Code: BIE-MA2.21 Mathematical Analysis 2
Lecturer: doc. Antonella Marchesiello Ph.D. Weekly load: 3P+2C Completion: A, EX
Department: 18105 Credits: 6 Semester: W
Description:
The course completes the theme of analysis of real functions of a real variable initiated in BIE-MA1 by introducing the Riemann integral. Students will learn how to integrate by parts and use the substitution method.The next part of the course is devoted to number series, and Taylor polynomials and series. We apply Taylor?s theorem to the computation of elementary functions with a prescribed accuracy. Then we study the linear recurrence equations with constant coefficients, the complexity of recursive algorithms, and its analysis using the Master theorem. Finally, we introduce the student to the theory of multivariate functions. After establishing basic concepts of partial derivative, gradient, and Hessian matrix, we study the analytical method of localization of local extrema of multivariate functions as well as the numerical descent method. We conclude the course with the integration of multivariate functions.
Contents:
1. Primitive function and indefinite integral.
2. Integration by parts and the substitution method for the indefinite integral.
3. Riemann?s definite integral, Newton-Leibniz theorem, and generalized Riemann?s integral.
4. Integration by parts and the substitution method for the definite integral.
5. Numerical computation of the definite integral.
6. Number series, criteria of their convergence, estimates of asymptotic behaviour of their partial sums.
7. Taylor?s polynomials and series.
8. Taylor?s theorem and its application to computation of elementary functions with prescribed precision.
9. Homogeneous linear recurrence equations with constant coefficients.
10. Non-homogeneous linear recurrence equations with constant coefficients.
11. The complexity of recurrence algorithms, the Master theorem.
12. Multivariate functions, partial derivative, gradient, and Hessian matrix.
13. Various types of definiteness of matrices and methods of its determination.
14. The analytical method for finding local extrema of multivariate functions.
15. Principle of numerical descent methods for localization of local extrema of multivariate functions.
16. Riemann?s integral of multivariate function, Fubini?s theorem.
17. Substitution in Riemann?s integral of multivariate function.
Seminar contents:
1. Indefinite integral, integration by parts and the substitution method.
2. Definite integral, Newton-Leibniz theorem, integration by parts and the substitution method.
3. Number series, criteria of their convergence
4. Estimates of asymptotic behaviour of partial sums of series.
5. Taylor?s polynomials and series.
6. Taylor?s theorem and its application.
7. Linear recurrence equations.
8. The Master theorem.
9. Multivariate functions, partial derivative, gradient, and Hessian matrix.
10. The analytical method for finding local extrema of multivariate functions.
11. Riemann?s integral of multivariate function, Fubini?s theorem.
12. Substitution in Riemann?s integral of multivariate function.
Recommended literature:
1. Oberguggenberger M., Ostermann A. : Analysis for Computer Scientists. Springer, 2018. ISBN 978-0-85729-445-6.
2. Nagle R. K., Saff E. B., Snider A. D. : Fundamentals of Differential Equations (9th Edition). Pearson, 2017. ISBN 978-0321977069.
3. Graham R. L., Knuth D. E., Patashnik O. : Concrete Mathematics: A Foundation for Computer Science (2nd Edition). Addison-Wesley Professional, 1994. ISBN 978-0201558029.
Keywords:
Integration, series, Taylor polynomials, linear recurrence, multivariate functions.

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