Editing Bilinear interpolation (section)

===Polynomial fit===
An alternative way is to write the solution to the interpolation problem as a [[multilinear polynomial]]

:<math>f(x, y) \approx a_{00} + a_{10}x + a_{01}y + a_{11}xy,</math>
where the coefficients are found by solving the linear system

:<math>\begin{align}
\begin{bmatrix}
1 & x_1 & y_1 & x_1 y_1 \\
1 & x_1 & y_2 & x_1 y_2 \\
1 & x_2 & y_1 & x_2 y_1 \\
1 & x_2 & y_2 & x_2 y_2 
\end{bmatrix}\begin{bmatrix}
a_{00}\\a_{10}\\a_{01}\\a_{11}
\end{bmatrix} = \begin{bmatrix}
f(Q_{11})\\f(Q_{12})\\f(Q_{21})\\f(Q_{22})
\end{bmatrix},
\end{align}</math>

yielding the result
:<math>\begin{align}
\begin{bmatrix}
a_{00}\\a_{10}\\a_{01}\\a_{11}
\end{bmatrix} = \frac{1}{(x_2-x_1)(y_2-y_1)}\begin{bmatrix}
x_2y_2 & -x_2y_1 & -x_1y_2 & x_1y_1 \\
-y_2 & y_1 & y_2 & -y_1 \\
-x_2 & x_2 & x_1 & -x_1 \\
1 & -1 & -1 & 1 
\end{bmatrix}\begin{bmatrix}
f(Q_{11})\\f(Q_{12})\\f(Q_{21})\\f(Q_{22})
\end{bmatrix}.
\end{align}</math>