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# SCIENTIFIC COMPUTATION I MATH 661

UNC

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This 3 page Class Notes was uploaded by Dovie Hyatt on Sunday October 25, 2015. The Class Notes belongs to MATH 661 at University of North Carolina - Chapel Hill taught by Jingfang Huang in Fall. Since its upload, it has received 37 views. For similar materials see /class/228746/math-661-university-of-north-carolina-chapel-hill in Mathematics (M) at University of North Carolina - Chapel Hill.

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Date Created: 10/25/15

Numerical Differentiation and Integration I Determine the coefficients C71 C0 and C1 such that the following integration formula has the highest possible orderl 2h mm 7 01flt7hgt co MD 01 1 72h Also write out the leading error terml II lntegral Equation Method For the ODE two point boundary value problem u z 7 0 lt I lt1 u0 a ul I 1 Verify that 1 x7tl eTftdt aex 6671 0 satis es the ODE u 7 u 2i Derive the linear equation for a and B in order to satisfy the boundary conditions u0 a and ul 12 3 Using trapeziodal rule write out the convolution fol dt as a matrix vector product 13 An1xn1 Please give explicitly the entries am in the matrix Al 4 Clearly direct matrix vector product requires 0n2 workl Design an algorithm which only requires 0n work and write the pseudocodel III Let 0 10 lt 11 lt lt IN 1 be evenly spaced points h lN and In nhl Assume f and f are given at each point Let pn be the third order polynomial that interpolates f and f at In and zn1i Show that 7 pnn 0h4 Show that f n h 7 pmn h 00 H Find the equation for in terms of f zn fzn1 f zn1 F90 If you use this interpolation to construct a single interval integration scheme show that the resulting global integration scheme is fourth order 5 Write down the composite scheme as a sum of and IVConsider dividing the interval 01 into m intervals with points 0zoltzlltltzml For a function f de ned on 01 we want to approximate the integrals I 71fzdr for j 0l i m7li Consider approximating the function on the whole interval with a single polynomial that interpolates f at the points 10 i i i zml Then integrate this approximation on each interval to approximate I 11 1 This approximation can be written as a matrix vector action I Af where f fzoiufzmT and A is a matrix The matrix A can be computed by splitting the action The rst part is the polynomial interpolation T poymypml Pf where the interpolation polynomial is given by 101 100 1011 7mm The second part is the piecewise integration of the polynomial Integrate the above polynomial on each interval to approximate Iji This is a linear action on the polynomials and can be written in matrix form as I Vp0 i i i pmlTi Set up the matrix A by showing how to compute P and Vi Compute the entries when feasible only leave elementary matrix operations left to be evaluated What are the sizes of these matricesi E0 What are the two main numerical problems with this approach CA3 1 Consider the error vector I7I E Rmi If the points 11 i i i zm1 are chosen to be the Chebyshev points compute a bound for the standard two norm of the error vector The bound should only depend on the function f and on the size ml F Instead of using a single polynomial use a low order polynomial to approximate each entry For each interval zizi1 approximate by interpolating f at 1171 1 i i zHgi with obvious changes for the end intervalsi Integrate this polynomial to get an approximation for Iji The resulting action can again be written as a matrix operation Bf Compute the entries in El Compute a bound for the error I 7 Bfi V Let be a smooth real valued function 1 Show that W 7 u z ch2 0h4i Give 5 explicitlyi 2 Notice that you can also develop a similar second order nite difference approximation using uz 2h and uz 7 2h Write out the formula and nd the leading order error 3 Using 1 and 2 and the idea of Richardson extrapolation develop an approximation scheme to approximate 1 using uz 7 2h uz 7 h uz h and uz 2hi Write out the leading order errori VI 1 Show that the evenly spaced composite trapezoidal rule is always exact for the integral I 02W w ere 2ak coskz bk sinkz 160 if using N grid points with N gt n Here I E 027rli 2 Is this true for N lt n For example N 2 Explain why VII For a function 6 01 consider the composite midpoint rule for computing 1m fltzgtdz hzmi 7 9h an

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