On the numerical solution of one-dimensional integral and differential equations
On the numerical solution of one-dimensional integral and differential equations
Matrix computations (3rd ed.)
Fast reliable algorithms for matrices with structure
Fast reliable algorithms for matrices with structure
Fast and Stable Algorithms for Banded Plus Semiseparable Systems of Linear Equations
SIAM Journal on Matrix Analysis and Applications
Two fast algorithms for solving diagonal-plus-semiseparable linear systems
Journal of Computational and Applied Mathematics - Special Issue: Proceedings of the 10th international congress on computational and applied mathematics (ICCAM-2002)
A QR–method for computing the singular values via semiseparable matrices
Numerische Mathematik
An Orthogonal Similarity Reduction of a Matrix into Semiseparable Form
SIAM Journal on Matrix Analysis and Applications
Simulation smoothing for state-space models: A computational efficiency analysis
Computational Statistics & Data Analysis
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In this paper, we will derive a solver for a symmetric strongly nonsingular higher order generator representable semiseparable plus band matrix. The solver we will derive is based on the Levinson algorithm, which is used for solving strongly nonsingular Toeplitz systems. In the first part an O(p2n) solver for a semiseparable matrix of semiseparability rank p is derived, and in a second part we derive an O(l2n) solver for a band matrix with bandwidth 2l + 1. Both solvers are constructed in a similar way: firstly a Yule-Walker-like equation needs to be solved, and secondly this solution is used for solving a linear equation with an arbitrary right-hand side. Finally, a combination of the above methods is presented to solve linear systems with semiseparable plus band coefficient matrices. The overall complexity of this solver is 6(l+ p)2n plus lower order terms. In the final section numerical experiments are performed. Attention is paid to the timing and the accuracy of the described methods.