Igor Simunec
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View article: Estimation of spectral gaps for sparse symmetric matrices
Estimation of spectral gaps for sparse symmetric matrices Open
In this paper we propose and analyze an algorithm for identifying spectral gaps of a real symmetric matrix $A$ by simultaneously approximating the traces of spectral projectors associated with multiple different spectral slices. Our method…
View article: A Sketch-and-Select Arnoldi Process
A Sketch-and-Select Arnoldi Process Open
A sketch-and-select Arnoldi process to generate a well-conditioned basis of a Krylov space at low cost is proposed. At each iteration the procedure utilizes randomized sketching to select a limited number of previously computed basis vecto…
View article: Error bounds for the approximation of matrix functions with rational Krylov methods
Error bounds for the approximation of matrix functions with rational Krylov methods Open
We obtain an expression for the error in the approximation of and with rational Krylov methods, where is a symmetric matrix, is a vector and the function admits an integral representation. The error expression is obtained by linking the ma…
View article: A low-memory Lanczos method with rational Krylov compression for matrix functions
A low-memory Lanczos method with rational Krylov compression for matrix functions Open
In this work we introduce a memory-efficient method for computing the action of a Hermitian matrix function on a vector. Our method consists of a rational Lanczos algorithm combined with a basis compression procedure based on rational Kryl…
View article: Error bounds for the approximation of matrix functions with rational Krylov methods
Error bounds for the approximation of matrix functions with rational Krylov methods Open
We obtain an expression for the error in the approximation of $f(A) \boldsymbol{b}$ and $\boldsymbol{b}^T f(A) \boldsymbol{b}$ with rational Krylov methods, where $A$ is a symmetric matrix, $\boldsymbol{b}$ is a vector and the function $f$…
View article: A sketch-and-select Arnoldi process
A sketch-and-select Arnoldi process Open
A sketch-and-select Arnoldi process to generate a well-conditioned basis of a Krylov space at low cost is proposed. At each iteration the procedure utilizes randomized sketching to select a limited number of previously computed basis vecto…
View article: Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods
Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods Open
We consider the problem of approximating the von Neumann entropy of a large, sparse, symmetric positive semidefinite matrix $A$, defined as $\operatorname{tr}(f(A))$ where $f(x)=-x\log x$. After establishing some useful properties of this …
View article: Computation of generalized matrix functions with rational Krylov methods
Computation of generalized matrix functions with rational Krylov methods Open
We present a class of algorithms based on rational Krylov methods to compute the action of a generalized matrix function on a vector. These algorithms incorporate existing methods based on the Golub-Kahan bidiagonalization as a special cas…
View article: Computation of generalized matrix functions with rational Krylov methods
Computation of generalized matrix functions with rational Krylov methods Open
We present a class of algorithms based on rational Krylov methods to compute the action of a generalized matrix function on a vector. These algorithms incorporate existing methods based on the Golub-Kahan bidiagonalization as a special cas…