general approximation techniques developed from exercise sheets

4 files changed, 106 insertions, 0 deletions

diff --git a/general/euler.m b/general/euler.m
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+++ b/general/euler.m
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+function [xa, ya] = euler(derivative, x_initial, y_initial, step, x_final)
+	% Define intial values for x and y
+	%x_initial=0; 
+	%y_initial=2;
+
+	%% Set the step size between calculations
+	%step=0.0003;
+
+	%% Define an endpoint
+	%x_final=1;
+
+	%% Define a dy/dx
+	%derivative=@(x,y) x+y; 
+
+	% Determine how many times to step forward
+	N=round((x_final-x_initial)/step); % use this determine size
+
+	% Define and initialise arrays to contain coordinates
+	ya=zeros(1,N); xa=zeros(1,N);
+	xa(1)=x_initial; ya(1)=y_initial;
+	
+	for j=1:N-1 % loop for N-1 steps
+		ya(j+1)=ya(j) + step*feval(derivative, xa(j),ya(j)); % next value
+		xa(j+1)=xa(j)+step; % increase x by stepsize
+	end
diff --git a/general/heun.m b/general/heun.m
new file mode 100644
index 0000000..e3d91b0
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+++ b/general/heun.m
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+function [xa, ya] = heun(derivative, x_initial, y_initial, step, x_final)
+	% Define intial values for x and y
+	%x_initial=0; 
+	%y_initial=2;
+
+	%% Set the step size between calculations
+	%step=0.0003;
+
+	%% Define an endpoint
+	%x_final=1;
+
+	%% Define a dy/dx
+	%derivative=@(x,y) x+y; 
+
+	% Determine how many times to step forward
+	N=round((x_final-x_initial)/step); % use this determine size
+
+	% Define and initialise arrays to contain coordinates
+	ya=zeros(1,N); xa=zeros(1,N);
+	xa(1)=x_initial; ya(1)=y_initial;
+	
+	% da is the value of the derivative at xa, ya
+	for j=1:N-1 % loop for N-1 steps
+		da = feval(derivative, xa(j), ya(j));
+		ya(j+1)=ya(j) + step/2*(da + feval(derivative, xa(j)+step, ya(j)+step*da)) ; 
+		xa(j+1)=xa(j)+step; % increase x by stepsize
+	end
diff --git a/general/midpoint.m b/general/midpoint.m
new file mode 100644
index 0000000..cecef44
--- /dev/null
+++ b/general/midpoint.m
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+function [xa, ya] = midpoint(derivative, x_initial, y_initial, step, x_final)
+	% Define intial values for x and y
+	%x_initial=0; 
+	%y_initial=2;
+
+	%% Set the step size between calculations
+	%step=0.0003;
+
+	%% Define an endpoint
+	%x_final=1;
+
+	%% Define a dy/dx
+	%derivative=@(x,y) x+y; 
+
+	% Determine how many times to step forward
+	N=round((x_final-x_initial)/step); % use this determine size
+
+	% Define and initialise arrays to contain coordinates
+	ya=zeros(1,N); xa=zeros(1,N);
+	xa(1)=x_initial; ya(1)=y_initial;
+	
+	% da is the value of the derivative at xa, ya
+	for j=1:N-1 % loop for N-1 steps
+		da = feval(derivative, xa(j), ya(j));
+		ya(j+1)=ya(j) + step*feval(derivative, xa(j)+step/2, ya(j)+step/2*da); 
+		xa(j+1)=xa(j)+step; % increase x by stepsize
+	end
diff --git a/general/ralston.m b/general/ralston.m
new file mode 100644
index 0000000..2d18571
--- /dev/null
+++ b/general/ralston.m
@@ -0,0 +1,27 @@
+function [xa, ya] = ralston(derivative, x_initial, y_initial, step, x_final)
+	% Define intial values for x and y
+	%x_initial=0; 
+	%y_initial=2;
+
+	%% Set the step size between calculations
+	%step=0.0003;
+
+	%% Define an endpoint
+	%x_final=1;
+
+	%% Define a dy/dx
+	%derivative=@(x,y) x+y; 
+
+	% Determine how many times to step forward
+	N=round((x_final-x_initial)/step); % use this determine size
+
+	% Define and initialise arrays to contain coordinates
+	ya=zeros(1,N); xa=zeros(1,N);
+	xa(1)=x_initial; ya(1)=y_initial;
+	
+	% da is the value of the derivative at xa, ya
+	for j=1:N-1 % loop for N-1 steps
+		da = feval(derivative, xa(j), ya(j));
+		ya(j+1)=ya(j) + step/4*(da + 3*feval(derivative, xa(j)+2/3*step, ya(j)+2/3*step*da)) ; 
+		xa(j+1)=xa(j)+step; % increase x by stepsize
+	end