The finite precision computation and the nonconvergence of difference scheme
State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics (LASG), Institute of Atmospheric Physics, Chinese Academy of Sciences（中国科学院大气物理研究所数值模拟大气科学和地球流体力学国家重点实验室）
The authors show that the round-off error can break the consistency which is the premise of using the difference equation to replace the original differential equations. We therefore proposed a theoretical approach to investigate this effect, and found that the difference scheme can not guarantee the convergence of the actual compute result to the analytical one. A conservation scheme experiment is applied to solve a simple linear differential equation satisfing the LAX equivalence theorem in a finite precision computer. The result of this experiment is not convergent when time step-size decreases trend to zero, which proves that even the stable scheme can't guarantee the numerical convergence in finite precision computer. Further the relative convergence concept is introduced.
Efficient Multi-Precision Computation of Bessel Functions for Real Orders and Complex Arguments with Fortran Implementation -- Part I: The Modified Bessel Function of the First Kind, $I_ν(z)$
United Arab Emirates University（阿联酋大学）；Massachusetts Institute of Technology（麻省理工学院）
这项工作是Alazah等人[M. Alazah，S. N. Wilde和S. La Porte，Numerische Mathematik，128（4）：635-661，2014]。我们将菲涅耳积分的计算分为三种情况：截断泰勒级数、修正梯形法则和分别对小、中、大宗量的渐近展开。这些特殊函数可以精确有效地计算到任意精度。提供了误差估计，我们给出了一个系统的方法，在选择所需的精度的各种参数。我们说明了这种方法，并用双精度数值验证。
Performance Evaluation of Mixed-Precision Runge-Kutta Methods
在文献[8]中提出并分析了为在混合精度计算中保持高精度解而设计的附加龙格-库塔法。这些特殊设计的方法使用降低的精度或隐式计算和全精度的显式计算。在IBM POWER 9和Intel x86 64芯片上开发了一个用MP-ARK方法求解非线性系统常微分方程的FORTRAN程序。这些方法的收敛性，准确性，运行时，和能源消耗进行了探讨。我们表明，这些MP-ARK方法有效地产生准确的解决方案，显着减少运行时间（和扩展的能源消耗）。
Charles University（查理大学）；Sorbonne Université, CNRS（索邦大学，法国国家科学研究中心）；Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）
On the relation between reliable computation time, float-point precision and the Lyapunov exponent in chaotic systems
Center for Monsoon System Research, Institute of Atmospheric Physics, Chinese Academy of Sciences（中国科学院大气物理研究所季风系统研究中心）；State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences（中国科学院大气物理研究所大气数值模拟和地球流体力学国家重点实验室）；College of Global Change and Earth System Science, Beijing Normal University（北京师范大学全球变化与地球系统科学学院）
Mixed precision explicit numerical methods for ordinary differential equations
Univ. Rennes（雷恩大学）；Université Claude Bernard Lyon 1, CNRS, École Centrale de Lyon, INSA Lyon, Université Jean Monnet, ICJ UMR5208, Inria（里昂第一大学、法国国家科学研究中心、里昂中央理工学院、里昂国立应用科学学院、让·莫内大学）；Inria（法国国家信息与自动化研究所）；CITI, INSA Lyon, CNRS, Inria, LIRIS UMR5205, Université Claude Bernard Lyon 1, ECL, Université Lumière Lyon 2（里昂国立应用科学学院、法国国家科学研究中心、法国国家信息与自动化研究所、里昂第一大学、里昂第二大学）
Efficient Multi-Precision Computation of Bessel Functions for Real Orders and Complex Arguments with Fortran Implementation -- Part II: The Modified Bessel Function of the Second Kind, $K_ν(z)$
United Arab Emirates University（阿联酋大学）；Massachusetts Institute of Technology（麻省理工学院）
On the influence of stochastic roundoff errors and their bias on the convergence of the gradient descent method with low-precision floating-point computation
Eindhoven University of Technology（埃因霍温理工大学）；Università di Pisa（比萨大学）
Efficient Multi-Precision Computation of Bessel Functions for Real Orders and Complex Arguments with a Fortran Implementation -- Part III: Regular Bessel Functions of the First and Second Kinds $J_ν(z)$ and $Y_ν(z)$
United Arab Emirates University（阿联酋大学）；Massachusetts Institute of Technology（麻省理工学院）
