Math Problem Statement

Solve the following equation by Gauss-Seidel iteration method (up to 4 decimal places without roundoff): 3x + 2y = 4.5 2x + 3y - z = 5 -y + 2z = 0.5 Initial approximations: x = 0.4, y = 1.6, z = 0.4

Solution

The system of equations to be solved using the Gauss-Seidel method is:

  1. 3x+2y=4.53x + 2y = 4.5
  2. 2x+3yz=52x + 3y - z = 5
  3. y+2z=0.5-y + 2z = 0.5

Initial approximations:

  • x=0.4x = 0.4
  • y=1.6y = 1.6
  • z=0.4z = 0.4

Rearrange the equations for the Gauss-Seidel method:

We need to solve each equation for the variable on the left side. The rearranged system will look like this:

  1. x=4.52y3x = \frac{4.5 - 2y}{3}
  2. y=52x+z3y = \frac{5 - 2x + z}{3}
  3. z=y+0.52z = \frac{y + 0.5}{2}

1st Iteration:

Using the initial approximations x=0.4x = 0.4, y=1.6y = 1.6, and z=0.4z = 0.4:

  1. Update x: x=4.52(1.6)3=4.53.23=1.33=0.4333x = \frac{4.5 - 2(1.6)}{3} = \frac{4.5 - 3.2}{3} = \frac{1.3}{3} = 0.4333

  2. Update y: y=52(0.4333)+0.43=50.8666+0.43=4.53333=1.5111y = \frac{5 - 2(0.4333) + 0.4}{3} = \frac{5 - 0.8666 + 0.4}{3} = \frac{4.5333}{3} = 1.5111

  3. Update z: z=1.5111+0.52=2.01112=1.0056z = \frac{1.5111 + 0.5}{2} = \frac{2.0111}{2} = 1.0056

So after the 1st iteration:

  • x=0.4333x = 0.4333
  • y=1.5111y = 1.5111
  • z=1.0056z = 1.0056

2nd Iteration:

Now, use the updated values from the 1st iteration:

  1. Update x: x=4.52(1.5111)3=4.53.02223=1.47783=0.4926x = \frac{4.5 - 2(1.5111)}{3} = \frac{4.5 - 3.0222}{3} = \frac{1.4778}{3} = 0.4926

  2. Update y: y=52(0.4926)+1.00563=50.9852+1.00563=5.02043=1.6735y = \frac{5 - 2(0.4926) + 1.0056}{3} = \frac{5 - 0.9852 + 1.0056}{3} = \frac{5.0204}{3} = 1.6735

  3. Update z: z=1.6735+0.52=2.17352=1.0868z = \frac{1.6735 + 0.5}{2} = \frac{2.1735}{2} = 1.0868

So after the 2nd iteration:

  • x=0.4926x = 0.4926
  • y=1.6735y = 1.6735
  • z=1.0868z = 1.0868

Would you like a detailed explanation of the method or help with further iterations?

Related Questions:

  1. What is the convergence criterion for the Gauss-Seidel method?
  2. How does the Gauss-Seidel method differ from the Jacobi method?
  3. Why is the system of equations rearranged for the Gauss-Seidel method?
  4. Can the Gauss-Seidel method fail to converge for certain types of systems?
  5. How does the choice of initial approximations affect the iterations in Gauss-Seidel?

Tip: Always check the diagonal dominance of the matrix for better convergence in the Gauss-Seidel method.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Gauss-Seidel Method
Linear Algebra
Iterative Methods

Formulas

x = (4.5 - 2y) / 3
y = (5 - 2x + z) / 3
z = (y + 0.5) / 2

Theorems

Gauss-Seidel Iterative Method
Convergence Theorems for Iterative Methods

Suitable Grade Level

College/University Level