How can I find the unknown parameters of a non-linear equation ??

I am having a equation z^2=x(1)*a+x(2)*b+x(3)*c+x(4)*a^2+x(5)*b^2+x(6)*c^2+x(7)*a*b+x(8)*a*c+x(9)*c*b

Value of a b c and z

a=[27 26.4 29.19 22.22 27.5 21.33 23.84 31.11 19.5 28.44 22.5 36.08]

b=[95 87 81 72 76 76 95 82 89 89 88 83 ]

c=[171.3 191.9 105.9 167.2 172.8 112.7 106.9 122.2 124.9 98.8 120.1 161.6 ]

z=[0.76 0.33 0.51 0.85 0.81 1.02 0.36 0.86 0.28 0.59 0.73 0.38 ]

can anyone explain the steps for finding x(1)...........x(9)

Answer 1

[Walter Roberson]

A = [a(:), b(:), c(:), a(:).^2, b(:).^2, c(:).^2, a(:).*b(:), a(:).*c(:), b(:).*c(:)];
B = z(:).^2;

x = A\B;

Your equation is linear in your x, so you can just construct the coefficient matrix and use the \ operator.

You will only get meaningful results if your a, b, c, and z are at least 9 elements long.

Answer 2

[chetan meena]

but i am having function which is not linear so what should i do in that case

 i have this  
f = 0; for i =1:236 
f = f + (z(i) - (a(i)^3*x(1)+  a(i)^2*b(i)*x(2)+  a(i)^2*c(i)*x(3)+ a(i)*b(i)*c(i)*x(4)+  a(i)*b(i)*x(5)+  a(i)*c(i)*x(6)+  a(i)*x(7)+  b(i)^3*x(8)+  b(i)^2*a(i)*x(9)+  b(i)^2*c(i)*x(10)+ c(i)^3*x(11)+  c(i)^2*b(i)*x(12)+  c(i)^2*a(i)*x(13)+  b(i)*x(14)+  c(i)*x(15)))^2

[Walter Roberson]

For that, I would suggest http://www.mathworks.com/help/optim/ug/lsqnonlin.html

[chetan meena]

thanks