Approximation by short exponential sums with geometric error decay based on Gauss quadrature

Abstract

We present new short exponential sum approximations of length $N$ for $f_1(x)=\frac{1}{a+x}$ with $a>0$ on $[0,\infty )$ and for $f_2(x)={\mathrm{e}}^{-x^2/2\ddot{I}ƒ}$ with $\ddot{I}ƒ>0$ on $\mathbb{R}$ with geometric error decay $\ddot{I}{}^{-2N}$ for user-defined $N\ge 2$ and $\ddot{I}>1$. The approximations are built over consecutive intervals $[b_j,b_{j+1})\subset [0,\infty )$, $j\in {\mathbb{N}}_0$, with interval lengths that depend on $\ddot{I}$ and grow exponentially for $f_1$ and are equidistant for $f_2$. All parameters determining the exponential sum approximations on $[b_j,b_{j+1})$ are easily computed from the initial parameters on $[b_0,b_1)$, ensuring numerical stability. Our method is based on Gauss-Laguerre and Gauss-Hermite quadrature, respectively, applied to suitable parametric integral representations of $f_1$ and $f_2$. This technique ensures consistent relative errors across all intervals. Using the obtained exponential sum approximations, we achieve highly accurate approximations of $\log (x)$ on $[1,\infty )$ and of the error function $\mathrm{erf}(x)$ with predictable geometric error decay. Numerical examples for $N=8$ and $N=10$ clearly illustrate the theoretical error estimates.

25 pages