The linear system. The problem: solve

Similar documents
Chapter Gauss-Seidel Method

ITERATIVE METHODS FOR SOLVING SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS

ME 501A Seminar in Engineering Analysis Page 1

12 Iterative Methods. Linear Systems: Gauss-Seidel Nonlinear Systems Case Study: Chemical Reactions

Review of Linear Algebra

Matrix. Definition 1... a1 ... (i) where a. are real numbers. for i 1, 2,, m and j = 1, 2,, n (iii) A is called a square matrix if m n.

CHAPTER 5 Vectors and Vector Space

CS321. Numerical Analysis

Analele Universităţii din Oradea, Fascicula: Protecţia Mediului, Vol. XIII, 2008

CS537. Numerical Analysis

minimize c'x subject to subject to subject to

Stats & Summary

Chapter 1 Vector Spaces

Problem Set 4 Solutions

1 4 6 is symmetric 3 SPECIAL MATRICES 3.1 SYMMETRIC MATRICES. Defn: A matrix A is symmetric if and only if A = A, i.e., a ij =a ji i, j. Example 3.1.

Lecture 3-4 Solutions of System of Linear Equations

this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]

Exercise # 2.1 3, 7, , 3, , -9, 1, Solution: natural numbers are 3, , -9, 1, 2.5, 3, , , -9, 1, 2 2.5, 3, , -9, 1, , -9, 1, 2.

Keywords: Heptic non-homogeneous equation, Pyramidal numbers, Pronic numbers, fourth dimensional figurate numbers.

Advanced Algorithmic Problem Solving Le 3 Arithmetic. Fredrik Heintz Dept of Computer and Information Science Linköping University

2. Elementary Linear Algebra Problems

An Alternative Method to Find the Solution of Zero One Integer Linear Fractional Programming Problem with the Help of -Matrix

Rendering Equation. Linear equation Spatial homogeneous Both ray tracing and radiosity can be considered special case of this general eq.

The Algebraic Least Squares Fitting Of Ellipses

COMPLEX NUMBERS AND DE MOIVRE S THEOREM

Level-2 BLAS. Matrix-Vector operations with O(n 2 ) operations (sequentially) BLAS-Notation: S --- single precision G E general matrix M V --- vector

CS473-Algorithms I. Lecture 3. Solving Recurrences. Cevdet Aykanat - Bilkent University Computer Engineering Department

Global Journal of Engineering and Technology Review

International Journal of Scientific and Research Publications, Volume 3, Issue 5, May ISSN

xl yl m n m n r m r m r r! The inner sum in the last term simplifies because it is a binomial expansion of ( x + y) r : e +.

Asymptotic Dominance Problems. is not constant but for n 0, f ( n) 11. 0, so that for n N f

Maximize: x (1.1) Where s is slack variable vector of size m 1. This is a maximization problem. Or (1.2)

10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n

FREE Download Study Package from website: &

MATRIX AND VECTOR NORMS

CHAPTER 7 SOLVING FUZZY LINEAR FRACTIONAL PROGRAMMING PROBLEM BY USING TAYLOR'S SERIES METHOD

CBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find

SOLUTION OF TWO DIMENSIONAL FRACTIONAL ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

3. REVIEW OF PROPERTIES OF EIGENVALUES AND EIGENVECTORS

Methods to Invert a Matrix

INTRODUCTION ( ) 1. Errors

5 - Determinants. r r. r r. r r. r s r = + det det det

Numerical Differentiation and Integration

Chapter 2 Solving Linear Equation

Dual-Matrix Approach for Solving the Transportation Problem

Second degree generalized gauss-seidel iteration method for solving linear system of equations. ABSTRACT

CS 4758 Robot Kinematics. Ashutosh Saxena

Fibonacci and Lucas Numbers as Tridiagonal Matrix Determinants

( ) ( ) ( ( )) ( ) ( ) ( ) ( ) ( ) = ( ) ( ) + ( ) ( ) = ( ( )) ( ) + ( ( )) ( ) Review. Second Derivatives for f : y R. Let A be an m n matrix.

CURVE FITTING LEAST SQUARES METHOD

Optimality of Strategies for Collapsing Expanded Random Variables In a Simple Random Sample Ed Stanek

The Z-Transform in DSP Lecture Andreas Spanias

MATH 104: INTRODUCTORY ANALYSIS SPRING 2009/10 PROBLEM SET 8 SOLUTIONS. and x i = a + i. i + n(n + 1)(2n + 1) + 2a. (b a)3 6n 2

The z-transform. LTI System description. Prof. Siripong Potisuk

In Calculus I you learned an approximation method using a Riemann sum. Recall that the Riemann sum is

Riemann Integral Oct 31, such that

DATA FITTING. Intensive Computation 2013/2014. Annalisa Massini

An Introduction to Robot Kinematics. Renata Melamud

MATH 104: INTRODUCTORY ANALYSIS SPRING 2008/09 PROBLEM SET 10 SOLUTIONS. f m. and. f m = 0. and x i = a + i. a + i. a + n 2. n(n + 1) = a(b a) +

SOLVING SYSTEMS OF EQUATIONS, DIRECT METHODS

Band structures calculations

The limit comparison test

PAIR OF STRAIGHT LINES. will satisfy L1 L2 0, and thus L1 L. 0 represent? It is obvious that any point lying on L 1

MTH 146 Class 7 Notes

ON THE LAWS OF LARGE NUMBERS FOR DEPENDENT RANDOM VARIABLES

Analyzing Control Structures

Numerical Analysis Topic 4: Least Squares Curve Fitting

Numerical Methods. Lecture 5. Numerical integration. dr hab. inż. Katarzyna Zakrzewska, prof. AGH. Numerical Methods lecture 5 1

More Regression Lecture Notes CE 311K - McKinney Introduction to Computer Methods Department of Civil Engineering The University of Texas at Austin

M098 Carson Elementary and Intermediate Algebra 3e Section 10.2

Abhilasha Classes Class- XII Date: SOLUTION (Chap - 9,10,12) MM 50 Mob no

Introduction to Matrix Algebra

Chapter Linear Regression

An Indian Journal FULL PAPER ABSTRACT KEYWORDS. Trade Science Inc. Matrix eigenvalues of jacobi iterative method for solving

POWERS OF COMPLEX PERSYMMETRIC ANTI-TRIDIAGONAL MATRICES WITH CONSTANT ANTI-DIAGONALS

Geometric Correction or Georeferencing

SOLUTION OF SYSTEM OF LINEAR EQUATIONS. Lecture 4: (a) Jacobi's method. method (general). (b) Gauss Seidel method.

Chapter 7 Infinite Series

Math 1313 Final Exam Review

Chapter 2 Intro to Math Techniques for Quantum Mechanics

Available online through

Mathematically, integration is just finding the area under a curve from one point to another. It is b

IMPROVED THREE PHASE POWER FLOW METHOD FOR CALCULATION OF POWER LOSSES IN UNBALANCED RADIAL DISTRIBUTION NETWORK

Current Programmed Control (i.e. Peak Current-Mode Control) Lecture slides part 2 More Accurate Models

Let. Then. k n. And. Φ npq. npq. ε 2. Φ npq npq. npq. = ε. k will be very close to p. If n is large enough, the ratio n

1. Twelve less than five times a number is thirty three. What is the number

Recent Progresses on the Simplex Method

MONOPOLISTIC COMPETITION MODEL

APPLICATION OF THE CHEBYSHEV POLYNOMIALS TO APPROXIMATION AND CONSTRUCTION OF MAP PROJECTIONS

Roberto s Notes on Integral Calculus Chapter 4: Definite integrals and the FTC Section 2. Riemann sums

A multi-domain boundary element analysis technique based on a row elimination-backsubstitution method for solving large-scale engineering problems

PubH 7405: REGRESSION ANALYSIS REGRESSION IN MATRIX TERMS

Sequences and summations

MATRIX ALGEBRA, Systems Linear Equations

SOME ASPECTS ON SOLVING A LINEAR FRACTIONAL TRANSPORTATION PROBLEM

0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.

Density estimation II

Moments of Generalized Order Statistics from a General Class of Distributions

Implicit Runge-Kutta method for Van der pol problem

A METHOD FOR THE RAPID NUMERICAL CALCULATION OF PARTIAL SUMS OF GENERALIZED HARMONICAL SERIES WITH PRESCRIBED ACCURACY

Transcription:

The ler syste The prole: solve Suppose A s vertle, the there ests uue soluto How to effetly opute the soluto uerlly??? A A A

evew of dret ethods Guss elto wth pvotg Meory ost: O^ Coputtol ost: O^ C oly e used for sll whe A s dese, e.g. < deoposto Meory ost: O^ Coputtol ost: O^ C oly e used for sll whe A s dese, e.g. < Good for prole to solve the ler syste wth dfferet rght hd oplete, Cholesy deoposto;

evew of dret ethods For tr-dgol tr Thos lgorth sed o Crout ftorzto Meory ost: O & Coputtol ost: O C e eteded to d-lted tr For ler syste fro dsretzto of Posso euto y FM ret Posso solver sed o FFT Meory ost: O & Coputtol ost: O l For ler syste fro dsretzto of ellpt euto y FEM Multgrd ethod MG or Alger Multgrd ethod AMG Meory ost: O & Coputtol ost: O For ler syste fro dsretzto of Posso euto y tegrl forulto Fst Multpole ethod Meory ost: O & Coputtol ost: O

tertve ethods A: to solve lrge sprse ler syste Bs tertve ethods A o ethod Guss-Sedel ethod Suessve over-relto ethod SO Krylov suspe oder tertve ethods Steepest deet ethod Cougte grdet CG ethod GMES for osyetr ehtod φ T T α d φ : A

Bs tertve ethods ewrte A A

o tertve ethod The ler syste Euto for Mtr for, to for : A A &

o tertve ethod A eple The ethod tl guess 7 7 7 5 5 5 A A.4.4.4............

o tertve ethod The results Ths ge ot urretly e dsplyed. Ths ge ot urretly e dsplyed..9959.9959.9959.4...9744.9744.9744.64.64.64.84.84.84.4.4.4 5 6 4 5 4

Guss-Sedel ethod de: sed the ew vlues whe they re vlle Euto for Mtr for to for :

Guss-Sedel ethod A eple The ethod 5 A 5 5 7 7 7 A....4..4.6....7.896 tl guess...

Guss-Sedel ethod The results Ths ge ot urretly e dsplyed. Ths ge ot urretly e dsplyed.......9996.4.4.9964.9964.4.9968.896..4 4 5 4

SO ethod de: To prove the Guss-Sedel ethod y ler otoof the old vlue d ew vlue Euto for Mtr for to for : ] [ ] [ ] [ SO SO

Covergee lyss Geerl for of s tertve ethods Et soluto efe the error t the -th terto e Error eutos e e e e :

Covergee lyss Covergee e: For y sure tr, there ests osgulr tr T suh tht ord ol for e e { } { } r r C T T T T,, dg,, dg

Covergee lyss efto: Spetrl rdus of ρ Egevlues of e: For y sure tr, ρ < < Theore: The tertve ethod overges to the et soluto of A ρ ff < <

Covergee rte Th: For the tertve ethod suppose the The tertve ethod overges er overgee rte wth < Error oud : < &

Proof for overgee rte Ft Error oud Aother error oud Error oud

Covergee results f A s strtly row dgolly dot, the oth o d Guss-Sedel ethods overge. Guss-Sedel ethod overges f A s syetr postve defte The relto preter e, s the eessry odto for the overgee of SO ethod. ddto, f A s syetr postve defte, the the odto s lso suffet for the overgee of SO ethod

Covergee results efto: A s strtly row dgolly dot f Eples Th: f A s strtly row dgolly dot, t s vertle!,, < > 5 4 5 4 5 B A

Covergee results Th: f A s strtly row dgolly dot, the oth o d Guss-Sedel ethods overge. ft, Proof < : ρ det f det s strtly row dgolly dot f det det det det : < < < ρ

Covergee results Th: et A e syetr postve defte tr, the the Guss-Sedel ethod overges for y tl guess. Proof: See detls lss er: There re ler syste, for whh the o ethod overges, ut the Guss-Sedel ethod dverges, e.g. 6.785 6 > < A ρ ρ

Covergee results Th: For SO ethod, we hve Thus the relto preter e, s eessry for SO overge Proof: ρ SO SO det det det det ρ SO SO SO

Covergee results Th: f A s syetr postve defte, the for,. Tht s, SO overges for ll ρ < SO, Proof: See detls lss er: Over relto: der relto: < < < < Optl relto preter: opt ρ

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback