Entering Non-Linear System of Equations
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Hi All. I am fairly new to Matlab and still learning. I am wondering if one may explain to me how I should enter a system of non-linear equations. For instance the equations I have are: 4*x(1)-x(2)+x(3)-x(1)*x(4) = 0; -x(1)+3*x(2)-2*x(3)-x(2)*x(4) = 0; and x(1)^2+x(2)^2+x(3)^2-1 = 0. Basically I want to enter the equations and use the Newton Method to solve using the Jacobian Matrix. This is how I enetered them in Matlab:
clear all
clc
syms x
f1 = 4*x(1)-x(2)+x(3)-x(1)*x(4);
f2 = -x(1)+3*x(2)-2*x(3)-x(2)*x(4);
f3 = x(1)^2+x(2)^2+x(3)^2-1;
F = [f1;f2;f3];
If I tell the program to run at this point, this is the error I happen to get:
??? Error using ==> mupadmex
Error in MuPAD command: Index exceeds matrix dimensions.
Error in ==> sym.sym>sym.subsref at 1366
B = mupadmex('mllib::subsref',A.s,inds{:});
Error in ==> NewtonMethosNonLinear at 5
f1 = 4*x(1)-x(2)+x(3)-x(1)*x(4);
I would appreciate some explanation on this since I don't know how to move next. Thank you.
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Alexander
2012년 3월 20일
I am not sure what exactly your program is doing, but you could try this:
x = sym('x', [4, 1])
instead of syms x. This will generate a 4-by-1 vector of symbolic variables. If you want to substitute values later on, you can do it this way:
substituted_f1 = subs(f1, x, [1, 11, 2, 22])
% substitutes x(1) = 1, x(2) = 11, x(3) = 2, x(4) = 22
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Alexander
2012년 3월 20일
Which version of MATLAB do you have? This feature has been introduced in R2010b. A quick search on Google suggests that you have an older version. If so, you could try this trick to get the vector:
x = sym(zeros(1, 4));
for k = 1:4
x(k) = sym(sprintf('x%d', k));
end
If you don't need to be flexible on the size of the vector, you can do it even more easy:
x = [sym('x1') sym('x2') sym('x3') sym('x4')]
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Rohit
2024년 7월 31일 10:10
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.
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