Page 18-59
If p = n-1, X = V
n
.
If p < n-1, then remove columns p+2, …, n-1, n from V
n
to form X.
If p > n-1, then add columns n+1, …, p-1, p+1, to V
n
to form matrix X.
In step 3 from this list, we have to be aware that column i (i= n+1, n+2, …,
p+1) is the vector [x
1
i
x
2
i
… x
n
i
]. If we were to use a list of data values for x
rather than a vector, i.e., x = { x
1
x
2
… x
n
}, we can easily calculate the
sequence { x
1
i
x
2
i
… x
n
i
}. Then, we can transform this list into a vector and
use the COL menu to add those columns to the matrix V
n
until X is completed.
After X is ready, and having the vector y available, the calculation of the
coefficient vector b is the same as in multiple linear fitting (the previous matrix
application). Thus, we can write a program to calculate the polynomial fitting
that can take advantage of the program already developed for multiple linear
fitting. We need to add to this program the steps 1 through 3 listed above.
The algorithm
for the program, therefore, can be written as follows:
Enter vectors x and y, of the same dimension, as lists. (Note: since the
function VANDERMONDE uses a list as input, it is more convenient to enter
the (x,y) data as a list.) Also, enter the value of p.
• Determine n = size of vector x.
• Use the function VANDERMONDE to generate the Vandermonde
matrix V
n
for the list x entered.
• If p = n-1, then
X = V
n
,
Else If p < n-1
Remove columns p+2, …, n from V
n
to form X
(Use a FOR loop and COL-)
Else
Add columns n+1, …, p+1 to V
n
to form X
(FOR loop, calculate x
i
, convert to vector, use COL+)
• Convert y to vector
• Calculate b using program MTREG (see example on multiple linear
fitting above)
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