Lecture 4 - Computer Science Department - University of Kentucky

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Adaptive Simpson’s Algorithm Which sub‐interval or both to divide for refinement 46
Gaussian Quadrature Formulas A general numerical integration formula is ∫ b a f x ) dx ≈ A f ( x ) + A f ( x ) + L+ ( 0 0 1 1 A n f ( x n It suffices to know the nodes x 0 , x 1 ,…, x n and the weights A 0 , A 1 ,…, A n . For important special functions, they are listed in some reference books Suppose a set <strong>of</strong> nodes is given, how to find the weights. This can be done using Lagrange interpolation polynomial as n p( x) = f ( x ) l ( x) With l ( x ) i = ∑ i= 0 n ∏ j= o, j≠1 ⎛ ⎜ ⎝ i i i x − x − x x i j ⎟⎞ ⎟ ⎠ ) If p is a good approximate to f, we anticipate approximate to ∫ b f ( x ) dx a ∫ b a p ( x) dx is a good 47
Page 1 and 2: CS321 Numerical Analysis Lecture 4
Page 3 and 4: Numerical Integration
Page 5 and 6: Partition
Page 7 and 8: Upper and Lower Bounds
Page 9 and 10: Computation 1.) need a procedure to
Page 11 and 12: Trapezoid Rule
Page 13 and 14: Trapezoid Rule with Uniform Spacing
Page 15 and 16: Error Analysis (2) It follows that
Page 17 and 18: Error Analysis (4) Then using the r
Page 19 and 20: Example 1 Show that a h ∫ + f ( x
Page 21 and 22: Estimate Grid Spacing Example. If t
Page 23 and 24: Recursive Trapezoid Formula Given a
Page 25 and 26: Recursive Formula Hence, we have R(
Page 27 and 28: Romberg Algorithm Recursive composi
Page 29 and 30: More Romberg Algorithm We could app
Page 31 and 32: General Extrapolation - Cont. Subtr
Page 33 and 34: Basic Simpson’s Rule A three poin
Page 35 and 36: Basic Simpson’s Rule Using Taylor
Page 37 and 38: Basic Simpson’s Rule (1) Subtract
Page 39 and 40: Adaptive Simpson’s Algorithm Give
Page 41 and 42: Adaptive Simpson - (III) Subtractin
Page 43 and 44: Adaptive Simpson - (IV) Numerical i
Page 45: Computational Procedure The interva
Page 49 and 50: An Example Determine a quadrature f
Page 51 and 52: Gaussian Quadrature Theorem Let q b
Page 53: Gaussian Theorem Proof (II) Since t

Lecture 4 - Computer Science Department - University of Kentucky

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