A note on evaluation of coefficients in the polynomials of Padé approximants by solving systems of linear equations

Abstract

This is a sequel to my previous paper concerning the determination of the coefficients in the polynomials which define Padé fractions, where the coefficients are found by solving systems of linear equations. The present note uses the same models and computer as before, but the computations are far more extensive so as to reveal more pointedly the effects of round off error in the coefficients as the order of the system increases. Only the main diagonal Padé entries are studied numerically. The numerics are achieved using two routines in LINPACK one of which evaluates a condition number for the matrix. This is advantageous if one suspects ill conditioning. In our previous paper, it was shown that though the relative errors in the numerator and denominator polynomials increase as the order of the system increases, the effect on the Padé approximant is virtually nil at least for the size of the variable x and the order n considered. Of course, ultimately so as to render the without bound, we should expect so much contamination due to round off so as to render the results meaningless. All of this is illustrated with numerics. Heuristic procedures based on the numerical data developed are presented to warn of deterioration due to round off.