Tr 3923 Week 2 Math Problem Solving Research

TR 3923 Programming Design in Solving Biology Problems Semester 1, 2011/2012

Elankovan Sundararajan School of Information Technology Faculty of Information Science and Technology
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Lecture 3
System of Linear Equations

TR 3923

Elankovan Sundararajan

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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.

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The methods
The pivot element is not zero. When the pivot element is zero, the elimination step leads to division by zero. ii. The pivot element is not close to zero. If the size (the magnitude) of the pivot element is small compared to the other elements, then in the course of our computation, rounding errors will result, which affect the accuracy of our solution. This situation can be rectified by introducing the pivoting strategy strategy.
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• With partial pivoting we can search for the largest coefficient (in the absolute sense) in the given linear system of equations and the interchange rows so that the largest element is that pivot element. • The Gaussian elimination process then follows: Example Solve the system 0.0003x1+3.0000x2=2.0001, 3.0000x 2.0001, 1.0000x1+1.0000x2=1.0000. (Exact solution is x1=1/3 , x2=2/3.) a) without pivoting, b) with partial pivoting.
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• Without pivoting The i Th given system i t is: R1: 0 0003x1+3 0000x2=2 0001 0.0003x +3.0000x =2.0001, R2: 1.0000x1+1.0000x2=1.0000. Note that our pivot is a11=0.0003, which is “close” to zero. Carry out the following