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Week 3.5 - Deriving 4th order approximation of a 2nd order derivative using Taylor Table method

Aim : To solve the 4th order approxiamtion of 2nd order derivative using : 1. Central difference 2. Skewed right-sided difference 3. Skewed left-sided difference 1. Central difference method: We take information from both sides of a point. f(i) f'(i)* $Δ x$ $Deltax$ f''(i)* $Δ x^{2}$ $Deltax^2$ f'''(i)* $Δ x^{3}$ $Deltax^3$ …

chetankumar nadagoud
updated on 04 Feb 2022

Aim : To solve the 4th order approxiamtion of 2nd order derivative using :

1. Central difference

2. Skewed right-sided difference

3. Skewed left-sided difference

1. Central difference method: We take information from both sides of a point.

	f(i)	f'(i)* $Δ x$ $Deltax$	f''(i)* $Δ x^{2}$ $Deltax^2$	f'''(i)* $Δ x^{3}$ $Deltax^3$	f''''(i)* $Δ x^{4}$ $Deltax^4$
af(i-2)	a	-2a	4a/2	-8a/6	16a/24
bf(i-1)	b	-b	b/2	-b/6	b/24
cf(i)	c	0	0	0	0
df(i+1)	d	d	d/2	d/6	d/24
ef(i+2)	e	2e	4e/2	8e/6	16e/24
	0	0	1	0	0

We create matrix for the above table to solve for a,b,c,d and e

A = $[\begin{matrix} 1 & 1 & 1 & 1 & 1 \\ - 2 & 1 & 0 & 1 & 2 \\ 2 & \frac{1}{2} & 0 & \frac{1}{2} & 2 \\ - \frac{4}{3} & - \frac{1}{6} & 0 & \frac{1}{6} & \frac{4}{3} \\ \frac{2}{3} & \frac{1}{24} & 0 & \frac{1}{24} & \frac{2}{3} \end{matrix}]$ $[[1,1,1,1,1],[-2,1,0,1,2],[2,1/2,0,1/2,2],[-4/3,-1/6,0,1/6,4/3],[2/3,1/24,0,1/24,2/3]]$

B = $[\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{matrix}]$ $[[0],[0],[1],[0],[0]]$

X = rats(linsolve(A,B))

We use rats to convert decimal to fractions we get:

where: a = -1/12

b = 4/3

c = -5/2

d = 4/3

e = -1/12

2.Right skew method: Here we take values right side of the point.

	f(i)	f'(i)* $Δ x$ $Deltax$	f''(i)* $Δ x^{2}$ $Deltax^2$	f'''(i)* $Δ x^{3}$ $Deltax^3$	f''''(i)* $Δ x^{4}$ $Deltax^4$	f'''''(i)* $Δ x^{5}$ $Deltax^5$
af(i)	a	0	0	0	0	0
bf(i+1)	b	b	b/2	b/6	b/24	b/120
cf(i+2)	c	2c	4c/2	8c/6	16c/24	32c/120
df(i+3)	d	3d	9d/2	27d/6	81d/24	243d/120
ef(i+4)	e	4e	16e/2	64e/6	256e/24	1024e/120
gf(i+5)	g	5g	25g/2	125g/6	625g/24	3125g/120
	0	0	1	0	0	0

We create matrix for the above table to solve for a,b,c,d,e and g

A = $[\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 & 4 & 5 \\ 0 & \frac{1}{2} & \frac{4}{2} & \frac{9}{2} & \frac{16}{2} & \frac{25}{2} \\ 0 & \frac{1}{6} & \frac{8}{6} & \frac{27}{6} & \frac{64}{6} & \frac{125}{6} \\ 0 & \frac{1}{24} & \frac{16}{24} & \frac{81}{24} & \frac{256}{24} & \frac{625}{24} \\ 0 & \frac{1}{120} & \frac{32}{120} & \frac{243}{120} & \frac{1024}{120} & \frac{3125}{120} \end{matrix}]$ $[[1,1,1,1,1,1],[0,1,2,3,4,5],[0,1/2,4/2,9/2,16/2,25/2],[0,1/6,8/6,27/6,64/6,125/6],[0,1/24,16/24,81/24,256/24,625/24],[0,1/120,32/120,243/120, 1024/120, 3125/120]]$

B = $[\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{matrix}]$ $[[0],[0],[1],[0],[0],[0]]$

X = rats(linsolve(A,B))

We use rats to convert decimal to fractions we get:

3. Left skew method : Here we take points on the left side of the point.

	f(i)	f'(i)* $Δ x$ $Deltax$	f''(i)* $Δ x^{2}$ $Deltax^2$	f'''(i)* $Δ x^{3}$ $Deltax^3$	f''''(i)* $Δ x^{4}$ $Deltax^4$	f'''''(i)* $Δ x^{5}$ $Deltax^5$
af(i)	a	0	0	0	0	0
bf(i-1)	b	-b	b/2	-b/6	b/24	-b/120
cf(i-2)	c	-2c	4c/2	-8c/6	16c/24	-32c/120
df(i-3)	d	-3d	9d/2	-27d/6	81d/24	-243d/120
ef(i-4)	e	-4e	16e/2	-64e/6	256e/24	-1024e/120
gf(i-5)	g	-5g	25g/2	-125g/6	625g/24	-3125g/120
	0	0	1	0	0	0

A = $[\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & - 1 & - 2 & - 3 & - 4 & - 5 \\ 0 & \frac{1}{2} & \frac{4}{2} & \frac{9}{2} & \frac{16}{2} & \frac{25}{2} \\ 0 & - \frac{1}{6} & - \frac{8}{6} & - \frac{27}{6} & - \frac{64}{6} & - \frac{125}{6} \\ 0 & \frac{1}{24} & \frac{16}{24} & \frac{81}{24} & \frac{256}{24} & \frac{625}{24} \\ 0 & - \frac{1}{120} & - \frac{32}{120} & - \frac{243}{120} & - \frac{1024}{120} & - \frac{3125}{120} \end{matrix}]$ $[[1,1,1,1,1,1],[0,-1,-2,-3,-4,-5],[0,1/2,4/2,9/2,16/2,25/2],[0,-1/6,-8/6,-27/6,-64/6,-125/6],[0,1/24,16/24,81/24,256/24,625/24],[0,-1/120,-32/120,-243/120, -1024/120, -3125/120]]$

B = $[\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{matrix}]$ $[[0],[0],[1],[0],[0],[0]]$

X = rats(linsolve(A,B))

We use rats to convert decimal to fractions we get:

Steps in codeing :

1. Create a function for $f (x) = \cos x \cdot e (x)$ $f(x)=cosx*e(x)$

function f = func(x)
f = cos(x)*exp(x);
end

2. Insert the coefficients in a matrix and solve them using linsolve

% coefficients for Left skew method
clear all
close all
clc

A = [1 1 1 1 1 1;
    0 -1 -2 -3 -4 -5;
    0 1/2 4/2 9/2 16/2 25/2;
    0 -1/6 -8/6 -27/6 -64/6 -125/6;
    0 1/24 16/24 81/24 256/24 625/24;
    0 -1/120 -32/120 -243/120 -1024/120 -3125/120];
B = [0;0;1;0;0;0];

X = rats(linsolve(A,B))


%Coefficients for Right skew method

clear all
close all
clc

A = [1 1 1 1 1 1;
    0 1 2 3 4 5;
    0 1/2 4/2 9/2 16/2 25/2;
    0 1/6 8/6 27/6 64/6 125/6;
    0 1/24 16/24 81/24 256/24 625/24;
    0 1/120 32/120 243/120 1024/120 3125/120];
B = [0;0;1;0;0;0];

X = rats(linsolve(A,B))

%coefficienst for Central difference method

clear all
close all
clc

A = [1 1 1 1 1;
    -2 -1 0 1 2;
    4/2 1/2 0 1/2 4/2;
    -8/6 -1/6 0 1/6 8/6;
    16/24 1/24 0 1/24 16/24]

B = [0;0;1;0;0]

X = rats(linsolve(A,B))

3. Create the functions for Central difference,Right skew and Left skew and substitute the values obtained from above program in it.

%central difference
function out = centraldiff(x,dx)
a=-1
b=16
c=-30
d=16
e=-1
analyticalsol= -2*exp(x)*sin(x);
central = (a*func(x-2*dx)+b*func(x-dx)+c*func(x)+d*func(x+dx)+e*func(x+2*dx))/(12*(dx^2));
out = abs(central-analyticalsol);
end

%Right skew
function out = rightskew(x,dx)

analyticalsol = -2*exp(x)*sin(x);
rightskew = (45*func(x)-154*func(x+dx)+214*func(x+2*dx)-156*func(x+3*dx)+61*func(x+4*dx)-10*func(x+5*dx))/(12*(dx^2))
out = abs(analyticalsol-rightskew);
end

%Left skew
function out = leftskew(x,dx)

analyticalsol = -2*exp(x)*sin(x);
leftskew= (45*func(x)-154*func(x-dx)+214*func(x-2*dx)-156*func(x-3*dx)+61*func(x-4*dx)-10*func(x-5*dx))/(12*(dx^2))
out = abs(analyticalsol-leftskew);
end

4. Now write main code involving all these functions and plot the graph.

Main code:

clear all
close all
clc

x = pi/3;
dx = linspace(pi/4,pi/400,100);


%central
for i = 1:length(dx)
    central_error(i) = centraldiff(x,dx(i));
end

%right
for i = 1:length(dx)
    right_error(i)=rightskew(x,dx(i));
end

%left
for i = 1:length(dx)
    left_error(i)=leftskew(x,dx(i));
end

%plotting
figure(1)
loglog(dx,central_error,'LineWidth',1,'Marker','pentagram','MarkerSize',2)
hold on
loglog(dx,right_error,'LineWidth',1,'Marker','pentagram','MarkerSize',2)
loglog(dx,left_error,'LineWidth',1,'Marker','pentagram','MarkerSize',2)
legend('central difference','right-skew difference','left-skew difference')
xlabel('dx')
ylabel('Error')
grid on

Plots and Results :