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# lect_7 - Introduction to Numerical Analysis for Engineers...

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Introduction to Numerical Analysis for Engineers Systems of Linear Equations – Cramer’s Rule – Gaussian Elimination • Numerical implementation • Numerical stability – Partial Pivoting – Equilibration – Full Pivoting • Multiple right hand sides • Computation count • LU factorization • Error Analysis for Linear Systems – Condition Number • Special Matrices – Iterative Methods • Jacobi’s method • Gauss-Seidel iteration • Convergence Numerical Marine Hydrodynamics Lecture 7 2.29

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Linear Systems of Equations Tri-diagonal Systems Forced Vibration of a String Finite Difference y(x,t) x i f(x,t) 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 Numerical Marine Hydrodynamics Lecture 7 2.29
Linear Systems of Equations Tri-diagonal Systems General Tri-diagonal Systems LU Factorization Numerical Marine Hydrodynamics Lecture 7 2.29

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Linear Systems of Equations Tri-diagonal Systems LU Factorization Reduction Forward Substitution Back Substitution LU Factorization: 2*(n-1) operations Forward substitution: n-1 operations Back substitution: n-1 operations Total: 4(n-1) ~ O(n) operations Numerical Marine Hydrodynamics Lecture 7 2.29
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lect_7 - Introduction to Numerical Analysis for Engineers...

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