Google Colab w/ Python

vector addition

vector subtraction

scaling / scale-vector multipliction

Vector "dot product"

** will be asked about

turns vectors into a scalar

well defined operation

vectors are the same size

max(x)

value of biggest element in a vector, matrix, or tensor

argmax(x)

which element is the biggest? element 1? element 2? in a vector, matrix, or tensor

Mathematical Universe Hypothesis

The physical universe is not merely described by mathematics, but is mathematics

→ coding allows us to model the universe

Python

• Founded in 1991 by Guido van Rossum at Centrum Wiskunde & Informatica, Netherlands

• Python 2.0 09/16/00, Python 3.0 12/03/08

Used a lot bc:

• FREE!!!

• Versatile

• Portable

• High-level

• Easy-to-use

• Interpreted → makes Python kinda slow… BUT libraries available to translate Python code

• Dynamically typed

• Object-oriented and procedural

• Extensive libraries

• Rapid development cycle Interfaces well with other languages

• One of Google's "official languages"

• Extremely widely used in scientific computing

why python > others?

Python is becoming a de facto standard for scientific and numerical computing in academia and industry MATLAB is a powerful tool that is a good alternative to Python but is slow and less widely used in the community Java is a good language for building apps, but is not well suited to numerical or scientific computing C/C++ is the best choice for high-performance scientific software but are more involved and prototyping is slow

Python Libraries

NumPy

SciPy

matplotlib

plotting

SymPy

calculus

pandas

used for data analysis of csv or excel file

scikit-learn

PyTorch, Keras, TensorFlow

neural networks

Django

share code + get other ppl to edit

Cython

compile python code so it can run as fast as if written in C or C++

Numba

translates Python to machine code using industry standard LLVM library

numerical accuracy

• Numbers on computers cannot be infinitely large or small or specified to infinite accuracy

• IEEE 754 standard upon which Python is built stipulates:

  • largest float = 21024 = 1.79769×10308

  • smallest float = 2-1022 = 2.22507×10-308

  • exceeding these limits results in overflow / underflow

• Integers have arbitrarily high precision (up to memory limit)

Numerical speed

• modern computers are not infinitely fast

• Important to have estimates of computational cost and complexity (i.e., scaling with problem size) of an algorithm → tells us if we need new algorithm or if problem is too big

• For hard problems we typically must balance trade-off between speed vs. accuracy