Posted: 08/23/21

Due: 08/30/21, 11:55PM

1. Construct a convergence plot in logarithmic coordinates for the Leibniz series approximation of $\pi$. Estimate the rate and order of convergence.

   Solution. The convergence of the Leibniz series partial sum sequence ${\left\{L_n\right\}}_{n\in \mathbb{N}}$

   ${\displaystyle L_n=\sum _{k=0}^n\frac{{(-1)}^k}{2k+1},{lim}_{n\to \infty }{L}_n=\frac{\pi }{4},}$

   is shown in Fig. 1, and can be visually estimated to converge with order $p=1$. The rate of convergence is

   ${\displaystyle r={lim}_{n\to \infty }\frac{L_{n+1}}{L_n}={lim}_{n\to \infty }(1+\frac{{(-1)}^{n+1}}{(2n+1)L_n})=1.}$

   $•$

   Figure 1. Convergence of Leibniz series

   ∴	N=200; n=20:20:N;

   ∴	function Leibniz(n) k=0:n sum((-1).^k ./ (2*k.+1)) end

   Leibniz

   ∴	err=log.(abs.(Leibniz.(n) .- π/4));

   ∴	clf(); plot(log.(n),err,"-o"); xlabel("log(n)"); ylabel("log(err)"); title("Leibniz series convergence"); grid("on"); plot([3,4],[-5,-7],"-"); plot([3,4],[-5,-6],"-");

   ∴	savefig(homedir()*"/courses/MATH661/images/H01Fig01.eps")

   ∴

2. Apply Aitken acceleration to the Leibniz partial sums sequence. Construct the convergence plot. Estimate the rate and order of convergence.

   Solution. The Aitken-accelerated sequence ${\left\{A_n\right\}}_{n\in \mathbb{N}}$ has general term

   ${\displaystyle A_n=L_n-\frac{{(L_n-L_{n-1})}^2}{L_n-2L_{n-1}+L_{n-2}},}$

   with convergence plot shown in Fig. 2, indicating faster than quadratic growth.

   $•$

   Figure 2. Convergence of Aitken acceleration of Leibniz series

   ∴	N=40; n=1:N; L=Leibniz.(n); A=copy(L);

   ∴	for n=3:N ΔL = L[n] - L[n-1] Δ2L = L[n] - 2*L[n-1] + L[n-2] A[n] = L[n] - ΔL^2/Δ2L end

   ∴	errL=log.(abs.(L .- π/4)); errA=log.(abs.(A .- π/4));

   ∴	clf(); plot(log.(n),errL,"-o"); plot(log.(n),errA,"-x"); xlabel("log(n)"); ylabel("log(err)"); title("Aitken-accelerated Leibniz series convergence"); grid("on"); plot([2,4],[-6,-8],"-"); plot([2,4],[-6,-10],"-");

   ∴	savefig(homedir()*"/courses/MATH661/images/H01Fig02.eps")

   ∴

3. Construct a convergence plot for the Ramanujan series approximation of $\pi$. Estimate the rate and order of convergence.

   Solution. The general term

   ${\displaystyle r_k=\frac{\left(4k\right)!(1103+26390k)}{{(k!)}^4{396}^{4k}}}$

   in the Ramanujuan series

   ${\displaystyle R_n=\frac{2\sqrt{2}}{9801}\sum _{k=0}^n\frac{\left(4k\right)!(1103+26390k)}{{(k!)}^4{396}^{4k}},{lim}_{n\to \infty }{R}_n=\frac{1}{\pi },}$

   decreases rapidly. Applying Stirling's formula $\ln n!\cong n\ln -n$ (valid for large $n$), gives

   ${\displaystyle \ln r_k\cong \left(4k\right)\ln 4k-4k+\ln (1103+26390k)-4(k\ln k-k)-4k\ln 396,}$ ${\displaystyle \ln r_k\cong -4k\ln 99+\ln (1103+26390k)\Rightarrow }$ ${\displaystyle r_k={99}^{-4k}(1103+26390k)}$

   Since $r_0=1103\cong {10}^3$, terms smaller than ${10}^3ϵ$ ($ϵ$ is machine epsilon) would not appreaciably contribute to the sum

   ∴	ε=eps(Float64); kmax = log(10^3*ε)/(-4*log(99))

   $1.5851544170976106$

   ∴

   The above computation indicates that 2 terms of the sum should already give an approximation within machine precision.

   ∴	r(k) = factorial(4*k)*(1103.0+26390*k)/(factorial(k))^4/396^(4*k)

   r

   ∴

   r.(1:5)

   ${\displaystyle \left[\begin{array}{c}2.6831974348925308e-5\\ -3.383976671091512e-11\\ 3.2409901671873466e-9\\ 8.916491639979272e-7\\ -0.00021134339512742683\end{array}\right]}$

   (1)

   ∴

   (4k)! (1103+26390k)

   (k!)4 3964k

   $•$

   Figure 3. Convergence of Ramanujan series

   ∴	N=200; n=20:20:N;

   ∴	function Leibniz(n) k=0:n sum((-1).^k ./ (2*k.+1)) end

   Leibniz

   ∴	err=log.(abs.(Leibniz.(n) .- π/4));

   ∴	clf(); plot(log.(n),err,"-o"); xlabel("log(n)"); ylabel("log(err)"); title("Leibniz series convergence"); grid("on"); plot([3,4],[-5,-7],"-"); plot([3,4],[-5,-6],"-");

   ∴	savefig(homedir()*"/courses/MATH661/images/H01Fig01.eps")

   ∴

4. Construct a convergence plot for the continued fraction approximation of $\pi$. Estimate the rate and order of convergence.

5. Completely state the mathematical problem of computing the absolute value

   ${\displaystyle \left|x\right|,x\in ℝ.}$

   Find the absolute and relative condition numbers.

   Solution. According to Demmel [?]

6. Completely state the mathematical problem of computing a difference.

   ${\displaystyle x_1-x_2,x_1,x_2\in ℝ.}$

   Find the absolute and relative condition numbers.

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