An ' li 2 error expression for the Lanczos procedure *
نویسنده
چکیده
The Lancws procedure is widely used in model reduction of large-scale djnamical systems. In this note we introduce an exact expression for the Hz norm of the error system in the Lanczos procedure. To our knowledge, this is the first such global result fur the Lanczos procedure. Then we tnm our attention to the rational Krylov method, which has the disadvantage that the selection of interpolation points is an ad-hoc process. The error expression for the Lanczos procedure suggests that the interpolation points he chosen at the negati\e of the poles of the to-be-reduced model. Numerical examples raeal that this choice of the interpolation points improves the approximation and reduces the error significantly.
منابع مشابه
Error Estimation of the Padé Approximation of Transfer Functions via the Lanczos Process
Krylov subspace based moment matching algorithms, such as PVL (Padé approximation Via the Lanczos process), have emerged as popular tools for efficient analyses of the impulse response in a large linear circuit. In this work, a new derivation of the PVL algorithm is presented from the matrix point of view. This approach simplifies the mathematical theory and derivation of the algorithm. Moreove...
متن کاملSharpness in rates of convergence for the symmetric Lanczos method
The Lanczos method is often used to solve a large and sparse symmetric matrix eigenvalue problem. There is a well-established convergence theory that produces bounds to predict the rates of convergence good for a few extreme eigenpairs. These bounds suggest at least linear convergence in terms of the number of Lanczos steps, assuming there are gaps between individual eigenvalues. In practice, o...
متن کاملA Restarted Lanczos Approximation to Functions of a Symmetric Matrix
Abstract. In this paper, we investigate a method for restarting the Lanczos method for approximating the matrix-vector product f(A)b, where A ∈ Rn×n is a symmetric matrix. For analytic f we derive a novel restart function that identifies the error in the Lanczos approximation. The restart procedure is then generated by a restart formula using a sequence of these restart functions. We present an...
متن کاملSharpness in Rates of Convergence For CG and Symmetric Lanczos Methods
Conjugate Gradient (CG) method is often used to solve a positive definite linear system Ax = b. Existing bounds suggest that the residual of the kth approximate solution by CG goes to zero like [( √ κ− 1)/(√κ + 1)], where κ ≡ κ(A) = ‖A‖2‖A−1‖2 is A’s spectral condition number. It is well-known that for a given positive definite linear system, CG may converge (much) faster, known as superlinear ...
متن کامل2-Norm Error Bounds and Estimates for Lanczos Approximations to Linear Systems and Rational Matrix Functions
The Lanczos process constructs a sequence of orthonormal vectors vm spanning a nested sequence of Krylov subspaces generated by a hermitian matrix A and some starting vector b. In this paper we show how to cheaply recover a secondary Lanczos process, starting at an arbitrary Lanczos vector vm and how to use this secondary process to efficiently obtain computable error estimates and error bounds...
متن کامل