Newton Raphson Algorithm MATLAB is used for identifying roots of real-valued function. Newton-Raphson algorithm is examined as a robust approach. For well-mannered operations, this method intersects rapidly and is an iterative technique. Our experts stay updated on trending areas we have all leading tools and resources to get your work done. We suggest a procedural instruction to apply the Newton-Raphson method in MATLAB:
Procedural Implementation
xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n – \frac{f(x_n)}{f'(x_n)}xn+1=xn−f′(xn)f(xn)
An initial guess x0x_0x0 has to be selected. Till the solution intersects to the preferred tolerance, focus on repeating the process with the aid of this formula.
Instance: Finding the root of f(x)=x2−2f(x) = x^2 – 2f(x)=x2−2
The following is an entire instance in MATLAB:
Define the function and its derivative
Initially, by means of the below content, we plan to develop a function file newton_raphson.m.
function root = newton_raphson(f, df, x0, tol, max_iter)
% NEWTON_RAPHSON finds the root of a function using the Newton-Raphson method.
% f: function handle
% df: derivative of the function
% x0: initial guess
% tol: tolerance for stopping criteria
% max_iter: maximum number of iterations
% Initialize variables
x = x0;
iter = 0;
% Iterate using the Newton-Raphson formula
while iter < max_iter
% Compute the next value
x_next = x – f(x) / df(x);
% Check for convergence
if abs(x_next – x) < tol
root = x_next;
fprintf(‘Converged to root: %f in %d iterations\n’, root, iter);
return;
end
% Update for next iteration
x = x_next;
iter = iter + 1;
end
% If maximum iterations are reached without convergence
error(‘Newton-Raphson method did not converge within the maximum number of iterations’);
end
Use the function
As a means to call the newton_raphson function, our team aims to utilize the command window or develop a script:
% Define the function and its derivative
f = @(x) x^2 – 2;
df = @(x) 2 * x;
% Initial guess, tolerance, and maximum number of iterations
x0 = 1;
tol = 1e-6;
max_iter = 100;
% Find the root using the Newton-Raphson method
root = newton_raphson(f, df, x0, tol, max_iter);
disp([‘Root found: ‘, num2str(root)]);
Description
Output
The number of iterations obtained to converge and the root identified are demonstrated in the script.
Hints for Using Newton-Raphson Method
If you are choosing a project based on Newton-Raphson method, you must prefer efficient and crucial project topics. We recommend some projects that encompasses supplementary techniques or methods to improve the solution and the process of implementing the Newton-Raphson algorithm to address certain issues in an effective manner:
Instance Project: Power Flow Analysis in Electrical Grids
% Define the power flow equations and their Jacobian
function [f, J] = power_flow_equations(V, theta, Ybus, P, Q)
% V: voltage magnitudes
% theta: voltage angles
% Ybus: admittance matrix
% P, Q: active and reactive power demands
% Initialize the mismatch equations
f = zeros(2*length(V), 1);
J = zeros(2*length(V), 2*length(V));
% Compute the mismatch equations and Jacobian
for i = 1:length(V)
for j = 1:length(V)
if i ~= j
f(i) = f(i) + V(i)*V(j)*abs(Ybus(i,j))*cos(theta(i) – theta(j) – angle(Ybus(i,j)));
f(length(V) + i) = f(length(V) + i) + V(i)*V(j)*abs(Ybus(i,j))*sin(theta(i) – theta(j) – angle(Ybus(i,j)));
end
end
f(i) = f(i) + V(i)^2*real(Ybus(i,i)) – P(i);
f(length(V) + i) = f(length(V) + i) – V(i)^2*imag(Ybus(i,i)) – Q(i);
J(i,i) = 2*V(i)*real(Ybus(i,i)) + sum(V(j)*abs(Ybus(i,j))*cos(theta(i) – theta(j) – angle(Ybus(i,j))));
J(length(V) + i, length(V) + i) = -2*V(i)*imag(Ybus(i,i)) – sum(V(j)*abs(Ybus(i,j))*sin(theta(i) – theta(j) – angle(Ybus(i,j))));
end
end
% Define the main function to solve power flow
function [V, theta] = solve_power_flow(Ybus, P, Q, V0, theta0, tol, max_iter)
% Ybus: admittance matrix
% P, Q: active and reactive power demands
% V0, theta0: initial guesses for voltage magnitudes and angles
% tol: tolerance for convergence
% max_iter: maximum number of iterations
% Initialize variables
V = V0;
theta = theta0;
iter = 0;
% Iterate using the Newton-Raphson method
while iter < max_iter
% Compute the mismatch equations and Jacobian
[f, J] = power_flow_equations(V, theta, Ybus, P, Q);
% Check for convergence
if norm(f) < tol
fprintf(‘Converged in %d iterations\n’, iter);
return;
end
% Update the variables
delta = -J \ f;
V = V + delta(1:length(V));
theta = theta + delta(length(V)+1:end);
% Increment the iteration counter
iter = iter + 1;
end
% If maximum iterations are reached without convergence
error(‘Newton-Raphson method did not converge within the maximum number of iterations’);
end
% Example usage
% Define the admittance matrix, power demands, and initial guesses
Ybus = [5 – 2j, -3 + 1j; -3 + 1j, 4 – 1j];
P = [1; -1];
Q = [0.5; -0.5];
V0 = [1; 1];
theta0 = [0; 0];
% Solve the power flow equations
[V, theta] = solve_power_flow(Ybus, P, Q, V0, theta0, 1e-6, 100);
% Display the results
disp(‘Voltage magnitudes:’);
disp(V);
disp(‘Voltage angles:’);
disp(theta);
Encompassing gradual instructions, instance MATLAB code, explanation, hints, and some project concepts, we have provided a detailed note on Newton-Raphson algorithm which can be beneficial for you in developing such kinds of projects.
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