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.
Building upon earlier work by Golub, Meurant, Strakos, and Tichy, we derive new a posteriori error bounds for Krylov subspace approximations to f(A)b, the action of a function f of a real symmetric or complex Hermitia...
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Building upon earlier work by Golub, Meurant, Strakos, and Tichy, we derive new a posteriori error bounds for Krylov subspace approximations to f(A)b, the action of a function f of a real symmetric or complex Hermitian matrix A on a vector b. To this purpose we assume that a rational function in partial fraction expansion form is used to approximate f, and the Krylov subspace approximations are obtained as linear combinations of Galerkin approximations to the individual terms in the partial fraction expansion. Our error estimates come at very low computational cost. In certain important special situations they can be shown to actually be lower bounds of the error. Our numerical results include experiments with the matrix exponential, as used in exponential integrators, and with the matrix sign function, as used in lattice quantum chromodynamics simulations, and demonstrate the accuracy of the estimates. The use of our error estimates within acceleration procedures is also discussed.
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.
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.
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.
Krylov subspace methods for approximating a matrix function f(A) times a vector v are analyzed in this paper. For the Arnoldi approximation to e (-tau A) v, two reliable a posteriori error estimates are derived from t...
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Krylov subspace methods for approximating a matrix function f(A) times a vector v are analyzed in this paper. For the Arnoldi approximation to e (-tau A) v, two reliable a posteriori error estimates are derived from the new bounds and generalized error expansion we establish. One of them is similar to the residual norm of an approximate solution of the linear system, and the other one is determined critically by the first term of the error expansion of the Arnoldi approximation to e (-tau A) v due to Saad. We prove that each of the two estimates is reliable to measure the true error norm, and the second one theoretically justifies an empirical claim by Saad. In the paper, by introducing certain functions I center dot (k) (z) defined recursively by the given function f(z) for certain nodes, we obtain the error expansion of the Krylov-like approximation for f(z) sufficiently smooth, which generalizes Saad's result on the Arnoldi approximation to e (-tau A) v. Similarly, it is shown that the first term of the generalized error expansion can be used as a reliable a posteriori estimate for the Krylov-like approximation to some other matrix functions times v. Numerical examples are reported to demonstrate the effectiveness of the a posteriori error estimates for the Krylov-like approximations to e (-tau A) v, cos(A)v and sin(A)v.
matrix power series may slowly converge or even diverge if some eigenvalues of the matrix are near the boundary or outside the disk of convergence. In this case it is proposed to apply suitably chosen summability meth...
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matrix power series may slowly converge or even diverge if some eigenvalues of the matrix are near the boundary or outside the disk of convergence. In this case it is proposed to apply suitably chosen summability methods to accelerate or generate convergence; special attention is paid to Euler methods. The matrix logarithm appearing in connection with stationary Markov chains is considered as an example.
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