Keynote: Tricks and Tips in Numerical Computing | Nick Higham | JuliaCon 2018 скачать в хорошем качестве

Keynote: Tricks and Tips in Numerical Computing | Nick Higham | JuliaCon 2018

Nick Higham is Royal Society Research Professor and Richardson Professor of Applied Mathematics at the University of Manchester. His research is in numerical linear algebra and ranges from theory to the development of algorithms and software, with a focus on accuracy and stability. Contents 00:00 Welcome 00:33 Introducing the speaker 01:14 What are tricks and tips? 02:14 Differentiation with(out) a difference 03:20 V-shape curve is a result of floating-point evaluation (cancellation) errors dominating truncation errors 04:00 "Automatic differentiation " 04:15 Complex step method 06:27 Example: derivative of atan(x)/(1 + e^(-x^2)) at x = 2 07:19 Computing principal logarithm in a complex plane, a multi-valued function 08:03 Computing the principle logarithm in the 1960s 10:35 Logarithm of the product of numbers, complex case 12:32 Arcsin and Arccos in complex plane 13:06 Unwinding number 14:33 Round trip relations 16:19 Accurate difference 18:02 Low rank updated of n x n real matrix A 19:13 Why Sherman-Morrison formula holds? 20:45 World's Most Fundamental Matrix Equation 21:33 Computing a product 22:31 Matrix chain multiplication problem (MCMP) 23:10 Chain rule of differentiation and MCMP 24:46 Randomization 27:17 1985 IEEE Standard 754 and it 2008 Revision 28:10 Model for rounding errors analysis 28:48 This model is weaker than what IEEE Standard actually says 29:16 Model vs correctly rounded result 30:19 Prevision versus accuracy 31:24 Accuracy is not limited by the precision 32:08 Photocopying errors 32:53 Typing errors 33:25 Low precision arithmetic 35:32 Applications of half-precision (fp16, floating point 16 bits) 36:50 Error analysis in low precision arithmetic 37:46 What you can do to reduce error in fp16? 40:11 Can we obtain more information bounds? 41:24 Conclusions 42:48 Q&A: how to avoid the case when randomization makes the problem worse? 43:27 Q&A: how to choose between methods like contour integral and higher precision arithmetic? 45:03 Q&A: does half-precision allow a brute force analysis of the distribution of operations? 46:17 Q&A: can you comment on low precision and power consumption? S/O to https://github.com/KZiemian for the video timestamps! Want to help add timestamps to our YouTube videos to help with discoverability? Find out more here: https://github.com/JuliaCommunity/You... Interested in improving the auto generated captions? Get involved here: https://github.com/JuliaCommunity/You...