Schemes using preliminary treatment of coefficients for polynomial calculation. A program for automatic determination of parameters

Abstract In the approximate calculation of functions it is often necessary to find the value of the same polynomial with real coefficients a0 ≠ 0 at many points x. Horner's scheme Pn(x) = ((a0x + a1) x+a2) x + … +an could be used for calculation, with n multiplications and n additions for each value of the polynomial. But when n ⩾ 4 another method, first mentioned in [1], is more suitable. The polynomial is first rewritten in a more convenient form. This single transformation or preliminary treatment of coefficients involves some amount of work, but on the other hand the multiply repeated calculations later are simplified.