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### lecture36

Course: CMPSC 451, Spring 2011
School: Penn State
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451 Numerical CMPSC/MATH Computations Lecture 36 Nov 14, 2011 Prof. Kamesh Madduri Class Overview Numerical Differentiation Finite difference schemes Richardson Extrapolation Slides from textbook follow. 2 Numerical Integration Numerical Differentiation Richardson Extrapolation Numerical Differentiation Finite Difference Approximations Automatic Differentiation Numerical Differentiation Differentiation is...

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451 Numerical CMPSC/MATH Computations Lecture 36 Nov 14, 2011 Prof. Kamesh Madduri Class Overview Numerical Differentiation Finite difference schemes Richardson Extrapolation Slides from textbook follow. 2 Numerical Integration Numerical Differentiation Richardson Extrapolation Numerical Differentiation Finite Difference Approximations Automatic Differentiation Numerical Differentiation Differentiation is inherently sensitive, as small perturbations in data can cause large changes in result Differentiation is inverse of integration, which is inherently stable because of its smoothing effect For example, two functions shown below have very similar denite integrals but very different derivatives Michael T. Heath Scientic Computing 47 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Numerical Differentiation Finite Difference Approximations Automatic Differentiation Numerical Differentiation, continued To approximate derivative of function whose values are known only at discrete set of points, good approach is to t some smooth function to given data and then differentiate approximating function If given data are sufciently smooth, then interpolation may be appropriate, but if data are noisy, then smoothing approximating function, such as least squares spline, is more appropriate < interactive example > Michael T. Heath Scientic Computing 48 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Numerical Differentiation Finite Difference Approximations Automatic Differentiation Finite Difference Approximations Given smooth function f : R R, we wish to approximate its rst and second derivatives at point x Consider Taylor series expansions f (x) 2 f (x) 3 h+ h + 2 6 f (x) 2 f (x) 3 h h + f (x h) = f (x) f (x)h + 2 6 Solving for f (x) in rst series, obtain forward difference approximation f (x + h) = f (x) + f (x)h + f (x + h) f (x) f (x) f (x + h) f (x) h + h 2 h which is rst-order accurate since dominant term in remainder of series is O(h) f (x) = Michael T. Heath Scientic Computing 49 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Numerical Differentiation Finite Difference Approximations Automatic Differentiation Finite Difference Approximations, continued Similarly, from second series derive backward difference approximation f (x) f (x h) f (x) + h + h 2 f (x) f (x h) h which is also rst-order accurate f (x) = Subtracting second series from rst series gives centered difference approximation f (x + h) f (x h) f (x) 2 h + 2h 6 f (x + h) f (x h) 2h which is second-order accurate f (x) = Michael T. Heath Scientic Computing 50 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Numerical Differentiation Finite Difference Approximations Automatic Differentiation Finite Difference Approximations, continued Adding both series together gives centered difference approximation for second derivative f (x + h) 2f (x) + f (x h) f (4) (x) 2 h + h2 12 f (x + h) 2f (x) + f (x h) h2 which is also second-order accurate f (x) = Finite difference approximations can also be derived by polynomial interpolation, which is less cumbersome than Taylor series for higher-order accuracy or higher-order derivatives, and is more easily generalized to unequally spaced points < interactive example > Michael T. Heath Scientic Computing 51 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Numerical Differentiation Finite Difference Approximations Automatic Differentiation Automatic Differentiation Computer program expressing function is composed of basic arithmetic operations and elementary functions, each of whose derivatives is easily computed Derivatives can be propagated through program by repeated use of chain rule, computing derivative of function step by step along with function itself Result is true derivative of original function, subject only to rounding error but suffering no discretization error Software packages are available implementing this automatic differentiation (AD) approach Michael T. Heath Scientic Computing 52 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Richardson Extrapolation Romberg Integration Richardson Extrapolation In many problems, such as numerical integration or differentiation, approximate value for some quantity is computed based on some step size Ideally, we would like to obtain value limiting as step size approaches zero, but we cannot take step size arbitrarily small because of excessive cost or rounding error Based on values for nonzero step sizes, however, we may be able to estimate value for step size of zero One way to do this is called Richardson extrapolation Michael T. Heath Scientic Computing 53 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Richardson Extrapolation Romberg Integration Richardson Extrapolation, continued Let F (h) denote value obtained with step size h If we compute value of F for some nonzero step sizes, and if we know theoretical behavior of F (h) as h 0, then we can extrapolate from known values to obtain approximate value for F (0) Suppose that F (h) = a0 + a1 hp + O(hr ) as h 0 for some p and r, with r > p Assume we know values of p and r, but not a0 or a1 (indeed, F (0) = a0 is what we seek) Michael T. Heath Scientic Computing 54 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Richardson Extrapolation Romberg Integration Richardson Extrapolation, continued Suppose we have computed F for two step sizes, say h and h/q for some positive integer q Then we have F (h) = a0 + a1 hp + O(hr ) F (h/q ) = a0 + a1 (h/q )p + O(hr ) = a0 + a1 q p hp + O(hr ) This system of two linear equations in two unknowns a0 and a1 is easily solved to obtain a0 = F (h) + F (h) F (h/q ) + O(hr ) q p 1 Accuracy of improved value, a0 , is O(hr ) Michael T. Heath Scientic Computing 55 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Richardson Extrapolation Romberg Integration Richardson Extrapolation, continued Extrapolated value, though improved, is still only approximate, not exact, and its accuracy is still limited by step size and arithmetic precision used If F (h) is known for several values of h, then extrapolation process can be repeated to produce still more accurate approximations, up to limitations imposed by nite-precision arithmetic Michael T. Heath Scientic Computing 56 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Richardson Extrapolation Romberg Integration Example: Richardson Extrapolation Use Richardson extrapolation to improve accuracy of nite difference approximation to derivative of function sin(x) at x=1 Using rst-order accurate forward difference approximation, we have F (h) = a0 + a1 h + O(h2 ) so p = 1 and r = 2 in this instance Using step sizes of h = 0.5 and h/2 = 0.25 (i.e., q = 2), we obtain sin(1.5) sin(1) F (h) = = 0.312048 0.5 sin(1.25) sin(1) F (h/2) = = 0.430055 0.25 Michael T. Heath Scientic Computing 57 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Richardson Extrapolation Romberg Integration Example, continued Extrapolated value is then given by F (0) = a0 = F (h)+ F (h) F (h/2) = 2F (h/2)F (h) = 0.548061 (1/2) 1 For comparison, correctly rounded result is cos(1) = 0.540302 < interactive example > Michael T. Heath Scientic Computing 58 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Richardson Extrapolation Romberg Integration Example: Romberg Integration As another example, evaluate /2 sin(x) dx 0 Using composite trapezoid rule, we have F (h) = a0 + a1 h2 + O(h4 ) so p = 2 and r = 4 in this instance With h = /2, F (h) = F (/2) = 0.785398 With q = 2, F (h/2) = F (/4) = 0.948059 Michael T. Heath Scientic Computing 59 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Richardson Extrapolation Romberg Integration Example, continued Extrapolated value is then given by F (h) F (h/2) 4F (h/2) F (h) = = 1.002280 22 1 3 which is substantially more accurate than values previously computed (exact answer is 1) F (0) = a0 = F (h)+ < interactive example > Michael T. Heath Scientic Computing 60 / 61 Numerical Integration Numerical Differentiation Richardson Extrapolation Richardson Extrapolation Romberg Integration Romberg Integration Continued Richardson extrapolations using composite trapezoid rule with successively halved step sizes is called Romberg integration It is capable of producing very high accuracy (up to limit imposed by arithmetic precision) for very smooth integrands It is often implemented in automatic (though nonadaptive) fashion, with extrapolations continuing until change in successive values falls below specied error tolerance Michael T. Heath Scientic Computing 61 / 61
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