This work is concerned with computing low-rank approximations of a matrix function f(A) for a large symmetric positive semidefinite matrix A, a task that arises in, e.g., statistical learning and inverse problems. The...
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This work is concerned with computing low-rank approximations of a matrix function f(A) for a large symmetric positive semidefinite matrix A, a task that arises in, e.g., statistical learning and inverse problems. The application of popular randomized methods, such as the randomized singular value decomposition or the Nystrom approximation, to f(A) requires multiplying f(A) with a few random vectors. A significant disadvantage of such an approach, matrix-vector products with f(A) are considerably more expensive than matrix-vector products with A, even when carried out only approximately via, e.g., the Lanczos method. In this work, we present and analyze funNystrom, a simple and inexpensive method that constructs a low-rank approximation of f(A) directly from a Nystrom approximation of A, completely bypassing the need for matrix-vector products with f(A). It is sensible to use funNystrom whenever f is monotone and satisfies f(0) = 0. Under the stronger assumption that f is operator monotone, which includes the matrix square root A(1/2) and the matrix logarithm log(I + A), we derive probabilistic bounds for the error in the Frobenius, nuclear, and operator norms. These bounds confirm the numerical observation that funNystrom tends to return an approximation that compares well with the best low-rank approximation of f(A). Furthermore, compared to existing methods, funNystrom requires significantly fewer matrix-vector products with A to obtain a low-rank approximation of f(A), without sacrificing accuracy or reliability. Our method is also of interest when estimating quantities associated with f(A), such as the trace or the diagonal entries of f(A). In particular, we propose and analyze funNystrom++, a combination of funNystrom with the recently developed Hutch++ method for trace estimation.
The paper presents new approach to the evaluation of matrix functions operating over tensors that are essential part of formulation of complex nonlinear material models in mechanics of solids. A method is presented ho...
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The paper presents new approach to the evaluation of matrix functions operating over tensors that are essential part of formulation of complex nonlinear material models in mechanics of solids. A method is presented how to automatically derive numerically efficient closed-form representation of an arbitrary matrix function and its first and second derivatives for 3 x 3 matrices with real eigenvalues. The method offers an unique solution to the standard problem of ill-conditioning in the vicinity of multiple equal eigenvalues which is characteristic for all closed-form representations. A compiled library of subroutines with derived closed-form representation of most commonly used matrix functions along with their first and second derivatives has been created and is available for the use in general finite element environments. Consequently, the matrix functions can become as accurate, efficient and commonly available as their scalar counterparts, resulting in more common use of advanced strain and stress measures, such as Hencky strain measure which have so far been considered difficult for implementation. Accuracy and efficiency of the derived closed-form representations was compared with corresponding truncated series expansion and a speed up between 20 and 80 times has been observed depending on the matrix. The proposed methodology was tested on a set of selected nonlinear material models where matrix functions play an essential part in nonlinear finite element formulation. (C) 2015 Elsevier B.V. All rights reserved.
The objective of this paper is to present an example in which matrix functions are used to solve a modern control exercise. Specifically, the solution for the equation of state, which is a matrix differential equation...
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The objective of this paper is to present an example in which matrix functions are used to solve a modern control exercise. Specifically, the solution for the equation of state, which is a matrix differential equation is calculated. To resolve this, two different methods are presented, first using the properties of the matrix functions and by other side, using the classical method of Laplace transform.
A complete solution is given to a first-order pole-zero meromorphic matrix function interpolation problem on a closed Riemann surface. The solution to the interpolation problem is constructed from the solution to a na...
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A complete solution is given to a first-order pole-zero meromorphic matrix function interpolation problem on a closed Riemann surface. The solution to the interpolation problem is constructed from the solution to a natural linear homogeneous system.
Many researchers are now working on computing the product of a matrix function and a vector, using approximations in a Krylov subspace. We review our results on the analysis of one implementation of that approach for ...
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Many researchers are now working on computing the product of a matrix function and a vector, using approximations in a Krylov subspace. We review our results on the analysis of one implementation of that approach for symmetric matrices, which we call the spectral lanczos decomposition method (SLDM). We have proved a general convergence estimate, relating SLDM error bounds to those obtained through approximation of the matrix function by a part of its Chebyshev series. Thus, we arrived at effective estimates for matrix functions arising when solving parabolic, hyperbolic and elliptic partial differential equations. We concentrate on the parabolic case, where we obtain estimates that indicate superconvergence of SLDM. For this case a combination of SLDM and splitting methods is also considered and some numerical results are presented. We implement our general estimates to obtain convergence bounds of Lanczos approximations to eigenvalues in the internal part of the spectrum. Unlike Kaniel-Saad estimates, our estimates are independent of the set of eigenvalues between the required one and the nearest spectrum bound. We consider an extension of our general estimate to the case of the simple Lanczos method (without reorthogonalization) in finite computer arithmetic which shows that for a moderate dimension of the Krylov subspace the results, proved for the exact arithmetic, are stable up to roundoff.
Let A be a symmetric matrix and let f be a smooth function defined on an interval containing the spectrum of A. Generalizing a well-known result of Demko, Moss and Smith on the decay of the inverse we show that when A...
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Let A be a symmetric matrix and let f be a smooth function defined on an interval containing the spectrum of A. Generalizing a well-known result of Demko, Moss and Smith on the decay of the inverse we show that when A is banded, the entries of f(A) are bounded in an exponentially decaying manner away from the main diagonal. Bounds obtained by representing the entries of f(A) in terms of Riemann-Stieltjes integrals and by approximating such integrals by Gaussian quadrature rules are also considered. Applications of these bounds to preconditioning are suggested and illustrated by a few numerical examples.
Suppose f = p/q is a quotient of two polynomials and that p has degree r(p) and q has degree r(q). Assume that f(A) and f(A + uv(T)) are defined where A is an element of R-nxn, u is an element of R-n, and v is an elem...
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Suppose f = p/q is a quotient of two polynomials and that p has degree r(p) and q has degree r(q). Assume that f(A) and f(A + uv(T)) are defined where A is an element of R-nxn, u is an element of R-n, and v is an element of R-n are given and set r = max{r(p), r(q)}. We show how to compute f(A + uv(T)) in O(rn(2)) flops assuming that f(A) is available together with an appropriate factorization of the denominator matrix q(A). The central result can be interpreted as a generalization of the well-known Sherman-Morrison formula. For an application we consider a Jacobian computation that arises in an inverse problem involving the matrix exponential. With certain assumptions the work required to set up the Jacobian matrix can be reduced by an order of magnitude by making effective use of the rank-1 update formulae developed in this paper.
The simultaneous null solutions of the two complex Hermitian Dirac operators are focused on in Hermitian Clifford analysis, where the Hermitian Cauchy integral was constructed and will play an important role in the fr...
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The simultaneous null solutions of the two complex Hermitian Dirac operators are focused on in Hermitian Clifford analysis, where the Hermitian Cauchy integral was constructed and will play an important role in the framework of circulant (2 x 2) matrix functions. Under this setting we will present the half Dirichlet problem for circulant (2 x 2) matrix functions on the unit ball of even dimensional Euclidean space. We will give the unique solution to it merely by using the Hermitian Cauchy transformation, get the solution to the Dirichlet problem on the unit ball for circulant (2 x 2) matrix functions and the solution to the classical Dirichlet problem as the special case, derive a decomposition of the Poisson kernel for matrix Laplace operator, and further obtain the decomposition theorems of solution space to the Dirichlet problem for circulant (2 x 2) matrix functions. (C) 2010 Elsevier Inc. All rights reserved.
We investigate the sensitivity of matrix functions to random noise in their input. We propose the notion of a stochastic condition number, which determines, to first order, the sensitivity of a matrix function to rand...
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We investigate the sensitivity of matrix functions to random noise in their input. We propose the notion of a stochastic condition number, which determines, to first order, the sensitivity of a matrix function to random noise. We derive an upper bound on the stochastic condition number that can be estimated efficiently by using "small-sample" estimation techniques. The bound can be used to estimate the median, or any other quantile, of the error in a function's output when its input is subjected to random noise. We give numerical experiments illustrating the effectiveness of our stochastic error estimate.
The conversion of a power series with matrix coefficients into an infinite product of certain elementary matrix factors is studied. The expansion of a power series with matrix coefficients as the inverse of an infinit...
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The conversion of a power series with matrix coefficients into an infinite product of certain elementary matrix factors is studied. The expansion of a power series with matrix coefficients as the inverse of an infinite product of elementary factors is also analyzed. Each elementary factor is the sum of the identity matrix and a certain matrix coefficient multiplied by a certain power of the variable. The two expansions provide us with representations of a matrix function and its inverse by infinite products of elementary factors. Estimates on the domain of convergence of the infinite products are given. (C) 2008 Elsevier Inc. All rights reserved.
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