Debugging accumulation of floating-point errors is hard; ideally, computer should track it automatically. Here we consider twofold approximation of an exact real with value + error pair of floating-point numbers. Normally, value + error sum is more accurate than value alone, so error can estimate deviation between value and its exact target. Fast summation algorithm, that provides twofold sum of x[1]+...+x[N] or dot product x[1]*y[1]+...+x[N]*y[N], can be same fast as direct summation sometimes if leveraging processor underused potential. This way, we can hit three goals: improve precision, track inaccuracy, and do this with little if any loss in performance.
A Fast Summation Method for Oscillatory Lattice Sums
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；National Institute of Standards and Technology（美国国家标准与技术研究院）；Yale University（耶鲁大学）
Fast summation of Stokes potentials using a new kernel-splitting in the DMK framework
Flatiron Institute, Simons Foundation（西蒙斯基金会平顿研究所）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Royal Institute of Technology（皇家理工学院）
Fast Ewald Summation with Prolates for Charged Systems in the NPT Ensemble
Center for Computational Mathematics, Flatiron Institute（西蒙斯基金会平顿研究所计算数学中心）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
Accelerating Molecular Dynamics Simulations using Fast Ewald Summation with Prolates
Center for Computational Mathematics, Flatiron Institute（计算数学中心，Flatiron研究所）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
Bashkir State University（巴什基尔国立大学）；Center for Micro and Nanoscale Dynamics of Dispersed Systems, Bashkir State University（巴什基尔国立大学分散系统微纳动力学中心）；Fantalgo, LLC.（Fantalgo有限责任公司）；Institute for Advanced Computer Studies（高级计算机研究所）；Department of Computer Science（计算机科学系）；University of Maryland, College Park（马里兰大学帕克分校）
