- Define a maximum number of iterations and a tolerance for convergence. The tolerance will help you determine when the value of "y" is close enough to 0 to stop the iterations.
- Use a loop to repeatedly apply the shooting method with new guesses for "z" based on the secant method until either the maximum number of iterations is reached or the convergence criterion is met.
- Update the values of z0, z1, y(z0), and y(z1) at the end of each iteration as per the secant method.
Shooting Method with Secant Method
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Hi all,
I am working on a problem to solve a second order differential equation. I am using the numerical method 'the shooting method' and I need to perform iterations to find the initial value of the slope. To do this I am using the secant method. My intial values to start the shooting method for z are -4 and 4. From this is get the corresponding y values of -65.025 (3dp) and 106.062 (3dp) respectively. I am aiming to find the value of z when y = 0. Thus, I proceed to the secant method to find the value of 'z' when y = 0;
The formula for secant method is:
z2 = z1 - (y(z1) - 0)*(z1 - z2)/(y(z1 - y(z0))
I want to make a loop where this updates: Hence:
z0 = z1
z1 = z2
y(z0) = y(z1)
y(z1) = y(z2)
So, the next iteration is:
z3 = z2 - (y(z2) - 0)*(z2 - z21/(y(z2 - y(z2))
The thing is I now need to compute y(z2). I can do this, but it would meaning copying and pasting another block of code. Thus, my question is how to make the algorithm in MATLAB so that I can perform the desired number of iterations I want within the for loop? Rather than copying and pasting the code below over and over again until I reach a satisfactory point of convergence?
Here is my code:
clear, clc, close all
format longG
% Structural Properties
E = 2.1E+08;
Ic = 22530/100^4;
Ib = 19460/100^4;
Ac = 168/100^2;
h = 0.5;
r = 1;
Gi = (1*E*Ib/10);
Ci = 1/2*(E*Ic/h);
g = Gi/Ci;
K = (6*g + 1 + r);
V = [0 0 0 0 0 35 0 0 0 0 0 25 0 0 0 0 0 15 0 0 0 0 0 5 0];
%Initial Conditions 1
y = 0
z = -4
for i = 1:24
if i == 6 || i == 12 || i == 18
F(i) = h*(V(i) + V(i+6))/(4*Ci);
z(i+1) = z(i) + ((F(i) - K*z(i) + y(i))/(-z(i)))*h;
y(i+1) = y(i) + z(i)*h;
else F(i) = 0;
z(i+1) = z(i) + ((F(i) - K*z(i) + y(i))/(-z(i)))*h;
y(i+1) = y(i) + z(i)*h;
end
if i == 24
F(i) = h*(V(i))/(4*Ci);
z(i+1) = z(i) + ((F(i) - K*z(i) + y(i))/(-z(i)))*h;
y(i+1) = y(i) + z(i)*h
end
q0 = y(end)
p0 = z(1)
end
q0 = y(end)
p0 = z(1)
%Initial Conditions 2
y = 0
z = 4
for i = 1:24
if i == 6 || i == 12 || i == 18
F(i) = h*(V(i) + V(i+6))/(4*Ci);
z(i+1) = z(i) + ((F(i) - K*z(i) + y(i))/(-z(i)))*h;
y(i+1) = y(i) + z(i)*h;
else F(i) = 0;
z(i+1) = z(i) + ((F(i) - K*z(i) + y(i))/(-z(i)))*h;
y(i+1) = y(i) + z(i)*h;
end
if i == 24
F(i) = h*(V(i))/(4*Ci);
z(i+1) = z(i) + ((F(i) - K*z(i) + y(i))/(-z(i)))*h;
y(i+1) = y(i) + z(i)*h
end
end
q1 = y(end)
p1 = z(1)
% Secant Method
p2 = p1 - (q1 - 0)*(p1 - p0)/(q1 -q0)
I have left the secant method formula outside the loop, because I am unsure how to include it in the loop to perform the iterations I previously explained.
I understand that this is quite a long question. So thank you in advance for any help. However, you guys on here are great, so I hoping its not too tedious for you.
Many thanks,
Scott
0 Comments
Accepted Answer
surya venu
on 20 Jun 2024
Edited: Torsten
on 20 Jun 2024
Hi,
To integrate the secant method into your MATLAB code and perform the iterations automatically, you can wrap the entire process in a loop. This way, you don't have to manually repeat the shooting method calculations for new guesses of "z". Here's how you can structure your code to do this:
Here's a modified version of your code incorporating these suggestions:
clear, clc, close all
format longG
% Structural Properties
E = 2.1E+08;
Ic = 22530/100^4;
Ib = 19460/100^4;
Ac = 168/100^2;
h = 0.5;
r = 1;
Gi = (1*E*Ib/10);
Ci = 1/2*(E*Ic/h);
g = Gi/Ci;
K = (6*g + 1 + r);
V = [0 0 0 0 0 35 0 0 0 0 0 25 0 0 0 0 0 15 0 0 0 0 0 5 0];
% Initial guesses and values
z0 = -4;
z1 = 4;
maxIters = 100; % Maximum number of iterations
tol = 1e-5; % Convergence tolerance
iter = 0;
while iter < maxIters
y = zeros(1,25); % Reset y for each shooting method iteration
z = zeros(1,25); % Reset z for each shooting method iteration
z(1) = z0; % Set initial condition for z
% Shooting method for z0
for i = 1:24
if i == 6 || i == 12 || i == 18
F(i) = h*(V(i) + V(i+6))/(4*Ci);
else
F(i) = 0;
end
z(i+1) = z(i) + ((F(i) - K*z(i) + y(i))/(-z(i)))*h;
y(i+1) = y(i) + z(i)*h;
end
q0 = y(end);
% Reset y and z for the next initial condition
y = zeros(1,25);
z = zeros(1,25);
z(1) = z1; % Set initial condition for z
% Shooting method for z1
for i = 1:24
if i == 6 || i == 12 || i == 18
F(i) = h*(V(i) + V(i+6))/(4*Ci);
else
F(i) = 0;
end
z(i+1) = z(i) + ((F(i) - K*z(i) + y(i))/(-z(i)))*h;
y(i+1) = y(i) + z(i)*h;
end
q1 = y(end);
% Check for convergence
if abs(q1) < tol
fprintf('Converged to y = 0 with z = %f after %d iterations\n', z1, iter);
break;
end
% Update for next iteration using secant method
p2 = z1 - (q1 - 0)*(z1 - z0)/(q1 - q0);
% Prepare for next iteration
z0 = z1;
z1 = p2;
q0 = q1;
iter = iter + 1;
end
if iter == maxIters
fprintf('Max iterations reached without convergence.\n');
end
Hope it helps.
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