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V. G. Kurbatov

Publications and source records attributed to V. G. Kurbatov.

14 recordsLinked to original sources
arXiv2024

A complete pair of solvents of a quadratic matrix pencil

Let $B$ and $C$ be square complex matrices. The differential equation \begin{equation*} x''(t)+Bx'(t)+Cx(t)=f(t) \end{equation*} is considered. A solvent is a matrix solution $X$ of the equation $X^2+BX+C=\mathbf0$. A pair of solvents $X$ and $Z$ is called complete if the matrix $X-Z$ is invertible. Knowing a complete pair of solvents $X$ and $Z$ allows us to reduce the solution of the initial value problem to the calculation of two matrix exponentials $e^{Xt}$ and $e^{Zt}$. The problem of finding a complete pair $X$ and $Z$, which leads to small rounding errors in solving the differential equation, is discussed.

math.NA↗
arXiv2023

An approximation of matrix exponential by a truncated Laguerre series

The Laguerre functions $l_{n,τ}^α$, $n=0,1,\dots$, are constructed from generalized Laguerre polynomials. The functions $l_{n,τ}^α$ depend on two parameters: scale $τ>0$ and order of generalization $α>-1$, and form an orthogonal basis in $L_2[0,\infty)$. Let the spectrum of a square matrix $A$ lie in the open left half-plane. Then the matrix exponential $H_A(t)=e^{At}$, $t>0$, belongs to $L_2[0,\infty)$. Hence the matrix exponential $H_A$ can be expanded in a series $H_A=\sum_{n=0}^\infty S_{n,τ,α,A}\,l_{n,τ}^α$. An estimate of the norm $\Bigl\lVert H_A-\sum_{n=0}^N S_{n,τ,α,A}\,l_{n,τ}^α\Bigr\rVert_{L_2[0,\infty)}$ is proposed. Finding the minimum of this estimate over $τ$ and $α$ is discussed. Numerical examples show that the optimal $α$ is often almost 0, which essentially simplifies the problem.

math.NA↗
arXiv2021

An estimate of approximation of an analytic function of a matrix by a rational function

Let $A$ be a square complex matrix; $z_1$, ..., $z_{N}\in\mathbb C$ be arbitrary (possibly repetitive) points of interpolation; $f$ be an analytic function defined on a neighborhood of the convex hull of the union of the spectrum $σ(A)$ of the matrix $A$ and the points $z_1$, ..., $z_{N}$; and the rational function $r=\frac uv$ (with the degree of the numerator $u$ less than $N$) interpolates $f$ at these points (counted according to their multiplicities). Under these assumptions estimates of the kind $$ \bigl\Vert f(A)-r(A)\bigr\Vert\le \max_{t\in[0,1];μ\in\text{convex hull}\{z_1,z_{2},\dots,z_{N}\}}\biggl\VertΩ(A)[v(A)]^{-1} \frac{\bigl(vf\bigr)^{(N)} \bigl((1-t)μ\mathbf1+tA\bigr)}{N!}\biggr\Vert, $$ where $Ω(z)=\prod_{k=1}^N(z-z_k)$, are proposed. As an example illustrating the accuracy of such estimates, an approximation of the impulse response of a dynamic system obtained using the reduced-order Arnoldi method is considered, the actual accuracy of the approximation is compared with the estimate based on this paper.

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arXiv2021

Calculating a function of a matrix with a real spectrum

Let $T$ be a square matrix with a real spectrum, and let $f$ be an analytic function. The problem of the approximate calculation of $f(T)$ is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that $T$ is triangular and its diagonal entries $t_{ii}$ are arranged in increasing order. To avoid calculations using the differences $t_{ii}-t_{jj}$ with close (including equal) $t_{ii}$ and $t_{jj}$, it is proposed to represent $T$ in a block form and calculate the two main block diagonals using interpolating polynomials. The rest of the $f(T)$ entries can be calculated using the Parlett recurrence algorithm. It is also proposed to perform scalar operations (such as the building of interpolating polynomials) with an enlarged number of decimal digits.

math.NA↗
arXiv2020

Inverse-closedness of the subalgebra of locally nuclear operators

Let $X$ be a Banach space and $T$ be a bounded linear operator acting in $l_p(\mathbb Z^c,X)$, $1\le p\le\infty$. The operator $T$ is called \emph{locally nuclear} if it can be represented in the form \begin{equation*} (Tx)_k=\sum\limits_{m\in\mathbb Z^c} b_{km}x_{k-m},\qquad k\in\mathbb Z^c, \end{equation*} where $b_{km}:\,X\to X$ are nuclear, \begin{equation*} \lVert b_{km}\rVert_{\mathfrak S_1}\leβ_{m},\qquad k,m\in\mathbb Z^c, \end{equation*} $\lVert\cdot\rVert_{\mathfrak S_1}$ is the nuclear norm, $β\in l_{1}(\mathbb Z^c,\mathbb C)$ or $β\in l_{1,g}(\mathbb Z^c,\mathbb C)$, and $g$ is an appropriate weight on $\mathbb Z^c$. It is established that if $T$ is locally nuclear and the operator $\mathbf1+T$ is invertible, then the inverse operator $(\mathbf1+T)^{-1}$ has the form $\mathbf1+T_1$, where $T_1$ is also locally nuclear. This result is refined for the case of operators acting in $L_p(\mathbb R^c,\mathbb C)$.

math.FA↗
arXiv2019

An estimate of approximation of a matrix-valued function by an interpolation polynomial

Let $A$ be a square complex matrix, $z_1$, ..., $z_{n}\in\mathbb C$ be (possibly repetitive) points of interpolation, $f$ be analytic in a neighborhood of the convex hull of the union of the spectrum of $A$ and the points $z_1$, ..., $z_{n}$, and $p$ be the interpolation polynomial of $f$, constructed by the points $z_1$, ..., $z_{n}$. It is proved that under these assumptions $$\Vert f(A)-p(A)\Vert\le\frac1{n!} \max_{t\in[0,1];\,μ\in\text{co}\{z_1,z_{2},\dots,z_{n}\}}\bigl\VertΩ(A)f^{(n)} \bigl((1-t)μ\mathbf1+tA\bigr)\bigr\Vert,$$ where $Ω(z)=\prod_{k=1}^n(z-z_k)$.

math.NA↗
arXiv2018

Inverse-closedness of subalgebras of integral operators with almost periodic kernels

The integral operator of the form $$\bigl(Nu\bigr)(x)=\sum_{k=1}^\infty e^{i\langleω_k,x\rangle} \int_{\mathbb R^c}n_k(x-y)\,u(y)\,dy$$ acting in $L_p(\mathbb R^c)$, $1\le p\le\infty$, is considered. It is assumed that $ω_k\in\mathbb R^c$, $n_k\in L_1(\mathbb R^c)$, and $$\sum_{k=1}^\infty\lVert n_k\rVert_{L_1}<\infty.$$ We prove that if the operator $\mathbf1+N$ is invertible, then $(\mathbf1+N)^{-1}=\mathbf1+M$, where $M$ is an integral operator possessing the analogous representation.

math.FA↗
arXiv2018

Green's function of the problem of bounded solutions in the case of a block triangular coefficient

It is known that the equation $x'(t)=Ax(t)+f(t)$, where $A$ is a bounded linear operator, has a unique bounded solution $x$ for any bounded continuous free term~$f$ if and only if the spectrum of the coefficient $A$ does not intersect the imaginary axis. The solution can be represented in the form \begin{equation*} x(t)=\int_{-\infty}^{\infty}\mathcal G(s)f(t-s)\,ds. \end{equation*} The kernel $\mathcal G$ is called Green's function. In this paper, the case when $A$ admits a representation by a block triangular operator matrix is considered. It is shown that the blocks of $\mathcal G$ are sums of special convolutions of Green's functions of diagonal blocks of $A$.

math.FA↗
arXiv2017

Computation of Green's function of the bounded solutions problem

It is well known that the equation $x'(t)=Ax(t)+f(t)$, where $A$ is a square matrix, has a unique bounded solution $x$ for any bounded continuous free term $f$, provided the coefficient $A$ has no eigenvalues on the imaginary axis. This solution can be represented in the form \begin{equation*} x(t)=\int_{-\infty}^{\infty}\mathcal G(t-s)x(s)\,ds. \end{equation*} The kernel $\mathcal G$ is called Green's function. In the paper, a representation of Green's function in the form of the Newton interpolating polynomial is used for approximate calculation of $\mathcal G$. An estimate of the sensitivity of the problem is given.

math.NA↗
arXiv2016

Analytic functional calculus for two operators

Properties of the mappings \begin{align*} C&\mapsto\frac1{(2πi)^2}\int_{Γ_1}\int_{Γ_2}f(λ,μ)\,R_{1,\,λ}\,C\, R_{2,\,μ}\,dμ\,dλ, C&\mapsto\frac1{2πi}\int_Γg(λ)R_{1,\,λ}\,C\, R_{2,\,λ}\,dλ\end{align*} are discussed; here $R_{1,\,(\cdot)}$ and $R_{2,\,(\cdot)}$ are pseudo-resolvents, i.~e., resolvents of bounded, unbounded, or multivalued linear operators, and $f$ and $g$ are analytic functions. Several applications are considered: a representation of the impulse response of a second order linear differential equation with operator coefficients, a representation of the solution of the Sylvester equation, and an exploration of properties of the differential of the ordinary functional calculus.

math.FA↗
arXiv2015

Inverse-closedness of the set of integral operators with $L_1$-continuously varying kernels

Let $N$ be an integral operator of the form $\bigl(Nu\bigr)(x)=\int_{\mathbb R^c}n(x,x-y)\,u(y)\,dy$ acting in $L_p(\mathbb R^c)$ with a measurable kernel $n$ satisfying the estimate $|n(x,y)|\leβ(y)$, where $β\in L_1$. It is proved that if the function $t\mapsto n(t,\cdot)$ is continuous in the norm of $L_1$ and the operator $\mathbf1+N$ has an inverse, then $(\mathbf1+N)^{-1}=\mathbf1+M$, where $M$ is an integral operator possessing the same properties.

math.FA↗
arXiv2014

Computation of a function of a matrix with close eigenvalues by means of the Newton interpolating polynomial

An algorithm for computing an analytic function of a matrix $A$ is described. The algorithm is intended for the case where $A$ has some close eigenvalues, and clusters (subsets) of close eigenvalues are separated from each other. This algorithm is a modification of some well known and widely used algorithms. A novel feature is an approximate calculation of divided differences for the Newton interpolating polynomial in a special way. This modification does not require to reorder the Schur triangular form and to solve Sylvester equations.

math.NA↗