Lecture 4 - Computer Science Department - University of Kentucky

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Adaptive Simpson’s Algorithm Reduce the size <strong>of</strong> the intervals to get more accurate approximations 38
Adaptive Simpson’s Algorithm Given an interval [a,b], we can use the basic Simpson’s rule to compute an approximation to the integral as where the approximation part is I ∫ f ( x) dx = S( a, b) + E( a, b) ≡ b a ( b − a) ⎡ ⎛ a + b ⎞ ⎤ S( a, b) = ⎢ f ( a) + 4 f ⎜ ⎟ + f ( b) 6 ⎥ ⎣ ⎝ 2 ⎠ ⎦ and the error term is E( a, b) = − b−a 5 (4) ( ) f ( ) 1 ξ 90 2 For simplicity, we assume f (4) (x) remains constant on (a,b). Let h = b – a, we have (1) (1) I = S + E For the first step approximation with (1) S = S( a, b) 39
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: Basic Simpson’s Rule (1) Subtract
Page 41 and 42: Adaptive Simpson - (III) Subtractin
Page 43 and 44: Adaptive Simpson - (IV) Numerical i
Page 45 and 46: Computational Procedure The interva
Page 47 and 48: Gaussian Quadrature Formulas A gene
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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