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lect_8_karimi - Introd uction to Num erical Analysis for...

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Lecture 7 Introduction to Num erical Analysis for Eng ineers System s of Linear Equations Cram er’s Rule Gaussian Elim ination • Num erical im plem entation • Num erical stability: Partial Pivoting , Equilibration, Full Pivoting • Multiple right hand sides • Com putation count • LU factorization Error Analysis for Linear System s Condition Num ber • Special Matrices Iterative Methods • Jacobi’s m ethod • Gauss-Seidel iteration • Convergence • Successive Overrelaxation Method • Gradient Methods 2.29 Numerical Marine Hydrodynamics Lecture 8
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Lecture 7 Linear System s of Equations Iterative Methods Sparse, Full-bandwidth Systems Rewrite Equations 0 x x 0 x 0 0 x 0 x 0 x x 0 0 x x 0 x Iterative, Recursive Methods 0 0 0 0 0 0 Gauss-Seidels’s Method Jacobi’s Method Numerical Marine Hydrodynamics Lecture 8 2.29
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Linear System s of Equations Iterative Methods Convergence Jacobi’s Method Numerical Marine Hydrodynamics 2.29 Iteration – Matrix form Decompose Coefficient Matrix with / / Iteration Matrix form Convergence Analysis Sufficient Convergence Condition Note: NOT LU-factorization Lecture 8
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Lecture 7 Linear System s of Equations Iterative Methods Sufficient Convergence Condition Stop Criterion for Iteration Sufficient Convergence Condition Jacobi’s Method Diagonal Dominance + _ 2.29 Numerical Marine Hydrodynamics Lecture 8
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Lecture 7 Linear System s of Equations Tri-diag onal System s Forced Vibration of a String Finite Difference Harmonic excitation f(x,t) = f(x) cos( ω t) Differential Equation Boundary Conditions Discrete Difference Equations Matrix Form Tridiagonal Matrix Symmetric, positive definite: No pivoting needed y(x,t) x i f(x,t) kh < 1 or kh > 3 y i ! 1 + (( kh ) 2 ! 2) y i + y i + 1 = h 2 f ( x i ) Numerical Marine Hydrodynamics Lecture 8 2.29
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