Numerical Linear Algebra

Transcription

1 Schedule Prerequisite Preliminaries Errors and Algorithms Numerical Linear Algebra Kim, Hyun-Min Department of Mathematics, Pusan National University Phone: , 2596, Office: 612 Kim, Hyun-Min Numerical Linear Algebra

2 Schedule Prerequisite Preliminaries Errors and Algorithms Schedule for Mid-term Exam Introduction to Numerical Linear Algebra - What is the numerical linear algebra? - Why? - How? Linear Systems - What are linear systems? - Existence and uniqueness of solution - Why? - How? Kim, Hyun-Min Numerical Linear Algebra

3 Schedule Prerequisite Preliminaries Errors and Algorithms Direct methods - Gaussian elimination - LU, QR Factorizations - Special matrices Iterative Methods - Norms of vectors and matrices - Iterative methods - Iterative methods for special matrices Decompositions - Diagonalization - Jordan canonical form - Schur decompositions - Singular decomposition Mid-term Exam Kim, Hyun-Min Numerical Linear Algebra

4 Schedule Prerequisite Preliminaries Errors and Algorithms Linear Algebra Math. Linear Algebra by S. Friedberg, A. Insel and L. Spence Chapters 1, 2, 3, 4, 5, 6, 7 Kim, Hyun-Min Numerical Linear Algebra

5 Schedule Prerequisite Preliminaries Errors and Algorithms Linear Algebra Math. Linear Algebra by S. Friedberg, A. Insel and L. Spence Chapters 1, 2, 3, 4, 5, 6, 7 Eng. Elementary Linear Algebra by H. Anton and C. Rorres Chapters 1, 2, 3, 4, 5, 6, 7, 8 Kim, Hyun-Min Numerical Linear Algebra

6 Schedule Prerequisite Preliminaries Errors and Algorithms Mathematical Programming Math. Mathematica, Maple, MATLAB Kim, Hyun-Min Numerical Linear Algebra

7 Schedule Prerequisite Preliminaries Errors and Algorithms Mathematical Programming Math. Mathematica, Maple, MATLAB Eng. Fortran, C, MATLAB Kim, Hyun-Min Numerical Linear Algebra

8 Schedule Prerequisite Preliminaries Errors and Algorithms Numerical Analysis What is the mathematics? What is the numerical analysis? Kim, Hyun-Min Numerical Linear Algebra

9 Schedule Prerequisite Preliminaries Errors and Algorithms Definition The study of quantitative approximations to the solutions of mathematical problems including consideration of the errors and bounds to the errors involved. - Webster s New Collegiate Dictionary (1973) The study of methods of approximation and their accuracy, etc. - Chambers 20th Century Dictionary (1983) The branch of mathematics concerned with the development and use of numerical methods for solving problems - Concise Oxford Dictionary 10th Edition (1999) Kim, Hyun-Min Numerical Linear Algebra

10 Schedule Prerequisite Preliminaries Errors and Algorithms New Definition Numerical analysis is the study of rounding errors. bad one Kim, Hyun-Min Numerical Linear Algebra

11 Schedule Prerequisite Preliminaries Errors and Algorithms New Definition Numerical analysis is the study of rounding errors. bad one Numerical analysis is the study of algorithms for the problems of continuous mathematics. good definition - Lloyd N. Trefethen (1993) Kim, Hyun-Min Numerical Linear Algebra

12 Schedule Prerequisite Preliminaries Errors and Algorithms Errors Definition x: the true value x : an approximation to x x x : Absolute Error x x x (x 0): Relative Error Kim, Hyun-Min Numerical Linear Algebra

13 Schedule Prerequisite Preliminaries Errors and Algorithms Errors Definition x: the true value x : an approximation to x x x : Absolute Error x x x (x 0): Relative Error Which one is better? Kim, Hyun-Min Numerical Linear Algebra

14 Schedule Prerequisite Preliminaries Errors and Algorithms Errors Definition x: the true value x : an approximation to x x x : Absolute Error x x x (x 0): Relative Error Which one is better? Why? Kim, Hyun-Min Numerical Linear Algebra

15 Schedule Prerequisite Preliminaries Errors and Algorithms Sources of Errors Errors in mathematical modelling: Simplifying and Assumptions Kim, Hyun-Min Numerical Linear Algebra

16 Schedule Prerequisite Preliminaries Errors and Algorithms Sources of Errors Errors in mathematical modelling: Simplifying and Assumptions Blunders Kim, Hyun-Min Numerical Linear Algebra

17 Schedule Prerequisite Preliminaries Errors and Algorithms Sources of Errors Errors in mathematical modelling: Simplifying and Assumptions Blunders (Prgramming Errors): Large programmes, Subprogrammes Kim, Hyun-Min Numerical Linear Algebra

18 Schedule Prerequisite Preliminaries Errors and Algorithms Sources of Errors Errors in mathematical modelling: Simplifying and Assumptions Blunders (Prgramming Errors): Large programmes, Subprogrammes Errors in input: Errors in data transfer, uncertainties associated with measurements Kim, Hyun-Min Numerical Linear Algebra

19 Schedule Prerequisite Preliminaries Errors and Algorithms Sources of Errors Errors in mathematical modelling: Simplifying and Assumptions Blunders (Prgramming Errors): Large programmes, Subprogrammes Errors in input: Errors in data transfer, uncertainties associated with measurements Machine errors by computer (Floating point arithmetic): Rounding and Chopping, Underflow and Overflow Kim, Hyun-Min Numerical Linear Algebra

20 Schedule Prerequisite Preliminaries Errors and Algorithms Arithmetic In 1985 IEEE(Institute for Electrical and Electronic Engineers) report: Binary Floating Point Arithmetic Standard 754. Kim, Hyun-Min Numerical Linear Algebra

21 Schedule Prerequisite Preliminaries Errors and Algorithms Arithmetic In 1985 IEEE(Institute for Electrical and Electronic Engineers) report: Binary Floating Point Arithmetic Standard 754. Single, Double and Extended Precisions Kim, Hyun-Min Numerical Linear Algebra

22 Schedule Prerequisite Preliminaries Errors and Algorithms Algorithms Examining approximation procedures involving finite sequence of calculations. Stable: Small changes in the initial data Kim, Hyun-Min Numerical Linear Algebra

23 Schedule Prerequisite Preliminaries Errors and Algorithms Algorithms Examining approximation procedures involving finite sequence of calculations. Stable: Small changes in the initial data Unstable: Otherwise Kim, Hyun-Min Numerical Linear Algebra

24 Schedule Prerequisite Preliminaries Errors and Algorithms Algorithms Examining approximation procedures involving finite sequence of calculations. Stable: Small changes in the initial data Unstable: Otherwise Conditionally Stable: Stable only for certain of initial data Kim, Hyun-Min Numerical Linear Algebra

25 Schedule Prerequisite Preliminaries Errors and Algorithms Convergence {α n } n=1, {β n} n=1 : sequences lim β n = 0, n lim α n = α. n If K > 0 s. t. α n α β n for large n then {α n } n=1 converges to α with the rate of convergence O(β n ). α n = α + O(β n ) Kim, Hyun-Min Numerical Linear Algebra

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