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Chapter 9.1 = LU Decomposition

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Chapter 9.1 = LU Decomposition. MATH 264 Linear Algebra. Introduction:. LU Decomposition is very usefu l when we have large matrices n x n and if we use gauss- jordan or the other methods, we can get errors.

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Chapter 9.1 = LU Decomposition
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  1. Chapter 9.1 = LU Decomposition MATH 264 Linear Algebra

  2. Introduction: • LU Decomposition is very useful when we have large matrices n x n and if we use gauss-jordan or the other methods, we can get errors. • Computers use this method because of roundoff errors, memory usage, and speed concerns

  3. Method of LU Decomposition: • Rewrite the system Ax = b as LUx = b • Define a new n x 1 matrix y by Ux = y • Use Ux = y to rewrite LUx = b as Ly = b and solve the system for y • Substitute y in Ux = y and solve for x

  4. Explanation on the method This procedure replaces the single linear system ax = b by a pair of linear systems that must be solved in succession.However, since each of these systems has a triangular coefficient matrix, it generally turns out to involve no more computation to solve the two systems than to solve the original system directly.

  5. Finding LU Decompositions • The previous example shows that once an LU-decomposition of A is obtained, a linear system Ax = b can be solved by one forward substitution and one backward substitution. • The main advantage of this method over Gaussian and Gauss-Jordan elimination is that it “decouples” Afrom b so that for solveing a linear systems with same coefficient matrix A the work in factoring that matrix need only be performed once.

  6. Finding LU Decomposition CONTINUED… • Such sequences occur in problems in which the matrix A remains fixed but the matrix b varies over time. • Not every square matrix has an LU-decomposition • However, if it is possible to reduce a square matrix A to REF by Gaussian elimination (without performing any row exchanges) then Awill have an LU-decomposition.

  7. Solution To obtain an LU-decomposition A = LUwe will reduce A to REF form U using Gaussian elimination and then calculate L from it. Steps are shown below: Continued next slide 

  8. Continued next slide 

  9. Continued next slide 

  10. Continued next slide 

  11. Constructing LU-Decomposition: • Reduce A to a REF form U by Gaussian elinmination without row exchanges, keeping track of the multipliers used to introduce the leading 1s and multipliers used to introduce the zeros below the leading 1s • In each position along the main diagonal of L place the reciprocal of the multiplier that introduced the leading 1 in that position in U • In each position below the main diagonal of L place negative of the multiplier used to introduce the zero in that position in U • Form the decomposition A = LU

  12. Continued next slide 

  13. LU-Decompositions are not unique

  14. Computer Application: PLU-Decomposition

  15. Questions to Get Done Suggested practice problems (11th edition) • Section 9.1 #1-7 odd questions