LU decomposition Method

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Margery Edwards
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1 LU decomposition Method Gss eimintion ecomes inefficient when soving eqtions with the sme coefficients for [A] t with different s. LU decomposition seprtes the time consming eimintion of [A] form the mniption of{ }., the decomposed [A] cod e sed with sever{ } s in n efficient mnner. LU decomposition is sed on the fct tht ny sqre mtrix [A] cn e written s prodct of two mtrices s: [A][L][U] Where [L] is ower tringr mtrix nd [U] is n pper tringr mtrix. Crot s method To istrte the Crot s method for LU decomposition, consider the foowing mtrix: ( ) ( ) ( + ) ( + ) ( ) ( ) Eements of the mtrices [L] nd [U] cn e fond y eqting the two ove mtrices: ; ;, hence +, hence +, hence, hence +, hence + +, hence

2 For gener n n mtrix, the foowing expressions cn e ppied to find the LU decomposition of mtrix [A]: j ij ij ik k i ij ik k ij ii kj nd ; i,,,n ii kj ; i j; i,,,n ; i<j; j,,,n Note: It is etter to foow certin order when compting the terms of the [L] nd [U] mtrices. This order is: i, j ; i, j; ; i,n-, n-,,j ; nn. Properties of LU mtrices of ALU () The L mtrices: s in digon, sme mtipiers ij s in the eimintion mtrices () When row of A strts with eros, so does tht row of L (c) When comn of A strts with eros, so does tht comn of U Exmpe Find the LU decomposition of the mtrix Sotion [ A ] [ L][ U ] [ ] The U 8 mtrix is the sme s fond t the end of the forwrd eimintion of Nïve Gss eimintion method, tht is [ U ].6.7 To find nd, find the mtipier tht ws sed to mke the nd eements ero in the first step of forwrd eimintion of the Nïve Gss eimintion method. It ws 64. 6

3 To find, wht mtipier ws sed to mke [ ] mde ero in the second step of forwrd eimintion. The A second step of forwrd eimintion ws So 6.8. [ L] [ ][ ] [ ] Confirm L U A. [ L][ U ] eement ero? Rememer eement ws mtrix t the eginning of the Another exmpe Perform LU decomposition of the foowing mtrix - -6 Mtipy the first row y f -/ -. nd strct the rest from the second row to eiminte the term. Then, mtipy the first row y f /. nd strct the rest from the third row to eiminte the term. The rest is Mtipy the second row y f.8/(-.4) nd strct the rest from the third row to eiminte the term

4 Therefore, the LU decomposition is [ L][ U ] Mtipying [L] nd [U] yieds the origin mtrix s verified y the foowing MATLAB session, >> L [ ;-. ; ]; >> U [ -; -.4.7;.8]; >> A L*U A Sotion of iner eqtions y LU decomposition Now to sove the system of iner eqtions, the initi system cn e expressed s: [ A ]{ x} { } Under the foowing form [ A ]{ x} [ L][ U ]{ x} { } To find the sotion{ x }, the first vector{ } cn e defined: { } [ U ]{ x} Or initi system ecomes, then: [ L ]{ } { } As [L] is ower tringr mtrix the { } cn e compted strting y nti n. Then the ves of { x } cn e fond sing the eqtion: { } [ U ]{ x} s [U] is n pper tringr mtrix, it is possie to compte{ x } sing ck sstittion process strting x n nti x. [Yo wi etter nderstnd with n exmpe ] The gener form to sove system of iner eqtions sing LU decomposition is: 4

5 i i ik k k i ; i,,..., n And ii x n x i i n n ik k i+ x k ; i n, n,...,, Exmpe Use the LU decomposition method to sove the foowing simtneos iner eqtions Sotion Rec tht [ A ][ X ] [ C] nd if [ A ] [ L][ U ] then first soving [ L ][ Z ] [ C] nd then [ U ][ X ] [ Z ] gives the sotion vector [ X ]. Now in the previos exmpe, we showed A L U [ ] [ ][ ].6.76 First sove [ L ][ Z] [ C] to give

6 Forwrd sstittion strting from the first eqtion gives [ Z ] ( 96.8) This mtrix is sme s the right hnd side otined t the end of the forwrd eimintion steps of Nïve Gss eimintion method. Is this coincidence? Now sove [ U ][ X ] [ Z] From the third eqtion Sstitting the ve of in the second eqtion, Sstitting the ve of nd in the first eqtion, 6

7 the sotion vector is Inverse of sqre mtrix sing LU decomposition? A mtrix [ B ] is the inverse of if A B I B A [ ][ ] [ ] [ ][ ]. How cn we se LU decomposition to find the inverse of the mtrix? Assme the first comn of [ B ] (the inverse of [ A ]) is T [ n ] Then from the ove definition of n inverse nd the definition of mtrix mtipiction n Simiry the second comn of [ ] n Simiry, comns of [ B] B is given y cn e fond y soving n different sets of eqtions with the comn of the right hnd side eing the n comns of the identity mtrix. Exmpe Use LU decomposition to find the inverse of Sotion Knowing tht [ A ] [ L][ U ] 8 7

8 We cn sove for the first comn of [ B] First sove [ L ][ Z] [ C] tht is, to give y soving for Forwrd sstittion strting from the first eqtion gives. 6.6( ) ( ).(.6).76. [ Z ].6. Now sove [ U ][ X ] [ Z] tht is

9 Bckwrd sstittion strting from the third eqtion gives (4.7).94 (.94) the first comn of the inverse of [ ] Simiry y soving nd soving gives gives A is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If