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Lecture on Numerical Methods Unit 5

SOLUTIONS TO LINEAR ALGEBRIC

EQUATIONS (Part I)

Prepared By: Er. Sarbesh Chaudhary H.O.D (Electronics Department) MMP

• Basic Fundamentals
• Systems of Linear equations
• Solutions of Linear Equations
• Direct or EliminationMethods
  • Basic Gauss Elimination
  • Gauss Elimination with partial pivoting
  • Gauss Jordon method
  • LU decomposition methods
    • Do Little Algorithm
    • Crout Algorithm
  • Matrix Inversion Method
  • Cholesky’s Method

CONTENTS

• MATRICES

, Tridiagonal

diagonal

identitymatrix

zeromatrix

Examples :

Matrix:a two dimensionalarray of numbers

BASIC FUNDAMENTALS

• Upper and Lower Triangular Matrix
• Upper Triangular Matrix
• It is a square matrix in which all the items

under the main diagonal are zero.

BASIC FUNDAMENTALS

• Lower Triangular Matrix
• It is a square matrix in which all the elements above the main diagonal are zero.

• Transpose Matrix

Let any matrix A=(aij) of mxn order, then AT^ is its transpose if the A rows are the AT^ columns

BASIC FUNDAMENTALS

• Symmetric Matrix

It is a square matrix in which the elements are symmetric about the main

diagonal

Example:

If A is a symmetric matrix, then:

a. The product A x AT^ is defined and is a symmetric

matrix.

b. The sum of symmetric matrices is a symmetric matrix.

c. The product of two symmetric matrices is a symmetric matrix if the

matrices commute

BASIC FUNDAMENTALS

det

Example :

BASIC FUNDAMENTALS

• Addition of Matrices
• Multiplication of Matrices

BASIC FUNDAMENTALS

C A B c a b i , j

theyhave the samesize

The addition of two matricesA and Bis possibleonlyif

   ij  ij  ij 

i j

m

k

C AB c a b ,

The productC ABisdefinedonlyif m p

Multiplicationof two matricesA(n m)and B(p q)

1

  ij  ik kj 

 

 



Standard form Matrix form

in differentforms

A systemof linear equationscan bepresented

3

2

1

1 3

1 2 3

1 2 3

x

x

x

x x

x x x

x x x

SYSTEMS OF LINEAR EQUATIONS

2 5

3

isasolution tothefollowingequations: 2

1

1 2

1 2

2

1

 

 

 

  

   

  



x x

x x

x

x

SOLUTIONS OF LINEAR EQUATIONS

• A set of equations is inconsistent if there exists no solution to the system of equations:

These equationsareinconsistent

1 2

1 2

 

x x

x x

• There are two classes of methods for solving system of linear, algebraic equations:

1.Direct methods.

2. Indirect or Iterative methods

• Direct methods
• They transform the original equation into equivalent equations (equations that have

the same solution) that can be solved more easily.

• The transformation is carried out by applying certain operations.
• The solution does not contain any truncation errors but the round off errors is

introduced due to floating point operations.

SOLUTIONS OF LINEAR EQUATIONS

• Indirect methods
• Iterative or indirect methods , start with a guess of the solution x , and then repeatedly refine

the solution until a certain convergence criterion is reached.

• Generally less efficient than direct methods since, large number of operations or iterations

required.

• Iterative procedures are self-correcting, meaning that round off errors (or even arithmetic

mistakes) in one iteration cycle are corrected in subsequent cycles.

• The solution contains truncation error.
• A serious drawback of iterative methods is that they do not always converge to the solution.

SOLUTIONS OF LINEAR EQUATIONS

DIRECT METHODS

• Gaussian Elimination Method procedure:

1. Write the augmented matrix for the system of linear equations.
2. Use elementary row operations on the augmented

matrix [A|B] to transform A into upper triangular form. If

a zero is located on the diagonal, switch the rows until a

nonzero is in that place. If you are unable to do so, stop;

the system has either infinite or no solutions.

3. Use back substitution to find the solution of the problem.

GAUSS ELIMINATION METHOD

Karl Friedrich Gauss Great mathematician 19th century