Tricks and Tips in Numerical Computing - Keynote

Tricks and Tips in Numerical Computing - Keynote

The Julia Programming Language via YouTube Direct link

Roundtrip relations

14 of 40

14 of 40

Roundtrip relations

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Tricks and Tips in Numerical Computing - Keynote

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  1. 1 Welcome
  2. 2 Introducing the speaker
  3. 3 What are tricks and tips?
  4. 4 Differentiation with(out) a difference
  5. 5 V-shape curve is a result of floating-point evaluation (cancelation) errors dominating truncation errors
  6. 6 "Automatic differentiation "
  7. 7 Complex step method
  8. 8 Example: derivative of atan(x)/(1 + e^(-x^2)) at x = 2
  9. 9 Computing principal logarithm in a complex plane, a multi-valued function
  10. 10 Computing the principle logarithm in the 1960s
  11. 11 Logarithm of the product of numbers, complex case
  12. 12 Arcsin and Arccos in complex plane
  13. 13 Unwinding number
  14. 14 Roundtrip relations
  15. 15 Accurate difference
  16. 16 Low rank updated of n x n real matrix A
  17. 17 Why Sherman-Morrison formula holds?
  18. 18 World's Most Fundamental Matrix Equation
  19. 19 Computing a product
  20. 20 Matrix chain multiplication problem (MCMP)
  21. 21 Chain rule of differentiation and MCMP
  22. 22 Randomization
  23. 23 1985 IEEE Standard 754 and it 2008 Revision
  24. 24 Model for rounding errors analysis
  25. 25 This model is weaker than what IEEE Standard actually says
  26. 26 Model vs correctly rounded result
  27. 27 Prevision versus accuracy
  28. 28 Accuracy is not limited by the precision
  29. 29 Photocopying errors
  30. 30 Typing errors
  31. 31 Low precision arithmetic
  32. 32 Applications of half-precision (fp16, floating point 16 bits)
  33. 33 Error analysis in low precision arithmetic
  34. 34 What you can do to reduce error in fp16?
  35. 35 Can we obtain more information bounds?
  36. 36 Conclusions
  37. 37 Q&A: how to avoid the case when randomization makes the problem worse?
  38. 38 Q&A: how to choose between methods like contour integral and higher precision arithmetic?
  39. 39 Q&A: does half-precision allow a brute force analysis of the distribution of operations?
  40. 40 Q&A: can you comment on low precision and power consumption?

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