O(n) Algorithms for Banded Plus Semiseparable Matrices
نویسندگان
چکیده
We present a new representation for the inverse of a matrix that is a sum of a banded matrix and a semiseparable matrix. In particular, we show that under certain conditions, the inverse of a banded plus semiseparable matrix can also be expressed as a banded plus semiseparable matrix. Using this result, we devise a fast algorithm for the solution of linear systems of equations involving such matrices. Numerical results show that the new algorithm competes favorably with existing techniques in terms of computational time.
منابع مشابه
Fast and Stable Algorithms for Banded Plus Semiseparable Systems of Linear Equations
We present fast and numerically stable algorithms for the solution of linear systems of equations, where the coefficient matrix can be written in the form of a banded plus semiseparable matrix. Such matrices include banded matrices, banded bordered matrices, semiseparable matrices, and block-diagonal plus semiseparable matrices as special cases. Our algorithms are based on novel matrix factoriz...
متن کاملA Levinson-like algorithm for symmetric strongly nonsingular higher order semiseparable plus band matrices
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 a first part an O(p 2 n) solver for a semiseparable matrix of semiseparability rank p is derived, and in a second part we d...
متن کاملComputing the Condition Number of Tridiagonal and Diagonal-Plus-Semiseparable Matrices in Linear Time
For an n × n tridiagonal matrix we exploit the structure of its QR factorization to devise two new algorithms for computing the 1-norm condition number in O(n) operations. The algorithms avoid underflow and overflow, and are simpler than existing algorithms since tests are not required for degenerate cases. An error analysis of the first algorithm is given, while the second algorithm is shown t...
متن کاملGegenbauer Polynomials and Semiseparable Matrices
In this paper, we develop a new O(n logn) algorithm for converting coefficients between expansions in different families of Gegenbauer polynomials up to a finite degree n. To this end, we show that the corresponding linear mapping is represented by the eigenvector matrix of an explicitly known diagonal plus upper triangular semiseparable matrix. The method is based on a new efficient algorithm ...
متن کاملA Levinson-like algorithm for symmetric positive definite semiseparable plus diagonal matrices
In this paper a Levinson-like algorithm is derived for solving symmetric positive definite semiseparable plus diagonal systems of equations. In a first part we solve a Yule-Walker-like system of equations. Based on this O(n) solver an algorithm for a general right-hand side is derived. The new method has a linear complexity and takes 19n − 13 operations. The relation between the algorithm and a...
متن کامل