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Module description - Advanced Calculus
(Vertiefung Analysis)

Number
vana
ECTS 3.0
Level intermediate
Overview In this module you will deepen some of the aspects and techniques you have learned in the introduction into calculus (eana), generalize them and learn some related numerical procedures relevant in practice.

    Non-linear Equations
  • Some results about continuous functions such as the intermediate value theorem.
  • Numerical solutions of non-linear equations such as Regula Falsi or Newton's method.
  • Numerical determination of the roots of polynomials.

    Multi-Variable Calculus
  • Partial derivatives, gradient.
  • Optimization algorithms, gradient descent and some of its variants.
  • Applications in data science, particularly in machine learning.

    Approximation and Interpolation
  • Mean value theorem.
  • Approximation with estimate of the remainder e.g. Taylor-approximation, regression.
  • Interpolation (e.g. splines) and their applications e.g. in image processing.

    Selected Topics in Numerics and Applications in Computer Science
  • Numerical calculation of integrals.
  • Integral transforms such as the Fourier transform and their application in image and signal processing.
  • Differential equations and their numerical solutions.
Learning objectives Non-linear Equations in a Single Variable
Students understand the notion of continuous functions, the contents of the intermediate value theorem and understand how it can guarantee the existence of solutions of non-linear equations. Students know how to apply selected numerical procedures for solving non-linear equations and know the prerequisites for their applicability.

Multi-Variable Calculus
Students know how to calculate partial derivatives and gradients for func-tions in many variables. They understand the notion of critical or extremal points and their significance for multi-dimensional optimization problems. Students can apply selected numerical procedures for solving non-linear equations in many variables or for determining extremal points in multi-dimensional optimization problems.

Approximation and Interpolation
Students know selected approximation and interpolations procedures and can apply them in example problems.

Selected Topics in Numerics and Applications in Computer Science
Students will be introduced to another application in numerics. Examples are numerical integration, integral transforms (Fourier) and their applications to image and signal processing.
Previous knowledge eana, lag
Exam format Continuous assessment grade, weighting 100 %
Additional information Also suited as preparation for MSE study in data science or computer science.

Studium
Module
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