Jacobs University, BremenSchool of Engineering and ScienceProf. Dr. Lars Linsen, Orif IbrogimovSpring Term 2010Homework 9120202: ESM4A - Numerical MethodsHomework Problems9.1.Least squares methodLetf(x) =a(x+ 1)2+bsinπx+ccosπx3withf(0) = 0, f(12) = 1 andf(1) = 1.(a) Derive the normal equations for the best approximative solution toa,b, andcin theleast-squares sense.(b) Find the least-squares solution fora,b, andc.(? points)9.2.Differentation&Richardson extrapolation(a) Compute estimates for the derivative of functionf(x) = (x-1)(x-2)(x-3)(x-4)(x-5) atx= 0 using forward and central differencing as well as using the estimate you obtain whenapplying two iterations of Richardson extrapolation to the central differencing estimate.Chooseh= 0.1. Compare the absolute errors.(b) Derive an estimate for computing the second-order derivative of a continuous functionfwith error term beingO(h4).(c) Derive the error term for the approximation:f(x)≈f(x)-2f(x+h)+f(x+2h)h2(d) Derive a numerical differentation formula of orderO(h4) by applying Richardson’s ex-trapolation tof(x) =f(x+h)-f(x-h)2h-h26f(3)(x)-h44!f(5)(x)-. . .Give the error term of
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