From cd346f38c11134dbb455b0feef1c2fbe666aa684 Mon Sep 17 00:00:00 2001 From: moritzleucke Date: Fri, 4 Oct 2024 16:07:29 +0200 Subject: [PATCH] add symmetry documentation in header --- docs/gx_ac.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/gx_ac.md b/docs/gx_ac.md index 3f65750c..c3b2b4a4 100644 --- a/docs/gx_ac.md +++ b/docs/gx_ac.md @@ -36,7 +36,7 @@ g_p(z_i) = \begin{dcases} f(z_i) & p=1 \\\;\\ \frac{g_{p-1}(z_{p-1})-g_{p-1}(z_ ``` Padé approximants are known to be [numerical instable](https://doi.org/10.1093/imamat/25.3.267). The GX-AC component uses two strategies to numerically stabilize the interpolation. -First, it incorporates a [greedy algorithm for Thiele Padé approximants](https://pubs.acs.org/doi/full/10.1021/acs.jctc.3c00555) that minimizes the numerical error by reordering of the reference points. A validation of the greedy algorithm can be found in this [reference](https://pubs.acs.org/doi/full/10.1021/acs.jctc.3c00555). Additionally it is possible to use the component with a higher internal numerical floating point precision. This helps reducing the numerical noise caused by [catastrophic cancellation](https://doi.org/10.1145/103162.103163). Catastrophic cancellation occurs when rounding errors are amplified through the subtraction of rounded numbers, such as double-precision floating-point numbers commonly used in most programs. This is implemented using the [GNU Multiple Precision (GMP) library](https://gmplib.org/) which allows floating-point operations with customizable precision. Furthermore, it is possible to impose various symmetries onto the Padé model using the GX-AC component. To maximize performance, the evaluation of the Padé model uses the [Wallis algorithm](https://numerical.recipes/book.html), which minimizes the number of divisions, an operation that is computationally expensive, especially for complex floating-point numbers and even more so for higher-precision complex numbers. +First, it incorporates a [greedy algorithm for Thiele Padé approximants](https://pubs.acs.org/doi/full/10.1021/acs.jctc.3c00555) that minimizes the numerical error by reordering of the reference points. A validation of the greedy algorithm can be found in this [reference](https://pubs.acs.org/doi/full/10.1021/acs.jctc.3c00555). Additionally it is possible to use the component with a higher internal numerical floating point precision. This helps reducing the numerical noise caused by [catastrophic cancellation](https://doi.org/10.1145/103162.103163). Catastrophic cancellation occurs when rounding errors are amplified through the subtraction of rounded numbers, such as double-precision floating-point numbers commonly used in most programs. This is implemented using the [GNU Multiple Precision (GMP) library](https://gmplib.org/) which allows floating-point operations with customizable precision. Furthermore, it is possible to impose various symmetries onto the Padé model using the GX-AC component. To maximize performance, the evaluation of the Padé model uses the [Wallis algorithm](https://numerical.recipes/book.html), which minimizes the number of divisions, an operation that is computationally expensive, especially for complex floating-point numbers and even more so for higher-precision complex numbers. The component also allows symmetry-constraint Padé models, for a full list of supported symmetries see the section "Usage". # Benchmarks