Many real-world problems rely on finding eigenvalues and eigenvectors of a matrix. The power iteration algorithm is a simple method for determining the largest eigenvalue and associated eigenvector of a general matrix. This algorithm relies on the idea that repeated multiplication of a randomly chosen vector x by the matrix A gradually amplifies the component of the vector along the eigenvector of the largest eigenvalue of A while suppressing all other components. Unfortunately, the power iteration algorithm may demonstrate slow convergence. In this report, we demonstrate an exponential speed up in convergence of the power iteration algorithm with only a polynomial increase in computation by taking advantage of the commutativity of matrix multiplication.
翻译:许多现实世界的问题都依赖于找到一个矩阵的源值和源值。 动力迭代算法是确定一个总矩阵的最大源值和相关源值的简单方法。 这种算法所依赖的理念是,由矩阵A 随机选择的矢量 x 的重复倍增会沿A 最大源值的源值递增矢量,同时抑制所有其他组成部分。 不幸的是, 动力迭代算法可能显示缓慢的趋同。 在本报告中, 我们展示了功率迭代算法的指数加速趋同, 利用矩阵倍增的共通性在计算中只增加了多数值。