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

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 Gauss Elimination Forward Elimination

Last Time Gauss Elimination Back Substitution

Last Time GE – 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

Eliminate Column 1 PIVOTS

Eliminate Column 1

Eliminate Column 2 PIVOTS

Eliminate Column 2

LU Decomposition PIVOTS Column 1 PIVOTS Column 2

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

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

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

LU Decomposition Lyb

Lyb

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