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Integration

Integration. COS 323. Numerical Integration Problems. Basic 1D numerical integration: Given ability to evaluate f ( x ) for any x , find Goal: best accuracy with fewest samples Other problems (future lectures): Improper integration Multi-dimensional integration

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Integration

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  1. Integration COS 323

  2. Numerical Integration Problems • Basic 1D numerical integration: • Given ability to evaluate f(x) for any x, find • Goal: best accuracy with fewest samples • Other problems (future lectures): • Improper integration • Multi-dimensional integration • Ordinary differential equations • Partial differential equations

  3. f(b) f(a) a b Trapezoidal Rule • Approximate function by trapezoid

  4. f(b) f(a) a b Trapezoidal Rule

  5. f(b) f(a) a b a b Extended Trapezoidal Rule Divide into segments of width h:

  6. Trapezoidal Rule Error Analysis • How accurate is this approximation? • Start with Taylor series for f(x) around a

  7. Trapezoidal Rule Error Analysis • Expand LHS: • Expand RHS

  8. Trapezoidal Rule Error Analysis • So, • In general, error for a single segment proportional to h3 • Error for subdividing entire ab interval proportional to h2 • “Cubic local accuracy, quadratic global accuracy”

  9. Determining Step Size • Change in integral when reducing step sizeis a reasonable guess for accuracy • For trapezoidal rule, easy to go from h  h/2without wasting previous samples a b

  10. Simpson’s Rule f(b) • Approximate integral byparabola throughthree points • Better accuracy for same # of evaluations f(a) a b

  11. Richardson Extrapolation • Better way of getting higher accuracy for agiven # of samples • Suppose we’ve evaluated integral for step sizeh and step size h/2: • Then

  12. Richardson Extrapolation • This treats the approximation as a function of h and “extrapolates” the result to h=0 • Can repeat: –1/3 –1/15 4/3 16/15 –1/63 64/63

  13. Open Methods • Trapezoidal rule won’t work if function undefined at one of the points where evaluating • Most often: function infinite at one endpoint • Open methods only evaluate function on the open interval (i.e., not at endpoints)

  14. a b Midpoint Rule • Approximate function by rectangle evaluated at midpoint

  15. a b a b Extended Midpoint Rule Divide into segments of width h:

  16. Midpoint Rule Error Analysis • Following similar analysis to trapezoidal rule,find that local accuracy is cubic,quadratic global accuracy • Formula suitable for adaptive method,Richardson extrapolation,but can’t halve intervals withoutwasting samples

  17. Discontinuities • All the above error analyses assumed nice (continuous, differentiable) functions • In the presence of a discontinuity, all methodsrevert to accuracy proportional to h • Locally-adaptive methods: do not subdivideall intervals equally, focus on those with large error(estimated from change with a single subdivision)

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