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?
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