We consider a new algorithm in light of the min-max Collatz-Wielandt formalism to compute the principal eigenvalue and the eigenvector (eigen-function) for a class of positive Perron-Frobenius-like operators. Such operators are natural generalizations of the usual nonnegative primitive matrices. These have nontrivial applications in PDE problems such as computing the principal eigenvalue of Dirichlet Laplacian operators on general domains. We rigorously prove that for general initial data the corresponding numerical iterates converge globally to the unique principal eigenvalue with quadratic convergence. We show that the quadratic convergence is sharp with compatible upper and lower bounds. We demonstrate the effectiveness of the scheme via several illustrative numerical examples.
翻译:我们考虑一种新的算法,根据最低数值 Collatz-Wielandt的正规主义来计算某类正的Perron-Frobenius型操作员的本值和原值(egen-功能),这些操作员是通常的非阴性原始矩阵的自然概括,在PDE问题上具有非三边性应用,例如计算Dirichlet Laplacian操作员在一般领域的本值。我们严格证明,对于一般的初步数据,相应的数字轴数会聚集到全球独特的原值本值和二次趋同之间。我们表明,四边形趋同与相容的上下界是尖锐的。我们通过几个数字示例来证明这个办法的有效性。