I have an assignment to do in matlab. I have to implement the modified richardson iteration. I couldn't really understand the algorithm but i came up with this:
A = [9 1 1;
2 10 3;
3 4 11];
b = [10;
19;
0];
x = [0;
0;
0];
G=eye(3)-A; %I-A
z = [0,x'];
for k=1:30
x = G*x + b;
z = [k,x'];
fprintf('Number of Iterations: %d \n', k);
display(z);
end
The output i recieve is wrong and i don't really know why. Any help is well recieved. Thanks!
You are missing the omega parameter. From the wiki page, the iteration is:
x(k+1) = x(k) + omega*( b - A*x(k) )
= (I - omega*A)*x(k) + omega*b
where omega is a scalar parameter that has to be chosen appropriately.
So you need to change your calculation of G to:
G = eye(3)-omega*A;
and the calculation of x inside the loop to:
x = G*x + omega*b;
The wiki page discusses how the value of omega can be chosen. For your particular case, omega = 0.1 seems to work well.
Related
I can't seem to find a fix to my infinite loop. I have coded a Jacobi solver to solve a system of linear equations.
Here is my code:
function [x, i] = Jacobi(A, b, x0, TOL)
[m n] = size(A);
i = 0;
x = [0;0;0];
while (true)
i =1;
for r=1:m
sum = 0;
for c=1:n
if r~=c
sum = sum + A(r,c)*x(c);
else
x(r) = (-sum + b(r))/A(r,c);
end
x(r) = (-sum + b(r))/A(r,c);
xxx end xxx
end
if abs(norm(x) - norm(x0)) < TOL;
break
end
x0 = x;
i = i + 1;
end
When I terminate the code it ends at the line with xxx
The reason why your code isn't working is due to the logic of your if statements inside your for loops. Specifically, you need to accumulate all values for a particular row that don't belong to the diagonal of that row first. Once that's done, you then perform the division. You also need to make sure that you're dividing by the diagonal coefficient of A for that row you're concentrating on, which corresponds to the component of x you're trying to solve for. You also need to remove the i=1 statement at the beginning of your loop. You're resetting i each time.
In other words:
function [x, i] = Jacobi(A, b, x0, TOL)
[m n] = size(A);
i = 0;
x = [0;0;0];
while (true)
for r=1:m
sum = 0;
for c=1:n
if r==c %// NEW
continue;
end
sum = sum + A(r,c)*x(c); %// NEW
end
x(r) = (-sum + b(r))/A(r,r); %// CHANGE
end
if abs(norm(x) - norm(x0)) < TOL;
break
end
x0 = x;
i = i + 1;
end
Example use:
A = [6 1 1; 1 5 3; 0 2 4]
b = [1 2 3].';
[x,i] = Jacobi(A, b, [0;0;0], 1e-10)
x =
0.048780487792648
-0.085365853612062
0.792682926806031
i =
20
This means it took 20 iterations to achieve a solution with tolerance 1e-10. Compare this with MATLAB's built-in inverse:
x2 = A \ b
x2 =
0.048780487804878
-0.085365853658537
0.792682926829268
As you can see, I specified a tolerance of 1e-10, which means we are guaranteed to have 10 decimal places of accuracy. We can certainly see 10 decimal places of accuracy between what Jacobi gives us with what MATLAB gives us built-in.
In Matlab I want to create the partial derivative of a cost function called J(theta_0, theta_1) (in order to do the calculations necessary to do gradient descent).
The function J(theta_0, theta_1) is defined as:
Lets say h_theta(x) = theta_1 + theta_2*x. Also: alpha is fixed, the starting values of theta_1 and theta_2 are given. Let's say in this example: alpha = 0.1 theta_1 = 0, theta_2 = 1. Also I have all the values for x and y in two different vectors.
VectorOfX =
5
5
6
VectorOfX =
6
6
10
Steps I took to try to solve this in Matlab: I have no clue how to solve this problem in matlab. So I started off with trying to define a function in Matlab and tried this:
theta_1 = 0
theta_2 = 1
syms x;
h_theta(x) = theta_1 + t2*x;
This worked, but is not what I really wanted. I wanted to get x^(i), which is in a vector. The next thing I tried was:
theta_1 = 0
theta_2 = 1
syms x;
h_theta(x) = theta_1 + t2*vectorOfX(1);
This gives the following error:
Error using sym/subsindex (line 672)
Invalid indexing or function definition. When defining a
function, ensure that the body of the function is a SYM
object. When indexing, the input must be numeric, logical or
':'.
Error in prog1>gradientDescent (line 46)
h_theta(x) = theta_1 + theta_2*vectorOfX(x);
I looked up this error and don't know how to solve it for this particular example. I have the feeling that I make matlab work against me instead of using it in my favor.
When I have to perform symbolic computations I prefer to use Mathematica. In that environment this is the code to get the partial derivatives you are looking for.
J[th1_, th2_, m_] := Sum[(th1 + th2*Subscript[x, i] - Subscript[y, i])^2, {i, 1, m}]/(2*m)
D[J[th1, th2, m], th1]
D[J[th1, th2, m], th2]
and gives
Coming back to MATLAB we can solve this problem with the following code
%// Constants.
alpha = 0.1;
theta_1 = 0;
theta_2 = 1;
X = [5 ; 5 ; 6];
Y = [6 ; 6 ; 10];
%// Number of points.
m = length(X);
%// Partial derivatives.
Dtheta1 = #(theta_1, theta_2) sum(2*(theta_1+theta_2*X-Y))/2/m;
Dtheta2 = #(theta_1, theta_2) sum(2*X.*(theta_1+theta_2*X-Y))/2/m;
%// Loop initialization.
toll = 1e-5;
maxIter = 100;
it = 0;
err = 1;
theta_1_Last = theta_1;
theta_2_Last = theta_2;
%// Iterations.
while err>toll && it<maxIter
theta_1 = theta_1 - alpha*Dtheta1(theta_1, theta_2);
theta_2 = theta_2 - alpha*Dtheta2(theta_1, theta_2);
it = it + 1;
err = norm([theta_1-theta_1_Last ; theta_2-theta_2_Last]);
theta_1_Last = theta_1;
theta_2_Last = theta_2;
end
Unfortunately for this case the iterations does not converge.
MATLAB is not very flexible for symbolic computations, however a way to get those partial derivatives is the following
m = 10;
syms th1 th2
x = sym('x', [m 1]);
y = sym('y', [m 1]);
J = #(th1, th2) sum((th1+th2.*x-y).^2)/2/m;
diff(J, th1)
diff(J, th2)
Not sure what I am doing wrong here;
I am trying to make a for loop with conditional statements for the following functions. I want to make it though so h is not a vector. I am doing this for 1 through 5 with increment 0.1.
Y = f(h) = h^2 if h <= 2 or h >= 3
Y = f(h) = 45 otherwise
my code is
for h = 0:0.1:5
if h <= 2;
Y = h^2;
elseif h >= 3;
Y = h^2;
else;
h = 45;
end
end
This could be done easier, but with a for loop i think you could use:
h=0:0.1:5;
y=zeros(1,length(h));
for i=1:length(h)
if or(h(i) <= 2, h(i) >= 3)
y(i) = h(i)^2;
else
y(i) = 45;
end
end
Why do you want to avoid making h an array? MATLAB specializes in operations on arrays. In fact, vectorized operations in MATLAB are generally faster than for loops, which I found counter-intuitive having started coding in C++.
An example of a vectorized verison of your code could be:
h = 0:0.1:5;
inds = find(h > 2 & h < 3); % grab indices where Y = 45
Y = h.^2; % set all of Y = h^2
Y(inds) = 45; % set only those entries for h between 2 and 3 to 45
The period in the .^2 operator broadcasts that operator to every element in the h array. This means that you end up squaring each number in h individually. In general, vectorized operation like this are more efficient in MATLAB, so it is probably best to get in the habit of vectorizing your code.
Finally, you could reduce the above code a bit by not storing your indices:
h = 0:0.1:5;
Y = h.^2; % set all of Y = h^2
Y(find(h > 2 & h < 3)) = 45; % set only those entries for h between 2 and 3 to 45
This blog series seems to be a good primer on vectorizing your MATLAB code.
Hi everyone this is What I did to carry out an iteration method(gauss seidel) and I want when iteration number greater than 30 it will stop and generate the corresponding result up to 30 iteration. But I wonder why the output result were so weird and I try to check the value on the command window by typing x_ans(:,1) it gives me the correct value. It really made me frustrated why the generate result were not the same. Or any other circumstance or function can be used to set for not converging condition. Sincerely thanks in advance for every single help.
clear;clc
A = [2 8 3 1;0 2 -1 4;7 -2 1 2;-1 0 5 2]
B = [-2;4;3;5]
Es = 1e-5
n = length(B);
x = zeros(n,1);
Ea = ones(n,1);
iter = 0;
while max(Ea) >= Es
if iter <= 30
iter = iter + 1;
x_old = x;
for i = 1:n
j = 1:n;
j(i) = [];
x_cal = x;
x_cal(i) = [];
x(i) = (B(i) - sum(A(i,j) * x_cal)) / A(i,i);
end
else
break
end
x_ans(:,iter) = x;
Ea(:,iter) =abs(( x - x_old) ./ x);
end
result = [1:iter; x_ans; Ea]'
I've gone through the formulas and they are all OK. On a side note, the sum is not necessary. The problem lies with your input data - try reordering! check for example the following, which works
A = [7 -2 1 2;
2 8 3 1;
-1 0 5 2;
0 2 -1 4;]
B = [3;-2;5;4]
see the wiki under convergence.
Hey there, I've been having difficulty writing the matlab equivalent of the conv(x,y) function. I cant figure out why this gives the incorrect output. For the arrays
x1 = [1 2 1] and x2 = [3 1 1].
Here's what I have
x1 = [1 2 1];
x2 = [3 1 1];
x1len = leng(x1);
x2len = leng(x2);
len = x1len + x2len - 1;
x1 = zeros(1,len);
x2 = zeros(1,len);
buffer = zeros(1,len);
answer = zeros(1,len);
for n = 1:len
buffer(n) = x(n);
answer(n) = 0;
for i = 1:len
answer(n) = answer(n) + x(i) * buffer(i);
end
end
The matlab conv(x1,x2) gives 3 7 6 3 1 as the output but this is giving me 3 5 6 6 6 for answer.
Where have I gone wrong?
Also, sorry for the formatting I am on opera mini.
Aside from not having x defined, and having all zeroes for your variables x1, x2, buffer, and answer, I'm not certain why you have your nested loops set up like they are. I don't know why you need to reproduce the behavior of CONV this way, but here's how I would set up a nested for-loop solution:
X = [1 2 1];
Y = [3 1 1];
nX = length(X);
nY = length(Y);
nOutput = nX+nY-1;
output = zeros(1,nOutput);
for indexY = 1:nY
for indexX = 1:nX
indexOutput = indexY+indexX-1;
output(indexOutput) = output(indexOutput) + X(indexX)*Y(indexY);
end
end
However, since this is MATLAB, there are vectorized alternatives to looping in this way. One such solution is the following, which uses the functions SUM, SPDIAGS, and FLIPUD:
output = sum(spdiags(flipud(X(:))*Y));
In the code as given, all vectors are zeroed out before you start, except for x which is never defined. So it's hard to see exactly what you're getting at. But a couple of things to note:
In your inner for loop you are using values of buffer which have not yet been set by your outer loop.
The inner loop always covers the full range 1:len rather than shifting one vector relative to the other.
You might also want to think about "vectorizing" some of this rather than nesting for loops -- eg, your inner loop is just calculating a dot product, for which a perfectly good Matlab function already exists.
(Of course the same can be said for conv -- but I guess you're reimplementing either as homework or to understand how it works?)