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

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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

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Last Time Linear Equations in Matrix Form

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

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

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

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

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

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Last Time Gauss Elimination Example Forward Elimination

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Last Time-Correction Gauss Elimination -

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Substitute 2 nd eq with new

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Last Time-Correction Gauss Elimination -

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Substitute 3rd eq with new

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Last Time-Correction Gauss Elimination -

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Substitute 3rd eq with new

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Last Time-Correction Gauss Elimination

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Last Time Gauss Elimination Forward Elimination

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Last Time-Correction Gauss Elimination Back Substitution

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Gauss Elimination – Potential Problem Forward Elimination

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

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Gauss Elimination – Potential Problem OK!!

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Gauss Elimination – Potential Problem Pivoting

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

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Partial Pivoting

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

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EXAMPLE We will work directly on the coefficient matrix

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