A Lanczos bidiagonalization algorithm for Hankel matrices
نویسندگان
چکیده
This paper presents a fast algorithm for bidiagonalizing a Hankel matrix. An m×n Hankel matrix is reduced to a real bidiagonal matrix in O((m+ n)n log(m+ n)) floating-point operations (flops) using the Lanczos method with modified partial orthogonalization and reset schemes to improve its stability. Performance improvement is achieved by exploiting the Hankel structure, as fast Hankel matrix–vector multiplication is used. The accuracy and efficiency of the algorithm are demonstrated by our numerical experiments. © 2008 Elsevier Inc. All rights reserved.
منابع مشابه
Evaluation of a Fast Algorithm for the Eigen-Decomposition of Large Block Toeplitz Matrices with Application to 5D Seismic Data Interpolation
We present a fast 5D (frequency and 4 spatial axes) reconstruction method that uses Multichannel Singular Spectrum Analysis / Cazdow algorithm. Rather than embedding the 4D spatial volume in a Hankel matrix, we propose to embed the data into a block Toeplitz form. Rank reduction is carried out via Lanczos bidiagonalization with fast block Toeplitz matrix-times-vector multiplications via 4D Fast...
متن کاملA harmonic Lanczos bidiagonalization method for computing interior singular triplets of large matrices
This paper proposes a harmonic Lanczos bidiagonalization method for computing some interior singular triplets of large matrices. It is shown that the approximate singular triplets are convergent if a certain Rayleigh quotient matrix is uniformly bounded and the approximate singular values are well separated. Combining with the implicit restarting technique, we develop an implicitly restarted ha...
متن کاملAn Implicitly Restarted Block Lanczos Bidiagonalization Method Using Leja Shifts
In this paper, we propose an implicitly restarted block Lanczos bidiagonalization (IRBLB) method for computing a few extreme or interior singular values and associated right and left singular vectors of a large matrix A. Our method combines the advantages of a block routine, implicit shifting, and the application of Leja points as shifts in the accelerating polynomial. The method neither requir...
متن کاملThe Lanczos Algorithm and Hankel Matrix Factorization
In 1950 Lanczos [22] proposed a method for computing the eigenvalues of symmetric and nonsymmetric matrices. The idea was to reduce the given matrix to a tridiagonal form, from which the eigenvalues could be determined. A characterization of the breakdowns in the Lanczos algorithm in terms of algebraic conditions of controllability and observability was addressed in [6] and [26]. Hankel matrice...
متن کاملA Refined Harmonic Lanczos Bidiagonalization Method and an Implicitly Restarted Algorithm for Computing the Smallest Singular Triplets of Large Matrices
The harmonic Lanczos bidiagonalization method can be used to compute the smallest singular triplets of a large matrix A. We prove that for good enough projection subspaces harmonic Ritz values converge if the columns of A are strongly linearly independent. On the other hand, harmonic Ritz values may miss some desired singular values when the columns of A are almost linearly dependent. Furthermo...
متن کامل