The Theory of Matrices in Numerical AnalysisCourier Corporation, 18.06.2013 - 272 Seiten This text explores aspects of matrix theory that are most useful in developing and appraising computational methods for solving systems of linear equations and for finding characteristic roots. Suitable for advanced undergraduates and graduate students, it assumes an understanding of the general principles of matrix algebra, including the Cayley-Hamilton theorem, characteristic roots and vectors, and linear dependence. An introductory chapter covers the Lanczos algorithm, orthogonal polynomials, and determinantal identities. Succeeding chapters examine norms, bounds, and convergence; localization theorems and other inequalities; and methods of solving systems of linear equations. The final chapters illustrate the mathematical principles underlying linear equations and their interrelationships. Topics include methods of successive approximation, direct methods of inversion, normalization and reduction of the matrix, and proper values and vectors. Each chapter concludes with a helpful set of references and problems. |
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Acad Akad algorithm Amer Angew applied approximation arbitrary bounds characteristic polynomial characteristic roots consider convergence convex body defined determinant diagonal elements diagonal matrix Duke Math eigenvalues Euclidean exists factorization field of values finding finite first column fixed follows Hence Hermitian matrix Hessenberg form implies inclusion region Indust inequality invariant subspace inverse irreducible iterative methods Jacobi method Jordan normal form Krylov sequence Lanczos algorithm linear combination linear equations linearly independent lubK matrix norm Matrizen Mech minimal polynomial modulus Moreover Nauk SSSR nonnegative nonnull nonsingular matrix nonvanishing normal matrix null obtained orthogonal particular principal submatrix Proc proper values proper vector belonging rows satisfies scalar Section semidefinite separation polynomial singular values solution square sufficient Suppose symmetric Tables Aids Comput theorem Uber unit lower triangular unit upper triangular unitary matrix upper triangular matrix uravnenii vanish verified Wielandt