Course Catalog 2010-2011
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Basic Pori International Postgraduate Open University

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Course Catalog 2010-2011

MAT-31107 Numerical Analysis, 4 cr

Person responsible

Robert Piche

Lessons

Study type P1 P2 P3 P4 Summer Implementations Lecture times and places
Excercises

 

 
 2 h/week

 

 
MAT-31107 2010-01  

Requirements

Exam
Completion parts must belong to the same implementation

Learning outcomes

Numerical analysis is the design and study of algorithms (usually implemented as computer programs) to solve mathematical problems that involve real and complex numbers: solving equations, approximating functions, calculating integrals, solving differential equations, etc. The theory deals with questions such as the accuracy of approximation, the rate of convergence, and the computational complexity of algorithms. A solid understanding of this theory is essential for the effective use of existing computer codes and for the development of new ones. After studying this course, the student is able to describe the derivation of standard algorithms and can use them to solve simple mathematical problems with a non-programmable calculator. He/she can demonstrate his/her understanding of theory by estimating the error of a given approximation formula, deriving new variants of algorithms, explaining results that are produced by computer programs, determining the convergence or accuracy of a given algorithm, and comparing the strengths and weaknesses of different algorithms for a certain kind of problem.

Content

Content Core content Complementary knowledge Specialist knowledge
1. sources of error; descriptions of approximation: absolute error, relative error, bounding error, correct decimals, significant digits; error propagation in arithmetic, in univariate functions, and in multivariate functions; cancellation and its mitigation via algebraic reformulation or series approximation; floating point concepts: base, mantissa, exponent, IEEE single and double, spacing, machine epsilon, unit roundoff, arithmetic model; forward error analysis of simple expressions  round to nearest even; additional error caused by rounding; derivation of error propagation formulas; roots of quadratic polynomial; special floating point numbers (0, inf, NaN, subnormal); forward and backward error of summation; roundoff in Matlab  decoding IEEE numbers 
2. root-finding algorithms (bisection, Newton-Raphson, secant): derivation, advantages, disadvantages; convergence test for single-variable fixed-point iteration; stopping criteria; Horner's algorithm for p and p'; sensitivity of polynomial roots; Newton-Raphson method for multivariate problem  multiple roots; Brent's method; cobweb diagram; quadratic convergence of Newton-Raphson; computing square roots; attainable accuracy; root deflation; convergence test for multivariate fixed-point iteration; solving equations using Matlab  proof of convergence of fixed-point algorithm; convergence of Newton-Raphson to multiple root; attainable accuracy of multiple root; chaotic iterations; companion matrix; 3 iterations suffice to compute square root 
3. polynomial interpolation: definition, uniqueness, existence; Newton polynomial using divided difference tables; stability of linear interpolation; Runge's example; derivation of cubic Hermite interpolation  proof of uniqueness and existence; proof of error formula; Neville's method; interpolation with Matlab; truncation error and stability of cubic Hermite polynomial; interpolation of data using Matlab  uniform convergence of sine function interpolation; derivation of Newton polynomial; divided difference properties; derivation of Hermite error formula 
4. numerical integration: Newton-Cotes formula derivations using interpolation and using undetermined coefficients; truncation error of trapezoid rule and Simpson's rule; composite rules: trapezoid, Simpson's; Romberg formula; adaptive quadrature; dealing with singularities and infinite intervals: change of variables, series approximation  proof of accuracy of Simpson's rule; error estimate for "cutting off the tail"; recursive composite trapezoid rule; equivalence of first 3 columns of Romberg to Newton-Cotes; calculation of integrals with Matlab  Euler-Maclaurin summation formula 
5. least squares approximation theory: weighted euclidean norm and scalar product for vectors and continuous functions, orthogonality theorem, normal equations; orthogonal polynomial construction using undetermined coefficients, best-approximation, and 3-term recurrence  uniqueness of least squares approximation; repeated abscissas; least-squares approximation of data using Matlab  Clenshaw's algorithm 
6. ordinary differential equation initial value problems (IVP): conversion to standard form; Euler method derivation, use, local truncation error; Heun and Runge-Kutta methods formula and accuracy; adaptive solution; stability test; stiff problems; trapezoidal method  uniqueness of IVP solution; proof of local truncation error of Heun's method; solving IVP's with Matlab   

Evaluation criteria for the course

Exam. Bonus points (up to 20%) can be earned by active participation in PC labs and in weekly exercises.

Assessment scale:

Numerical evaluation scale (1-5) will be used on the course

Partial passing:

Completion parts must belong to the same implementation

Study material

Type Name Author ISBN URL Edition, availability, ... Examination material Language
Book   Introduction to Numerical Computation   Lars Eldén et al.            English  

Prerequisite relations (Requires logging in to POP)



Correspondence of content

Course Corresponds course  Description 
MAT-31107 Numerical Analysis, 4 cr MAT-31106 Numerical Analysis 1, 3 cr  

Additional information

Suitable for postgraduate studies

More precise information per implementation

Implementation Description Methods of instruction Implementation
MAT-31107 2010-01        

Last modified28.12.2010