Polynomial Fitting - HP 50g User Manual

Compare these fitted values with the original data as shown in the table below:

Polynomial fitting

Consider the x-y data set {(x
to fit a polynomial or order p to this data set. In other words, we seek a fitting
of the form y = b
least-square approximation to the values of the coefficients b = [b
... b ], by putting together the matrix X
p
_
1
1
1
.
.
1
_
Then, the vector of coefficients is obtained from b = (X
the vector y = [y
In Chapter 10, we defined the Vandermonde matrix corresponding to a vector
x = [x x ... x
1 2 m
interest to the polynomial fitting, but having only n, rather than (p+1) columns.
We can take advantage of the VANDERMONDE function to create the matrix X
if we observe the following rules:
If p = n-1, X = V
If p < n-1, then remove columns p+2, ..., n-1, n from V
x x
1 2
1.20 3.10
2.50 3.10
3.50 4.50
4.00 4.50
6.00 5.00
,y ), (x
1 1
2
+ b x + b x
0 1 2
2
x
x
1
1
2
x
x
2
2
2
x
x
3
3
. .
. .
2
x
x
n
n
T
y ... y ] .
1 2 n
] . The Vandermonde matrix is similar to the matrix X of
.
n
x y
3
2.00 5.70
2.50 8.20
2.50 5.00
3.00 8.20
3.50 9.50
,y ), ..., (x ,y
2 2 n
3
+ b x + ... + b
3
3
...
x
1
3
...
x
2
3
...
x
3
.
. .
3
...
x
n
y-fitted
5.63
8.25
5.03
8.22
9.45
)}. Suppose that we want
n
p
x . You can obtain the
p
0
p-1 p
x y
1 1
p-1 p
x y
2 2
p-1 p
x y
3 3
. .
. .
p-1 p
x y
n n
T -1 T
X) X y, where y is
to form X.
n
b b b
1 2 3
_
_
Page 18-59