Write the Pade approximant for the truncated Taylor series for which it is to represent. The approximant should be in the form sigma(Ak*x^k, for 0,…,N)/sigma(Bk*x^k, for 0,…,M). Here, “sigma” refers to sigma notation, which sums the inside terms. Ak and Bk are the coefficients of the Pade approximant. “M” and “N” can be any numbers so long as the conform to the following truncated Taylor series: sigma(Ck*x^k, for 0,…,M+N).
Evaluate the Pade approximant. Write the sigma functions in long form (as the addition of terms). For example, sigma(Ak*x^k, for 0,…,2) will become A0 + A1*x + A2*x^2. The solution will be a series of sums divided by a series of sums (a fraction, in essence).
Evaluate the Taylor series. Write the sigma function out as an addition problem. For example, if the truncated Taylor series for which the Pade approximant corresponds is sigma(Ck*x^k, for 0,…,5), write C0 + C1*x + C2*x^2, …, C5*x^5.
Equate the Pade approximant to the Taylor series. You will now have an equation equating a fraction of sums to a series of sums.
Solve for the Ak coefficients. Use basic algebra. For example, you should find that A0 = C0 and A1 = C1 + C0*B1.
Solve for Bk coefficients numerically. Using algebra, move the right hand side of the equation that equates the Pade approximant to the Taylor series to the left hand side of the equation. You will be left with a series of equations that all equate to zero. This will be a series of linear equations that you can solve using matrix methods (for large values of M and N, solving this by hand is intractable, so you should use matrix software to find the numerical values of Bk).
Substitute the numerical values of Bk into the Ak coefficient solutions. Your Ak coefficients should now be numbers and not equations. At this point you have all of the coefficients calculated.