Matlab - Least squares approximation of f(x) = e^x

Hello,

I'm having some trouble with finding the least squares approximation (with given n) of in [−1, 1] using matlab. The approximation should be of form .

My Matlab code:

Code:

---

function z = leastsq(n)

---

• Plot

Code:

---

x = (-1.5:0.001:1.5);
y = exp(x);
plot(x, y);
xlim([-1.5 1.5])

---

• Integral of

Code:

---

matrixf = zeros(n,1);
    for (grade = 0:n)
        matrixf(grade+1,1) = quad(@(x) exp(x)*exp(grade),-1,1);
    end

---

• Problem: matrix with coefficients of

Code:

---

    for (count = 0:n)
        if (count == 0 && mod((1+gradex),2) ~= 0)
            matrixa(gradex+1, 1) = 2/(count+1+gradex);
        elseif (count ~= 0)
            matrixa(gradex+1, count+1) = quad(@(x) exp((count)*x)*exp(gradex),-1,1);
        end
    end

---

The code I used here succesfully calculates the coefficients for . I can't find how to make it work for ...

The next step would be to calculate the coefficients to :

Code:

---

coef = matrixa\matrixf

---

Finally the function should be constructed and plotted.

Any hints or tips would be greatly appreciated.

What algorithm are you trying to implement here?

CB

Here is some code which I believe may be for the algorithm (linear regression) you are trying to use:

Code:

---

>> x=linespace(-1,1,10);  %sample points for the fit
>> n=4;                  %order of polynomial
>> xx=zeros(length(x),n); %coefficient array
>> for idx=0:3;xx(:,idx+1)=(x.').^(idx);end;
>> a=exp(x.');            %right hand side vector
>> coeffs=xx\a;          %linear least squares solution for coeffs
>> pp=zeros(size(x));    %evaluate the polynomial
>> for idx=1:n;pp=pp+coeffs(idx)*x.^(idx-1);end;
>> plot(x,pp);
>> hold on;plot(x,exp(x));hold off

---

I'm trying to find the solution for



(So I'm not trying to approximate a vector of data points! For which I would use )

I'm trying to find the miminum solution for



I already wrote the code to take the integrals of the n amount of f_j's.

These are all placed in matrixf.

Next I should calculate the coefficient matrix (matrixa) and solve the system by using

Code:

---

coef = matrixa\matrixf

---

coef will then be a vector of length n+1 containing the coefficients to . These will be used to describe the approximating polynomial

I hope I made it a bit clearer. My apologies, all those lines of code make it hard to express myself properly. (Sleepy)

Quote:

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Originally Posted by CaptainBlack

Here is some code which I believe may be for the algorithm (linear regression) you are trying to use:

---

Thanks! I believe that is what I'm trying to do... (Happy)
I'll try it out!

It worked! Thank you for your help!

(PS: linespace should be linspace)