Let $XYZ$ be a matrix: $$XYZ = \left(\begin{matrix} 200.0&1.05&-0.0021\\ 400.0&2.16&-0.0043 \\ 600&3.32&-0.0067\\ 800&4.55&-0.0092\\ \end{matrix}\right)$$
I am trying to express 3rd column($Z$) in terms of corresponding first($X$) and second($Y$) column coefficients using the equation:
$$ z = ax + b\sin(ky)x + cx^2 + d\sin(ky)y^2 \qquad\qquad\qquad(1) $$
where $k = 9.816250011216700\times 10^{-04}$ a constant.
I want to know the values of $a,b,c$ and $d$.
To this end, I have formed four linear equations by substituting the values of $x$ and $y$.
The matrix-vector equations will be $Ap = q$, where $A$ is the coefficient matrix, $p$ is vector $[a,b,c,d]'$ and $q$ is the third column of matrix $XYZ$.
I have solved these equations using LU-factorization as well as matrix inverse method. $A$ is nonsingular in this case.
$$A = \left(\begin{matrix} 0.000991554& -2.42414\times10^{-5}& -0.000292566& 0.000108656\\ -36.893& 53.7078& -34.7174& 8.4074\\ 0.000182553& -0.000269718& 0.000176237& -4.29571\times10^{-5}\\ 0.00496607& -0.00673969& 0.00402018& -0.000886811\\ \end{matrix}\right) $$
I have got the coefficients ($p$) as:
$$\left(\begin{matrix} -1.01747\times 10^{-6} \\ 0.00179026\\ -9.15726\times 10^{-9}\\ -2.24656\times 10^{-7}\\ \end{matrix}\right)$$
With these coefficients, I have reconstructed the third column of $XYZ$ using equation (1). I got the values:
$$\left(\begin{matrix} -0.00201115\\ -0.00356892\\ -0.00417168\\ -0.00303998\\ \end{matrix}\right)$$
which are different from the third column of $XYZ$. Why this happen? What would be the proper way to solve these type of equations?
