Editing Bilinear interpolation (section)

== A simplification of terms ==

This example is of tabularised pressure (columns) vs temperature (rows) data as a lookup against some variable:

:<math>\begin{array}{|c|ccc|}
\bcancel{{}_T \quad {}^P}
     &    P_1 & P_x    & P_2
\\
\hline
 T_1 & V_{11} & V_{1x} & V_{12} 
\\
 T_x &        & V_{xx} &
\\
 T_2 & V_{21} & V_{2x} & V_{22} 
\end{array}</math>

The following standard calculation by parts has 27 operations:
:<math>\begin{align}
I_{T_1, P_1-P_2} &= \frac{P_2 - P_x}{P_2 - P_1} \cdot V_{11} + \frac{P_x - P_1}{P_2 - P_1} \cdot V_{12} = V_{1x},\\
I_{T_2, P_1-P_2} &= \frac{P_2 - P_x}{P_2 - P_1} \cdot V_{21} + \frac{P_x - P_1}{P_2 - P_1} \cdot V_{22} = V_{2x},\\
I_{P_x, T_1-T_2} &= \frac{T_2 - T_x}{T_2 - T_1} \cdot V_{1x} + \frac{T_x - T_1}{T_2 - T_1} \cdot V_{2x} = V_{xx}.
\end{align}</math>
The above has several repeated operations, e.g., <math>(P_2 - P_1)</math>, <math>(P_x - P_1)</math>, <math>(P_x - P_1)</math>, <math>(T_2 - T_1)</math>, as well as some ratios. These repetitions can be assigned temporary variables whilst computing a single interpolation, which will reduce the number of operations to 19.

This can all be simplified from the initial 19 individual operations to 17 individual operations as such:
:<math>V_{xx} = \frac{[(P_2 - P_x) \cdot V_{11} + (P_x - P_1) \cdot V_{12}] \cdot (T_2 - T_x) + [(P_2 - P_x) \cdot V_{21} + (P_x - P_1) \cdot V_{22}] \cdot (T_x - T_1)}{(P_2 - P_1) \cdot (T_2 - T_1)}.</math>

Simplification of terms is good practice for application of mathematical methodology to engineering applications and can reduce computational and energy requirements for a process.{{cn|date=December 2024}}