In this paper we propose an efficiently preconditioned Newton method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices. A sequence of preconditioners based on the BFGS update formula is proposed, for the Preconditioned Conjugate Gradient solution of the linearized Newton system to solve A u = q(u) u, q(u) being the Rayleigh Quotient. We give theoretical evidence that the sequence of preconditioned Jacobians remains close to the identity matrix if the initial preconditioned Jacobian is so. Numerical results onto matrices arising from various realistic problems with size up to one million unknowns account for the efficiency of the proposed algorithm which reveals competitive with the Jacobi-Davidson method on all the test problems.
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机译：在本文中，我们提出了一种有效的预处理牛顿法，用于计算大型和稀疏对称正定矩阵的最左本征对。针对线性牛顿系统的预处理共轭梯度解，提出了一个基于BFGS更新公式的预处理器序列，以求解A u = q（u）u，q（u）为瑞利商。我们提供了理论证据，如果初始的预条件雅可比行列是如此，则预条件的雅可比行列的序列仍然接近恒等矩阵。由大小不超过一百万个未知数的各种现实问题产生的矩阵上的数值结果说明了所提出算法的效率，该算法显示出与Jacobi-Davidson方法在所有测试问题上的竞争力。
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