# Newton for a 4x4 system of nonlinear eqns

• Moly
In summary, a sample MATLAB code for the Newton-Raphson Method for a nonlinear 4x4 system of equations was requested and the user was struggling with setting it up. Later, they shared a program for a linear set of equations and mentioned that the method was supposed to converge in one iteration, but for them it took three iterations. A code was then provided, showing the use of symbolic variables and a for loop to solve the system of equations.
Moly
Hi All.
Can somebody give me a sample MATLAB code for Newton-Raphson Method for Nonlinear 4x4 System of Equations. I'm trying to set a very ugly one up and haven't seen NR before the beginning of this week. I've figured the NR for a single eqn but this is driving me nuts.

Hi All. I think i figured it out... i am attaching the program below for a linear set of eqns... that was just to verify that all is working well... however, i was told that the method should converge in one iteration for a linear system, and for me it happens in 3.. am i doing something wrong here:

clear all
close all

syms x y z k
f1=x-2*y+0*z+3*k+2;
f2=0*x+y+0*z-4*k-7;
f3=0*x-0*y+z+0*k-6;
f4=0*x-0*y+0*z+k+3;

Jk=zeros(4,4);
Jk=[1 -2 0 3;
1 -1 0 -4;
1 -2 1 3;
1 -2 1 4];

x1=2; x2=-5; x3=-4; x4=-1;
xxold=[2;-5;-4;-1]
for i=1:4
F=[subs(f1, {x,y,z,k},{x1,x2,x3,x4});
subs(f2, {x,y,z,k},{x1,x2,x3,x4});
subs(f3, {x,y,z,k},{x1,x2,x3,x4});
subs(f4, {x,y,z,k},{x1,x2,x3,x4})];
xx=xxold-inv(Jk)*F;
x1=xx(1); x2=xx(2);x3=xx(3);x4=xx(4);
xxold=xx
end

## 1. What is Newton's method for solving a 4x4 system of nonlinear equations?

Newton's method is an algorithm used to find the roots or solutions to a system of nonlinear equations. It involves iteratively updating initial guesses for the solutions using the derivative of the equations until a desired level of accuracy is reached.

## 2. How does Newton's method work for a 4x4 system of nonlinear equations?

Newton's method starts with an initial guess for the solutions to the system of equations. Then, the derivative of the equations is calculated at that point, and the solution is updated using the Newton-Raphson formula. This process is repeated until the desired level of accuracy is achieved.

## 3. What are the advantages of using Newton's method for a 4x4 system of nonlinear equations?

One advantage of Newton's method is that it is a fast and efficient algorithm for finding the solutions to a system of nonlinear equations. It also allows for a high degree of accuracy and can handle complex equations with multiple variables.

## 4. What are the limitations of Newton's method for a 4x4 system of nonlinear equations?

One limitation of Newton's method is that it may not always converge to the correct solution. This can happen if the initial guess is too far from the true solution or if the equations have multiple roots. In addition, calculating the derivatives can be computationally expensive for large systems of equations.

## 5. How can I apply Newton's method to a 4x4 system of nonlinear equations?

To apply Newton's method, you will need to know the equations in the system and choose an initial guess for the solutions. Then, you can use a computer program or manually calculate the derivatives and update the solutions until the desired level of accuracy is reached. There are also many online resources and software packages available for implementing Newton's method.

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