Assignmnt 4
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The document discusses two interpolation methods - Newton forward difference method and backward method. It provides the code to calculate the forward difference table using Newton forward difference method. It also provides functions to implement backward interpolation method and Stirling's interpolation formula.
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![Sajid Ali
081220-097
INTERPOLATION Methods:
NEWTON FORWARD DIFFERENCE METHOD:
n = input('nEnter number of data points = ');
h = input('nEnter step size (h) = ')
x(1) = input('nx0 = ');
y(1) = input('y0 = ');
for i=2:n
x(i)=x(i-1)+h;
fprintf('nX%d = %f',i,x(i));
fprintf('ttY%d: ',i);
y(i) = input('');
end
x_reqd = input('nEnter X for which value of Y is sought: ');
s=(x_reqd-x(1))/h;
for i=1:n
diff(i,1)=y(i);
end
%% Calculate Forward Differance Table
for j=2:n
for i=1:n-j+1
diff(i,j)=diff(i+1,j-1)-diff(i,j-1);
end
end
%% Print Forward Differance Table
fprintf('ntXt Forward Differance Table');
for i=1:n
fprintf('n %.4f',x(i));
for j=1:n-i+1
fprintf('t%.4f',diff(i,j));
end
end
ans=y(1);
for i=1:n-1
term=1;
for j=1:i
term=term*(s+j-1)/j;
end
ans=ans+term*diff(1,i+1);
end
Backward Methed:
function Forward()
x=[0;1;2;3;4;5;6];](https://image.slidesharecdn.com/assignmnt4-120515015138-phpapp01/85/Assignmnt-4-1-320.jpg)
![y=[2;3;10;29;66;127;218];
dy=[y(2,1)-y(1,1);y(3,1)-y(2,1);y(4,1)-y(3,1);y(5,1)-y(4,1);y(6,1)-
y(5,1);y(7,1)-y(6,1)];
d2y=[dy(2,1)-dy(1,1);dy(3,1)-dy(2,1);dy(4,1)-dy(3,1);dy(5,1)-
dy(4,1);dy(6,1)-dy(5,1)];
d3y=[d2y(2,1)-d2y(1,1);d2y(3,1)-d2y(2,1);d2y(4,1)-d2y(3,1);d2y(5,1)-
d2y(4,1)];
d4y=[d3y(2,1)-d3y(1,1);d3y(3,1)-d3y(2,1);d3y(4,1)-d3y(3,1)];
d5y=[d4y(2,1)-d4y(1,1);d4y(3,1)-d4y(2,1)];
d6y=[d5y(2,1)-d5y(1,1)];
p=(.5-0)/1;
Y=y(1,1)+(((p*dy(1,1)))+((p*(p-1))/2)*d2y(1,1))+(((p*(p-1)*(p-
2))/6)*d3y(1,1))+(((p*(p-1)*(p-2)*(p-3))/24)*d4y(1,1))+(((p*(p-1)*(p-2)*(p-
3)*(p-4))/120)*d5y(1,1))+(((p*(p-1)*(p-2)*(p-3)*(p-4)*(p-5))/720)*d6y(1,1))
end
Striling
function Forward()
x=[0;1;2;3;4;5;6];
y=[2;3;10;29;66;127;218];
dy=[y(2,1)-y(1,1);y(3,1)-y(2,1);y(4,1)-y(3,1);y(5,1)-y(4,1);y(6,1)-
y(5,1);y(7,1)-y(6,1)];
d2y=[dy(2,1)-dy(1,1);dy(3,1)-dy(2,1);dy(4,1)-dy(3,1);dy(5,1)-
dy(4,1);dy(6,1)-dy(5,1)];
d3y=[d2y(2,1)-d2y(1,1);d2y(3,1)-d2y(2,1);d2y(4,1)-d2y(3,1);d2y(5,1)-
d2y(4,1)];
d4y=[d3y(2,1)-d3y(1,1);d3y(3,1)-d3y(2,1);d3y(4,1)-d3y(3,1)];
d5y=[d4y(2,1)-d4y(1,1);d4y(3,1)-d4y(2,1)];
d6y=[d5y(2,1)-d5y(1,1)];
p=(.5-0)/1;
Y=y(1,1)+(((p*dy(1,1)))+((p*(p-1))/2)*d2y(1,1))+(((p*(p-1)*(p-
2))/6)*d3y(1,1))+(((p*(p-1)*(p-2)*(p-3))/24)*d4y(1,1))+(((p*(p-1)*(p-
2)*(p-3)*(p-4))/120)*d5y(1,1))+(((p*(p-1)*(p-2)*(p-3)*(p-4)*(p-
5))/720)*d6y(1,1))
end](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Incorporate the SOR method in the multigridTest.m and apply the multigridTest.m, to
determine the effects of changes in
Pre-smoothing iterations
Post-smoothing iterations
Multigrid cycle type
Iterative method
Number of grids (or grid hierarchy)
% SOR (Successive Over-Relaxation)
n = input('Enter number of equations, n: ');
A = zeros(n,n+1);
x1 = zeros(1,n);
A=[5 -2 3 -1;-3 9 1 2;2 -1 -7 3];
x1 = [0 0 0];
tol = input('Enter tolerance, tol: ');
m = input('Enter maximum number of iterations, m: ');
w = input('Enter the parameter w (omega): ');
k = 1;
while k <= m
err = 0;
for i = 1 : n
s = 0;
for j = 1 : n
s = s-A(i,j)*x1(j);
end
s = w*(s+A(i,n+1))/A(i,i);
if abs(s) > err
err = abs(s);
end
x1(i) = x1(i)+s;
end
if err <= tol
break;
else
k = k+1;
end
end
fprintf('The solution vector after %d iterations is :\n', k);
for i = 1 : n
fprintf(' %11.8f \n', x1(i));
end
%% Multigrid test case with visulization
% Author: Ben Beaudry
clc; clear; close all
load A_b.mat;
A = full(A); % Unsparce matrix to show full power of Multigrid
pre = 2; % Number of presmoothing iterations
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grids = 5; % Max grids in cycle
maxit = 10; % Max iterations of solver
tol = 1e-02; % Tolerance of solver
%% Solvers
% solve directly
t=cputime;
x_D = A\b;
fprintf('Direct Solve Time: %g Seconds\n',cputime-t)
% V-Cycle
t=cputime;
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fprintf('V-Cycle Solve Time: %g Seconds\n',cputime-t)
% W-Cycle
t=cputime;
[x_W,res_W,it_W] = multigrid(A,b,pre,post,2,smooth,grids,maxit,tol);
fprintf('W-Cycle Solve Time: %g Seconds\n',cputime-t)
% F-Cycle
t=cputime;
[x_F,res_F,it_F] = multigrid(A,b,pre,post,3,smooth,grids,maxit,tol);
fprintf('F-Cycle Solve Time: %g Seconds\n',cputime-t)
% max it for iterative methods
it = 100;
% Gauss-Siedel
t=cputime;
L = tril(A);
U = triu(A,1);
x_G = zeros(length(b),1);
res_G= zeros(it,1);
for g = 1:it
x_G = L\(b-U*x_G);
r = b - A * x_G;
res_G(g) = norm(r);
if res_G(g) < tol
break;
end
end
fprintf('Gauss-Seidel Solve Time: %g Seconds\n',cputime-t)
% Jacobi
t=cputime;
d = diag(A);
dinv = 1./d;
LU = tril(A,-1)+triu(A,1);
U = triu(A,1);
x_J = zeros(length(b),1);
res_J= zeros(it,1);
for j = 1:it
x_J = dinv.*(b-(LU*x_J));
r = b - A * x_J;
res_J(j) = norm(r);
if res_J(j) < tol
break;
end
end
fprintf('Jacobi Solve Time: %g Seconds\n',cputime-t)
%% Plotting
figure;
hold on
plot(1:g,res_G(1:g),'o-','DisplayName','Guass-Seidel')
plot(1:j,res_J(1:j),'o-','DisplayName','Jacobi')
plot(1:it_V,res_V(1:it_V),'o-','DisplayName','V-Cycle Multigrid')
plot(1:it_F,res_F(1:it_F),'o-','DisplayName','F-Cycle Multigrid')
plot(1:it_W,res_W(1:it_W),'o-','DisplayName','W-Cycle Multigrid')
set(gca,'XLim',[0 100]);
grid on
legend('location','best')
ylabel('Relative Convergance')
xlabel('Iterations')
title('Convergance of Numerical System')
hold off
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