(7.9 - 5y_{1}) = \frac{1}{3}(7.9 - 5(0.3)) = 1.3\) Iteration 3: \(x_{3} = \frac{1}{4}(y_{2} - 9.4) = \frac{1}{4}(1.3 - 9.4) = -2.025\) \(y_{3} = \frac{1}{3}(7.9 - 5y_{2}) = \frac{1}{3}(7.9 - 5(1.3)) = 0.9\) Iteration 4: \(x_{4} = \frac{1}{4}(y_{3} - 9.4) = \frac{1}{4}(0.9 - 9.4) = -2.125\) \(y_{4} = \frac{1}{3}(7.9 - 5y_{3}) = \frac{1}{3}(7.9 - 5(0.9)) = 1.1\) Iteration 5: \(x_{5} = \frac{1}{4}(y_{4} - 9.4) = \frac{1}{4}(1.1 - 9.4) = -2.075\) \(y_{5} = \frac{1}{3}(7.9 - 5y_{4}) = \frac{1}{3}(7.9 - 5(1.1)) = 1\) After five iterations, we have the following approximate solutions for x and y: \(x_{5} = -2.075\) \(y_{5} = 1\)

Key concepts

Iterative Methods

Iterative methods are a core part of numerical analysis, especially when solving systems of linear equations. These methods, as the name suggests, involve iterating, or repeating, a sequence of calculations. The aim is to gradually converge on a solution through successive approximations.

For example, with Jacobi's method, initial guesses for the values of the variables in the system are made, and these are used to calculate new guesses. Each guess is based on the previous one, hence the process is 'iterative'. This is ideal for large systems where using direct methods like Gaussian elimination can be computationally expensive.

The method requires a few key decisions: a good choice of initial guess, which can drastically affect the speed of convergence, and a stopping criterion, typically a threshold below which the difference between subsequent approximations is considered negligible. It’s also important to have a system that has a unique solution and is diagonally dominant, or the method may not converge.

By applying an iterative method, students can develop an intuition for the behavior of linear systems and the impact of different variables and configurations on the solution process.

Systems of Linear Equations

In mathematics, systems of linear equations are collections of two or more linear equations involving the same set of variables. The solution to a system is the set of values that satisfies all equations simultaneously.

Linear systems can be represented in multiple ways, including graphically where the solution is the point(s) at which the lines or planes intersect. Algebraically, they might appear in matrix form. These systems underpin a vast number of applications, from engineering to economics.

Solutions are determined by the system's characteristics—consistent and independent systems have a single unique solution, underdetermined systems have infinitely many solutions, and inconsistent systems have no solution. For the Jacobi's method exercise, we're dealing with two equations in two variables (x and y). Our goal is to find the values of x and y that work for both equations. We transform the system into a form that allows for iterative refinement, spotlighting the intersection between iterative methods and linear systems.

Numerical Analysis

Numerical analysis is a branch of mathematics dealing with algorithms that obtain numerical solutions to continuous problems. This discipline is important for mathematical problems that cannot be solved analytically or where approximate solutions are acceptable due to constraints.

In the context of solving linear equations, numerical analysis provides tools like Jacobi's method to find approximations to the solutions. These methods are particularly useful when the equations are too complicated or too resource-intensive to solve with direct analytical methods.

Methods in numerical analysis may vary in efficiency, accuracy, and ease of implementation. Factors like rounding error, truncation error, and convergence are key considerations when designing and choosing an algorithm. Students should understand that numerical analysis is a balance of precision and practicality, often employing iterative methods to approximate solutions within a margin of error that is deemed suitable for the problem at hand.