ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 15 Solution of Systems of Equations

Objectives Introduction to Matrix Algebra Express System of Equations in Matrix Form Introduce Methods for Solving Systems of Equations Advantages and Disadvantages of each Method

Last Time Linear Equations in Matrix Form

# Equations = # Unknowns = n Square Matrix n x n

Last Time Solution of Linear Equations Express In Matrix Form Upper Triangular What is the characteristic? Solution by Back Substitution

Last Time Solution of Linear Equations Objective Can we express any system of equations in a form 0

Last Time Background Consider (Eq 1) (Eq 2) Solution 2*(Eq 1) (Eq 2) Solution !!!!!! Scaling Does Not Change the Solution

Last Time Background Consider (Eq 1) (Eq 2)-(Eq 1) Solution !!!!!! (Eq 1) (Eq 2) Solution Operations Do Not Change the Solution

Last Time Gauss Elimination Example Forward Elimination

Last Time-Correction Gauss Elimination -

Substitute 2 nd eq with new

Last Time-Correction Gauss Elimination -

Substitute 3rd eq with new

Last Time-Correction Gauss Elimination -

Substitute 3rd eq with new

Last Time-Correction Gauss Elimination

Last Time Gauss Elimination Forward Elimination

Last Time-Correction Gauss Elimination Back Substitution

Gauss Elimination – Potential Problem Forward Elimination

Gauss Elimination – Potential Problem Division By Zero!! Operation Failed

Gauss Elimination – Potential Problem OK!!

Gauss Elimination – Potential Problem Pivoting

Partial Pivoting a 32 >a 22 a l2 >a 22 NO YES

Partial Pivoting

Full Pivoting In addition to row swaping Search columns for max elements Swap Columns Change the order of x i Most cases not necessary

EXAMPLE We will work directly on the coefficient matrix