How to fit the data with passing specific point and specific function

Please could you me some tips for the below question

To do the fitting, I have searched the polyfit and curve fitting. but i could find the fitting method like below

I would like fit the data by using the specific function like below

y= A (x^2) (x-1)^2 ( I want find A and the function should pass (0,0) and (1,0))

and data is like this:

x=[0 0.166666667 0.333333333 0.5 0.666666667 0.833333333 1]

y=[0 1.09162 1.71904 2.52173 1.20849 2.47433 0]

Therefore, from the data , I would like to get A values from y= A (x^2) (x-1)^2 and data point

And here is constraints that the fiiting function must pass the (0,0) and (1,0)

for example like this:

[Alex Sha]

Hi, is this what you want:

a=38.9351344970728

Answer 1

[KSSV]

x=[0	0.166666667	0.333333333	0.5	0.666666667	0.833333333	1]' ;
y=[0	1.09162	1.71904	2.52173	1.20849	2.47433	0]' ;

n = 3; % Degree of polynomial to fit

V(:,n+1) = ones(length(x),1);   % Vandermonde matrix 
for j = n:-1:1
     V(:,j) = x.*V(:,j+1);
end
% Constraints 
Aeq = x([1 end]).^(n:-1:0);
beq = [y(1) y(end)];
p = lsqlin(V, y,[],[],Aeq,beq);

Minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

xnew = linspace(x(1),x(end)) ; 
ynew = polyval(p,xnew);

plot(x,y,'.b-') 
hold on
% Plot constraints 
plot(x([1 end]),y([1 end]),'gx','linewidth',4) 
% Plot fitted data
plot(xnew,ynew,'r','linewidth',2) 
hold off

[L world]

Thanks so much KSSV

your reply would be a good! guide to me .

Could I ask something more?

In your code red line is fitted line.

But, that i want to know is A value from y= A (x^2) (x-1)^2 < 4th polynomial

y= A( x^4 - 2x^2 + x^2)

the fitted line shoul be y= A (x^2) (x-1)^2 format

because I will use this shpae of function y= A (x^2) (x-1)^2

If A would be calculated , then i use this function format with calculated A.

I highly appreciate your reply!

Thanks

[L world]

Could I get A value by keeping the the shape of A(x^2)*(x^2-1) like below ?

the shpae is symmetry at the center of x=0.5 (this graph has A=5 as an example)

[KSSV]

As of now I have no idea. Will think over it and get back.