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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