Lecture 17 - Intermediate Lecture

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Topics: Recalling important subjects of the course till lecture 17; questions

Check list lecture 01 - 17:

 - Error Analysis
     - Calculating into and with IEEE double precision floating point numbers
     - Calculating the absolute and relative error
     - Being aware of loss of significancy in calculations
     - Horner's method
 
 - Root finding problem:
      - What is it?
      - How to guarantee a solution?
      - Bisection method (scheme, convergence behavior, ... )
      - What is the error if a solution is correct within p decimal places?
      - Convergence and convergence order
      - Backward and forward error
      - Multiple roots
      
  - Fixed point
      - What is a fixed point?
      - Existence and uniqueness of a fixed point.
      - What is a fixed-point iteration?
      - Convergence of fixed-point iterations
      - Contraction
      - Banach's fixed-point theorem 
      - Stopping criteria
      
  - Newton's method
      - Scheme
      - Convergence of Newton's method
      
  - Linear Systems
      - Existence and uniqueness of a solution
      - Back and forward substitution
      - Counting number of operations
      - Gaussian elimination
      - LU factorization
      - What is a norm?
      - Vector and matrix norms
      - Equivalence of norms
      - Calculating with norms
      - (Relative) forward and backward error
      - Perturbation analysis
      - Condition number
      - Computing the inverse of a 2x2 matrix
      - Pivoting (partial and scaled partial)
      - LU decomposition with pivoting
      - What is a permutation matrix?
      - Solving linear systems with Gaussian elimination and LU decomposition (with and without pivoting).
      - Jacobi method (scheme, convergence, ... )
      - Gauss-Seidel method (scheme, convergence, ... )
      
  - Interpolation
      - Interpolation problem 
      - Polynomial interpolation
      - Existence and uniqueness
      - Monomial basis
      - Lagrange basis
      - Newton basis
      - Newton's divided differences
      - Interpolation error
      - Chebyshev nodes
      - Piecewise interpolation
      - Splines
      - Types of cubic splines
      - Bernstein Polynomials
      - Bezier curve
      
  - Least squares problem
      - Normal equation
      - Existence and uniqueness of a solution
      - Fitting data with least squares
      - Fitting periodic data with least squares
      - Data linearization
      - Solving least squares via QR decomposition
      
  - QR decomposition
      - Gram-Schmidt orthogonalization
      - Orthogonal matrix
      - Calculating with projection matrices