COURSE CONTENT:
Errors in numerical computing, floating-point number system and arithmetics. Evaluation of polynomials and their derivatives, Horner algorithm. Three-term recurrence relations: properties and application to summation of series, Miller's algorithm. Some results from mathematical analysis. Polynomial interpolation: Lagrange and barycentric form. Newton's form of interpolation polynomial and the error estimate. Approximations of functions: L2 and max-norm, vector and unitary spaces, orthonormal basis. Minimal distance and abstract problem of best approximation. Applications: Chebishev and Fourier approximation, discrete approximation. Numerical differentiation, Richardson extrapolation. Numerical integration (basic and extended formulas). Basics of numerical linear algebra (systems of linear equations, matrix condition number).
LEARNING OUTCOMES:
Students will be able to:
explain floating-point number system, explain and compare rounding vs. cut-off error.
evaluate polynomials, calculate sum of series of functions and identify possible numerical problems.
approximately solve nonlinear equations in one unknown.
approximate functions by applying the least squares method.
numerically evaluate derivative and integral of function and estimate the associated error.
Conditions for Obtaining a Signature:
Regular class attendance and at least 50% completion of all three homework assignments.
Method of Knowledge Assessment and Examination:
Two midterms with a possible oral exam or a written and oral exam.
Each midterm consists of three tasks, with an equal emphasis on theory and practical problems, and lasts 75 minutes. Each midterm has a total of 30 points, and a minimum of 15 points is required to pass. A student who successfully passes both midterms may take the oral exam to potentially achieve a higher grade (the next one).
Students who do not pass the midterms must take both a written and an oral exam. The written exam consists of four tasks (two theoretical and two practical) and lasts 120 minutes. The total number of points is 40, and at least 20 points are needed to pass. The oral exam consists of two to three questions covering the entire syllabus.
Grading for Midterms and Written Exam is based on the percentage of points achieved:
50 - 62%: Sufficient (2)
63 - 75%: Good (3)
76 - 88%: Very Good (4)
89 - 100%: Excellent (5)
The final grade is based on the grades from the midterms or from both the written and oral parts of the exam.
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