*Conducted in terms:*2020L, 2021L

*Erasmus code:*11.1

*ISCED code:*0541

*ECTS credits:*7.5

*Language:*Polish

*Organized by:*Faculty of Mathematics, Informatics, and Mechanics

*Related to study programmes:*

# Computational Mathematics 1000-114bMOBa

1/ Elements of fl rounding error analysis

2/Interpolation

2.1 Polynomial interpolation

- specifying Lagrange interpolation problem

- existence and uniqueness of a solution

- finite differences algorithm

- approximation error estimates

2.2 Spline approximation

- definition of spline spaces

- linear splines

- cubic splines

3/Approximation

3.1 approximation in Hilbert spaces

- existence and uniqueness

- algorithms including the Gram-Schmidt process

- orthogonal polynomials - properties and application to the polynomial approximation problem

-Chebyshev polynomials and their properties

- Linear Least Square problem as a special case of an approximation problem

3.2 Uniform approximation - (optional if time permits)

4/Numerical integration

4.1 Interpolation quadratures

-Gauss quadratures

4.2 Quadrature rules: trapezoidal and Simpson rules

5/Numerical methods of solving system of algebraic equations

- LU decompositions with partial pivoting

- QR orthogonal decomposition: Housholder method

- condition of the matrices and their influence on rounding error analysis of LU decomposition

- an application of QR factorization of M x N matrix to Linear Least Square problems

6/ Roots of a nonlinear equation

- bisection method

- Newton method

- Secant method

- Banach iteration method

- order of convergence

7/Numerical Eigenproblem (optional if time permits)

- Power Method

- Inverse Power Method

## Type of course

## Course coordinators

## Bibliography

David Kincaid and Ward Cheney, Numerical analysis. Mathematics of scientific computing. 2nd ed., Brooks/Cole Publishing Co., Pacific Grove, CA, 1996.

## Additional information

Information on *level* of this course, *year of study* and semester when the course
unit is delivered, types and amount of *class hours* - can be found in course structure
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