11—Numerical Analysis 35111.20 Rederive the first Gauss integration formula Eq. ( 24 ) without assuming the symmetry of the result
∫+h
−hf(x)dx≈αf(β) +γf(δ).11.21 Derive the coefficients for the stable two-point Adams method.
11.22 By putting in one more parameter in the differentiation algorithm for noisy data, it is possible both to
minimize the variance inf′and to eliminate the error terms in h^2 f′′′. Find such a 6-point formula for the
derivatives halfway between data points OR one for the derivatives at the data points (with errors and variance).
11.23 In the same spirit as the method for differentiating noisy data, how do you interpolate noisy data?
That is, use some extra points to stabilize the interpolation against random variations in the data. To be
specific, do a midpoint interpolation for equally spaced points. Compare the variance here to that in Eq. ( 3 ).
Ans:f(0)≈[f(− 3 k) +f(−k) +f(k) +f(3k)]/ 4 , σ^2 is 4. 8 times smaller
11.24 Find the dispersion resulting from the use of a four point formula foruxin the numerical solution of the
PDEut+cux= 0.
11.25 Find the exact dispersion resulting from the equation
ut=−c[
u(t,x+ ∆x)−u(t,x−∆x)]
/2∆x.That is, don’t do the series expansion on∆x.
11.26 Compute the dispersion and the dissipation in the Lax-Friedrichs and in the Lax-Wendroff methods.
11.27 In the simple iteration method of Eq. ( 66 ), if the grid points are denotedx=m∆x,t=n∆t, wheren
andmare integers (−∞< n,m <+∞), the result is a linear, constant-coefficient, partial difference equation.
Solve subject to the initial condition
u(0,m) =eikm∆x.
11.28 Lobatto integration is like Gaussian integration, except that you require the end-points of the interval to
be included in the sum. The interior points are left free. Three point Lobatto is the same as Simpson; find the
four point Lobatto formula. The points found are roots ofPn′− 1.