2 <20190916 Mon> 
Summation. 


Assignment 2 
Consider the Taylor series $$\cos(x)=\sum_{k=0}^{\infty}\frac{(1)^kx^{2k}}{(2k)!}.$$ 



Write a routine to evaluate \(\cos(x)\) with the following truncated serise: 

$$s_n=\sum_{k=0}^n\frac{(1)^kx^{2k}}{(2k)!}$$ 

in single precision at \(x=1.5708\). 



You should determine the truncation \(n\) by bounding the relative error 

$$\lvert (s_n\cos(x))/\cos(x)\rvert$$ 

under threshold RelTol . Generate the exact answer, \(\cos(x)\), the exact 

truncated sum, \(s_n\) and the exact series in double precision. 

NOTE: Computation done in double precision is consider exact w.r.t single precision. 



You routine should evaluate the sum, \(\hat{s}_n\), in different strategies: 

1. Accumulate in decreasing order of the magnitude. 

2. Accumulate in increasing order of the magnitude. 

3. Accumulate the positive and nagative parts in decreasing order of magnitude separately. 

4. Accumulate the positive and nagative parts in increasing order of magnitude separately. 



Check the final error you achieved \(\lvert(\hat{s}_n\cos(x))/\cos(x) \rvert\) as well as 

the error w.r.t to the truncated sum \(\lvert (\hat{s}_ns_n)/s_n \rvert\). Discuss your observation. 

Do you find certain strategies under certain RelTol perform terrible? In what sense? 






3 <20190930 Mon> 
Newton method. 


Assignemnt 3 
Implement Newton method, i.e., the iteration 

$$x_{k+1}=x_k\frac{f(x_k)}{f'(x_{k+1})}$$ 

to solve the problem 

$$x^3x+0.384900179=0$$ 

in IEEE double precision system. 



Your routine should terminate when \(x_{k+1}\) gets no update, i.e. \(x_{k+1}=x_{k}\). Use the last 

\(x_{k}\) as the true solution \(x_*\), i.e. \(x_*=x_{k_{\max}}\). Do the loglog plot of 

\(\lvert x_kx_*\rvert\) vs \(k\). Comment on the quadratic behavior and possibly some outliers. 



Note that in this simple case, you may not obtain a nice straight line for convergent order. 

Quadratic convergence in 1 dimension problem is way too fast therefore convergent sequence 

is short and the numeric error comes in too early. You can still comment on magnitude, 

which indicates quadratic convergence. 

