Numerical Methods for Engineers. Week # 6. Chapter 10

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Week # 6
Chapter 10
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Today’s Quote:

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Learning Outcomes
On completion you should be able to . . .
Find an LU decomposition of simple matrices and apply it to solve
systems of equations
Determine when an LU decomposition is unavailable and
when it is possible to avoid the problem

4. LU Decomposition and Matrix Inversion Chapter 10

• Provides an efficient way to compute matrix
inverse by separating the time consuming
elimination of the Matrix [A] from
manipulations of the right-hand side {B}.
• Gauss elimination, in which the forward
elimination comprises the bulk of the
computational effort, can be implemented as
an LU decomposition.
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If
L- lower triangular matrix
U- upper triangular matrix
Then,
[A]{X}={B} can be decomposed into two matrices [L] and
[U] such that
[L][U]=[A]
[L][U]{X}={B}
Similar to first phase of Gauss elimination, consider
[U]{X}={D}
[L]{D}={B}
– [L]{D}={B} is used to generate an intermediate vector {D}
by forward substitution
– Then, [U]{X}={D} is used to get {X} by back substitution.
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Fig.10.1
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LU decomposition
• requires the same total FLOPS as for Gauss
elimination.
• Saves computing time by separating timeconsuming elimination step from the
manipulations of the right hand side.
• Provides efficient means to compute the matrix
inverse
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18. Example

From the 3rd equation
0.7a3 0.735
0.735
0.7
a3 1.050
a3
Substituting in a3 and using the
second equation
4.8a2 1.56a3 96.21
a2
a2
96.21 1.56a3
4.8
96.21
1.56 1.050
a2 19.70
4.8

19. Example

Substituting in a3 and a2 using
the first equation
25a1 5a2 a3 106.8
a1
106.8 5a2 a3
25
106.8
5 19.70 1.050
0.2900
25