Jacobi Iterative Method
The Jacobi iterative method is a specific implementation of the Jacobian method. It assumes that the system of linear equations can be written in the form Ax = b, where A is the coefficient matrix, x is the vector of unknown variables, and b is the vector of constants. The Jacobi method proceeds as follows:
- Assumption 1: The system of linear equations has a unique solution.
- Assumption 2: The coefficient matrix A has no zeros on its main diagonal.
Steps for Jacobi Iterative Method
We can use the following steps:
- Rewrite the system of linear equations in the form: [Tex]x_i = (b_i – ∑_j≠i a_ij x_j) / a_ii[/Tex]
- Make an initial guess of the solution, x(0) = (x1(0), x2(0), …, xn(0)).
- Compute the first approximation, x(1), by substituting the initial guess into the rewritten equations.
- Repeat step 3, using the previous approximation to compute the next approximation, until the desired accuracy is achieved.
Let’s consider an example for better understanding:
Example:Consider the system of linear equations:
2x + y + z = 6
x + 3y – z = 0
-x + y + 2z = 3
Solution:
Rewriting the system in the Jacobi form, we get:
x = (6 – y – z) / 2
y = (0 – x + z) / 3
z = (3 + x – y) / 2
Starting with the initial guess x(0) = (0, 0, 0), the first approximation is:
x(1) = (6 – 0 – 0) / 2 = 3
y(1) = (0 – 0 + 0) / 3 = 0
z(1) = (3 + 0 – 0) / 2 = 1.5
Continuing the iterations, we obtain the following approximations:
x(2) = 1.5, y(2) = 0.5, z(2) = 1.0
x(3) = 1.75, y(3) = 0.25, z(3) = 1.25
The iterations continue until the desired accuracy is achieved.
Jacobian Method
Jacobian Method, also known as the Jacobi Iterative Method, is a fundamental algorithm used to solve systems of linear equations. This method, named after the mathematician Carl Gustav Jacob Jacobi, is particularly useful when dealing with large systems where direct methods are computationally expensive.
The Jacobian Method works by breaking down a complex set of equations into simpler parts, making it easier to approximate the solutions. Here’s a simple way to understand it:
Imagine you have a set of equations, and you need to find the values of certain variables that make all these equations true simultaneously. Instead of solving everything at once, the Jacobian Method starts with an initial guess for these values. It then repeatedly refines this guess, step by step, getting closer to the actual solution with each iteration.
Table of Content
- What is the Jacobian Method?
- Basics of Jacobian Method
- Jacobi Iterative Method
- Steps for Jacobi Iterative Method
- Jacobian Method in Matrix Form
- Difference between Jacobi method and Gauss-Seidel Method
- Conclusion
- Solved Examples on Jacobian Method
- FAQs on the Jacobian Method