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

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

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Gauss Seidel

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First Iteration: Better Estimate

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Gauss Seidel Second Iteration: Better Estimate

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Gauss Seidel Iteration Error: Convergence Criterion:

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Jacobi Iteration

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First Iteration: Better Estimate

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Jacobi Iteration Second Iteration: Better Estimate

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Jacobi Iteration Iteration Error:

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Example

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Determinants Are composed of same elements Completely Different Mathematical Concept

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Determinants Defined in a recursive form 2x2 matrix

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Determinants Defined in a recursive form 3x3 matrix

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Determinants Minor a 11

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Determinants Minor a 12

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Determinants Minor a 13

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

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

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

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

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

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

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

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Determinants and LU Decomposition {x} is not affected

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Determinants and LU Decomposition {x} is not affected

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Determinants and LU Decomposition

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Example

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After Elimination [A] becomes

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