Numerical performance of hyperplane constrained method and its hybrid method for singular value decomposition

Abstract

The hyperplane constrained method has been proposed in Yadani et al. (Appl Math Comp 216:779–790, 2010) computing singular value decomposition (SVD) of matrix. In the method, the SVD is replaced with solving nonlinear systems whose solutions are constrained on hyperplane, and then their solutions are computed with the help of Newton’s iterative method. In this paper, we present a new convergence theorem concerning the hyperplane constrained method in finite arithmetic. We also clarify the numerical performance of the hyperplane constrained method. In numerical experiments, we first show that the computed singular values and singular vectors are with high accuracy, even if the target matrix of SVD has small singular values, almost the same singular values, not small condition number. Though the hyperplane constrained method requires not small amount of computations, it fastens by combining other fast singular value decomposition method. We next propose a hybrid method which adopts the singular vectors computed by other fast method as the initial guess of the Newton type iteration in order to decrease the iteration number. By numerical experiments, we can see that the hybrid method runs faster than the original hyperplane constrained method with almost same accuracy.