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NM5A_ludecom - LU Decomposition The solution of linear...

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2002 1 LU Decomposition The solution of linear equations How to find [L] and [U]
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2002 2 The basic questions of: [A] {x} ={b} are: 1. Does a solution {x} exist (existence) 2. If a solution exists, is it unique (one solution or an infinity of solutions, uniqueness)
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2002 3 For example, if the equation set is one equation and one unknown: a x =b i) if a 0 then for any b , x = b/a ii) if a=0 & b=0 , then there exists an infinity of solutions iii) if a=0 & b 0 then there is no solution
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2002 4 For coupled sets of equations: [A] {x} ={b} all alternatives are possible except replace a 0 with [a] is invertible (non-singular). Now the question is: How to find {x} ?
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2002 5 If two matrices, [L] & [U], can be found (easily) such that: [L][U]=[A] Then the linear equations: [A] {x} = {b} can be solved for {x} by solving two triangular equations. Known Unknown Known Unknown [L] {c} ={b} [U] {x} ={c}
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2002 6 Which is true because if the equation: [U]{x}={c} is pre-multiplied by [L], then [L][U]{x}=[L]{c} [A] {b}
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2002 7 For example, to find [L] & [U], start with the equation set: 2 u + v + w =5 4 u -6 v +0 w =-2 -2 u +7 v +2 w =9 or, in matrix form: 2 1 1 4 6 0 2 7 2 5 2 9 - - = - u v w
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2002 8 Permissible Operations There are three operations that do not alter the solution to a set of equations. They are: Change the order of the equations Any equation can be multiplied or divided by a nonzero constant It is permissible to add two equations together and use the resulting equation to replace either of the two original equations
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2002 9 Foundation of LU decomposition LU decomposition is an elimination/substitution technique Elimination techniques rely heavily on the second and third permissible operations
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