Question

Could someone explain these four promblems on matlab and if you do, could you write what you wrote on matlab I.e. on the command window or script. Also if you have written anything by hand can you write neatly. Also an explain of how you did it would be greatly appreciated.

1] 5 points) Write the following set of equations in Matrix form and use Matlab to find the solution. 50 -5x3-6x2 2x2 + 7x3-30 x1-7x3-50-3x2 + 5x1 [2] (10 points) Given the equations 2x1-6x2-x,--38 -3x1 x27x3-34 -8x1 +x2-2x3-20 Solve by Gaussian Elimination Using partial pivoting (show the steps by hand) 3] (10 points) Apply two iterations of the Newton Raphson Method to the following set of equations. Use initial guesses ofx =y= 1. Do all your work in MATLAB including defining the functions and their respective partial derivatives. Please show what the answer is after two iterations. Include all your code. [4] (15 points) a) Solve the following set of linear equations using a built-in MATLAB function. b) Show two iterations of the Gauss Siedel Method to solve the following system of linear equations. Show your calculations in MATLAB. What is the true initial guess of x2x30. Comment on what test you can do to make sure whether the algorithm will merge or not. percentage error after two iterations for x x2 and x.Use arn 10x1 + 2x2-x,-27 123 -3x 6x2 +2x3 43 x1 +x2+Sx3

Answer 1

clc;
%1)
%Define the system of equations
%Matrix A
A = [0 6 -5; %Coefficient of x1 = 0, x2 = 6, x3 = -5
0 2 7; %Coefficient of x1 = 0, x2 = 2, x3 = 7
-4 3 -7]; %Coefficient of x1 = -4, x2 = 3, x3 = -7

%Matrix b
b = [-50; -30; 50];

%find the solution using inverse A * b
x = A\b

Output:

2)

clc;
%2)
%Define the matrices
A = [2 -6 -1;
-3 -1 7;
-8 1 -2];
b = [-38; -34; -20];

%create the augmented matrix A|B
Aug=[A b];
n=rank(A);
%initialize the nrow vector
for i=1:n
nrow(i)=i;
end
nrow=nrow';
for k=1:n-1
max=0;
index=0;
%find the maximum value in the column under the current checked element and
%return its row position
for j=k:n
if abs(Aug(nrow(j),k))>max
max=abs(Aug(nrow(j),k));
index=j;
end
end
%perform row exchange in the nrow vector
if nrow(k)~=nrow(index)
ncopy=nrow(k);
nrow(k)=nrow(index);
nrow(index)=ncopy;
  
else
  
end
%Gaussian elimination
for i=(k+1):n
m(nrow(i),k)=Aug(nrow(i),k)/Aug(nrow(k),k);

for j=k:n+1
Aug(nrow(i),j)=Aug(nrow(i),j)-m(nrow(i),k)*Aug(nrow(k),j);
end
  
end
end
%backward subsitution
x(n)=0;
x=x';
x(n)=Aug(nrow(n),n+1)/Aug(nrow(n),n);
i=n-1;
while i>0
x(i)=(Aug(nrow(i),n+1)-Aug(nrow(i),i+1:n)*x(i+1:n))/(Aug(nrow(i),i));
i=i-1;
end
x_soln=x
A_aug=Aug;

Output:

3)