Find the best fit 2 piecewise polynomial function for a data set with 1 optimal breakpoint location

I have a 2D data set (2 variables x and y). I would like to use 2 piecewise polynomial functions with 1 breakpoint to best fit the dataset.

For example, function 1 = polynomial1 for mini_data < t < a and function 2 = polynomial2 for a<t<max_data. Also, I need to find the optimal breakpoint a value.

What maths tool/algorithm should I use, please?

Thanks in advance!

[Torsten]

Degree of the polynomials is given ?

Any conditions for the polynomials at the breakpoint (continuity, continuity of the first derivative, ...) ?

[Yutao Wang]

Thanks, @Torsten.

Any degree of polynomial is fine, perhaps degree = 2 to 6 is preferred.

At breapoint, it needs to be continuous + continuity of the first derivative. An example of the data is given here. Thanks in advance!

[Torsten]

min: sum_{i=1}^{i=ndata} (p(xdata(i),a) - ydata(i))^2

subject to

p1(a,a) = p2(a,a)

p1'(a,a) = p2'(a,a)

xdata(1) <= a <= xdata(ndata)

with

p(x,a) = p1(x,a) for x<=a

p(x,a) = p2(x,a) for x>=a

Parameters to be optimized are a and the coefficients of the polynomials p1 and p2.

A typical application for Matlab's "fmincon".