This paper considers an approximate factorisation of three bivariate Bernstein basis polynomials that are defined in a triangular domain. This problem is important for the computation of the intersection points of curves in computer-aided design systems, and it reduces to the determination of an approximate greatest common divisor ($\text{AGCD}$) $d\left(y\right)$ of the polynomials. The Sylvester matrix and its subresultant matrices of these three polynomials are formed and it is shown that there are four forms of these matrices. The most difficult part of the computation is the determination of the degree of $d\left(y\right)$ because it reduces to the determination of the rank loss of these matrices. This computation is made harder by the presence of trinomial terms in the Bernstein basis functions because they cause the entries of the matrices to span many orders of magnitude. The adverse numerical effects of this wide range of magnitudes of the entries of the four forms of the Sylvester matrix and its subresultant matrices are mitigated by processing the polynomials before these matrices are formed. It is shown that significantly improved results are obtained if the polynomials are processed before computations are performed on their Sylvester matrices and subresultant matrices.

本文考虑了在三角域中定义的三个双变量Bernstein基多项式的近似因式分解。这个问题对于计算机辅助设计系统中曲线的交点的计算很重要，并且可以简化为确定最大近似公约数（$\text{AGCD}$） $d（ÿ）$多项式 形成了这三个多项式的Sylvester矩阵及其子结果矩阵，结果表明这些矩阵有四种形式。计算中最困难的部分是确定$d（ÿ）$因为它减少了对这些矩阵的秩损失的确定。由于伯恩斯坦基函数中存在三项式项，因此使计算变得更加困难，因为它们会导致矩阵的项跨越多个数量级。通过在形成这些矩阵之前处理多项式，可以减轻四种形式的Sylvester矩阵及其次结果矩阵的条目的幅度幅度的不利数值影响。结果表明，如果在对Sylvester矩阵和次结果矩阵进行计算之前对多项式进行处理，则可获得明显改善的结果。