This document discusses iterative methods for solving systems of linear equations. It introduces the Gauss-Jacobi and Gauss-Seidel methods. For Gauss-Jacobi, each new approximation is calculated from the previous approximations. For Gauss-Seidel, the most recent approximation is used. The document provides examples of applying each method and discusses conditions for convergence. As homework, students are asked to use the two methods to approximate solutions for a given system of equations.
The well-known methods' that utilized in the numerical techniques. Several examples illustrated to make the numerical methods that presented more sensible. The Excel program is utilized in this paper.
Least Square Plane and Leastsquare Quadric Surface Approximation by Using Mod...IOSRJM
Now a days Surface fitting is applied all engineering and medical fields. Kamron Saniee ,2007 find a simple expression for multivariate LaGrange’s Interpolation. We derive a least square plane and least square quadric surface Approximation from a given N+1 tabular points when the function is unique. We used least square method technique. We can apply this method in surface fitting also.
Chebyshev Collocation Approach for a Continuous Formulation of Implicit Hybri...IOSR Journals
In this paper, an implicit one-step method for numerical solution of second order Initial Value
Problems of Ordinary Differential Equations has been developed by collocation and interpolation technique.
The one-step method was developed using Chebyshev polynomial as basis function and, the method was
augmented by the introduction of offstep points in order to bring about zero stability and upgrade the order of
consistency of the new method. An advantage of the derived continuous scheme is that it can produce several
outputs of solution at the off-grid points without requiring additional interpolation. Numerical examples are
presented to portray the applicability and the efficiency of the method.
Adomian decomposition Method and Differential Transform Method to solve the H...IJERA Editor
In this paper, we consider the AdomianDecomposition Method (ADM) and the Differential Transform Method (DTM) for finding approximate and exact solution of the heat equation with a power nonlinearity. Moreover, the reliability and performance of ADM and DTM. Numerical results show that these methods are powerful tools for solving heat equation with a power nonlinearity.
Comparative analysis of x^3+y^3=z^3 and x^2+y^2=z^2 in the Interconnected Sets Vladimir Godovalov
This paper introduces an innovative technique of study z^3-x^3=y^3 on the subject of its insolvability in integers. Technique starts from building the interconnected, third degree sets: A3={a_n│a_n=n^3,n∈N}, B3={b_n│b_n=a_(n+1)-a_n }, C3={c_n│c_n=b_(n+1)-b_n } and P3={6} wherefrom we get a_n and b_n expressed as figurate polynomials of third degree, a new finding in mathematics. This approach and the results allow us to investigate equation z^3-x^3=y in these interconnected sets A3 and B3, where z^3∧x^3∈A3, y∈B3. Further, in conjunction with the new Method of Ratio Comparison of Summands and Pascal’s rule, we finally prove inability of y=y^3. After we test the technique, applying the same approach to z^2-x^2=y where we get family of primitive z^2-x^2=y^2 as well as introduce conception of the basic primitiveness of z^'2-x^'2=y^2 for z^'-x^'=1 and the dependant primitiveness of z^'2-x^'2=y^2 for co-prime x,y,z and z^'-x^'>1.
The well-known methods' that utilized in the numerical techniques. Several examples illustrated to make the numerical methods that presented more sensible. The Excel program is utilized in this paper.
Least Square Plane and Leastsquare Quadric Surface Approximation by Using Mod...IOSRJM
Now a days Surface fitting is applied all engineering and medical fields. Kamron Saniee ,2007 find a simple expression for multivariate LaGrange’s Interpolation. We derive a least square plane and least square quadric surface Approximation from a given N+1 tabular points when the function is unique. We used least square method technique. We can apply this method in surface fitting also.
Chebyshev Collocation Approach for a Continuous Formulation of Implicit Hybri...IOSR Journals
In this paper, an implicit one-step method for numerical solution of second order Initial Value
Problems of Ordinary Differential Equations has been developed by collocation and interpolation technique.
The one-step method was developed using Chebyshev polynomial as basis function and, the method was
augmented by the introduction of offstep points in order to bring about zero stability and upgrade the order of
consistency of the new method. An advantage of the derived continuous scheme is that it can produce several
outputs of solution at the off-grid points without requiring additional interpolation. Numerical examples are
presented to portray the applicability and the efficiency of the method.
Adomian decomposition Method and Differential Transform Method to solve the H...IJERA Editor
In this paper, we consider the AdomianDecomposition Method (ADM) and the Differential Transform Method (DTM) for finding approximate and exact solution of the heat equation with a power nonlinearity. Moreover, the reliability and performance of ADM and DTM. Numerical results show that these methods are powerful tools for solving heat equation with a power nonlinearity.
Comparative analysis of x^3+y^3=z^3 and x^2+y^2=z^2 in the Interconnected Sets Vladimir Godovalov
This paper introduces an innovative technique of study z^3-x^3=y^3 on the subject of its insolvability in integers. Technique starts from building the interconnected, third degree sets: A3={a_n│a_n=n^3,n∈N}, B3={b_n│b_n=a_(n+1)-a_n }, C3={c_n│c_n=b_(n+1)-b_n } and P3={6} wherefrom we get a_n and b_n expressed as figurate polynomials of third degree, a new finding in mathematics. This approach and the results allow us to investigate equation z^3-x^3=y in these interconnected sets A3 and B3, where z^3∧x^3∈A3, y∈B3. Further, in conjunction with the new Method of Ratio Comparison of Summands and Pascal’s rule, we finally prove inability of y=y^3. After we test the technique, applying the same approach to z^2-x^2=y where we get family of primitive z^2-x^2=y^2 as well as introduce conception of the basic primitiveness of z^'2-x^'2=y^2 for z^'-x^'=1 and the dependant primitiveness of z^'2-x^'2=y^2 for co-prime x,y,z and z^'-x^'>1.
2022 ملزمة الرياضيات للصف السادس التطبيقي الفصل الثالث - تطبيقات التفاضلanasKhalaf4
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حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
2022 ملزمة الرياضيات للصف السادس الاحيائي الفصل الثالث تطبيقات التفاضلanasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
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Brief description of current state of drones and some future challenges.
The presentation is prepared for delivery in the "Interact with Today's World" conference held in Bibliotica Alexandria 5-6 August 2016
1. 1
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Linear Algebraic Equations
Gauss Elimination
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Objectives
• Knowing how to solve a system of linear
equations
• Understanding how to implement Gauss
elimination method
• Understanding the concepts of singularity
and ill-conditioning
2. 2
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
A System of Linear Equations
1823 21 =+ xx
22 21 =+− xx
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Solution
Subtract second equation
from first
16*04 21 =+ xx
41 =x
32 =x
3. 3
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
In Matrix Form
=
− 2
18
21
23
2
1
x
x
1823 21 =+ xx
22 21 =+− xx
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Elimination
=
− 2
18
21
23
2
1
x
x
Using row operations
=
− 2
16
21
04
2
1
x
x
4. 4
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Elimination
Using row operations
=
− 2
16
21
04
2
1
x
x
=
6
16
20
04
2
1
x
x
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Naïve Elimination Routine!
5. 5
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Back Substitution
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Special Cases
6. 6
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Conclusions
• In this lecture, we revised the process of
Gauss elimination
• A clear algorithm for the elimination
process and the back substitution was
presented
• The different cases of no solution, infinite
number of solutions, and ill conditioning
were graphically presented
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Linear Algebraic Equations
Iterative Solutions
7. 7
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Objectives
• Recognize the need for iterative solutions
• Understand the difference between
different iterative methods for solving
systems of linear equations
• Apply iterative methods to solve a system
of linear equations
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Why Iterative Methods?
• When system of equations is sparse; too many
zero elements. Such systems are produced
when approximately solving differential
equations; finite difference, finite element, etc…
• We already have sources of error in the solution;
model errors, approximation errors, truncation
errors, etc…, so why not use approximate
method any way
• Saves on time!
8. 8
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Gauss-Jacobi
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Gauss-Jacobi: Example
1642
302
1124
321
21
321
=++
=++−
=++
xxx
xx
xxx ( )
( )
( ) 4/216
2/3
4/211
213
12
321
xxx
xx
xxx
−−=
+=
−−=
( )
( )
( ) 4/216
2/3
4/211
21
1
3
1
1
2
32
1
1
kkk
kk
kkk
xxx
xx
xxx
−−=
+=
−−=
+
+
+
9. 9
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Is that really going to work?!!!
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Let’s try it!
{ }
=
1
1
1
0
x
( )
( )
( ) 25.34/1216
22/13
24/1211
1
3
1
2
1
1
=−−=
=+=
=−−=
x
x
x
( )
( )
( ) 5.24/22*216
5.22/23
9375.04/25.32*211
2
3
2
2
2
1
=−−=
=+=
=−−=
x
x
x ( )
( )
( ) 9063.24/5.2875.116
9688.12/9375.03
875.04/5.2511
3
3
3
2
3
1
=−−=
=+=
=−−=
x
x
x
10. 10
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
In General!
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
A system of linear equations
=
nnnnnn
n
n
b
b
b
x
x
x
aaa
aaa
aaa
MM
L
MOMM
L
L
2
1
2
1
21
22221
11211
Let’s examine one equation!
11212111 ... bxaxaxa nn =+++
( )
11
12121
1
...
a
xaxab
x nn++−
=
11. 11
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
For any equation
−−= ∑∑ +=
−
=
+
n
ij
k
jij
i
j
k
jiji
ii
k
i xaxab
a
x
1
1
1
1 1
( )
ii
niniiiiiiii
i
a
xaxaxaxab
x
+++++−
= ++−− ...... 11,11,11
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Can it be any better?
12. 12
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Gauss-Seidel
−−= ∑∑ +=
−
=
+
n
ij
k
jij
i
j
k
jiji
ii
k
i xaxab
a
x
1
1
1
1 1
−−= ∑∑ +=
−
=
++
n
ij
k
jij
i
j
k
jiji
ii
k
i xaxab
a
x
1
1
1
11 1
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Gauss-Seidel: Example
1642
302
1124
321
21
321
=++
=++−
=++
xxx
xx
xxx ( )
( )
( ) 4/216
2/3
4/211
213
12
321
xxx
xx
xxx
−−=
+=
−−=
( )
( )
( ) 4/216
2/3
4/211
1
2
1
1
1
3
1
1
1
2
32
1
1
+++
++
+
−−=
+=
−−=
kkk
kk
kkk
xxx
xx
xxx
13. 13
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Let’s try it!
{ }
=
1
1
1
0
x
( )
( )
( ) 375.24/5.22*216
5.22/23
24/1211
1
3
1
2
1
1
=−−=
=+=
=−−=
x
x
x
( )
( )
( ) 0586.34/9531.1906.0*216
9531.12/90625.03
90625.04/375.2511
2
3
2
2
2
1
=−−=
=+=
=−−=
x
x
x
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Convergence Condition
• For the iterative solutions presented to
converge, the matrix must be diagonally
dominant.
∑
≠
=
>
n
ij
j
ijii aa
1
14. 14
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Error!
{ } { } { }kkk
xxe −= −1
k
i
k
ik
a
x
e
max=ε
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Algorithm
1. If the system is not diagonally dominant; end
2. Start with any initial solution {x}
3. Apply the steps for Gauss-Seidal method to
evaluate the next iteration
4. If the maximum approximate relative error < εs;
end
5. Let the old solution vector equal the new
solution vector
6. Goto step 3
15. 15
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Homework #3
• For the given system of simultaneous equations
• Write down the system of equation in a form that can be used for
iterative methods for solving systems of equations
• Use four iterations using Gauss-Jacobi method to find an
approximate solution using initial values {0,0,0}
• Use four iterations using Gauss-Seidel method to find an
approximate solution using initial values {0,0,0}
1642
1124
32
321
321
21
=++
=++
=+−
xxx
xxx
xx
ENEM602 Spring 2007
Dr. Eng. Mohammad Tawfik
Homework #3 cont’d
• Chapter 11, p 303, numbers:
11.8,11.9,11.10
• Due Next week