Resolve system of equations with 10th degree polynomial, LSM

Question

I have numerical problem with resolve system of equations (polynomial 10th degree) using ordinary LSM (Least Square Method). I obtained parameters with huge and very small values - therefore I can't inverse matrix constructed in this method - precision is to low even in extended variables. I tried do this in C++,Matlab,Delphi. Can somebody know application instruments which can I do this with enough accurancy or numerical tips do get good results. Standard calculation on matrix is unfortunatly elusive.

Answer 1

I think that your problem comes from the fact that you are using 10th order polynomials, which quite often lead to numerical problems:

• First of all, they can be unsuitable because of large oscillations. Even when interpolating a simple function, these oscillations can be present, see the famous Runge's example.
• Secondly, the fitting of the high order polynomials can lead to hill-conditioned linear systems, which is why you could not invert the matrix (which you should anyway not do). I made a simple experiment: I took 11 equidistant points (on the interval [0,1]) and assembled the matrix of the linear system to solve. Matlab gives me a condition number of about 1e8, so the least square matrix has a condition number of 1e16. So your matrix is 'close to singular' and this means that all the numerical precision is lost.

So, the best way to get rid of your problem is to get rid of the 10th order polynomial. You should maybe consider lower order polynomials, splines or piecewise polynomial approximations.

If you really need 10th order polynomials (e.g. if you know that your data have been generated by such a polynomial), then do not invert the matrix. Use a good preconditioner and an iterative method to solve the system without inverting the matrix.