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Numerical Analysis

last update: 2021-10-03

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(CourseId 327.010, 2 hours per week, Semester 3)

Lecturer: O.Univ.-Prof. Dr. Ulrich Langer

Appointments:

Link appointments

Results of the exercises:

Link results

Time and room:

Tue, 2006-10-03	11:15 - 12:45 Room: T 041	Lecture 1
Tue, 2006-10-10	11:15 - 12:45 Room: T 041	Lecture 2
Tue, 2006-10-17	11:15 - 12:45 Room: T 041	Lecture 3
Tue, 2006-10-24	11:15 - 12:45 Room: T 041	Lecture 4
Tue, 2006-10-31	11:15 - 12:45 Room: T 041	Lecture 5
Tue, 2006-11-07	11:15 - 12:45 Room: T 041	Lecture 6
Tue, 2006-11-14	11:15 - 12:45 Room: T 041	Lecture 7
Tue, 2006-11-21	11:15 - 12:45 Room: T 041	Lecture 8
Tue, 2006-11-28	11:15 - 12:45 Room: T 041	Lecture 9
Tue, 2006-12-05	11:15 - 12:45 Room: T 041	Lecture 10
Tue, 2006-12-12	11:15 - 12:45 Room: T 041	Lecture 11
Tue, 2007-01-09	11:15 - 12:45 Room: T 041	Lecture 12
Tue, 2007-01-16	11:15 - 12:45 Room: T 041	Lecture 13
Tue, 2007-01-23	11:15 - 12:45 Room: T 041	Lecture 14
Tue, 2007-01-30	11:15 - 12:45 Room: T 041	Lecture 15

Lecturer: O.Univ.-Prof. Dr. Ulrich Langer

EXERCISES	HANDOVER DATE	FILE FORMAT
Exercise 1	2007-01-15	pdf
Exercise 2	2007-02-13	pdf
Exercise 3	2007-03-02	pdf
MATLAB	Introduction	pdf

Note: Rating of the exercises is only announced as of March 5!

Transparency 01: b/w	1.1. Process to solve problems
Transparency 02: colour	1.2. Example I
Transparency 03: colour	1.2. Example II
Transparency 04: colour	1.2. Example III
Transparency 05: colour	1.2. Example IV
Transparency 06: b/w	1.2. Example V
Transparency 07: b/w	1.3. Problems
Transparency 08: colour	2.1. Number representation I
Transparency 09: b/w	2.1. Number representation II
Transparency 10: colour	2.1. Number representation III
Transparency 11: b/w	2.2. Floating point mathematics I
Transparency 12: b/w	2.2. Floating point mathematics II
Transparency 13: b/w	2.3. Computation rate
Transparency 14: b/w	2.4. Error Analysis
Transparency 15: b/w	2.4.1. Data Error Analysis I
Transparency 16: b/w	2.4.1. Data Error Analysis II
Transparency 17: b/w	2.4.2. Roundoff Error Analysis I
Transparency 18: colour	2.4.2. Roundoff Error Analysis II
Transparency 19: colour	2.4.2. Roundoff Error Analysis III
Transparency 20: colour	2.4.2. Roundoff Error Analysis IV
Transparency 21: colour	Example 3.8.
Transparency 22: colour	Gauss-Algorithm
Transparency 23: b/w	3.5. Backward Error Analysis I
Transparency 24: colour	3.5. Backward Error Analysis II
Transparency 25: colour	3.5. Backward Error Analysis III
Transparency 26: b/w	3.5. Backward Error Analysis IV
Transparency 27: colour	3.5. Backward Error Analysis V
Transparency 28: b/w	3.5. Backward Error Analysis V
Transparency 29: b/w	3.7. Special systems of equations
Transparency 30: colour	3.7.1. Band matrices
Transparency 31: colour	3.7.2. Cholesky
Transparency 32: colour	3.8. Additional information
Transparency 33: b/w	4. Iterative Methods
Transparency 34: b/w	4.1.1. Poisson Equation
Transparency 35: b/w	4.1.1. FD-discretization I
Transparency 36: b/w	4.1.1. FD-discretization II
Transparency 37: b/w	4.1.2. Properties
Transparency 38: b/w	Theorem 4.3.
Transparency 39: b/w	Proof Theorem 4.3.
Transparency 40: colour	Choice of Preconditioners
Transparency 41: colour	Improvements
Transparency 42: colour	PCG
Transparency 43: colour	Theorem 4.6.
Transparency 44: b/w	5.1: FPI
Transparency 45: b/w	Theorem 5.2: Banach
Transparency 46: colour	5.2: Newton
Transparency 47: b/w	Theorem 5.5
Transparency 48: b/w	Quadratic Convergence
Transparency 49: colour	5.3. Options
Transparency 50: b/w	5.3.1. LSV
Transparency 51: b/w	5.3.1: Theorem 5.8
Transparency 52: colour	Homotopy
Transparency 53: colour	5.3.2: Outer-Inner
Transparency 54: colour	Sekants Method
Transparency 55: b/w	Broyden
Transparency 56: b/w	6.1: Basics I
Transparency 56a: color	Pkt. 6.1: Basics I (proof)
Transparency 57: b/w	6.1: Basics II
Transparency 58: colour	6.1: Basics III
Transparency 59: b/w	6.1: Basics IV
Transparency 60: b/w	6.1: Basics V
Transparency 61: b/w	6.2.1: QR-Alg.
Transparency 62: b/w	6.2.1: Theorem 6.13
Transparency 63: b/w	6.2.1: Remark I
Transparency 64: colour	6.2.1: Remark II
Transparency 65: colour	6.2.2: Definition 6.15
Transparency 66: b/w	6.2.2: Theorem 6.16
Transparency 67: b/w	6.2.2: F. 6.17, Remark 6.18
Transparency 68: b/w	6.3.1: Newton
Transparency 69: b/w	6.3.2: Direct VI
Transparency 70: b/w	6.3.2: Inverse VI (1)
Transparency 71: b/w	6.3.2: Inverse VI (2)
Transparency 72: b/w	7.1: Interpolation
Transparency 73: colour	7.1.1: Lagrange-Interpolation
Transparency 74: colour	7.1.1: Neville-Algorithm
Transparency 75: colour	7.1.1: Div. differences
Transparency 76: b/w	7.1.1: Hoerner scheme
Transparency 77: b/w	7.1.1: Interpolation error I
Transparency 78: colour	7.1.1: Interpolation error II
Transparency 79: colour	7.1.2: Splines
Transparency 80: b/w	7.1.2: Spline-Interpolation
Transparency 81: b/w	7.1.2: Konstruction
Transparency 82: b/w	7.1.2: Theorems 7.7 and 7.8
Transparency 83: b/w	7.2: Numer. Differentation I
Transparency 84: b/w	7.2: Numer. Differentation II
Transparency 85: colour	7.3: Numer. Integration
Transparency 86: colour	7.3.1: T. 7.11, Scale 7.12
Transparency 87: colour	7.3.1: Convergence of QF
Transparency 88: colour	7.3.2: Example 7.15, Error
Transparency 89: colour	7.3.2: Generalisation

1. Lindner E., Zulehner W.: Skriptum zur Vorlesung Numerische Analysis. Institut für Numerische Mathematik, Johannes Kepler Universität Linz, Wintersemester 2005/06. pdf-File

Classical Lecture Notes on Numerical Analysis:

1. Golub G.H., Van Loan C.F.: Matrix Computation, 2nd ed., John Hopkins University Press, Baltimore 1989.
2. Isaacson E., Keller H.B.: Analyse numerischer Verfahren. Edition Leipzig 1972.
3. Stoer J.: Einfuehrung in die Numerische Mathematik I, 6. Aufl., Springer-Verlag, Berlin - Heidelberg - New York - Tokyo 1993.
4. Stoer J., Bulirsch: Einfuehrung in die Numerische Mathematik II, 3. Aufl., Springer-Verlag, Berlin - Heidelberg - New York - Tokyo 1990.
5. Stoer J., Bulirsch: Introduction to Numerical Analysis, Springer-Verlag, New York - Berlin - Heidelberg 1980.

More recently published Lecture Notes on Numerical Analysis:

1. Deuflhard P., Hohmann A.: Numerische Mathematik: Eine algorithmisch orientierte Einfuehrung, Walter de Gruyter, Berlin - New York 1991.
2. Haemmerlin G., Hoffmann K.-H.: Numerische Mathematik, 2. Aufl., Springer-Verlag, Berlin - Heidelberg - New York 1991.
3. Herrmann M.: Numerische Mathematik, Oldenbourg Verlag, Muenchen - Wien 2001.
4. Schwarz H.R.: Numerische Mathematik, Teubner Verlag, Stuttgart 1988.

Journal Publications:

1. Gould N.: SIAM J. Matrix Anal. Appl., 12 (1991), 354-361
2. Edelman A.: Note to the Editor, SIAM J. Matrix Anal. Appl., 12 (1991).
3. Sautter W.: Fehlerfortpflanzung und Rundungsfehler bei der verallgemeinerten Inversion von Matrizen. Dissertation, TU Muenchen, Fakultaet fuer Allgemeine Wissenschaften, 1971.

Basic Knowledge in Linear Algebra:

1. Strang G.: Linear Algebra, Springer-Verlag, Berlin - Heidelberg - New York 2003.

Solution of Linear System of Algebraic Equations:

1. Steinbach O.: Loesungsverfahren fuer lineare Gleichungssysteme: Algorithmen und Anwendungen, Teubner, Wiesbaden 2005.
2. Meurant G.: Computer Solution of Large Linear Systems. Studies in Mathematics and its Applications, 28, North-Holland, Amsterdam 1999.
3. Saad Y.: Iterative Methods for Sparse Linear Systems, PWS Publishing, 1995.
4. Barrett R., Berry M., Chan T.F., Demmel J., Donato J, Dongarra J., Eijkhout V., Pozo R., Romine C., Van der Vorst H. (eds): Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition, SIAM, Philadelphia, PA, 1994. http://netlib2.cs.utk.edu/linalg/html_templates/Templates.html

Solution of Nonlinear System of Equations:

1. Deuflhard P.: Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer-Verlag, Berlin - Heidelberg - New York 2004.

Eigenvalue Problems:

1. Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, and Henk van der Vorst (eds): Templates for the Solution of Algebraic Eigenvalue Problems: a Practical Guide.

Required Previous Knowledge:

• Linear Algebra and Analytic Geometry 1 and 2
• Analysis 1 - 2
• Computer Science and Programming

Is required for:

• Numerical Methods for Partial Differential Equations
• Numerical Methods for Elliptic Partial Differential Equations
• Numerical Methods for Non-stationary Problems
• Special Topics in Computational Mathematics
• Special Seminars in Computational Mathematics

Objective:

Get knowledge in the analysis of basic numerical methods

Contents:

1. Introduction
2. Special Features of Numerical Computations
3. Direct Solvers for Linear Systems
4. Iterative Solvers for Linear Systems
5. Iterative Solvers for Non-linear Systems
6. Eigenvalue Problems
7. Interpolation, Numerical Differentiation and Integration

Additional Information:

• The lectures are accompanied by practical exercises.

Examination: Oral (mark = 1/3 exercises + 2/3 oral examination)

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