User:Egm6341.s10.team3.sa

Om Sri Guru Raghavendra

HW1 HW2 HW3 HW4 HW1-Updated HW2-Updated HW5 HW5-Updated HW6 HW7 <br\>

Mtg1

(1) Plotting a given function $f(x)$

Ref: Lecture Notes p.2-2

Problem Statement

$f(x)={\frac {e^{x}-1}{x}}x\in [0,1]$

(i) Find $\lim _{x\rightarrow 0}f(x)$

(ii)Plot $\displaystyle f(x)$

Solution

(i) By L'Hospital's rule,

$\lim _{x\rightarrow 0}f(x)=\lim _{x\rightarrow 0}{\frac {e^{x}-1}{x}}$

$=\lim _{x\rightarrow 0}{\frac {{\frac {d}{dx}}(e^{x}-1)}{{\frac {d}{dx}}(x)}}$

$=\lim _{x\rightarrow 0}{e^{x}}$

$\displaystyle =1$

(ii) MATLAB Code to generate the plot

clc;
clear all;
x=0:0.1:1;
for i=1:length(x)
    if x(i)==0
        y(i)=1;
    else
    y(i)=(exp(x(i)) - 1)/ x(i);
    end
end
plot(x,y,'r*-');
title('Plot of (exp(x)-1)/x  vs  x');
xlabel('x');
ylabel('f(x)');

Plot:   ${\frac {e^{x}-1}{x}}$ vs $\displaystyle x$

(7) Expressing a function $f(.)$ using Taylor Series

Ref: Lecture Notes p.6-1

Problem Statement

$f(x)=sinx,x\in [0,\pi ]$

(i)Construct a Taylor series of $f(x)$ around $x_{0}={\frac {\pi }{4}}$ for $\displaystyle n=0,1,2......,10.$

(ii)Plot the series for each $\displaystyle n$

(iii)Estimate the maximum $\displaystyle R(x)$ at $x={\frac {\pi }{2}}$

Solution

(i) $\displaystyle n^{th}$ order polynomial is given by,

$\displaystyle P_{n}(x)=f(x_{0})+{\frac {(x-x_{0})}{1!}}f^{1}(x_{0})+........+{\frac {(x-x_{0})^{n}}{n!}}f^{n}(x_{0})$

(a)$\displaystyle P_{n}(x)$ for $\displaystyle n=0$ and $\displaystyle x_{0}={\frac {\pi }{4}}$

$\displaystyle P_{0}(x)=f(x_{0})$

$\displaystyle P_{0}(x)=sin({\frac {\pi }{4}})={\frac {1}{\sqrt {2}}}$

(b)$\displaystyle P_{n}(x)$ for $\displaystyle n=1$ and $\displaystyle x_{0}={\frac {\pi }{4}}$

$\displaystyle P_{1}(x)=f(x_{0})+{\frac {(x-x_{0})}{1!}}f^{1}(x_{0})$

$\displaystyle =sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})}{1!}}cos({\frac {\pi }{4}})$

$\displaystyle P_{1}(x)={\frac {1}{\sqrt {2}}}{\Bigg [}1+{\frac {(x-{\frac {\pi }{4}})}{1!}}{\Bigg ]}$

(c)$\displaystyle P_{n}(x)$ for $\displaystyle n=2$ and $\displaystyle x_{0}={\frac {\pi }{4}}$

$\displaystyle P_{2}(x)=f(x_{0})+{\frac {(x-x_{0})}{1!}}f^{1}(x_{0})+{\frac {(x-x_{0})^{2}}{2!}}f^{2}(x_{0})$

$\displaystyle =sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})}{1!}}cos({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}sin({\frac {\pi }{4}})$

$\displaystyle P_{2}(x)={\frac {1}{\sqrt {2}}}{\Bigg [}1+{\frac {(x-{\frac {\pi }{4}})}{1!}}-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}{\Bigg ]}$

(d)$\displaystyle P_{n}(x)$ for $\displaystyle n=3$ and $\displaystyle x_{0}={\frac {\pi }{4}}$

$\displaystyle P_{3}(x)=f(x_{0})+{\frac {(x-x_{0})}{1!}}f^{1}(x_{0})+{\frac {(x-x_{0})^{2}}{2!}}f^{2}(x_{0})+{\frac {(x-x_{0})^{3}}{3!}}f^{3}(x_{0})$

$\displaystyle =sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})}{1!}}cos({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}sin({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{3}}{3!}}cos({\frac {\pi }{4}})$

$\displaystyle P_{3}(x)={\frac {1}{\sqrt {2}}}{\Bigg [}1+{\frac {(x-{\frac {\pi }{4}})}{1!}}-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}-{\frac {(x-{\frac {\pi }{4}})^{3}}{3!}}{\Bigg ]}$

(e)$\displaystyle P_{n}(x)$ for $\displaystyle n=4$ and $\displaystyle x_{0}={\frac {\pi }{4}}$

$\displaystyle P_{4}(x)=f(x_{0})+{\frac {(x-x_{0})}{1!}}f^{1}(x_{0})+{\frac {(x-x_{0})^{2}}{2!}}f^{2}(x_{0})+{\frac {(x-x_{0})^{3}}{3!}}f^{3}(x_{0})+{\frac {(x-x_{0})^{4}}{4!}}f^{4}(x_{0})$

$\displaystyle =sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})}{1!}}cos({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}sin({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{3}}{3!}}cos({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{4}}{4!}}sin({\frac {\pi }{4}})$

$\displaystyle P_{4}(x)={\frac {1}{\sqrt {2}}}{\Bigg [}1+{\frac {(x-{\frac {\pi }{4}})}{1!}}-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}-{\frac {(x-{\frac {\pi }{4}})^{3}}{3!}}+{\frac {(x-{\frac {\pi }{4}})^{4}}{4!}}{\Bigg ]}$

(f)$\displaystyle P_{n}(x)$ for $\displaystyle n=5$ and $\displaystyle x_{0}={\frac {\pi }{4}}$

$\displaystyle P_{5}(x)=f(x_{0})+{\frac {(x-x_{0})}{1!}}f^{1}(x_{0})+{\frac {(x-x_{0})^{2}}{2!}}f^{2}(x_{0})+{\frac {(x-x_{0})^{3}}{3!}}f^{3}(x_{0})+{\frac {(x-x_{0})^{4}}{4!}}f^{4}(x_{0})+{\frac {(x-x_{0})^{5}}{5!}}f^{5}(x_{0})$

$\displaystyle =sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})}{1!}}cos({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}sin({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{3}}{3!}}cos({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{4}}{4!}}sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{5}}{5!}}cos({\frac {\pi }{4}})$

$\displaystyle P_{5}(x)={\frac {1}{\sqrt {2}}}{\Bigg [}1+{\frac {(x-{\frac {\pi }{4}})}{1!}}-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}-{\frac {(x-{\frac {\pi }{4}})^{3}}{3!}}+{\frac {(x-{\frac {\pi }{4}})^{4}}{4!}}+{\frac {(x-{\frac {\pi }{4}})^{5}}{5!}}{\Bigg ]}$

(g)$\displaystyle P_{n}(x)$ for $\displaystyle n=6$ and $\displaystyle x_{0}={\frac {\pi }{4}}$

$\displaystyle P_{6}(x)=f(x_{0})+{\frac {(x-x_{0})}{1!}}f^{1}(x_{0})+{\frac {(x-x_{0})^{2}}{2!}}f^{2}(x_{0})+{\frac {(x-x_{0})^{3}}{3!}}f^{3}(x_{0})+{\frac {(x-x_{0})^{4}}{4!}}f^{4}(x_{0})+{\frac {(x-x_{0})^{5}}{5!}}f^{5}(x_{0})+{\frac {(x-x_{0})^{6}}{6!}}f^{6}(x_{0})$

$\displaystyle =sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})}{1!}}cos({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}sin({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{3}}{3!}}cos({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{4}}{4!}}sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{5}}{5!}}cos({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{6}}{6!}}sin({\frac {\pi }{4}})$

$\displaystyle P_{6}(x)={\frac {1}{\sqrt {2}}}{\Bigg [}1+{\frac {(x-{\frac {\pi }{4}})}{1!}}-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}-{\frac {(x-{\frac {\pi }{4}})^{3}}{3!}}+{\frac {(x-{\frac {\pi }{4}})^{4}}{4!}}+{\frac {(x-{\frac {\pi }{4}})^{5}}{5!}}-{\frac {(x-{\frac {\pi }{4}})^{6}}{6!}}{\Bigg ]}$

(h)$\displaystyle P_{n}(x)$ for $\displaystyle n=7$ and $\displaystyle x_{0}={\frac {\pi }{4}}$

$\displaystyle P_{7}(x)=f(x_{0})+{\frac {(x-x_{0})}{1!}}f^{1}(x_{0})+{\frac {(x-x_{0})^{2}}{2!}}f^{2}(x_{0})+{\frac {(x-x_{0})^{3}}{3!}}f^{3}(x_{0})+{\frac {(x-x_{0})^{4}}{4!}}f^{4}(x_{0})+{\frac {(x-x_{0})^{5}}{5!}}f^{5}(x_{0})+{\frac {(x-x_{0})^{6}}{6!}}f^{6}(x_{0})$                 $\displaystyle +{\frac {(x-x_{0})^{7}}{7!}}f^{7}(x_{0})$

$\displaystyle =sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})}{1!}}cos({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}sin({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{3}}{3!}}cos({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{4}}{4!}}sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{5}}{5!}}cos({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{6}}{6!}}sin({\frac {\pi }{4}})$                 $\displaystyle -{\frac {(x-{\frac {\pi }{4}})^{7}}{7!}}cos({\frac {\pi }{4}})$

$\displaystyle P_{7}(x)={\frac {1}{\sqrt {2}}}{\Bigg [}1+{\frac {(x-{\frac {\pi }{4}})}{1!}}-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}-{\frac {(x-{\frac {\pi }{4}})^{3}}{3!}}+{\frac {(x-{\frac {\pi }{4}})^{4}}{4!}}+{\frac {(x-{\frac {\pi }{4}})^{5}}{5!}}-{\frac {(x-{\frac {\pi }{4}})^{6}}{6!}}-{\frac {(x-{\frac {\pi }{4}})^{7}}{7!}}{\Bigg ]}$

(i)$\displaystyle P_{n}(x)$ for $\displaystyle n=8$ and $\displaystyle x_{0}={\frac {\pi }{4}}$

$\displaystyle P_{8}(x)=f(x_{0})+{\frac {(x-x_{0})}{1!}}f^{1}(x_{0})+{\frac {(x-x_{0})^{2}}{2!}}f^{2}(x_{0})+{\frac {(x-x_{0})^{3}}{3!}}f^{3}(x_{0})+{\frac {(x-x_{0})^{4}}{4!}}f^{4}(x_{0})+{\frac {(x-x_{0})^{5}}{5!}}f^{5}(x_{0})+{\frac {(x-x_{0})^{6}}{6!}}f^{6}(x_{0})$                 $\displaystyle +{\frac {(x-x_{0})^{7}}{7!}}f^{7}(x_{0})+{\frac {(x-x_{0})^{8}}{8!}}f^{8}(x_{0})$

$\displaystyle =sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})}{1!}}cos({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}sin({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{3}}{3!}}cos({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{4}}{4!}}sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{5}}{5!}}cos({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{6}}{6!}}sin({\frac {\pi }{4}})$                 $\displaystyle -{\frac {(x-{\frac {\pi }{4}})^{7}}{7!}}cos({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{8}}{8!}}sin({\frac {\pi }{4}})$

$\displaystyle P_{8}(x)={\frac {1}{\sqrt {2}}}{\Bigg [}1+{\frac {(x-{\frac {\pi }{4}})}{1!}}-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}-{\frac {(x-{\frac {\pi }{4}})^{3}}{3!}}+{\frac {(x-{\frac {\pi }{4}})^{4}}{4!}}+{\frac {(x-{\frac {\pi }{4}})^{5}}{5!}}-{\frac {(x-{\frac {\pi }{4}})^{6}}{6!}}-{\frac {(x-{\frac {\pi }{4}})^{7}}{7!}}+{\frac {(x-{\frac {\pi }{4}})^{8}}{8!}}{\Bigg ]}$

(j)$\displaystyle P_{n}(x)$ for $\displaystyle n=9$ and $\displaystyle x_{0}={\frac {\pi }{4}}$

$\displaystyle P_{9}(x)=f(x_{0})+{\frac {(x-x_{0})}{1!}}f^{1}(x_{0})+{\frac {(x-x_{0})^{2}}{2!}}f^{2}(x_{0})+{\frac {(x-x_{0})^{3}}{3!}}f^{3}(x_{0})+{\frac {(x-x_{0})^{4}}{4!}}f^{4}(x_{0})+{\frac {(x-x_{0})^{5}}{5!}}f^{5}(x_{0})+{\frac {(x-x_{0})^{6}}{6!}}f^{6}(x_{0})$                 $\displaystyle +{\frac {(x-x_{0})^{7}}{7!}}f^{7}(x_{0})+{\frac {(x-x_{0})^{8}}{8!}}f^{8}(x_{0})+{\frac {(x-x_{0})^{9}}{9!}}f^{9}(x_{0})$

$\displaystyle =sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})}{1!}}cos({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}sin({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{3}}{3!}}cos({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{4}}{4!}}sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{5}}{5!}}cos({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{6}}{6!}}sin({\frac {\pi }{4}})$                 $\displaystyle -{\frac {(x-{\frac {\pi }{4}})^{7}}{7!}}cos({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{8}}{8!}}sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{9}}{9!}}cos({\frac {\pi }{4}})$

$\displaystyle P_{9}(x)={\frac {1}{\sqrt {2}}}{\Bigg [}1+{\frac {(x-{\frac {\pi }{4}})}{1!}}-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}-{\frac {(x-{\frac {\pi }{4}})^{3}}{3!}}+{\frac {(x-{\frac {\pi }{4}})^{4}}{4!}}+{\frac {(x-{\frac {\pi }{4}})^{5}}{5!}}-{\frac {(x-{\frac {\pi }{4}})^{6}}{6!}}-{\frac {(x-{\frac {\pi }{4}})^{7}}{7!}}+{\frac {(x-{\frac {\pi }{4}})^{8}}{8!}}+{\frac {(x-{\frac {\pi }{4}})^{9}}{9!}}{\Bigg ]}$

(k)$\displaystyle P_{n}(x)$ for $\displaystyle n=10$ and $\displaystyle x_{0}={\frac {\pi }{4}}$

$\displaystyle P_{10}(x)=f(x_{0})+{\frac {(x-x_{0})}{1!}}f^{1}(x_{0})+{\frac {(x-x_{0})^{2}}{2!}}f^{2}(x_{0})+{\frac {(x-x_{0})^{3}}{3!}}f^{3}(x_{0})+{\frac {(x-x_{0})^{4}}{4!}}f^{4}(x_{0})+{\frac {(x-x_{0})^{5}}{5!}}f^{5}(x_{0})+{\frac {(x-x_{0})^{6}}{6!}}f^{6}(x_{0})$                 $\displaystyle +{\frac {(x-x_{0})^{7}}{7!}}f^{7}(x_{0})+{\frac {(x-x_{0})^{8}}{8!}}f^{8}(x_{0})+{\frac {(x-x_{0})^{9}}{9!}}f^{9}(x_{0})+{\frac {(x-x_{0})^{10}}{10!}}f^{10}(x_{0})$

$\displaystyle =sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})}{1!}}cos({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}sin({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{3}}{3!}}cos({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{4}}{4!}}sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{5}}{5!}}cos({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{6}}{6!}}sin({\frac {\pi }{4}})$                 $\displaystyle -{\frac {(x-{\frac {\pi }{4}})^{7}}{7!}}cos({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{8}}{8!}}sin({\frac {\pi }{4}})+{\frac {(x-{\frac {\pi }{4}})^{9}}{9!}}cos({\frac {\pi }{4}})-{\frac {(x-{\frac {\pi }{4}})^{10}}{10!}}sin({\frac {\pi }{4}})$

$\displaystyle P_{10}(x)={\frac {1}{\sqrt {2}}}{\Bigg [}1+{\frac {(x-{\frac {\pi }{4}})}{1!}}-{\frac {(x-{\frac {\pi }{4}})^{2}}{2!}}-{\frac {(x-{\frac {\pi }{4}})^{3}}{3!}}+{\frac {(x-{\frac {\pi }{4}})^{4}}{4!}}+{\frac {(x-{\frac {\pi }{4}})^{5}}{5!}}-{\frac {(x-{\frac {\pi }{4}})^{6}}{6!}}-{\frac {(x-{\frac {\pi }{4}})^{7}}{7!}}+{\frac {(x-{\frac {\pi }{4}})^{8}}{8!}}+{\frac {(x-{\frac {\pi }{4}})^{9}}{9!}}-{\frac {(x-{\frac {\pi }{4}})^{10}}{10!}}{\Bigg ]}$

(ii) MATLAB Code for generating the plots for :     $\displaystyle n=0,1......,10$

clc;
clear all;
syms x;
f = sin (x);
x_0 = pi/4;
y=0:0.01:pi;
n=0:1:10;
for j=1:length(n)
    for i=1:length(y)
        if n(j)== 0
            z(i,j)=((y(i)-x_0)^n(j) / (factorial(n(j))))* subs(diff(f,x,n(j)),x,x_0);
        else
            z(i,j)=z(i,j-1) + ((y(i)-x_0)^n(j) / (factorial(n(j))))* subs(diff(f,x,n(j)),x,x_0);
        end
    end
end

for k=1:5
    if n(k)==0
        plot (y,z(:,1),'r-');
        hold on;
    end
    hold on;
    if n(k)==1
        plot (y,z(:,2),'b-');
        hold on;
    end

    if n(k)==2
        plot (y,z(:,3),'g-');
        legend('n=2');
    end

    if n(k)==3
        plot (y,z(:,4),'k-');
        hold on;
    end

    if n(k)==4
        plot (y,z(:,5),'m-');
        hold on;
    end

    if n(k)==5
        plot (y,z(:,6),'c-');
        hold on;
    end

    legend('n=0','n=1','n=2','n=3','n=4','n=5');
end
    title('sin (x) vs x for n=0,1,2,3,4 and 5');
    xlabel('x');
    ylabel('sin (x)');
    hold on;

for k=6:length(n)
    if n(k)==6
        plot (y,z(:,7),'r-');
        hold on;
    end
    hold on;
    if n(k)==7
        plot (y,z(:,8),'b-');
        hold on;
    end

    if n(k)==8
        plot (y,z(:,9),'g-');
        legend('n=2');
    end

    if n(k)==9
        plot (y,z(:,10),'k-');
        hold on;
    end

    if n(k)==10
        plot (y,z(:,11),'m-');
        hold on;
    end

    legend('n=6','n=7','n=8','n=9','n=10');
end
    title('sin (x) vs x for n=6,7,8,9,and 10');
    xlabel('x');
    ylabel('sin (x)');
    hold on;

Plot: $\displaystyle f(x)=sin(x)$ for $\displaystyle n=0,1,2,3,4and5$

$\displaystyle f(x)=sin(x)$ for $\displaystyle n=6,7,8,9and10$

(iii)Error Estimation

$\displaystyle R_{n+1}(x)={\frac {(x-x_{0})^{n+1}}{(n+1)!}}f^{n+1}(\xi )$

$\displaystyle maxR_{n+1}(x)={\frac {(x-x_{0})^{n+1}}{(n+1)!}}max{\bigg |}f^{n+1}(\xi ){\bigg |}$

Since $\displaystyle f(x)=sin(x)$   $\displaystyle max{\bigg |}f^{n+1}(\xi ){\bigg |}=1$ for all $\displaystyle n$

(a)$\displaystyle R_{n+1}(x)$ for $\displaystyle n=0$  ,$\displaystyle x_{0}={\frac {\pi }{4}}$ and $\displaystyle x={\frac {\pi }{2}}$

$\displaystyle R_{n+1}(x)_{max}={\frac {x-x_{0}}{1!}}$

$\displaystyle ={\frac {{\frac {\pi }{2}}-{\frac {\pi }{4}}}{1!}}$

$\displaystyle R_{1}(x)_{max}=0.6168$

(b)$\displaystyle R_{n+1}(x)_{max}$ for $\displaystyle n=1$  ,$\displaystyle x_{0}={\frac {\pi }{4}}$ and $\displaystyle x={\frac {\pi }{2}}$

$\displaystyle R_{2}(x)_{max}={\frac {(x-x_{0})^{2}}{2!}}$

$\displaystyle ={\frac {({\frac {\pi }{2}}-{\frac {\pi }{4}})^{2}}{2!}}$

$\displaystyle R_{2}(x)_{max}=0.3084$

(c)$\displaystyle R_{n+1}(x)_{max}$ for $\displaystyle n=2$  ,$\displaystyle x_{0}={\frac {\pi }{4}}$ and $\displaystyle x={\frac {\pi }{2}}$

$\displaystyle R_{3}(x)_{max}={\frac {(x-x_{0})^{3}}{3!}}$

$\displaystyle ={\frac {({\frac {\pi }{2}}-{\frac {\pi }{4}})^{3}}{3!}}$

$\displaystyle R_{3}(x)_{max}=0.0807$

(d)$\displaystyle R_{n+1}(x)_{max}$ for $\displaystyle n=3$  ,$\displaystyle x_{0}={\frac {\pi }{4}}$ and $\displaystyle x={\frac {\pi }{2}}$

$\displaystyle R_{4}(x)_{max}={\frac {(x-x_{0})^{4}}{4!}}$

$\displaystyle ={\frac {({\frac {\pi }{2}}-{\frac {\pi }{4}})^{4}}{4!}}$

$\displaystyle R_{4}(x)_{max}=0.0158$

(e)$\displaystyle R_{n+1}(x)_{max}$ for $\displaystyle n=4$  ,$\displaystyle x_{0}={\frac {\pi }{4}}$ and $\displaystyle x={\frac {\pi }{2}}$

$\displaystyle R_{5}(x)_{max}={\frac {(x-x_{0})^{5}}{5!}}$

$\displaystyle ={\frac {({\frac {\pi }{2}}-{\frac {\pi }{4}})^{5}}{5!}}$

$\displaystyle R_{5}(x)_{max}=2.4903*10^{-3}$

(f)$\displaystyle R_{n+1}(x)_{max}$ for $\displaystyle n=5$  ,$\displaystyle x_{0}={\frac {\pi }{4}}$ and $\displaystyle x={\frac {\pi }{2}}$

$\displaystyle R_{6}(x)_{max}={\frac {(x-x_{0})^{6}}{6!}}$

$\displaystyle ={\frac {({\frac {\pi }{2}}-{\frac {\pi }{4}})^{6}}{6!}}$

$\displaystyle R_{6}(x)_{max}=3.2599*10^{-4}$

(g)$\displaystyle R_{n+1}(x)_{max}$ for $\displaystyle n=6$  ,$\displaystyle x_{0}={\frac {\pi }{4}}$ and $\displaystyle x={\frac {\pi }{2}}$

$\displaystyle R_{7}(x)_{max}={\frac {(x-x_{0})^{7}}{7!}}$

$\displaystyle ={\frac {({\frac {\pi }{2}}-{\frac {\pi }{4}})^{7}}{7!}}$

$\displaystyle R_{7}(x)_{max}=3.6576*10^{-5}$

(h)$\displaystyle R_{n+1}(x)_{max}$ for $\displaystyle n=7$  ,$\displaystyle x_{0}={\frac {\pi }{4}}$ and $\displaystyle x={\frac {\pi }{2}}$

$\displaystyle R_{8}(x)_{max}={\frac {(x-x_{0})^{8}}{8!}}$

$\displaystyle ={\frac {({\frac {\pi }{2}}-{\frac {\pi }{4}})^{8}}{8!}}$

$\displaystyle R_{8}(x)_{max}=3.5908*10^{-6}$

(i)$\displaystyle R_{n+1}(x)_{max}$ for $\displaystyle n=8$  ,$\displaystyle x_{0}={\frac {\pi }{4}}$ and $\displaystyle x={\frac {\pi }{2}}$

$\displaystyle R_{9}(x)_{max}={\frac {(x-x_{0})^{9}}{9!}}$

$\displaystyle ={\frac {({\frac {\pi }{2}}-{\frac {\pi }{4}})^{9}}{9!}}$

$\displaystyle R_{9}(x)_{max}=3.1336*10^{-7}$

(j)$\displaystyle R_{n+1}(x)_{max}$ for $\displaystyle n=9$  ,$\displaystyle x_{0}={\frac {\pi }{4}}$ and $\displaystyle x={\frac {\pi }{2}}$

$\displaystyle R_{10}(x)_{max}={\frac {(x-x_{0})^{10}}{10!}}$

$\displaystyle ={\frac {({\frac {\pi }{2}}-{\frac {\pi }{4}})^{10}}{10!}}$

$\displaystyle R_{10}(x)_{max}=2.4611*10^{-8}$

(k)$\displaystyle R_{n+1}(x)_{max}$ for $\displaystyle n=10$  ,$\displaystyle x_{0}={\frac {\pi }{4}}$ and $\displaystyle x={\frac {\pi }{2}}$

$\displaystyle R_{11}(x)_{max}={\frac {(x-x_{0})^{11}}{11!}}$

$\displaystyle ={\frac {({\frac {\pi }{2}}-{\frac {\pi }{4}})^{11}}{11!}}$

$\displaystyle R_{11}(x)_{max}=1.7572*10^{-9}$

--Subramanian Annamalai 23:58, 26 January 2010 (UTC)

(10)Kessler's code

Problem Statement

1. Run Kessler's code to reproduce the table of constants and the values of polynomials at t=1 obtained by Kessler.
2. Explain Kessler's code by giving comments for every line.
3. Obtain $\displaystyle (p_{2},p_{3})\;(p_{4},p_{5})\;(p_{6},p_{7})$ by understanding Kessler's code line by line.

Ref: Lecture Notes p.30-2 and p.37-1

Ref: Trapezoidal rule error

Solution

Part I : Generation of the Table
To generate the table given in Kessler's paper, the following modifications are to be carried out.

1. The first half of the code is the main code.
Give that a name called kessler.m

2. The second half of the code is a subroutine with the name fracsum.
This should not be part of the file kessler.m
We need to create a new file called fracsum.m in the same folder
that contains the file kessler.m

3. Set n=8. This is to ensure that we get the constants $c_{1}\;through\;c_{17}$
and polynomials $p_{2}(1)\;through\;p_{16}(1)$.

4. The most important part is that the subroutine fracsum should end with the line,
dsum=dsum/div;

5. The following lines have been appended to kessler.m to generate the table.

   format long;
   q_c = (cn./cd);    % Will generate the quotients of c
   q_p = (pn./pd);    % Will generate the quotients of p
   %Printing results
   disp(c); disp(q_c);
   disp(p); disp(q_p);

Result

The constants

Constant	Numerator	Denominator	Quotient
$c_{1}$	-1	1	-1.00000000000000
$c_{3}$	1	6	0.16666666666667
$c_{5}$	-7	360	-0.01944444444444
$c_{7}$	31	15120	0.00205026455026
$c_{9}$	-127	604800	-0.00020998677249
$c_{11}$	73	3421440	0.00002133604564
$c_{13}$	-1414477	653837184000	-0.00000216334744
$c_{15}$	8191	37362124800	0.00000021923271
$c_{17}$	-16931177	762187345920000	-0.00000002221393

The polynomials evaluated at t=1

Polynomial	Numerator	Denominator	Quotient
$p_{2}(1)$	-1	3	-0.33333333333333
$p_{4}(1)$	1	45	0.02222222222222
$p_{6}(1)$	-2	945	-0.00211640211640
$p_{8}(1)$	1	4725	0.00021164021164
$p_{10}(1)$	-2	93555	-0.00002137779916
$p_{12}(1)$	1382	638512875	0.00000216440428
$p_{14}(1)$	-4	18243225	-0.00000021925948
$p_{16}(1)$	3617	162820783125	0.00000002221461

Part II: Line by Line Explanation of Kessler's MATLAB Code - Adding Comments

function [c,p]=traperror(n) %traperror is a user-defined function which takes in 'n' as the input argument %and gives the constants and the coefficients of t in the polynomial p. f=1; g=2; cn=-1; cd=1; %cn is the numerator of the constants in the polynomial %cd is the denominator of the constants in the polynomial %cn is initially set to 1 and cd to -1 because c1 = -1, i.e,(-1/1). for k=1:n %The process is repeated for as many number of polynomials as desired. f=f.*g.*(g+1); % If we observe carefully we find that the denominators of the constants c1, c2, c3... are odd-numbered. % f generates these odd-numbered factorials. % For example, 1! =1 , 3! =6 and so on. [newcn,newcd]=fracsum(-1*cn,cd.*f); %The function fracsum is called to get the values of the numerator %and denominator of the constants. %Again on observing that, once if we find the constant c1, % the constant c3 can be found from the equation, c1/1! + c3/3! =0 % Then using c1 and c3, the constant c5 can be found using, % c1/5! + c3/3! + c1/1! = 0 and so on. % Hence the negative of the constant and the corresponding factorials are needed. % These are in fact the arguments of the function fracsum. cn=[cn;newcn]; cd=[cd;newcd]; % The current returned by the function fracsum and the % previous values of cn and cd are stored f=[f;1]; g=[2+g(1);g]; %The next value of f and g are generated. % The denominator factorials of the polynomials evaluated at t=1 differ from % the previous values of the constants only by 1. % Hence f and g are so adjusted to evaluate p(1) for all n. [newpn,newpd]=fracsum((g-1).*cn,f.*cd); %The function fracsum() returns the new values of the numerator and denominator %of the coefficient of t which is now stored in newpn and newpd respectively. pn(k,1)=newpn; pd(k,1)=newpd; % Here we assign p(1) of the numerator and denominator for varying n. end c=int64([cn cd]); %This gives the numerator and denominator of the constants. p=int64([pn pd]); %This gives the numerator and denominator of the coefficients of t. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [nsum,dsum]=fracsum(n,d) %fracsum is a user-defined function that takes in all the %previous values of the constants and generates the new values. div=gcd(round(n),round(d)); % The input arguments n and d are rounded of to the nearest integer values % This is carried out using the in-built "round" function %Then the greatest common divisor of n and d is found using the in-built 'gcd' function. % This is done in order that we have the constants in fractions - Only then can we call it as cn and cd. % Otherwise we will end up in a decimal. n=round(n./div); % The numerator is divided by the GCD to end up in a whole number d=round(d./div); % The numerator is divided by the GCD to end up in a whole number dsum=1; %The denominator of the constant is initialized to 1. for k=1:length(d) dsum=lcm(dsum,d(k)); %The least common factor of the new values of the denominator %of the constant and the previous value is calculated. %Here since we have to add all the fractions LCM is calculated. end %Then dsum gives the new value of the denominator of the constant. nsum=dsum*sum(n./d); div=gcd(round(nsum),round(dsum)); % Again the gratest common divisor is found between the numerator and denominator % of the constants using the 'gcd' and 'round' functions as before. % As before this is to ensure that we end up in a whole number and not a fraction. nsum=nsum/div; %nsum gives the new value of the numerator of the constant. dsum=dsum/div; %dsum gives the new value of the denominator of the constant.

Part III : Generating $\displaystyle (p_{2},p_{3}),(p_{4},p_{5}),(p_{6},p_{7})$

Recall the following:

<br\>

$\displaystyle p_{1}(t)=c_{1}t$

$\displaystyle (Eq.1)$

$\displaystyle p_{2}(t)=c_{1}{\frac {t^{2}}{2!}}+c_{3}$

$\displaystyle (Eq.2)$

$\displaystyle p_{3}(t)=c_{1}{\frac {t^{3}}{3!}}+c_{3}t$

$\displaystyle (Eq.3)$

$\displaystyle p_{4}(t)=c_{1}{\frac {t^{4}}{4!}}+c_{3}{\frac {t^{2}}{2!}}+c_{5}$

$\displaystyle (Eq.4)$

$\displaystyle p_{5}(t)=c_{1}{\frac {t^{5}}{5!}}+c_{3}{\frac {t^{3}}{3!}}+c_{5}t$

$\displaystyle (Eq.5)$

$\displaystyle p_{6}(t)=c_{1}{\frac {t^{6}}{6!}}+c_{3}{\frac {t^{4}}{4!}}+c_{5}{\frac {t^{2}}{2!}}+c_{7}$

$\displaystyle (Eq.6)$

$\displaystyle p_{7}(t)=c_{1}{\frac {t^{7}}{7!}}+c_{3}{\frac {t^{5}}{5!}}+c_{5}{\frac {t^{3}}{3!}}+c_{7}t$

$\displaystyle (Eq.7)$

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Hence we need to find out the constants, $\displaystyle c_{1},c_{3},c_{5}\;and\;c_{7}$

<br\>

So we start off with setting $c_{1}=-1$ And then we need to have n=3 ,because,

$\displaystyle n=1\;will\;yield\;c_{3}$

$\displaystyle n=2\;will\;yield\;c_{5}$

$\displaystyle n=3\;will\;yield\;c_{7}$

<br\>

$\displaystyle c_{1}=-1$ $\displaystyle f=1\,\,\,g=2$ $\displaystyle n=3;$

Consider the first iteration: i.e, k=1

$\displaystyle f=f.*g.*(g+1)$ yields f= 1*2*3 =6

Hence,

$\displaystyle f=6$

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$\displaystyle [newcn,newcd]=fracsum(-1*cn,cd.*f);$

$\displaystyle fracsum(-1*cn,cd.*f)$ yields fracsum(-1*1, 1*6) i.e, fracsum(-1,6)

Hence,we have

$\displaystyle fracsum(-1,6)$

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Now looking into the fracsum(n,d) function, $\displaystyle div=gcd(round(n),round(d))$

will yield div = 1. because, round(-1) = -1 round(6) = 6 and hence, gcd(-1,6) = 1.

<br\>

Now, $n=round(n./div);$ will give n=1 becasue round(1/1) =1

and,

$\displaystyle d=round(d./div);$ will give d=6 becasue round(6/1) =6

Thus,

$\displaystyle n=1\;and\;d=6$

$\displaystyle dsum=1;$ $\displaystyle fork=1:length(d)$ $\displaystyle dsum=lcm(dsum,d(k));$ $\displaystyle end$

Since length(d) = 1 and lcm (1,6) = 6, we have, dsum =6.

Thus,

$\displaystyle dsum=6$

$\displaystyle nsum=dsum*sum(n./d);$ $\displaystyle div=gcd(round(nsum),round(dsum));$

will yield,

nsum = 6* sum(1/6) = 1 and

<br\>

and,

div = 1 since, round(nsum) = round(1) = 1 round(dsum) = round(6) = 6 gcd(1,6) =1

Now,

$\displaystyle nsum=nsum/div;$ $\displaystyle dsum=dsum/div;$

will yield ,

nsum = 1 dsum = 6

Now going back, $\displaystyle [newcn\;newcd]=[1\;6]$

Thus,

$\displaystyle \therefore c_{3}={\frac {1}{6}}$

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Next, we have, $\displaystyle cn=[cn;newcn];cd=[cd;newcd]$ $\displaystyle f=[f;1];$

Hence, cn = [-1;1]; cd=[1;6]

and, So, f=[6;1]

This is followed by,

$\displaystyle g=[2+g(1);g]$ ; which means, g = [2+2, 2] = [4;2]

Now consider the second iteration - i.e, k=2

Consider the second iteration: i.e, k=2

$\displaystyle f=f.*g.*(g+1)$ yields f= (6*4*5; 1*2*3)

Hence,

$\displaystyle f=(120;6)$

<br\>

$\displaystyle [newcn,newcd]=fracsum(-1*cn,cd.*f);$

$\displaystyle fracsum(-1*cn,cd.*f)$ yields fracsum(-1*-1, 1*120) i.e, fracsum(1,120)

and,

$\displaystyle fracsum(-1*cn,cd.*f)$ yields fracsum(-1*1, 6*6) i.e, fracsum(-1,36)

Hence,we have

$\displaystyle fracsum(1,120)\;and\;fracsum(-1,36)$

<br\>

Now looking into the fracsum(1,120) function, $\displaystyle div=gcd(round(n),round(d))$

will yield div = 1. because, round(1) = 1 round(120) = 120 and hence, gcd(1,120) = 1.

<br\>

Now, $n=round(n./div);$ will give n=1 becasue round(1/1) =1

and,

$\displaystyle d=round(d./div);$ will give d=120 becasue round(36/1) =120

and,

<br\>

Now looking into the fracsum(-1,36) function, $\displaystyle div=gcd(round(n),round(d))$

will yield div = 1. because, round(-1) = -1 round(36) = 36 and hence, gcd(-1,36) = 1.

<br\>

Now, $n=round(n./div);$ will give n=-1 becasue round(-1/1) =-1

and,

$\displaystyle d=round(d./div);$ will give d=6 becasue round(36/1) =36

Thus,

$\displaystyle n=[1;-1]\;and\;d=[120;36]$

$\displaystyle dsum=1;$ $\displaystyle fork=1:length(d)$ $\displaystyle dsum=lcm(dsum,d(k));$ $\displaystyle end$

Since length(d) = 2 and For d(1), dsum = lcm (1,120) = 120, and once again for d(2), dsum = lcm (120,36) = 360 Finally we have, dsum = 360.

Thus,

$\displaystyle dsum=360$

$\displaystyle nsum=dsum*sum(n./d);$ $\displaystyle div=gcd(round(nsum),round(dsum));$

will yield,

nsum = 360* sum(1/120 - 1/36) = -7 and

<br\>

and,

div = 1 since, round(nsum) = round(-7) = -7 round(dsum) = round(360) = 360 gcd(-7,360) =1

Now,

$\displaystyle nsum=nsum/div;$ $\displaystyle dsum=dsum/div;$

will yield ,

nsum = -7 dsum = 360

Now going back, $\displaystyle [newcn\;newcd]=[-7\;360]$

Thus,

$\displaystyle \therefore c_{5}=-{\frac {7}{360}}$

<br\>

Next, we have, $\displaystyle cn=[cn;newcn];cd=[cd;newcd]$ $\displaystyle f=[f;1];$

Hence, cn = [-1;1;7]; cd=[1;6;360]

and, So, f=[120;6;1]

This is followed by,

$\displaystyle g=[2+g(1);g]$ ; which means, g = [2+4;4; 2] = [6;4;2]

<br\>

Again following the same procedure for Iteration 3, i.e for k=3 we will obtain,

$\displaystyle \therefore c_{7}={\frac {31}{15120}}$

<br\> <br\>

On substituting, the values of

$\displaystyle c_{1}\;c_{3}\;c_{5}\;and\;c_{7}$ in (Eq.1) through (Eq.7) we obtain,

$\displaystyle p_{1}(t)=-t$

$\displaystyle p_{2}(t)=-{\frac {t^{2}}{2!}}+{\frac {1}{6}}$

$\displaystyle p_{3}(t)=-{\frac {t^{3}}{3!}}+{\frac {t}{6}}$

$\displaystyle p_{4}(t)=-{\frac {t^{4}}{4!}}+{\frac {t^{2}}{12}}-{\frac {7}{360}}$

$\displaystyle p_{5}(t)=-{\frac {t^{5}}{5!}}+{\frac {t^{3}}{36}}-{\frac {7t}{360}}$

$\displaystyle p_{6}(t)=-{\frac {t^{6}}{6!}}+{\frac {t^{4}}{144}}-{\frac {7t^{2}}{720}}+{\frac {31}{15120}}$

$\displaystyle p_{7}(t)=-{\frac {t^{7}}{7!}}+{\frac {t^{5}}{720}}-{\frac {7t^{3}}{2160}}+{\frac {31t}{15120}}$

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