General information
Let us have N data points with coordinates xi, yi, zi.The main purpose of this calculation is to obtain a relatively simple formula of data fitting z=f(x,y). We apply the generalized Pade functions for the approximation of data in the following form:
where R(x, y), Q(x,y) are polynomial functions, Ri and Qi are their values calculated at i-th point.
Pade functions have the following advantages:
  • They have the same or better convergence compared to the power series.
  • They can include functions with singularities and asymptotic behaviour of a function
You can find an excellent summary of properties of Pade functions here.
The residual sum of squares (RSS) corresponding to (1) is the following
Minimization of S0 leads to a system of nonlinear equations with respect to coefficients of polynomial functions, therefore the equation (1) is presented in the equivalent form
with corresponding RSS
This new form of RSS yields a system of linear equations. This idea was efficiently used in paper [2] for 2D data fitting. For the case of 3D regression analysis we use the following approximation R(x)=a0+ a1x+ a2y+ a3x2+ a4xy+ a5y2 and Q(x)=b1x+ b2y+ b3x2+ b4xy+ b5y2 if N>11. In this case we have 11 free parameters defined in the calculation by minimization of S1 solving the corresponding linear equations.
In addition we try substitutions X=xqx, Y=yqy, X=exp(ex*x), and Y=exp(ey*y. These four parameters (qx, qy, ex, ey) are calculated by minimizing S0.
In all cases we check if the denominator is equal to 0 in the given domain [min{xi}, max{xi}] for variable x and [min{yi}, max{yi}] for y. We discard this solution if it is happened.
If N<12 the polynomials are reduced to the form R(x)=a0+ a1x+ a2y and Q(x)=b1x+ b2y
Standard deviation SD is calculated
as well as maximum local error

We suppose that this calculation will be useful for researchers and engineers looking for good approximation of data with simple formula. To check the efficiency of suggested calculation, please see examples.

  1. Baker C. , Graves-Morris P., Pade Approximation.
  2. Yevkin G., Yevkin O. 'On regression analysis with Pade approximants' (August, 2022, DOI: 10.48550/arXiv.2208.09945)