In the field of computer science, a graph is an abstract data type that is widely used to represent connections and relationships. This post gives an overview about a selection of definitions, terms, and algorithms, which are related to graphs. The content was put together during preparation for a theoretical computer science test at Cooperative State University Baden-Württemberg and is mostly taken from either Wikipedia or lecture notes. Continue reading Graph Theory Overview
Tag: Math
Least Squares Derivation
The least squares optimization problem searches for a vector, that minimizes the euclidean norm in the following statement as much as possible: $$x_\text{opt}=\arg\min_x\frac{1}{2}\left\lVert Ax-y\right\rVert^2_2\,.$$This article explains how $x_\text{opt}=(A^{\text{T}}A)^{-1}A^{\text{T}}y$, the solution to the problem, can be derived and how it can be used for regression problems. Continue reading Least Squares Derivation
Cubic Spline Interpolation
Cubic spline interpolation is a mathematical method commonly used to construct new points within the boundaries of a set of known points. These new points are function values of an interpolation function (referred to as spline), which itself consists of multiple cubic piecewise polynomials. This article explains how the computation works mathematically.
After an introduction, it defines the properties of a cubic spline, then it lists different boundary conditions (including visualizations), and provides a sample calculation. Furthermore, it acts as a reference for the mathematical background of the cubic spline interpolation tool on tools.timodenk.com which is introduced at the end of the article. Continue reading Cubic Spline Interpolation
Guess Solutions of Polynomials
For a given polynomial of $n$th degree
$$P_n(x)=\sum_{i=0}^n a_ix^i = a_nx^n+a_{n-1}x^{n-1}+…+a_1x+a_0$$
you can guess rational solutions $x$ for the corresponding problem $P_n(x)=0$ by applying the following two rules:
- $$x=\frac{p}{q}\text{, with } p \in \mathbb{Z} \land q \in \mathbb{N}\land p\mid a_0 \land q\mid a_n$$
- $$\lvert x\rvert\le2\cdot \max\left\lbrace \sqrt[k]{\frac{\lvert a_{n-k}\rvert}{\lvert a_n\rvert}}, k=1, …, n\right\rbrace$$
Trigonometric Functions Formulary
This formulary has been created during the online onboarding process at Baden-Wuerttemberg Cooperative State University (DHBW). It is suitable for the related online tests and might be helpful for other people, seeking for formulas in this field of mathematics.
Basics
$$\begin{array}{l} \tan x = \frac{{\sin x}}{{\cos x}}\\ \cot x = {\tan ^{ – 1}}x = \frac{{\cos x}}{{\sin x}} \end{array}$$ Continue reading Trigonometric Functions Formulary