System of Linear Equation

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  • Topic: System of linear equations, Numerical linear algebra, Pivot element
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TR 3923  Programming Design in Solving Biology  Problems Semester 1, 2011/2012

Elankovan Sundararajan School of Information Technology Faculty of Information Science and Technology TR 3923 Elankovan Sundararajan 1

Lecture 3
System of Linear Equations

TR 3923

Elankovan Sundararajan

2

Introduction
• Solving sets of linear equations is the most frequently used numerical procedure when real-world situations are modeled. modeled Linear equations are the basis for mathematical models of 1. 2. 2 3. 4. 5. Economics, Computational Biology Comp tational Biolog and Bioinformatics Bioinformatics, Weather prediction, Heat and mass transfer, Statistical analysis, and a myriad of other application.





TR 3923

The methods for solving ODEs and PDEs also depend on them.
Elankovan Sundararajan

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System of Linear Equations
• Consider the following general set of n equations in n unknowns: a11 x1  a12 x2    a1n xn  c1 : R1

a21 x1  a22 x2    a2 n xn  c2 ,    an 1,1 x1  an 1, 2 x2    an 1,n xn  cn 1 , an ,1 x1  an , 2 x2    an ,n xn  cn . Which can be written in matrix form as:

: R2

: R n-1 : Rn

A x  b.
~ ~
TR 3923 Elankovan Sundararajan 4

where, A is the nxn matrix, ,

 a11 a12  a1n     a21 a22  a2 n  A .       a an 2  ann   n1 

x iis th ( x 1) column vector, the (n l t
~

x   x1 , x2 ,, xn 
~

T

 x1     x2    .    x   n 5

TR 3923

Elankovan Sundararajan

• Containing the n unknowns which we seek, and g , the known (n x 1) column,

c
~

is

c  c1 , c2 ,, cn 
~

T

 c1     c2    .    c   n

• We shall be employing the Gauss elimination scheme to solve for the vector x . Here the forward elimination ~ process leads to an upper triangular system of equations and the solutions are recovered by means of backward substitution substitution.

TR 3923

Elankovan Sundararajan

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a)
i.

Forward Elimination
First Gauss elimination step Retain R1: a1,1x1+ a1,2x2+ … + a1,nxn=c1. Eliminate x1 from the second to the nth equations

a2,1  R1 , R2 : R2  a1,1
      a a a  a2,1  2,1  a1,1 x1   a2,2  2,1  a1,2 x2   a2,n  2,1  a1,n xn  R2 ~       a1,1 a1,1 a1,1       a  c2  2,1  c1. a1,1

    a a a a2,2  2,1  a1,2 x2  a2,n  2,1  a1,n xn  c2  2,1 c1.  i.e. R2 ~     a1,1 a1,1 a1,1    

TR 3923

 a2 , 2

Elankovan Sundararajan

 a 2 ,n

 c2

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     i.e. R2 ~ a2,2x2  a2,3x3 a2,n xn  c2 . Do the same row operations for R3, R4, … , Rn, i.e.

i.e.

a3,1  R3 : R3  R1 , a1,1 a4,1  R1 , R4 : R4  a1,1   Rn : Rn  an ,1 R1. a1,1

i.e.

ai ,1 Ri : Ri  R1 , a1,1 i  2, 3, , n. ai ,1 ci : ci  c1. a1,1

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Elankovan Sundararajan

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• Therefore at the end of the 1st Gauss elimination Therefore, step, we obtain the following reduced system: R 1 : a1,1 x1  a1, 2 x2  a1,3 x3    a1,n xn  c1 , R : 2 R : 3     a2, 2 x2  a2,3 x3    a2,n xn  c2 ,     a3, 2 x2  a3,3 x3    a3,n xn  c3 ,    R  -1 : n R : n TR 3923

    an 1, 2 x2  an 1,3 x3    an 1,n xn  cn 1 ,     an , 2 x2  an ,3 x3    an ,n xn  cn . Elankovan Sundararajan 9



The 1st equation is called the pivot equation and a1,1 which appears in the multiplier ai ,1 , i=2,3, … , n is a1,1 called the pivot pivot. ii. Second Gauss Elimination step i.e. Ri: Ri  a, 2 i  a2 , 2  R2 ,

i  3, 4, , n. ai, 2  ci : ci  c2 .  a2 , 2

Retain R1, Retain R2, p , and at the end of the 2nd elimination process, we get the equivalent system: TR 3923 Elankovan Sundararajan 10

R 1 : a1,1 x1  a1, 2 x2  a1,3 x3    a1,n xn  c1 , R : 2 R  : 3     a2, 2 x2  a2,3 x3    a2,n xn  c2 ,    a3,3 x3    a3,n xn  c3 ,    R -1 : n R  : n • •

   an1,3 x3    an1,n xn  cn1 ,    an,3 x3    an,n xn  cn ....
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