Autar Kaw Benjamin Rigsby. Transforming Numerical Methods Education for STEM Undergraduates

Transcription

1 Autar Kaw Benjamin Rigsy Transforming Numerical Methods Education for STEM Undergraduates

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3 LU Decomposition is another method to solve a set of simultaneous linear equations Which is etter, Gauss Elimination or LU Decomposition?

4 Method For most non-singular matrix [A] that one could conduct Naïve Gauss Elimination forward elimination steps, one can always write it as where [A] [L][U] [L] lower triangular matrix [U] upper triangular matrix

5 If solving a set of linear equations If [A] [L][U] then Multiply y Which gives Rememer [L] - [L] [I] which leads to Now, if [I][U] [U] then Now, let Which ends with and [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 How can this e used? Given [A][X] [C] Decompose [A] into [L] and [U] Solve [L][Z] [C] for [Z] Solve [U][X] [Z] for [X]

7 Solve [A][X] [B] T clock cycle time and n n sie of the matrix Forward Elimination CT FE 8n T + 8n ( 4n + n) CT BS T n Back Sustitution Decomposition to LU CT DE 8n T + 4n n Forward Sustitution ( 4n n) CT FS T 4 Back Sustitution ( 4n + n) CT BS T

8 Time taken y methods To solve [A][X] [B] Gaussian Elimination LU Decomposition T n 8 + n 4n + T T clock cycle time and n n sie of the matrix n 8 + n + 4n So oth methods are equally efficient

9 Time taken y Gaussian Elimination n ( CT + CT ) 8n T 4 FE + n + BS 4n Time taken y LU Decomposition CT LU n T + n CT + n FS + n CT n + BS

10 Time taken y Gaussian Elimination 8n T 4 + n + 4n Time taken y LU Decomposition n T + n n Tale Comparing computational times of finding inverse of a matrix using LU decomposition and Gaussian elimination n CT inverse GE / CT inverse LU For large n, CT inverse GE / CT inverse LU n/4

11 [ A] [ L][ U ] u u u u u u [U] is the same as the coefficient matrix at the end of the forward elimination step [L] is otained using the multipliers that were used in the forward elimination process

12 Using the Forward Elimination Procedure of Gauss Elimination Step : 64 5 ( ) 56; Row Row ( ) 576; Row Row

13 Matrix after Step : Step : ; Row Row ( 5) [ U ]

14 Using the multipliers used during the Forward Elimination Procedure From the first step of forward elimination a a 44 5 a a

15 From the second step of forward elimination a a [ L]

16 [ L][ U ] ?

17 Solve the following set of linear equations using LU Decomposition x x x Using the procedure for finding the [L] and [U] matrices [ ] [ ][ ] U L A

18 Set [L][Z] [C] Solve for [Z]

19 Complete the forward sustitution to solve for [Z] ( ) ( ) 5( 96) [ Z ]

20 Set [U][X] [Z] x x x Solve for [X] The equations ecome 5a 48a + 5a + a 56a 7a

21 From the rd equation Sustituting in a and using the second equation 7a a a a a a 4 8a 56 a a ( )

22 Sustituting in a and a using the first equation Hence the Solution Vector is: 5a + 5a + a 6 8 a 6 8 5a a ( ) 5 a a a

23 The inverse [B] of a square matrix [A] is defined as [A][B] [I] [B][A]

24 How can LU Decomposition e used to find the inverse? Assume the first column of [B] to e [ n ] T Using this and the definition of matrix multiplication First column of [B] Second column of [B] [ ] n A [ ] n A The remaining columns in [B] can e found in the same manner

25 Find the inverse of a square matrix [A] [ A] Using the decomposition procedure, the [L] and [U] matrices are found to e [ A] [ L][ U ]

26 Solving for the each column of [B] requires two steps ) Solve [L] [Z] [C] for [Z] ) Solve [U] [X] [Z] for [X] Step : [ ][ ] [ ] C Z L This generates the equations:

27 Solving for [Z] ( ) ( ) ( ) [ ] 56 Z

28 Solving [U][X] [Z] for [X]

29 Using Backward Sustitution ( 457) ( 954) So the first column of the inverse of [A] is:

30 Repeating for the second and third columns of the inverse Second Column Third Column

31 The inverse of [A] is [ A] To check your work do the following operation [A][A] - [I] [A] - [A]

32 For all resources on this topic such as digital audiovisual lectures, primers, textook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, logs, related physical prolems, please visit nhtml

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