Math 416  Abstract Linear Algebra
Fall 2011, section E1
Least squares solution
1. Curve fitting
The least squares solution can be used to fit certain functions through data points.
Example:
Find the best fit line through the points (1
,
0)
,
(2
,
1)
,
(3
,
1).
Solution:
We are looking for a line with equation
y
=
a
+
bx
that would ideally go through
all the data points, i.e. satisfy all the equations
a
+
b
(1) = 0
a
+
b
(2) = 1
a
+
b
(3) = 1
.
In matrix form, we want the unknown coefficients
a
b
to satisfy the system
1
1
1
2
1
3
a
b
=
0
1
1
but the system has no solution. Instead, we find the least squares fit, i.e. minimize the sum of
the squares of the errors
3
X
i
=1

(
a
+
bx
i
)

y
i

2
which is precisely finding the least squares solution of the system above. Writing the system as
A~
c
=
~
y
, the normal equation is
A
T
A~
c
=
A
T
~
y
and we compute
A
T
A
=
1
1
1
1
2
3
1
1
1
2
1
3
=
3
6
6
14
A
T
~
y
=
1
1
1
1
2
3
0
1
1
=
2
5
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 FRANKLAND
 Math, Linear Algebra, Algebra, Least Squares, Orthogonal matrix, Hilbert space, Inner product space

Click to edit the document details