Week 1

VL 1 - 14/10/25

• Orga

Intro

• computation of
  • mathematical expressions, e.g. \(\sqrt{2}\)
  • integrals:
  • equations
  • differential equations
• Computation Mathematical Expressions:

The Babylonian (Heron’s) method for \(\sqrt{2}\) uses the recurrence

\[ x_{n+1} = \frac{1}{2}\left(x_n + \frac{2}{x_n}\right), \qquad x_0>0. \]

A common choice is \(x_0=1\), giving the sequence \[ 1,; 1.5,; 1.416666\ldots,; 1.414215686\ldots,; 1.41421356237\ldots ;\to; \sqrt{2}. \]

• Scientific Computing:
• Interpolation:
  • corona-virus example: linear approaximatin of parameters
  • non-linear approximation

VL 1 - 14/10/25

• Floating point number representation
• Numerical data types in c++
• motivating example: computing \(e^x\)
• newer forms used in ML with less precision:
  • bfloat16
  • float16
  • fg8