Ch 8 - Solving Systems of Equations z Many Science &...

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Solving Systems of Equations + Many Science & Engineering problems require the solution to systems of equations + General Rule: Need as many (unique) equations as unknowns + Guide to putting equations in Natural Form: 1. Write the equations in their natural form 2. Identify Unknowns, and order them 3. Isolate the unknowns 4. Write Equations in Matrix form (Ax=b) 11 1 12 2 1 1 21 1 22 2 2 2 11 2 2 nn n n n n ax ax ax b !! !" ± ± ² ± Ax b " 1 In General: For Ax=b, we need same number of equations as unknowns (a square matrix) 11 12 13 21 22 23 31 32 33 aaa #$ %& '( 11 12 13 21 22 23 31 32 33 41 42 43 11 12 13 14 21 22 23 24 31 32 33 34 aaaa N equations = N unknowns M equations > N unknowns OVERDETERMINED M equations < N unknowns UNDERDETERMINED 2
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Special Matrices + Diagonal has zeros for all elements except i=j + Inverse has property that A -1 A=I and AA -1 =I (Impt for solving systems of equations) + Symmetric matrix is equal to its transpose 1 12 0 (, , , ) 0 n n c Cd i a g cc c c #$ %& "" '( ± ±² ³ ² ´ 11 1 1 A Ax Ab I x Ab ±± ± ± "* " 52 1 26 1 3 3 More Special Matrices + Tridiagonal matrices are square with nonzero entries on main diagonal, and the diagonals above and below the main diagonal + Positive Definite matrices have all positive eigenvalues + (A symmetric, diagonally dominant matrix is always positive definite) + Diagonally dominant means that the sum of the Absolute Value of the off diagonals is less than the absolute value of the main diagonals 1 n ii ij j ji aa " + ² - 11 12 1 21 22 n mm m n a a ± ³ 21 0 0 12 10 01 00 1 2 ± ./ ± 23 222 33 3 111 nnn nn ab cab ca b ca ±±± ³³ ³ 4
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Small Systems of Equations 1. Isolate Unknowns 2. Graphical Approach 3. Determinants and Cramer’s Rule 12 32 1 8 22 xx !" ±! " 1 2 18 3 2 x x ± " "± ! 4 5 1 2 2 18 3 12 3 2 2 2 x x x ± ./ 23 "" ± ±! 2 1 3 ( )4 3 x x " " 1 2 18 3 2 x x ± " 5 Requirements for a solution + Need as many ( unique ) equations as unknowns + Having a square matrix (n equations and n unknowns) isn’t good enough + The matrix A must be non-singular , or the rank(A) must equal n + The rows must be linearly independent 6
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Gauss Elimination is a Fundamental procedure for solving systems of linear equations + Diagonal systems are easy: + Consider the following: 100 1 030 , 6 005 1 5 Ab ± #$ # $ %& % & "" % & % & ± '( ' ( 1 2 3 1 36 51 5 x x x " x 3 = -15/5 = -3 x 2 = 6/3 = 2 x 1 = -1/1 = -1 Back Substitution 7 Solving Triangular Systems: Back Substitution + Upper triangular matrix has zeros in all positions below the main diagonal. Lower triangular matrix has zeros above the main diagonal 21 2 9 03 2 , 1 004 8 ± # $ % & " ± % & % & ' ( 12 3 23 3 22 9 32 1 48 xx x x !! " ± " ± " 3 8 2 4 x 45 13 1 33 ! " " 3 14 92 2 x ± " " ± ±± The Solution to the last equation: Now that x 3 is known, it is substituted into the 2 nd equation: Finally x 2 and x 3 are substituted into the 1 st equation: 8
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A few properties of Systems of Equations + You can write the equations in any order + You can always multiply both sides by a scalar + You can add/subtract two equations to replace an existing one 12 22 32 1 8 xx ±! " !" 1 8 4 5 23 6 1 8 1/3 4 5 45 2 6 8 08 3 x ±±±±±±±±± 4 5 2 6 8 3 x 9 Naïve Gaussian Elimination (No Pivoting) + Always write system in terms of an “augmented” matrix +
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Ch 8 - Solving Systems of Equations z Many Science &amp;...

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