uni_notes/DAS/Arithmetic.md

Asymptotic Equivalence Classes (Big-O)

The equivalence relation definition given in the task is asking you to find functions that grow at the exact same rate (also known as Big-Theta \Theta):

f asymp g <==> f in O(g) and g in O(f)

Notation of f in O(g) means "function f doesn't grow faster than $g$"

The "Dominant Term" Rule

To find which class a function belongs to, simplify it to its core growth rate:

1. Drop all lower-order terms: In n^3 + n^2, drop the n^2.
2. Drop all constant multipliers: 5n^2 and 1000n^2 both become just n^2.
3. Identify the highest rank: Factorials (n!) > Exponentials (2^n, e^n) > Polynomials (n^3, n^2) > Linear (n) > Logarithmic (log n) > Constant (1).

Euclidian Algorithm

Purpose is to find the GCD (Greatest Common Divisor).

Core Rule

Fill out this formula:

"Dividend" = ("Quotient" * "Divisor") + "Remainder"

1. Divide the bigger number by the smaller one
2. How many times does it fit -> "Quotient"
3. Find out whats leftover -> "Remainder"
4. Shift to left and repeat ("Old Divisor" -> "Dividend", "Remainder" -> "Divisor")

Result is the last "Remainder" that is not 0.

Bezout Coefficients

Goal: find a x and y so that "Divident" * x + "Divisor" * y = gcd("Dividend", "Divisor")

1. Rewrite Euclid (above) equations to solve for remainder ("Remainder" = "Old Remainder" - "Dividend" * "Divisor")
2. Substitute remainders
3. Simplify (Final Form: "GCD" = x * "Dividend" + "y * Divisor")

Extended Euclidian Algorithm

1. Find GCD using Euclidian Algorithm