ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 20 Solution of Linear System of Equations - Iterative Methods

Iterative Methods Recall Techniques for Root finding of Single Equations Initial Guess New Estimate Error Calculation Repeat until Convergence

Gauss Seidel

First Iteration: Better Estimate

Gauss Seidel Second Iteration: Better Estimate

Gauss Seidel Iteration Error: Convergence Criterion:

Jacobi Iteration

First Iteration: Better Estimate

Jacobi Iteration Second Iteration: Better Estimate

Jacobi Iteration Iteration Error:

Example

Determinants Are composed of same elements Completely Different Mathematical Concept

Determinants Defined in a recursive form 2x2 matrix

Determinants Defined in a recursive form 3x3 matrix

Determinants Minor a 11

Determinants Minor a 12

Determinants Minor a 13

Solution of Small Systems of Equations – Cramer’s Rule 1. Compute

Solution of Small Systems of Equations – Cramer’s Rule 2. Compute

Solution of Small Systems of Equations – Cramer’s Rule 3. Compute

Solution of Small Systems of Equations – Cramer’s Rule 4. Compute

Solution of Small Systems of Equations – Cramer’s Rule If D =0 solution does NOT exist

Singular Matrices If D=0 solution does NOT exist Regardless of Method

Singular Matrices For Example {x} does not exist [A] -1 does not exist

Determinants and LU Decomposition {x} is not affected

Determinants and LU Decomposition {x} is not affected

Determinants and LU Decomposition

Example

After Elimination [A] becomes