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Semelhante a Lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2 (13)
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Lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2
- 1. © 2012 Pearson Education, Inc.
Math 337-102
Lecture 6
LU Factorization
Computer Graphics
Determinants
- 2. Slide 2.2- 2© 2012 Pearson Education, Inc.
LU Factorization
Factor A = LU
A is mxn
A does NOT have to be square!!!
L is mxm (square) – Lower Triangular with 1’s on the
diagonal
U is mxn REF(A)
- 3. Using LU to Solve Ax = b
Ax = b
(LU)x = b
L(Ux) = b
Let y = Ux
Solve Ly = b
Then Ux = y
Slide 2.2- 3© 2012 Pearson Education, Inc.
- 5. Finding L and U
U is Row Echelon Form of A using row
replacement only
No interchanges or scaling!!!!
L entries are such that the same
sequence of row operations that reduce A
to U will reduce L to I.
Slide 2.2- 5© 2012 Pearson Education, Inc.
- 6. Finding L and U - Example
Slide 2.2- 6© 2012 Pearson Education, Inc.
- 7. Using LU to solve Ax = b
Factor A as LU
Solve Ly = b for y
Solve Ux = y for x
Slide 2.2- 7© 2012 Pearson Education, Inc.
- 17. Row Operations and Determinants
Thm 3-3: Let A be a square matrix.
a)If a multiple of one row is added to another
row to produce a matrix B, then |B| = |A|
b)If two rows are interchanged to produce B,
then |B| = -|A|
c)If one row of A is multiplied by k to produce
B, then |B| = k|A|
Slide 2.2- 17© 2012 Pearson Education, Inc.
- 18. Row Operations and Determinants
Row replacement does not change determinant
Row interchange negates the determinant
Scaling – think of as factoring
Slide 2.2- 18© 2012 Pearson Education, Inc.