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Register a new userCalculating a function of a matrix with a real spectrum
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Added by
Vitalii Kurbatov
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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.
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Let $A$ be a square complex matrix; $z_1$, ..., $z_{N}inmathbb C$ 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 $sigma(A)$ 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 $$ biglVert f(A)-r(A)bigrVertle max_{tin[0,1];muintext{convex hull}{z_1,z_{2},dots,z_{N}}}bigglVertOmega(A)[v(A)]^{-1} frac{bigl(vfbigr)^{{(N)}} bigl((1-t)mumathbf1+tAbigr)}{N!}biggrVert, $$ where $Omega(z)=prod_{k=1}^N(z-z_k)$, 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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Localized collocation methods based on radial basis functions (RBFs) for elliptic problems appear to be non-robust in the presence of Neumann boundary conditions. In this paper we overcome this issue by formulating the RBF-generated finite difference method in a discrete least-squares setting instead. This allows us to prove high-order convergence under node refinement and to numerically verify that the least-squares formulation is more accurate and robust than the collocation formulation. The implementation effort for the modified algorithm is comparable to that for the collocation method.
We present a new fast algorithm for computing the Boys function using nonlinear approximation of the integrand via exponentials. The resulting algorithms evaluate the Boys function with real and complex valued arguments and are competitive with previously developed algorithms for the same purpose.
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