MTH510_III System of Equations

# MTH510_III System - III System of Equations A]cfw_X =[B 1 Introduction In chapter II studied method of solving single Non-linear equations of the

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1 III System of Equations [ ] { } [ ] B X A =

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2 Introduction • In chapter II, studied method of solving single Non-linear equations of the form f(x) =0 (single-variable) • Many practical engineering problems described by a system of linear/ non-linear equations of the form 0 ) , , , ( 0 ) , , , ( 0 ) , , , ( 2 1 2 1 2 2 1 1 = = = n n n n x x x f x x x f x x x f K M M K K (Multi-Variable)
3 Introduction (Cont.) Objective : Study several algorithms to solve Linear system of Equations of the form constants ' ' , , , , , , 2 2 1 1 2 2 22 1 21 1 1 2 12 1 11 - = + = + = + s b s a b x a x a x a b x a x a x a b x a x a x a n n nn n n n n n n n K M M K K [A]{x}={b} Unknown Vector (nx1) Coefficient Matrix (nxn) RHS Vector (nx1)

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4 Matrix Operating Rules Addition/Subtraction Rule : (nxm) [A] & (nxm) [B] [C] =[A] ± [B] Elements of (nxm) [C] c ij = a ij ± b ij i=1,2,. ., n; j=1,2,…,m [A]+[B]=[B]+[A] Commutative law ([A]+[B])+[C]=[A]+([B]+[C]) Associative Law
5 Rules (Cont.) Multiplication Rule • Multiplication of Matrix by a constant k [C]=k[A] c ij =ka ij • Multiplication of Matrix ( nxm ) Matrix [A] by a ( mxn ) [B] matrix [C] =[A][B] [C] –(nxn) matrix Elements of [C] m- the column dim of [A] and the row dim of [B] kj m k ik ij b a c = = 1

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6 Rules (Cont.) • Matrix Multiplication- Associative ([A][B])[C]=[A]([B][C]) - Distributive Law [A]([B]+[C])=[A][B]+[A][C] Multiplication NOT Commutative [A][B] [B][A] Order of Multiplication Important * Matrix Division Opertaion is NOT Defined
7 Rules (Cont.) Transpose of [A]- [A] T Elements of the Transpose a ij = a ji of [A] Trace of a matrix Matrix Augmentation ] | [ ] [ ~ B A A = = = n i ii a A tr 1 ] [ (nx1) (nxn) (nx(n+1))

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8 3.1 Method of Solution: Small Number of Equations • Example 2 eqns. • Methods – Graphical Method • Little practical value beyond 3 equations • Useful in visualizing properties of the solutions – Cramer’s Rule • Eqns. >3- time consuming to evaluate by hand ) 2 ( ) 1 ( 2 2 22 1 21 1 2 12 1 11 b x a x a b x a x a = + = +
9 x 1 x 1 Intersection (Solution) Eqn 1 Eqn 2

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10 x 1 Parallel Eqn 1 Eqn 2 NO Solution x 2 x 1 Parallel InfiniteSolution x 2
11 3.2 Gauss Elimination Method with

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## This note was uploaded on 09/29/2010 for the course COMPUTER S cps615 taught by Professor Pro during the Spring '10 term at Randolph College.

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MTH510_III System - III System of Equations A]cfw_X =[B 1 Introduction In chapter II studied method of solving single Non-linear equations of the

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