Sistem bilangan yang sudah dikenal sebelumnya adalah sistem bilangan real, tetapi sistem bilangan real ternyata masih belum cukup untuk menyelesaikan semua bentuk permasalahan dalam berbagai operasi dan persamaan dalam matematika. Oleh karena itu, diperlukan sistem bilangan baru yaitu sistem bilangan kompleks. Sistem bilangan kompleks terdiri dari bilangan kompleks, fungsi analitik, fungsi elementer, integral fungsi kompleks, deret kompleks, dan metode pengintegralan residu.
Dalam sistem bilangan kompleks fungsi elementer sangat penting dan sebagai penunjang untuk mempelajari sistem bilangan kompleks yang lainnya. Fungsi elementer diantaranya, fungsi linear, fungsi pangkat, fungsi bilinear, fungsi eksponensial, fungsi logaritma, fungsi trigonometri dan fungsi hiperbola. Pemahaman tentang fungsi elementer sendiri sangat diperlukan dalam menganalisis suatu kurva secara geometris.
Dalam makalah ini akan dibahas tentang Fungsi Trigonometri dan Fungsi Hiperbolik dalam bilangan kompleks.
Sistem bilangan yang sudah dikenal sebelumnya adalah sistem bilangan real, tetapi sistem bilangan real ternyata masih belum cukup untuk menyelesaikan semua bentuk permasalahan dalam berbagai operasi dan persamaan dalam matematika. Oleh karena itu, diperlukan sistem bilangan baru yaitu sistem bilangan kompleks. Sistem bilangan kompleks terdiri dari bilangan kompleks, fungsi analitik, fungsi elementer, integral fungsi kompleks, deret kompleks, dan metode pengintegralan residu.
Dalam sistem bilangan kompleks fungsi elementer sangat penting dan sebagai penunjang untuk mempelajari sistem bilangan kompleks yang lainnya. Fungsi elementer diantaranya, fungsi linear, fungsi pangkat, fungsi bilinear, fungsi eksponensial, fungsi logaritma, fungsi trigonometri dan fungsi hiperbola. Pemahaman tentang fungsi elementer sendiri sangat diperlukan dalam menganalisis suatu kurva secara geometris.
Dalam makalah ini akan dibahas tentang Fungsi Trigonometri dan Fungsi Hiperbolik dalam bilangan kompleks.
Conformable Chebyshev differential equation of first kindIJECEIAES
In this paper, the Chebyshev-I conformable differential equation is considered. A proper power series is examined; there are two solutions, the even solution and the odd solution. The Rodrigues’ type formula is also allocated for the conformable Chebyshev-I polynomials.
Multidimensional integrals may be approximated by weighted averages of integrand values. Quasi-Monte Carlo (QMC) methods are more accurate than simple Monte Carlo methods because they carefully choose where to evaluate the integrand. This tutorial focuses on how quickly QMC methods converge to the correct answer as the number of integrand values increases. The answer may depend on the smoothness of the integrand and the sophistication of the QMC method. QMC error analysis may assumes the integrand belongs to a reproducing kernel Hilbert space or may assume that the integrand is an instance of a stochastic process with known covariance structure. These two approaches have interesting parallels. This tutorial also explores how the computational cost of achieving a good approximation to the integral depends on the dimension of the domain of the integrand. Finally, this tutorial explores methods for determining how many integrand values are needed to satisfy the error tolerance. Relevant software is described.
Gauss jordan and Guass elimination methodMeet Nayak
This ppt is based on engineering maths.
the topis is Gauss jordan and gauss elimination method.
This ppt having one example of both method and having algorithm.
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in GraphsIJERA Editor
In this paper we study about x-geodominating set, geodetic set, geo-set, geo-number of a graph G. We study the
binary operation, link vectors and some required results to develop algorithms. First we design two algorithms
to check whether given set is an x-geodominating set and to find the minimum x-geodominating set of a graph.
Finally we present another two algorithms to check whether a given vertex is geo-vertex or not and to find the
geo-number of a graph.
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in GraphsIJERA Editor
In this paper we study about x-geodominating set, geodetic set, geo-set, geo-number of a graph G. We study the
binary operation, link vectors and some required results to develop algorithms. First we design two algorithms
to check whether given set is an x-geodominating set and to find the minimum x-geodominating set of a graph.
Finally we present another two algorithms to check whether a given vertex is geo-vertex or not and to find the
geo-number of a graph.
Anti-differentiating approximation algorithms: A case study with min-cuts, sp...David Gleich
This talk covers the idea of anti-differentiating approximation algorithms, which is an idea to explain the success of widely used heuristic procedures. Formally, this involves finding an optimization problem solved exactly by an approximation algorithm or heuristic.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
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Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
1. Iterative Techniques in Matrix Algebra
Jacobi & Gauss-Seidel Iterative Techniques II
Numerical Analysis (9th Edition)
R L Burden & J D Faires
Beamer Presentation Slides
prepared by
John Carroll
Dublin City University
c
2011 Brooks/Cole, Cengage Learning
2. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 2 / 38
3. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 2 / 38
4. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 2 / 38
5. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
4 Application to the Jacobi & Gauss-Seidel Methods
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 2 / 38
6. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
4 Application to the Jacobi & Gauss-Seidel Methods
Numerical Analysis (Chapter 7) Jacobi & Gauss-Seidel Methods II R L Burden & J D Faires 3 / 38
7. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Looking at the Jacobi Method
A possible improvement to the Jacobi Algorithm can be seen by
re-considering
x(k)
i =
1
aii
2
Xn
664
j=1
j6=i
−aijx(k−1)
j
+ bi
3
775
, for i = 1, 2, . . . , n
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 4 / 38
8. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Looking at the Jacobi Method
A possible improvement to the Jacobi Algorithm can be seen by
re-considering
x(k)
i =
1
aii
2
Xn
664
j=1
j6=i
−aijx(k−1)
j
+ bi
3
775
, for i = 1, 2, . . . , n
The components of x(k−1) are used to compute all the
components x(k)
i of x(k).
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 4 / 38
9. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Looking at the Jacobi Method
A possible improvement to the Jacobi Algorithm can be seen by
re-considering
x(k)
i =
1
aii
2
Xn
664
j=1
j6=i
−aijx(k−1)
j
+ bi
3
775
, for i = 1, 2, . . . , n
The components of x(k−1) are used to compute all the
components x(k)
i of x(k).
But, for i 1, the components x(k)
1 , . . . , x(k)
i−1 of x(k) have already
been computed and are expected to be better approximations to
the actual solutions x1, . . . , xi−1 than are x(k−1)
1 , . . . , x(k−1)
i−1 .
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 4 / 38
10. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Instead of using
x(k)
i =
1
aii
2
Xn
664
j=1
j6=i
−aijx(k−1)
j
+ bi
3
775
, for i = 1, 2, . . . , n
it seems reasonable, then, to compute x(k)
i using these most recently
calculated values.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 5 / 38
11. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Instead of using
x(k)
i =
1
aii
2
Xn
664
j=1
j6=i
−aijx(k−1)
j
+ bi
3
775
, for i = 1, 2, . . . , n
it seems reasonable, then, to compute x(k)
i using these most recently
calculated values.
The Gauss-Seidel Iterative Technique
x(k)
i =
1
aii
2
4−
Xi−1
j=1
(aijx(k)
j ) −
Xn
j=i+1
(aijx(k−1)
j ) + bi
3
5
for each i = 1, 2, . . . , n.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 5 / 38
12. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Example
Use the Gauss-Seidel iterative technique to find approximate solutions
to
10x1 − x2 + 2x3 = 6
−x1 + 11x2 − x3 + 3x4 = 25
2x1 − x2 + 10x3 − x4 = −11
3x2 − x3 + 8x4 = 15
,
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 6 / 38
13. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Example
Use the Gauss-Seidel iterative technique to find approximate solutions
to
10x1 − x2 + 2x3 = 6
−x1 + 11x2 − x3 + 3x4 = 25
2x1 − x2 + 10x3 − x4 = −11
3x2 − x3 + 8x4 = 15
,
starting with x = (0, 0, 0, 0)t
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 6 / 38
14. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Example
Use the Gauss-Seidel iterative technique to find approximate solutions
to
10x1 − x2 + 2x3 = 6
−x1 + 11x2 − x3 + 3x4 = 25
2x1 − x2 + 10x3 − x4 = −11
3x2 − x3 + 8x4 = 15
,
starting with x = (0, 0, 0, 0)t and iterating until
kx(k) − x(k−1)k1
kx(k)k1
10−3
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 6 / 38
15. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Example
Use the Gauss-Seidel iterative technique to find approximate solutions
to
10x1 − x2 + 2x3 = 6
−x1 + 11x2 − x3 + 3x4 = 25
2x1 − x2 + 10x3 − x4 = −11
3x2 − x3 + 8x4 = 15
,
starting with x = (0, 0, 0, 0)t and iterating until
kx(k) − x(k−1)k1
kx(k)k1
10−3
Note: The solution x = (1, 2, −1, 1)t was approximated by Jacobi’s
method in an earlier example.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 6 / 38
16. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Solution (1/3)
For the Gauss-Seidel method we write the system, for each
k = 1, 2, . . . as
x(k)
1 =
1
10
x(k−1)
2 −
1
5
x(k−1)
3 +
3
5
x(k)
2 =
1
11
x(k)
1 +
1
11
x(k−1)
3 −
3
11
x(k−1)
4 +
25
11
x(k)
3 = −
1
5
x(k)
1 +
1
10
x(k)
2 +
1
10
x(k−1)
4 −
11
10
x(k)
4 = −
3
8
x(k)
2 +
1
8
x(k)
3 +
15
8
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 7 / 38
17. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Solution (2/3)
When x(0) = (0, 0, 0, 0)t , we have
x(1) = (0.6000, 2.3272, −0.9873, 0.8789)t .
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 8 / 38
18. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Solution (2/3)
When x(0) = (0, 0, 0, 0)t , we have
x(1) = (0.6000, 2.3272, −0.9873, 0.8789)t . Subsequent iterations give
the values in the following table:
k 0 1 2 3 4 5
x(k)
1 0.0000 0.6000 1.030 1.0065 1.0009 1.0001
x(k)
2 0.0000 2.3272 2.037 2.0036 2.0003 2.0000
x(k)
3 0.0000 −0.9873 −1.014 −1.0025 −1.0003 −1.0000
x(k)
4 0.0000 0.8789 0.984 0.9983 0.9999 1.0000
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 8 / 38
19. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Solution (3/3)
Because
kx(5) − x(4)k1
kx(5)k1
=
0.0008
2.000 = 4 × 10−4
x(5) is accepted as a reasonable approximation to the solution.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 9 / 38
20. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method
Solution (3/3)
Because
kx(5) − x(4)k1
kx(5)k1
=
0.0008
2.000 = 4 × 10−4
x(5) is accepted as a reasonable approximation to the solution.
Note that, in an earlier example, Jacobi’s method required twice as
many iterations for the same accuracy.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 9 / 38
21. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method: Matrix Form
Re-Writing the Equations
To write the Gauss-Seidel method in matrix form,
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 10 / 38
22. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method: Matrix Form
Re-Writing the Equations
To write the Gauss-Seidel method in matrix form, multiply both sides of
x(k)
i =
1
aii
2
4−
Xi−1
j=1
(aijx(k)
j ) −
Xn
j=i+1
(aijx(k−1)
j ) + bi
3
5
by aii and collect all kth iterate terms,
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 10 / 38
23. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method: Matrix Form
Re-Writing the Equations
To write the Gauss-Seidel method in matrix form, multiply both sides of
x(k)
i =
1
aii
2
4−
Xi−1
j=1
(aijx(k)
j ) −
Xn
j=i+1
(aijx(k−1)
j ) + bi
3
5
by aii and collect all kth iterate terms, to give
ai1x(k)
1 + ai2x(k)
2 + · · · + aiix(k)
i = −ai,i+1x(k−1)
i+1 − · · · − ainx(k−1)
n + bi
for each i = 1, 2, . . . , n.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 10 / 38
24. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method: Matrix Form
Re-Writing the Equations (Cont’d)
Writing all n equations gives
(k)
1 = −a12x
a11x
(k−1)
2 − a13x
(k−1)
3 − · · · − a1nx
(k−1)
n + b1
(k)
1 + a22x
a21x
(k)
2 = −a23x
(k−1)
3 − · · · − a2nx
(k−1)
n + b2
...
an1x(k)
1 + an2x(k)
2 + · · · + annx(k)
n = bn
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 11 / 38
25. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method: Matrix Form
Re-Writing the Equations (Cont’d)
Writing all n equations gives
(k)
1 = −a12x
a11x
(k−1)
2 − a13x
(k−1)
3 − · · · − a1nx
(k−1)
n + b1
(k)
1 + a22x
a21x
(k)
2 = −a23x
(k−1)
3 − · · · − a2nx
(k−1)
n + b2
...
an1x(k)
1 + an2x(k)
2 + · · · + annx(k)
n = bn
With the definitions of D, L, and U given previously, we have the
Gauss-Seidel method represented by
(D − L)x(k) = Ux(k−1) + b
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 11 / 38
26. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method: Matrix Form
(D − L)x(k) = Ux(k−1) + b
Re-Writing the Equations (Cont’d)
Solving for x(k) finally gives
x(k) = (D − L)−1Ux(k−1) + (D − L)−1b, for each k = 1, 2, . . .
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 12 / 38
27. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method: Matrix Form
(D − L)x(k) = Ux(k−1) + b
Re-Writing the Equations (Cont’d)
Solving for x(k) finally gives
x(k) = (D − L)−1Ux(k−1) + (D − L)−1b, for each k = 1, 2, . . .
Letting Tg = (D − L)−1U and cg = (D − L)−1b, gives the Gauss-Seidel
technique the form
x(k) = Tgx(k−1) + cg
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 12 / 38
28. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
The Gauss-Seidel Method: Matrix Form
(D − L)x(k) = Ux(k−1) + b
Re-Writing the Equations (Cont’d)
Solving for x(k) finally gives
x(k) = (D − L)−1Ux(k−1) + (D − L)−1b, for each k = 1, 2, . . .
Letting Tg = (D − L)−1U and cg = (D − L)−1b, gives the Gauss-Seidel
technique the form
x(k) = Tgx(k−1) + cg
For the lower-triangular matrix D − L to be nonsingular, it is necessary
and sufficient that aii6= 0, for each i = 1, 2, . . . , n.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 12 / 38
29. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
4 Application to the Jacobi Gauss-Seidel Methods
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 13 / 38
30. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (1/2)
To solve Ax = b given an initial approximation x(0):
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 14 / 38
31. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (1/2)
To solve Ax = b given an initial approximation x(0):
INPUT the number of equations and unknowns n;
the entries aij , 1 i, j n of the matrix A;
the entries bi , 1 i n of b;
the entries XOi , 1 i n of XO = x(0);
tolerance TOL;
maximum number of iterations N.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 14 / 38
32. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (1/2)
To solve Ax = b given an initial approximation x(0):
INPUT the number of equations and unknowns n;
the entries aij , 1 i, j n of the matrix A;
the entries bi , 1 i n of b;
the entries XOi , 1 i n of XO = x(0);
tolerance TOL;
maximum number of iterations N.
OUTPUT the approximate solution x1, . . . , xn or a message
that the number of iterations was exceeded.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 14 / 38
33. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k N) do Steps 3–6:
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 15 / 38
34. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k N) do Steps 3–6:
Step 3 For i = 1, . . . , n
set xi =
1
aii
2
4−
Xi−1
j=1
aijxj −
Xn
j=i+1
aijXOj + bi
3
5
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 15 / 38
35. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k N) do Steps 3–6:
Step 3 For i = 1, . . . , n
set xi =
1
aii
2
4−
Xi−1
j=1
aijxj −
Xn
j=i+1
aijXOj + bi
3
5
Step 4 If ||x − XO|| TOL then OUTPUT (x1, . . . , xn)
(The procedure was successful)
STOP
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 15 / 38
36. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k N) do Steps 3–6:
Step 3 For i = 1, . . . , n
set xi =
1
aii
2
4−
Xi−1
j=1
aijxj −
Xn
j=i+1
aijXOj + bi
3
5
Step 4 If ||x − XO|| TOL then OUTPUT (x1, . . . , xn)
(The procedure was successful)
STOP
Step 5 Set k = k + 1
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 15 / 38
37. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k N) do Steps 3–6:
Step 3 For i = 1, . . . , n
set xi =
1
aii
2
4−
Xi−1
j=1
aijxj −
Xn
j=i+1
aijXOj + bi
3
5
Step 4 If ||x − XO|| TOL then OUTPUT (x1, . . . , xn)
(The procedure was successful)
STOP
Step 5 Set k = k + 1
Step 6 For i = 1, . . . , n set XOi = xi
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 15 / 38
38. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm (2/2)
Step 1 Set k = 1
Step 2 While (k N) do Steps 3–6:
Step 3 For i = 1, . . . , n
set xi =
1
aii
2
4−
Xi−1
j=1
aijxj −
Xn
j=i+1
aijXOj + bi
3
5
Step 4 If ||x − XO|| TOL then OUTPUT (x1, . . . , xn)
(The procedure was successful)
STOP
Step 5 Set k = k + 1
Step 6 For i = 1, . . . , n set XOi = xi
Step 7 OUTPUT (‘Maximum number of iterations exceeded’)
STOP (The procedure was unsuccessful)
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 15 / 38
39. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm
Comments on the Algorithm
Step 3 of the algorithm requires that aii6= 0, for each
i = 1, 2, . . . , n.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 16 / 38
40. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm
Comments on the Algorithm
Step 3 of the algorithm requires that aii6= 0, for each
i = 1, 2, . . . , n. If one of the aii entries is 0 and the system is
nonsingular, a reordering of the equations can be performed so
that no aii = 0.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 16 / 38
41. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm
Comments on the Algorithm
Step 3 of the algorithm requires that aii6= 0, for each
i = 1, 2, . . . , n. If one of the aii entries is 0 and the system is
nonsingular, a reordering of the equations can be performed so
that no aii = 0.
To speed convergence, the equations should be arranged so that
aii is as large as possible.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 16 / 38
42. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm
Comments on the Algorithm
Step 3 of the algorithm requires that aii6= 0, for each
i = 1, 2, . . . , n. If one of the aii entries is 0 and the system is
nonsingular, a reordering of the equations can be performed so
that no aii = 0.
To speed convergence, the equations should be arranged so that
aii is as large as possible.
Another possible stopping criterion in Step 4 is to iterate until
kx(k) − x(k−1)k
kx(k)k
is smaller than some prescribed tolerance.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 16 / 38
43. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Gauss-Seidel Iterative Algorithm
Comments on the Algorithm
Step 3 of the algorithm requires that aii6= 0, for each
i = 1, 2, . . . , n. If one of the aii entries is 0 and the system is
nonsingular, a reordering of the equations can be performed so
that no aii = 0.
To speed convergence, the equations should be arranged so that
aii is as large as possible.
Another possible stopping criterion in Step 4 is to iterate until
kx(k) − x(k−1)k
kx(k)k
is smaller than some prescribed tolerance.
For this purpose, any convenient norm can be used, the usual
being the l1 norm.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 16 / 38
44. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
4 Application to the Jacobi Gauss-Seidel Methods
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 17 / 38
45. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Introduction
To study the convergence of general iteration techniques, we need
to analyze the formula
x(k) = Tx(k−1) + c, for each k = 1, 2, . . .
where x(0) is arbitrary.
The following lemma and the earlier Theorem on convergent
matrices provide the key for this study.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 18 / 38
46. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Lemma
If the spectral radius satisfies (T) 1, then (I − T)−1 exists, and
(I − T)−1 = I + T + T2 + · · · =
1X
j=0
Tj
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 19 / 38
47. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Lemma
If the spectral radius satisfies (T) 1, then (I − T)−1 exists, and
(I − T)−1 = I + T + T2 + · · · =
1X
j=0
Tj
Proof (1/2)
Because Tx = x is true precisely when (I − T)x = (1 − )x, we
have as an eigenvalue of T precisely when 1 − is an
eigenvalue of I − T.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 19 / 38
48. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Lemma
If the spectral radius satisfies (T) 1, then (I − T)−1 exists, and
(I − T)−1 = I + T + T2 + · · · =
1X
j=0
Tj
Proof (1/2)
Because Tx = x is true precisely when (I − T)x = (1 − )x, we
have as an eigenvalue of T precisely when 1 − is an
eigenvalue of I − T.
But || (T) 1, so = 1 is not an eigenvalue of T, and 0
cannot be an eigenvalue of I − T.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 19 / 38
49. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Lemma
If the spectral radius satisfies (T) 1, then (I − T)−1 exists, and
(I − T)−1 = I + T + T2 + · · · =
1X
j=0
Tj
Proof (1/2)
Because Tx = x is true precisely when (I − T)x = (1 − )x, we
have as an eigenvalue of T precisely when 1 − is an
eigenvalue of I − T.
But || (T) 1, so = 1 is not an eigenvalue of T, and 0
cannot be an eigenvalue of I − T.
Hence, (I − T)−1 exists.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 19 / 38
50. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (2/2)
Let
Sm = I + T + T2 + · · · + Tm
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 20 / 38
51. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (2/2)
Let
Sm = I + T + T2 + · · · + Tm
Then
(I −T)Sm = (1+T +T2+· · ·+Tm)−(T +T2+· · ·+Tm+1) = I −Tm+1
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 20 / 38
52. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (2/2)
Let
Sm = I + T + T2 + · · · + Tm
Then
(I −T)Sm = (1+T +T2+· · ·+Tm)−(T +T2+· · ·+Tm+1) = I −Tm+1
and, since T is convergent, the Theorem on convergent matrices
implies that
lim
m!1
(I − T)Sm = lim
(I − Tm+1) = I
m!1
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 20 / 38
53. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (2/2)
Let
Sm = I + T + T2 + · · · + Tm
Then
(I −T)Sm = (1+T +T2+· · ·+Tm)−(T +T2+· · ·+Tm+1) = I −Tm+1
and, since T is convergent, the Theorem on convergent matrices
implies that
lim
m!1
(I − T)Sm = lim
(I − Tm+1) = I
m!1
Thus, (I − T)−1 = limm!1 Sm = I + T + T2 + · · · =
P1j
=0 Tj
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 20 / 38
54. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Theorem
For any x(0) 2 IRn, the sequence {x(k)}1k
=0 defined by
x(k) = Tx(k−1) + c, for each k 1
converges to the unique solution of
x = Tx + c
if and only if (T) 1.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 21 / 38
55. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (1/5)
First assume that (T) 1.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 22 / 38
56. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (1/5)
First assume that (T) 1. Then,
x(k) = Tx(k−1) + c
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 22 / 38
57. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (1/5)
First assume that (T) 1. Then,
x(k) = Tx(k−1) + c
= T(Tx(k−2) + c) + c
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 22 / 38
58. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (1/5)
First assume that (T) 1. Then,
x(k) = Tx(k−1) + c
= T(Tx(k−2) + c) + c
= T2x(k−2) + (T + I)c
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 22 / 38
59. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (1/5)
First assume that (T) 1. Then,
x(k) = Tx(k−1) + c
= T(Tx(k−2) + c) + c
= T2x(k−2) + (T + I)c
...
= Tkx(0) + (Tk−1 + · · · + T + I)c
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 22 / 38
60. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (1/5)
First assume that (T) 1. Then,
x(k) = Tx(k−1) + c
= T(Tx(k−2) + c) + c
= T2x(k−2) + (T + I)c
...
= Tkx(0) + (Tk−1 + · · · + T + I)c
Because (T) 1, the Theorem on convergent matrices implies that T
is convergent, and
lim
k!1
Tkx(0) = 0
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 22 / 38
61. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (2/5)
The previous lemma implies that
lim
k!1
x(k) = lim
k!1
Tkx(0) +
0
@
1X
j=0
Tj
1
Ac
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 23 / 38
62. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (2/5)
The previous lemma implies that
lim
k!1
x(k) = lim
k!1
Tkx(0) +
0
@
1X
j=0
Tj
1
Ac
= 0 + (I − T)−1c
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 23 / 38
63. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (2/5)
The previous lemma implies that
lim
k!1
x(k) = lim
k!1
Tkx(0) +
0
@
1X
j=0
Tj
1
Ac
= 0 + (I − T)−1c
= (I − T)−1c
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 23 / 38
64. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (2/5)
The previous lemma implies that
lim
k!1
x(k) = lim
k!1
Tkx(0) +
0
@
1X
j=0
Tj
1
Ac
= 0 + (I − T)−1c
= (I − T)−1c
Hence, the sequence {x(k)} converges to the vector x (I − T)−1c
and x = Tx + c.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 23 / 38
65. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (3/5)
To prove the converse, we will show that for any z 2 IRn, we have
limk!1 Tkz = 0.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 24 / 38
66. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (3/5)
To prove the converse, we will show that for any z 2 IRn, we have
limk!1 Tkz = 0.
Again, by the theorem on convergent matrices, this is equivalent
to (T) 1.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 24 / 38
67. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (3/5)
To prove the converse, we will show that for any z 2 IRn, we have
limk!1 Tkz = 0.
Again, by the theorem on convergent matrices, this is equivalent
to (T) 1.
Let z be an arbitrary vector, and x be the unique solution to
x = Tx + c.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 24 / 38
68. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (3/5)
To prove the converse, we will show that for any z 2 IRn, we have
limk!1 Tkz = 0.
Again, by the theorem on convergent matrices, this is equivalent
to (T) 1.
Let z be an arbitrary vector, and x be the unique solution to
x = Tx + c.
Define x(0) = x − z, and, for k 1, x(k) = Tx(k−1) + c.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 24 / 38
69. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (3/5)
To prove the converse, we will show that for any z 2 IRn, we have
limk!1 Tkz = 0.
Again, by the theorem on convergent matrices, this is equivalent
to (T) 1.
Let z be an arbitrary vector, and x be the unique solution to
x = Tx + c.
Define x(0) = x − z, and, for k 1, x(k) = Tx(k−1) + c.
Then {x(k)} converges to x.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 24 / 38
70. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (4/5)
Also,
x − x(k) = (Tx + c) −
Tx(k−1) + c
= T
x − x(k−1)
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 25 / 38
71. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (4/5)
Also,
x − x(k) = (Tx + c) −
Tx(k−1) + c
= T
x − x(k−1)
so
x − x(k) = T
x − x(k−1)
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 25 / 38
72. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (4/5)
Also,
x − x(k) = (Tx + c) −
Tx(k−1) + c
= T
x − x(k−1)
so
x − x(k) = T
x − x(k−1)
= T2
x − x(k−2)
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 25 / 38
73. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (4/5)
Also,
x − x(k) = (Tx + c) −
Tx(k−1) + c
= T
x − x(k−1)
so
x − x(k) = T
x − x(k−1)
= T2
x − x(k−2)
=
...
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 25 / 38
74. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (4/5)
Also,
x − x(k) = (Tx + c) −
Tx(k−1) + c
= T
x − x(k−1)
so
x − x(k) = T
x − x(k−1)
= T2
x − x(k−2)
=
...
= Tk
x − x(0)
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 25 / 38
75. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (4/5)
Also,
x − x(k) = (Tx + c) −
Tx(k−1) + c
= T
x − x(k−1)
so
x − x(k) = T
x − x(k−1)
= T2
x − x(k−2)
=
...
= Tk
x − x(0)
= Tkz
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 25 / 38
76. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (5/5)
Hence
lim
k!1
Tkz = lim
k!1
Tk
x − x(0)
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 26 / 38
77. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (5/5)
Hence
lim
k!1
Tkz = lim
k!1
Tk
x − x(0)
= lim
k!1
x − x(k)
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 26 / 38
78. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (5/5)
Hence
lim
k!1
Tkz = lim
k!1
Tk
x − x(0)
= lim
k!1
x − x(k)
= 0
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 26 / 38
79. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Proof (5/5)
Hence
lim
k!1
Tkz = lim
k!1
Tk
x − x(0)
= lim
k!1
x − x(k)
= 0
But z 2 IRn was arbitrary, so by the theorem on convergent
matrices, T is convergent and (T) 1.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 26 / 38
80. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Corollary
kTk 1 for any natural matrix norm and c is a given vector, then the
sequence {x(k)}1k
=0 defined by
x(k) = Tx(k−1) + c
converges, for any x(0) 2 IRn, to a vector x 2 IRn, with x = Tx + c, and
the following error bounds hold:
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 27 / 38
81. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Corollary
kTk 1 for any natural matrix norm and c is a given vector, then the
sequence {x(k)}1k
=0 defined by
x(k) = Tx(k−1) + c
converges, for any x(0) 2 IRn, to a vector x 2 IRn, with x = Tx + c, and
the following error bounds hold:
(i) kx − x(k)k kTkkkx(0) − xk
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 27 / 38
82. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Corollary
kTk 1 for any natural matrix norm and c is a given vector, then the
sequence {x(k)}1k
=0 defined by
x(k) = Tx(k−1) + c
converges, for any x(0) 2 IRn, to a vector x 2 IRn, with x = Tx + c, and
the following error bounds hold:
(i) kx − x(k)k kTkkkx(0) − xk
(ii) kx − x(k)k kTkk
1−kTkkx(1) − x(0)k
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 27 / 38
83. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence Results for General Iteration Methods
Corollary
kTk 1 for any natural matrix norm and c is a given vector, then the
sequence {x(k)}1k
=0 defined by
x(k) = Tx(k−1) + c
converges, for any x(0) 2 IRn, to a vector x 2 IRn, with x = Tx + c, and
the following error bounds hold:
(i) kx − x(k)k kTkkkx(0) − xk
(ii) kx − x(k)k kTkk
1−kTkkx(1) − x(0)k
The proof of the following corollary is similar to that for the Corollary to
the Fixed-Point Theorem for a single nonlinear equation.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 27 / 38
84. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Outline
1 The Gauss-Seidel Method
2 The Gauss-Seidel Algorithm
3 Convergence Results for General Iteration Methods
4 Application to the Jacobi Gauss-Seidel Methods
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 28 / 38
85. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Using the Matrix Formulations
We have seen that the Jacobi and Gauss-Seidel iterative techniques
can be written
x(k) = Tjx(k−1) + cj and
x(k) = Tgx(k−1) + cg
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 29 / 38
86. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Using the Matrix Formulations
We have seen that the Jacobi and Gauss-Seidel iterative techniques
can be written
x(k) = Tjx(k−1) + cj and
x(k) = Tgx(k−1) + cg
using the matrices
Tj = D−1(L + U) and Tg = (D − L)−1U
respectively.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 29 / 38
87. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Using the Matrix Formulations
We have seen that the Jacobi and Gauss-Seidel iterative techniques
can be written
x(k) = Tjx(k−1) + cj and
x(k) = Tgx(k−1) + cg
using the matrices
Tj = D−1(L + U) and Tg = (D − L)−1U
respectively. If (Tj ) or (Tg) is less than 1, then the corresponding
sequence {x(k)}1k
=0 will converge to the solution x of Ax = b.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 29 / 38
88. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Example
For example, the Jacobi method has
x(k) = D−1(L + U)x(k−1) + D−1b,
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 30 / 38
89. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Example
For example, the Jacobi method has
x(k) = D−1(L + U)x(k−1) + D−1b,
and, if {x(k)}1k
=0 converges to x,
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 30 / 38
90. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Example
For example, the Jacobi method has
x(k) = D−1(L + U)x(k−1) + D−1b,
and, if {x(k)}1k
=0 converges to x, then
x = D−1(L + U)x + D−1b
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 30 / 38
91. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Example
For example, the Jacobi method has
x(k) = D−1(L + U)x(k−1) + D−1b,
and, if {x(k)}1k
=0 converges to x, then
x = D−1(L + U)x + D−1b
This implies that
Dx = (L + U)x + b and (D − L − U)x = b
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 30 / 38
92. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Example
For example, the Jacobi method has
x(k) = D−1(L + U)x(k−1) + D−1b,
and, if {x(k)}1k
=0 converges to x, then
x = D−1(L + U)x + D−1b
This implies that
Dx = (L + U)x + b and (D − L − U)x = b
Since D − L − U = A, the solution x satisfies Ax = b.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 30 / 38
93. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
The following are easily verified sufficiency conditions for convergence
of the Jacobi and Gauss-Seidel methods.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 31 / 38
94. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
The following are easily verified sufficiency conditions for convergence
of the Jacobi and Gauss-Seidel methods.
Theorem
If A is strictly diagonally dominant, then for any choice of x(0), both the
Jacobi and Gauss-Seidel methods give sequences {x(k)}1k
=0 that
converge to the unique solution of Ax = b.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 31 / 38
95. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Is Gauss-Seidel better than Jacobi?
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 32 / 38
96. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Is Gauss-Seidel better than Jacobi?
No general results exist to tell which of the two techniques, Jacobi
or Gauss-Seidel, will be most successful for an arbitrary linear
system.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 32 / 38
97. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Is Gauss-Seidel better than Jacobi?
No general results exist to tell which of the two techniques, Jacobi
or Gauss-Seidel, will be most successful for an arbitrary linear
system.
In special cases, however, the answer is known, as is
demonstrated in the following theorem.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 32 / 38
98. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
(Stein-Rosenberg) Theorem
If aij 0, for each i6= j and aii 0, for each i = 1, 2, . . . , n, then one
and only one of the following statements holds:
(i) 0 (Tg) (Tj ) 1
(ii) 1 (Tj ) (Tg)
(iii) (Tj ) = (Tg) = 0
(iv) (Tj ) = (Tg) = 1
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 33 / 38
99. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
(Stein-Rosenberg) Theorem
If aij 0, for each i6= j and aii 0, for each i = 1, 2, . . . , n, then one
and only one of the following statements holds:
(i) 0 (Tg) (Tj ) 1
(ii) 1 (Tj ) (Tg)
(iii) (Tj ) = (Tg) = 0
(iv) (Tj ) = (Tg) = 1
For the proof of this result, see pp. 120–127. of
Young, D. M., Iterative solution of large linear systems, Academic
Press, New York, 1971, 570 pp.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 33 / 38
100. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Two Comments on the Thoerem
For the special case described in the theorem, we see from part
(i), namely
0 (Tg) (Tj ) 1
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 34 / 38
101. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Two Comments on the Thoerem
For the special case described in the theorem, we see from part
(i), namely
0 (Tg) (Tj ) 1
that when one method gives convergence, then both give
convergence,
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 34 / 38
102. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Two Comments on the Thoerem
For the special case described in the theorem, we see from part
(i), namely
0 (Tg) (Tj ) 1
that when one method gives convergence, then both give
convergence, and the Gauss-Seidel method converges faster than
the Jacobi method.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 34 / 38
103. Gauss-Seidel Method Gauss-Seidel Algorithm Convergence Results Interpretation
Convergence of the Jacobi Gauss-Seidel Methods
Two Comments on the Thoerem
For the special case described in the theorem, we see from part
(i), namely
0 (Tg) (Tj ) 1
that when one method gives convergence, then both give
convergence, and the Gauss-Seidel method converges faster than
the Jacobi method.
Part (ii), namely
1 (Tj ) (Tg)
indicates that when one method diverges then both diverge, and
the divergence is more pronounced for the Gauss-Seidel method.
Numerical Analysis (Chapter 7) Jacobi Gauss-Seidel Methods II R L Burden J D Faires 34 / 38
105. Eigenvalues Eigenvectors: Convergent Matrices
Theorem
The following statements are equivalent.
(i) A is a convergent matrix.
(ii) limn!1 kAnk = 0, for some natural norm.
(iii) limn!1 kAnk = 0, for all natural norms.
(iv) (A) 1.
(v) limn!1 Anx = 0, for every x.
The proof of this theorem can be found on p. 14 of Issacson, E. and H.
B. Keller, Analysis of Numerical Methods, John Wiley Sons, New
York, 1966, 541 pp.
Return to General Iteration Methods — Introduction
Return to General Iteration Methods — Lemma
Return to General Iteration Methods — Theorem
106. Fixed-Point Theorem
Let g 2 C[a, b] be such that g(x) 2 [a, b], for all x in [a, b]. Suppose, in
addition, that g0 exists on (a, b) and that a constant 0 k 1 exists
with
|g0(x)| k, for all x 2 (a, b).
Then for any number p0 in [a, b], the sequence defined by
pn = g(pn−1), n 1
converges to the unique fixed point p in [a, b].
Return to the Corrollary to the Fixed-Point Theorem
107. Functional (Fixed-Point) Iteration
Corrollary to the Fixed-Point Convergence Result
If g satisfies the hypothesis of the Fixed-Point Theorem then
|pn − p|
kn
1 − k
|p1 − p0|
Return to the Corollary to the Convergence Theorem for General Iterative Methods