We describe algorithms for computing eigenpairs (eigenvalue--eigenvector) of a complex @math matrix @math . These algorithms are numerically stable, strongly accurate, and theoretically efficient (i.e., polynomial-time). We do not believe they outperform in practice the algorithms currently used for this computational problem. The merit of our paper is to give a positive answer to a long-standing open problem in numerical linear algebra.
The eigenvalue problem of a matrix A can be considered a system of polynomial equations @math in complex variables @math and @math . In this paper, a homotopy continuation algorithm for solving eigenvalue problems of real nonsymmetric matrices is developed based on this point. Different from current homotopy continuation methods for real nonsymmetric matrices, this algorithm makes use of the homotopy which consists of real polynomials. Hence when a complex path is followed, its conjugate path is obtained as a by-product. Moreover, techniques including a double step inverse iteration are developed to eliminate complex computations completely. A great amount of operations is saved with these complex-to-real conversions.The algorithm employed the strategy of “divide and conquer,” which makes most of the eigen-paths almost straight lines and extremely easy to follow. Although some of the paths may contain bifurcation, the situation is handled with a complexification process whic...  Abstract The homotopy method is used to find all eigenpairs of symmetric matrices. A special homotopy is constructed for Jacobi matrices. It is shown that there are exactly n distinct smooth curves connecting trivial solutions to desired eigenpairs. These curves are solutions of a certain ordinary differential equation with different initial values. Hence, they can be followed numerically. Incorporated with sparse matrix techniques, this method might be used to solve eigenvalue problems for large scale matrices. Abstract Generalized eigenvalue problems can be considered as a system of polynomials. The homotopy continuation method is used to find all the isolated zeros of the polynomial system which corresponds to the eigenpairs of the generalized eigenvalue problem. A special homotopy is constructed in such a way that there are exactly n distinct smooth curves connecting trivial solutions to desired eigenpairs. Since the curves followed by general homotopy curve following scheme are computed independently of one another, the algorithm is a likely candidate for exploiting the advantages of parallel processing to the generalized eigenvalue problems. Publisher Summary This chapter presents the numerical solution of polynomial systems by homotopy continuation methods. Solving polynomial systems is an area where numerical computations arise almost naturally. Given the complexity of the problem, e standard machine arithmetic to obtain efficient programs is used. The eigenvalues of a matix A are the zeros of its characteristic polynomial f[lambda] = det[A - [lambda]I]. With Hyman's method of determinant evaluation, a new homotopy continuation method, homotopy-determinant method, is developed in this paper for finding all eigenvalues of a real upper Hessenberg matrix. In contrast to other homotopy continuation methods, the homotopy-determinant method calculates eigenvalues without computing their corresponding eigenvectors. Like all homotopy methods, our method solves the eigenvalue problem by following eigenvalue paths of a real homotopy whose regularity is established to the extent necessary. The inevitable bifurcation and possible path jumping are handled by effective processes. 18 refs., 4 figs., 1 tab.