关键词: symmetric positive definite matrix, conjugate gradient, eigenvalues, condition number

Abstract: From the formulas of the conjugate gradient, a similarity between a symmetric positive definite (SPD) matrix A and a tridiagonal matrix B is obtained.The elements of the matrix B are determined by the parameters of the conjugate gradient.The computation of eigenvalues of A is then reduced to the case of the tridiagonal matrix B.The approximation of extreme eigenvalues of A can be obtained as a ‘by-product’ in the computation of the conjugate gradient if a computational cost of O(s) arithmetic operations is added, where s is the number of iterations This computational cost is negligible compared with the conjugate gradient.If the matrix A is not SPD, the approximation of the condition number of A can be obtained from the computation of the conjugate gradient on A^T A.Numerical results show that this is a convenient and highly efficient method for computing extreme eigenvalues and the condition number of nonsingular matrices.

Key words: symmetric positive definite matrix, conjugate gradient, eigenvalues, condition number