lect_13 - 2.29 Numerical Integration Error Analysis x f(x)...

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Numerical Marine Hydrodynamics Interpolation – Lagrange interpolation – Triangular families – Newton’s iteration method – Equidistant Interpolation Numerical Integration – Newton-Cotes Integration • Errors of numerical integration – Romberg Integration • Richardson Extrapolation • Romberg Integration Formula – Gaussian Quadrature • Gauss Legendre Quadrature Numerical Marine Hydrodynamics Lecture 13 2.29
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Numerical Interpolation Lagrange Polynomials f(x) 1 k-3 k-2 k-1 k k+1 k+2 x Difficult to program Difficult to estimate errors Divisions are expensive Important for numerical integration Numerical Marine Hydrodynamics Lecture 13 2.29
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Numerical Integration Lagrange Interpolation Equidistant Sampling Integration Weights (Cote’s Numbers) Properties f(x) a b x Numerical Marine Hydrodynamics Lecture 13 2.29
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Trapezoidal Rule Simpson’s Rule Numerical Integration f(x) n=1 h x f(x) n=2 h h Numerical Marine Hydrodynamics Lecture 13 2.29 x
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Lecture 13 Numerical Marine Hydrodynamics
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Unformatted text preview: 2.29 Numerical Integration Error Analysis x f(x) h h Trapezoidal Rule Local Absolute Error Global Error Simpsons Rule Local Error Global Error Local Error N Intervals Romberg Integration Richardson Extrapolation Integration Error I(h) Trapezoidal Rule Richardson Extrapolation I h 2 h 1 h Numerical Marine Hydrodynamics Lecture 13 2.29 Romberg Integration Romberg Integration Algorithm h I(h) I h 1 h 2 Order 2 k 1: O(h 2 ) 2: O(h 4 ) 3: O(h 6 ) 4: O(h 8 ) a. 0.172800 1.068800 b,. 0.172800 1.068800 1.484800 c. 0.172800 1.367467 1.640533 1.640533 j 1.068800 1.623467 1.640533 1.484800 1.639467 1.600800 Numerical Marine Hydrodynamics Lecture 13 1.367467 1.367467 1.640533 1.623467 2.29 a b f(x) Gaussian Quadrature Trapezoidal Rule Exact Integration of const and linear f x f(x) a b x Numerical Marine Hydrodynamics Lecture 13 2.29 Gaussian Quadrature f(x) a b x 2-point Gaussian Quadrature Numerical Marine Hydrodynamics Lecture 13 2.29...
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lect_13 - 2.29 Numerical Integration Error Analysis x f(x)...

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