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ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 17 Solution of Systems of Equations

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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 Gauss Elimination Forward Elimination

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Last Time Gauss Elimination Back Substitution

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Last Time GE – 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

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Eliminate Column 1 PIVOTS

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Eliminate Column 1

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Eliminate Column 2 PIVOTS

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Eliminate Column 2

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LU Decomposition PIVOTS Column 1 PIVOTS Column 2

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LU Decomposition As many as, and in the location of, zeros Upper Triangular Matrix U

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LU Decomposition PIVOTS Column 1 PIVOTS Column 2 Lower Triangular Matrix L

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LU Decomposition = This is the original matrix!!!!!!!!!!

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LU Decomposition Lyb

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Lyb

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Ax=b A=LU -LU Decomposition Ly=b- Solve for y Ux=y- Solve for x

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