Lecture 10 handout.pdf - Math 472 Lecture 10 Chapter 2...

Math 472 Lecture 10
Chapter 2 Systems of Equations
Ruoyu Wu
U Michigan
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2.3 Sources of error
There are two major sources of error for the Gaussian Elimination Algorithm:
1. ill-conditioning : The sensitivity of the solution to the input data ( A or b ),
as illustrated in Example 2.12 in the textbook.
2. swamping : A deficiency of the naive Gaussian Elimination.
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2.3 Sources of error
Definition 2.4 (Backward and Forward Error)
Consider the system Ax = b , and let x
a
be an approximate solution.
•
The backward error is k b - Ax
a
k
∞
(also written as k r k
∞
where r = b - Ax
a
is called the residual )
•
The relative backward error is k b - Ax
a
k
∞
/ k b k
∞
•
The forward error is k x - x
a
k
∞
•
The relative forward error is k x - x
a
k
∞
/ k x k
∞
•
Solving the system Ax = b is equivalent to solving the equation(s)
f ( y ) = 0 , where f ( y ) = Ay - b .
•
The solution (root) is x .
•
The backward error is k f ( x
a
) k
∞
= k Ax
a
- b k
∞
.
•
The forward error is k x
a
- x k
∞
.
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2.3 Sources of error
Example 2.11
Find the forward, backward, relative forward, and relative backward errors for
the approximate solution ( - 1 , 3 . 0001) of the system
x
1
+ x
2
= 2
1 . 0001 x
1
+ x
2
= 2 . 0001
☼
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