The Regular Falsi method, or False Position method, is useful for solving nonlinear equations where you expect the root to lie between two points. Here's a step-by-step explanation and MATLAB code to solve the equation \( x^2 - x + 4 = 0 \) using this method:
### MATLAB Code for Regular Falsi Method
```matlab
% Define the function
f = @(x) x.^2 - x + 4;
% Set the initial interval [a, b]
a = -10; % Lower bound of the interval
b = 10; % Upper bound of the interval
% Set tolerance and maximum number of iterations
tol = 1e-6; % Tolerance for stopping criterion
max_iter = 1000; % Maximum number of iterations
% Check if the function values at the endpoints of the interval have opposite signs
if f(a) * f(b) >= 0
error('Function values at the interval endpoints must have opposite signs.');
end
% Initialize iteration counter
iter = 0;
% Main iteration loop
while abs(b - a) > tol
% Calculate the false position
c = (a * f(b) - b * f(a)) / (f(b) - f(a));
% Evaluate the function at the new point
fc = f(c);
% Check if the root is found exactly
if fc == 0
break; % Solution found
elseif f(a) * fc < 0
b = c; % Update the upper bound
else
a = c; % Update the lower bound
end
% Update iteration counter
iter = iter + 1;
% Check for maximum iterations
if iter > max_iter
warning('Maximum number of iterations reached');
break;
end
end
% Display the result
fprintf('Root found at x = %.6f\n', c);
fprintf('Function value at x = %.6f is f(x) = %.6f\n', c, f(c));
```
### Explanation:
1. **Function Definition**: The function `f` represents the equation \( x^2 - x + 4 \) that you want to solve.
2. **Initial Interval**: The initial interval `[a, b]` should be chosen such that `f(a)` and `f(b)` have opposite signs. This ensures that the root lies within the interval.
3. **Tolerance and Maximum Iterations**: `tol` defines the acceptable error range for the result. `max_iter` limits the number of iterations to prevent infinite loops.
4. **Iteration Loop**: The loop continues until the interval width is less than the tolerance. The false position `c` is calculated each iteration.
5. **Updating the Interval**: Depending on the function value at `c`, the interval `[a, b]` is updated to ensure the root is bracketed.
6. **Result**: Once the loop terminates, the approximate root `c` and the function value at `c` are displayed.
### Important Note:
For the quadratic equation \( x^2 - x + 4 \), it's worth noting that this equation does not have real roots because the discriminant is negative. Hence, the Regular Falsi method will not find a real root but can still demonstrate the iterative process. You might want to choose an interval that ensures `f(a)` and `f(b)` have opposite signs for this method to be applicable in practice.
