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Schedule Prerequisite Preliminaries Errors and Algorithms
Numerical Linear Algebra
Kim, Hyun-Min
Department of Mathematics, Pusan National UniversityE-mail:[email protected]
Phone: 510-1060, 2596, 010-3833-8200Office: 612
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Schedule for Mid-term Exam
I Introduction to Numerical Linear Algebra
- What is the numerical linear algebra?- Why?- How?
I Linear Systems
- What are linear systems?- Existence and uniqueness of solution- Why?- How?
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
I Direct methods
- Gaussian elimination- LU, QR Factorizations- Special matrices
I Iterative Methods
- Norms of vectors and matrices- Iterative methods- Iterative methods for special matrices
I Decompositions
- Diagonalization- Jordan canonical form- Schur decompositions- Singular decomposition
I Mid-term Exam
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Linear Algebra
Math.Linear Algebra by S. Friedberg, A. Insel and L. SpenceChapters 1, 2, 3, 4, 5, 6, 7
Eng.Elementary Linear Algebra by H. Anton and C. RorresChapters 1, 2, 3, 4, 5, 6, 7, 8
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Linear Algebra
Math.Linear Algebra by S. Friedberg, A. Insel and L. SpenceChapters 1, 2, 3, 4, 5, 6, 7
Eng.Elementary Linear Algebra by H. Anton and C. RorresChapters 1, 2, 3, 4, 5, 6, 7, 8
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Mathematical Programming
Math.Mathematica, Maple, MATLAB
Eng.Fortran, C, MATLAB
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Mathematical Programming
Math.Mathematica, Maple, MATLAB
Eng.Fortran, C, MATLAB
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Numerical Analysis
I What is the mathematics?
I What is the numerical analysis?
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Definition
I The study of quantitative approximations to the solutions ofmathematical problems including consideration of the errorsand bounds to the errors involved.- Webster’s New Collegiate Dictionary (1973)
I The study of methods of approximation and their accuracy,etc.- Chambers 20th Century Dictionary (1983)
I The branch of mathematics concerned with the developmentand use of numerical methods for solving problems- Concise Oxford Dictionary 10th Edition (1999)
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
New Definition
I Numerical analysis is the study of rounding errors. bad one
I Numerical analysis is the study of algorithms for the problemsof continuous mathematics. good definition- Lloyd N. Trefethen (1993)
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
New Definition
I Numerical analysis is the study of rounding errors. bad one
I Numerical analysis is the study of algorithms for the problemsof continuous mathematics. good definition- Lloyd N. Trefethen (1993)
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Errors
Definitionx: the true valuex∗: an approximation to x
I |x− x∗|: Absolute Error
I|x− x∗||x|
(x 6= 0): Relative Error
Which one is better?
Why?
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Errors
Definitionx: the true valuex∗: an approximation to x
I |x− x∗|: Absolute Error
I|x− x∗||x|
(x 6= 0): Relative Error
Which one is better?
Why?
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Errors
Definitionx: the true valuex∗: an approximation to x
I |x− x∗|: Absolute Error
I|x− x∗||x|
(x 6= 0): Relative Error
Which one is better?
Why?
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Sources of Errors
I Errors in mathematical modelling: Simplifying andAssumptions
I Blunders (Prgramming Errors): Large programmes,Subprogrammes
I Errors in input: Errors in data transfer, uncertaintiesassociated with measurements
I Machine errors by computer (Floating point arithmetic):Rounding and Chopping, Underflow and Overflow
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Sources of Errors
I Errors in mathematical modelling: Simplifying andAssumptions
I Blunders
(Prgramming Errors): Large programmes,Subprogrammes
I Errors in input: Errors in data transfer, uncertaintiesassociated with measurements
I Machine errors by computer (Floating point arithmetic):Rounding and Chopping, Underflow and Overflow
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Sources of Errors
I Errors in mathematical modelling: Simplifying andAssumptions
I Blunders (Prgramming Errors): Large programmes,Subprogrammes
I Errors in input: Errors in data transfer, uncertaintiesassociated with measurements
I Machine errors by computer (Floating point arithmetic):Rounding and Chopping, Underflow and Overflow
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Sources of Errors
I Errors in mathematical modelling: Simplifying andAssumptions
I Blunders (Prgramming Errors): Large programmes,Subprogrammes
I Errors in input: Errors in data transfer, uncertaintiesassociated with measurements
I Machine errors by computer (Floating point arithmetic):Rounding and Chopping, Underflow and Overflow
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Sources of Errors
I Errors in mathematical modelling: Simplifying andAssumptions
I Blunders (Prgramming Errors): Large programmes,Subprogrammes
I Errors in input: Errors in data transfer, uncertaintiesassociated with measurements
I Machine errors by computer (Floating point arithmetic):Rounding and Chopping, Underflow and Overflow
Kim, Hyun-Min Numerical Linear Algebra
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
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
Schedule Prerequisite Preliminaries Errors and Algorithms
Algorithms
Examining approximation procedures involving finite sequenceof calculations.
I Stable: Small changes in the initial data
I Unstable: Otherwise
I Conditionally Stable: Stable only for certain of initial data
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Algorithms
Examining approximation procedures involving finite sequenceof calculations.
I Stable: Small changes in the initial data
I Unstable: Otherwise
I Conditionally Stable: Stable only for certain of initial data
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Algorithms
Examining approximation procedures involving finite sequenceof calculations.
I Stable: Small changes in the initial data
I Unstable: Otherwise
I Conditionally Stable: Stable only for certain of initial data
Kim, Hyun-Min Numerical Linear Algebra
Schedule Prerequisite Preliminaries Errors and Algorithms
Convergence
{αn}∞n=1, {βn}∞n=1: sequences
limn→∞
βn = 0, limn→∞
αn = α.
If ∃ K > 0 s. t. |αn−α| ≤ |βn| for large n then {αn}∞n=1 convergesto α with the rate of convergence O(βn).
αn = α+O(βn)
Kim, Hyun-Min Numerical Linear Algebra
