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LU Decomposition

LU Decomposition. Method If an nxn matrix A has an LU-factorization, then the solution of AX = b can be determined by a Forward substitution followed by a Back substitution [ A ] = [ L ][ U ] where [ L ] = lower triangular matrix all of whose diagonal

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LU Decomposition

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  1. LU Decomposition Method If an nxn matrix A has an LU-factorization, then the solution of AX = b can be determined by a Forward substitution followed by a Back substitution [A] = [L][U] where [L] = lower triangular matrix all of whose diagonal entries are different from zero. [U] = upper triangular matrix all of whose diagonal entries are different from zero.

  2. LU Decomposition How can this be used? • Given [A][X] = [b] • Decompose [A] into [L] and [U] • (LU)X = b • L(UX) = b ; Let UX = z • then LZ = b • Use Forward substitution to Solve [L][Z] = [b] for [Z] • Use Back substitution to Solve • [U][X] = [Z] for [X]

  3. Exercise 2.5 ; Page # 136; Qn # 1

  4. Ax = b(LU)X = bL(UX) = bLZ = b; Let UX = Z

  5. [L][Z] = [b] Solve for [Z] using Forward Substitution :

  6. Complete the forward substitution to solve for [Z]

  7. [U][X] = [Z] Solve for [X] using Back Substitution :

  8. The Solution is :

  9. Method: [A] Decompose to [L] and [U]Storage of Multipliers Scheme :

  10. Exercise 2.5; Page #136; Qn # 5 Finding the [U] matrix Step 1:

  11. Finding the [U] Matrix Matrix after Step 1: Step 2:

  12. Finding the [L] matrix http://numericalmethods.eng.usf.edu

  13. Finding the [L] Matrix

  14. [L][Z] = [b] Solve for [Z] using Forward Substitution :

  15. Complete the forward substitution to solve for [Z]

  16. [U][X] = [Z] Solve for [X] using Back Substitution :

  17. The Solution is :

  18. Limitations to use this Procedure : • Interchanging of any two rows not allowed . • Only elementary row operation permitted is the one that subtract a multiple of one row to another. • In matrix A, if then this procedure fails. • In matrix if , or in if then this procedure fails.

  19. Test for you?Exercise 2.5 ; Page # 136; Qn # 7

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