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We derive a new algorithm for computing the action $f(A)V$ of the cosine, sine, hyperbolic cosine, and hyperbolic sine of a matrix $A$ on a matrix $V$, without first computing $f(A)$. The algorithm can compute $cos(A)V$ and $sin(A)V$ simultaneously, and likewise for $cosh(A)V$ and $sinh(A)V$, and it uses only real arithmetic when $A$ is real. The algorithm exploits an existing algorithm texttt{expmv} of Al-Mohy and Higham for $mathrm{e}^AV$ and its underlying backward error analysis. Our experiments show that the new algorithm performs in a forward stable manner and is generally significantly faster than alternatives based on multiple invocations of texttt{expmv} through formulas such as $cos(A)V = (mathrm{e}^{mathrm{i}A}V + mathrm{e}^{mathrm{-i}A}V)/2$.
It is shown that generalized trigonometric functions and generalized hyperbolic functions can be transformed from each other. As an application of this transformation, a number of properties for one immediately lead to the corresponding properties fo
The paper gives a constructive method, based on greedy algorithms, that provides for the classes of functions with small mixed smoothness the best possible in the sense of order approximation error for the $m$-term approximation with respect to the trigonometric system.
We investigate the problem of approximating the matrix function $f(A)$ by $r(A)$, with $f$ a Markov function, $r$ a rational interpolant of $f$, and $A$ a symmetric Toeplitz matrix. In a first step, we obtain a new upper bound for the relative interp
We consider the matrix representation of the Eisenstein numbers and in this context we discuss the theory of the Pseudo Hyperbolic Functions. We develop a geometrical interpretation and show the usefulness of the method in Physical problems related to the anomalous scattering of light by crystals
The Evans function has been used extensively to study spectral stability of travelling-wave solutions in spatially extended partial differential equations. To compute Evans functions numerically, several shooting methods have been developed. In this