SelisOnelLectureNotes_SolLinEqnI_DirectMethods

SelisOnelLectureNotes_SolLinEqnI_DirectMethods - Selis nel...

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Selis Önel, PhD
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Solution of Linear systems Direct Methods -Elimination Methods -Inverse of a matrix - Cramer’s Rule -LU Decomposition Indirect Methods Iterative Methods 2 Selis Önel, PhD
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A x = y Works better for coefficient matrices with no or few zeros Coefficient matrix A is augmented with the y matrix Good if |A i,j |~|y i |, Not good if the elements of y are too different from those of A The goal is to: obtain an upper triangular matrix (lower triangular part all zeros) by forward elimination of the coefficients below the diagonal coefficients Get the values of x by back substitution 3 Selis Önel, PhD
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(1) a 1,1 x 1 + a 1,2 x 2 + …+ a 1,n x n = y 1 (2) a 2,1 x 1 + a 2,2 x 2 + …+ a 2,n x n = y 2 .. (n) a n,1 x 1 + a n,2 x 2 + …+ a n,n x n = y n Forward elimination: Eliminate the 1st term a n,1 x 1 of all rows below the 1st row, for n>1: Multiply Eq(1) by a n,1 /a 1,1 Subtract from all equations as n=2:n 4 Selis Önel, PhD
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First elimination: (1) a 1,1 x 1 + a 1,2 x 2 + … + a 1,n x n = y 1 (2)’ 0 + a’ 2,2 x 2 + … + a’ 2,n x n = y’ 2 (n)’ 0 + a’ n,2 x 2 + … + a’ n,n x n = y’ n ,1 , , 1, 1,1 1 ' ' i i j i j j i ii a a a a a a y y y a  For i >1 5 Selis Önel, PhD
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,2 , , 2, 2,2 2 ' '' ' ' ' ' ' ' ' i i j i j j i ii a a a a a a y y y a  Second elimination: (1) a 1,1 x 1 + a 1,2 x 2 + a 1,3 x 3 + … + a 1,n x n = y 1 (2)’ 0 + a’ 2,2 x 2 + a’ 2,3 x 3 + … + a’ 2,n x n = y’ 2 (3)’’ 0 + 0 + a’’ 3,3 x 3 + … + a’’ 3,n x n = y’ 3 (n) (n-1) 0 + 0 + 0 + … + a (n-1) n,n x n = y (n-1) n For i >2 6 Selis Önel, PhD
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Back substitution: Forward elimination Upper triangular matrix The last equation is a (n-1) n,n x n = y (n-1) n ( 1) ( 1) , ( 2) ( 1 1, 1 ( 1 1 2 1 1,1 ... n n n n nn n n n n n n n ii i y x a y a x x a y a x x a  7 Selis Önel, PhD
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Assume m≠n (1) a 1,1 x 1 + a 1,2 x 2 + …+ a 1,n x n = y 1 (2) a 2,1 x 1 + a 2,2 x 2 + …+ a 2,n x n = y 2 .. (m) a m,1 x 1 + a m,2 x 2 + …+ a m,n x n = y m Check if a 1,1 =0, if so reorganize, then Multiply Eqn. (1) by a i,1 /a 1,1 Subtract from all equations as i=2:m Repeat this procedure until the elements under a i,i are all 0 8 Selis Önel, PhD
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If the original equations are not all linearly independent, Reduced equations may be: (1) a 1,1 x 1 + a 1,2 x 2 + … … + a 1,n x n = y 1 (2) a 2,2 x 2 + … … + a 2,n x n = y 2 .. (m) a m,m x m + … + a m,n x n = y m Family of possible solutions with m variables 9 Selis Önel, PhD
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If elimination ends before the upper triangular matrix is obtained: (1) a 1,1 x 1 + a 1,2 x 2 + + a 1,n x n = y 1 (2) a 2,2 x 2 + + a 2,n x n = y 2 (..) (k) a k,k x k + a k,k+1 x k+1 +… + a k,n x n = y k (k+1) 0 = y k+1 (..) (m) 0 = y m Family of possible solutions if n>k and y k+1 , …, y m = 0 Unique solution if n=k and y k+1 , …, y m = 0 Inconsistent equations if at least one of y k+1 , …, y m ≠ 0 10 Selis Önel, PhD
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1 1 3 1 (1) 1 2 5 2 (2) 3 1 5 1 (3)    1 1 3 1 0 1 2 1 (2') 0 0 0 2 (3'') (3') 2(2') 1 1 3 1 0 1 2 1 (2) 0 2 4 4 (3') (3) 3(1)  Augmented matrix [A , y] First elimination Second elimination Result 0=-2 shows the equations are inconsistent Det(A)=0 11 Selis Önel, PhD
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If a diagonal element (or coefficient) a i,i
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This note was uploaded on 03/14/2012 for the course CHEMICAL E kmu 206 taught by Professor Onel,selis during the Spring '08 term at Hacettepe Üniversitesi.

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SelisOnelLectureNotes_SolLinEqnI_DirectMethods - Selis nel...

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