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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
Full Pivoting In addition to row swaping Search columns for max elements Swap Columns Change the order of x i Most cases not necessary
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
Ax=b A=LU -LU Decomposition Ly=b- Solve for y Ux=y- Solve for x
Scientific Computing Linear Systems – Gaussian Elimination.
Simultaneous Linear Equations
Grayson Ishihara Math 480 April 15, What is Partial Pivoting? What is the PA=LU Factorization? What kinds of things can we use these tools.
SOLVING SYSTEMS OF LINEAR EQUATIONS. Overview A matrix consists of a rectangular array of elements represented by a single symbol (example: [A]). An individual.
CISE301_Topic3KFUPM1 SE301: Numerical Methods Topic 3: Solution of Systems of Linear Equations Lectures 12-17: KFUPM Read Chapter 9 of the textbook.
Lecture 9: Introduction to Matrix Inversion Gaussian Elimination Sections 2.4, 2.5, 2.6 Sections 2.2.3, 2.3.
Part 3 Chapter 9 Gauss Elimination
Solution of linear system of equations
Chapter 9 Gauss Elimination The Islamic University of Gaza
Major: All Engineering Majors Author(s): Autar Kaw
Linear Algebraic Equations
Linear Systems What is the Matrix?. Outline Announcements: –Homework III: due Wed. by 5, by Office Hours: Today & tomorrow, 11-1 –Ideas for Friday?
The Islamic University of Gaza Faculty of Engineering Civil Engineering Department Numerical Analysis ECIV 3306 Chapter 10 LU Decomposition and Matrix.
ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 20 Solution of Linear System of Equations - Iterative Methods.
ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 14 Elimination Methods.
ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 15 Solution of Systems of Equations.
ECIV 520 Structural Analysis II Review of Matrix Algebra.
Dr. Jie Zou PHY Chapter 3 Solution of Simultaneous Linear Algebraic Equations: Lecture (III) Note: Besides the main textbook, also see Ref: Applied.
Math for CSLecture 31 LU Decomposition Pivoting Diagonalization Gram-Shcmidt Orthogonalization Lecture 3.
Algorithm for Gauss elimination 1) first eliminate for each eq. j, j=1 to n-1 for all eq.s k greater than j a) multiply eq. j by a kj /a jj b) subtract.
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