Some extensions of Bairstow's method

Abstract

u _xx +u _yy =u _t Bairstow's method for improving an approximate real quadratic factor (x ²−px−q) of a polynomial with real coefficients which leaves a remainderr(x), is to determine δp and δq to satisfy

$$0 = r\left( x \right) + \frac{{\partial r\left( x \right)}}{{\partial P}}\delta P + \frac{{\partial r\left( x \right)}}{{\partial q}}\delta q$$

.

One extension is to determine δp and δq when the three second-order terms

$$\frac{1}{2}\left[ {\frac{{\partial ^2 r\left( x \right)}}{{\partial P^2 }}\left( {\delta P} \right)^2 + 2\frac{{\partial ^2 r\left( x \right)}}{{\partial P \partial q}}\left( {\delta P} \right)\left( {\delta q} \right) + \frac{{\partial ^2 r\left( x \right)}}{{\partial q^2 }}\left( {\delta q} \right)^2 } \right]$$

are added to the right member. By taking advantage of polynomial congruences and the linearity inx of every$\frac{{\partial ^{i + k} r\left( x \right)}}{{\partial P^i \partial q^k }}$, only one extra division is needed besides the two required divisions ofBairstow's method. Another extension improves an approximate real quartic factor (x ⁴−px³−qx²−rx−s), considering only terms of the first order in δp, δq, δr and δs. This latter method may be immediately generalized for approximate real factors of any degree. By employing polynomial congruences, no more than two divisions are necessary in any case.