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# lect_24 - Numerical Marine Hydrodynamics Summary...

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Numerical Marine Hydrodynamics Summary Fundamentals of Digital Computing Error Analysis Roots of Non-linear Equations Systems of Linear Equations – Gaussian Elimination – Iterative Methods Optimization, Curve Fitting Interpolation – Numerical Integration – Numerical Differentiation Ordinary Differential Equations – Initial Value Problems – Boundary Value Problems Partial Differential Equations Finite Element and Spectral Methods Boundary Integral – Panel Methods Numerical Marine Hydrodynamics Lecture 24 2.29

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Digital Computer Models Continuum Model Differential Equation V Difference Equation System of Equations Linear System of Equations Eigenvalue Problems Non-trivial Solutions Root finding Differentiation Integration Solving linear equations x dx w = dt R Discrete Model t m t x n x n n m Accuracy and Stability => Convergence Numerical Marine Hydrodynamics Lecture 24 2.29
Error Analysis Number Representation Addition and Subtraction Absolute Error Relative Error Multiplication and Division Relative Error Unbounded Absolute Error Bounded Relative Error Shift mantissa of largest number Result has exponent of largest number Numerical Marine Hydrodynamics Lecture 24 2.29

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Recursion Heron’s Device a=26; n=10; MATLAB script g=1; Numerically evaluate square-root % Number of Digits heron.m dig=5; sq(1)=g; for i=2:n sq(i)= 0.5*radd(sq(i-1),a/sq(i-1),dig); end hold off plot([0 n],[sqrt(a) sqrt(a)],'b') hold on plot(sq,'r') plot(a./sq,'r-.') plot((sq-sqrt(a))/sqrt(a),'g') grid on Numerical Marine Hydrodynamics Lecture 24 2.29 Initial guess Test Mean of guess and its reciprocal Recursion Algorithm
Spherical Bessel Functions Generation by Recurrence Relations Forward Recurrence Unstable Forward Recurrence Backward Recurrence N ~ x+20 Numerical Marine Hydrodynamics Lecture 24 2.29 Miller’s algorithm Stable

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Error Propagation Absolute Errors Function of one variable General Error Propagation Formula Δ y~f ’(x) Δ x Δ x = x - x x x Numerical Marine Hydrodynamics Lecture 24 2.29
Error Propagation Condition Number y = f(x) x x y y x = x(1 + α ) y = y(1 + β ) Problem Condition Number Error cancellation example Problem ill-conditioned Well-conditioned problem Numerical Marine Hydrodynamics Lecture 24 2.29

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Roots of Nonlinear Equations General Method Example: Cube root % f(x) = x^3 - a = 0 % g(x) = x + C*(x^3 - a) cube.m a=2; Non-linear Equation n=10; g=1.0; C=-0.1; sq(1)=g; for i=2:n sq(i)= sq(i-1) + C*(sq(i-1)^3 -a); end hold off plot([0 n],[a^(1./3.) a^(1/3.)],'b') hold on plot(sq,'r') plot( (sq-a^(1./3.))/(a^(1./3.)),'g') grid on Goal: Converging series Rewrite Problem Example Iteration Numerical Marine Hydrodynamics Lecture 24 2.29
Roots of Nonlinear Equations General Method x y x y Convergent Divergent > x x x x 0 1 1 0 y=x y=x y=g(x) y=g(x) Convergence Convergence Mean-value Theorem Numerical Marine Hydrodynamics Lecture 24 2.29

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Systems of Linear Equations Gaussian Elimination Reduction Reduction Step n-1 Step k Back-Substitution Numerical Marine Hydrodynamics Lecture 24 2.29
Systems of Linear Equations Gaussian Elimination Reduction Partial Pivoting by Columns Step k New Row k Pivotal Elements New Row i Required at each step!

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