LU Decomposition. Mechanical Engineering Majors. Authors: Autar Kaw

Transcription

1 LU Decomposition Mechnicl Engineering Mjors Authors: Autr Kw Trnsforming Numericl Methods Eduction for STEM Undergrdutes //

2 LU Decomposition

3 LU Decomposition LU Decomposition is nother method to solve set of simultneous liner equtions Which is etter, Guss Elimintion or LU Decomposition? To nswer this, closer look t LU decomposition is needed

4 Method For most non-singulr mtrix [A] tht one could conduct Nïve Guss Elimintion forwrd elimintion steps, one cn lwys write it s where LU Decomposition [L] lower tringulr mtrix [U] upper tringulr mtrix [A] [L][U]

5 How does LU Decomposition work? If solving set of liner equtions If [A] [L][U] then Multiply y Which gives Rememer [L] - [L] [I] which leds to Now, if [I][U] [U] then Now, let Which ends with nd [A][X] [C] [L][U][X] [C] [L] - [L] - [L][U][X] [L] - [C] [I][U][X] [L] - [C] [U][X] [L] - [C] [L] - [C][Z] [L][Z] [C] () [U][X] [Z] ()

6 LU Decomposition How cn this e used? Given [A][X] [C] Decompose [A] into [L] nd [U] Solve [L][Z] [C] for [Z] Solve [U][X] [Z] for [X]

7 When is LU Decomposition etter thn Gussin Elimintion? Tle Time tken y methods To solve [A][X] [B] Gussin Elimintion LU Decomposition T n 8 + n 4n + where T clock cycle time nd n sie of the mtrix T n 8 + n 4n + So oth methods re eqully efficient

8 To find inverse of [A] Time tken y Gussin Elimintion n ( CT + CT ) 8n T 4 FE + n + BS 4n Time tken y LU Decomposition CT LU n T + n CT + n FS + n CT n + BS Tle Compring computtionl times of finding inverse of mtrix using LU decomposition nd Gussin elimintion n CT inverse GE / CT inverse LU 8 8 8

9 Method: [A] Decompose to [L] nd [U] [ A] [ L][ U ] u u u u u u [U] is the sme s the coefficient mtrix t the end of the forwrd elimintion step [L] is otined using the multipliers tht were used in the forwrd elimintion process

10 Finding the [U] mtrix Using the Forwrd Elimintion Procedure of Guss Elimintion Step : 64 ( ) 6; Row Row ( ) 76; Row Row

11 Finding the [U] Mtrix Mtrix fter Step : Step : ; Row Row ( ) [ U ]

12 Finding the [L] mtrix Using the multipliers used during the Forwrd Elimintion Procedure From the first step of forwrd elimintion

13 Finding the [L] Mtrix From the second step of forwrd elimintion [ L] 6 76

14 Does [L][U] [A]? 76 [ L][ U ] ?

15 Exmple: Therml Coefficient A trunnion of dimeter 6 hs to e cooled from room temperture of 8 F efore it is shrink fit into steel hu The eqution tht gives the dimetric contrction ΔD of the trunnion in dryice/lcohol (oiling temperture is 8 F is given y: 8 D 6 α( T ) dt 8 Figure Trunnion to e slid through the hu fter contrcting

16 Exmple: Therml Coefficient The expression for the therml expnsion coefficient, + T + T is otined using regression nlysis nd hence solving the following simultneous liner equtions: Find the vlues of,,nd using LU Decomposition

17 Exmple: Therml Coefficient Use Forwrd Elimintion to find the [U] mtrix ( ) ; Row Row ( ) ; Row Row Step

18 Exmple: Therml Coefficient ( ) ; Row Row [ ] U This is the mtrix fter the st step Step

19 Exmple: Therml Coefficient Use the multipliers from Forwrd Elimintion From the first step of forwrd elimintion

20 Exmple: Therml Coefficient From the second step of forwrd elimintion [ L] 97 9

21 Exmple: Therml Coefficient Does [L][U] [A]? [ L][ U ] ?

22 Exmple: Therml Coefficient Set [L][Z] [C] ( ) Solve for [Z]

23 Exmple: Therml Coefficient Solve for [Z] ( 97 ) ( 97 ) 7 4 [ Z ] ( 9 ) ( 9 ) 797

24 Exmple: Therml Coefficient Set [U][A] [Z] Solve for A The equtions ecome ( 86) ( 9997 )

25 Exmple: Therml Coefficient Solve for A ( 9997 ) ( 9997 ) ( 466 )

26 Exmple: Therml Coefficient ( 86) ( 86) ( 466 )

27 Exmple: Therml Coefficient The solution vector is The polynomil tht psses through the three dt points is then: α ( T ) + T + T T 466 T +

28 Finding the inverse of squre mtrix The inverse [B] of squre mtrix [A] is defined s [A][B] [I] [B][A]

29 Finding the inverse of squre mtrix How cn LU Decomposition e used to find the inverse? Assume the first column of [B] to e [ n ] T Using this nd the definition of mtrix multipliction First column of [B] Second column of [B] [ ] n A [ ] n A The remining columns in [B] cn e found in the sme mnner

30 Exmple: Inverse of Mtrix Find the inverse of squre mtrix [A] [ A] Using the decomposition procedure, the [L] nd [U] mtrices re found to e [ A] [ L][ U ]

31 Exmple: Inverse of Mtrix Solving for the ech column of [B] requires two steps ) Solve [L] [Z] [C] for [Z] ) Solve [U] [X] [Z] for [X] Step : [ ][ ] [ ] 76 6 C Z L This genertes the equtions:

32 Exmple: Inverse of Mtrix Solving for [Z] ( ) ( ) ( ) [ ] 6 Z

33 Exmple: Inverse of Mtrix Solving [U][X] [Z] for [X]

34 Exmple: Inverse of Mtrix Using Bckwrd Sustitution ( 47) ( 94) So the first column of the inverse of [A] is:

35 Exmple: Inverse of Mtrix Repeting for the second nd third columns of the inverse Second Column Third Column

36 Exmple: Inverse of Mtrix The inverse of [A] is [ A] To check your work do the following opertion [A][A] - [I] [A] - [A]

37 Additionl Resources For ll resources on this topic such s digitl udiovisul lectures, primers, textook chpters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MthCd nd MAPLE, logs, relted physicl prolems, plese visit /topics/lu_decomp ositionhtml

38 THE END