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Miscellaneous
No class next Monday, April 11th (Sechseläuten)
How to Write Fast Numerical Code Spring 2011 Lecture 13 Instructor: - - PowerPoint PPT Presentation
How to Write Fast Numerical Code Spring 2011 Lecture 13 Instructor: - - PowerPoint PPT Presentation
How to Write Fast Numerical Code Spring 2011 Lecture 13 Instructor: Markus Pschel TA: Georg Ofenbeck Miscellaneous No class next Monday, April 11 th (Sechseluten) Midterm exam: Friday, April 15 th Today Solving linear systems
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SLIDE 3
Today
Solving linear systems
Matrix inversion
PLU factorization
Determinant Chapter 2 in James W. Demmel, Applied Numerical Linear Algebra, SIAM, 1997
LAPACK BLAS
BLAS 1: vector-vector ops BLAS 2: matrix-vector ops BLAS 3: matrix-matrix ops Linear system solving Matrix inversion Singular value decomposition ... and more
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Gauss Elimination and LU Factorization
A = [ak,l]1≤k,l≤n
A
- a2,1/a1,1
- an,1/a1,1
a2,1 a1,1
- =
a1,1 a1,n
A’ 1 1 1
a1,1
- an,1 0
A
- a1,1
a1,n a’2,2 a’1,n
A’
a2,1 a1,1
- 1
1 1
a1,1
- an,1 0
A
- a’3,2
a’2,2
- 1
1 1
a’2,2
- a’n,2 0
- =
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after n-1 steps ….
U =
A
- A
U
a2,1 a1,1
1 1 1
a1,1 an,1 0 a’3,2 a’2,2
1 1 1
a’2,2 a’n,2 0
- =
A
U
a2,1 a1,1
1 1 1
a1,1 an,1 a’3,2 a’2,2 a’2,2 a’n,2
- =
= L ● U
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Summary
Gauss elimination is LU factorization
- We assume that the occurring diagonal elements a1,1, a’2,2, … are all ≠ 0
(otherwise the LU factorization does not exist)
- U = upper triangular
- L = lower triangular (1’s on diagonal)
- L contains the multipliers occurring in Gauss elimination
Now Ax = b is LUx = b and can be solved in two steps:
- Solve Ly = b
- Solve Ux = y
- Cost: n2 + O(n) for each step = 2n2 + O(n)
SLIDE 7
LU Factorization: Algorithm