First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 3. INTERPOLATION 105
In the first iteration (for loop, j =2), a 1 and a 2 are updated:
a 1 := a 1 − a 0
x 1 − x 0
= y 1 − y 0
x 1 − x 0
= f[x 0 ,x 1 ],a 2 := a 2 − a 1
x 2 − x 1
= y 2 − y 1
x 2 − x 1
.
In the last iteration (for loop, j =3), only a 2 is updated:
a 2 := a 2 − a 1
x 2 − x 0
=
y 2 −y 1
x 2 −x 1
− y 1−y 0
x 1 −x 0
x 2 − x 0
= f[x 0 ,x 1 ,x 2 ].
The final array a is
a =(y 0 ,f[x 0 ,x 1 ],f[x 0 ,x 1 ,x 2 ])
containing the divided differences needed to construct the Newton’s form of the polynomial.
Here is the Julia function diff which computes the divided differences. The code uses a
function we have not used before: reverse(collect(j:m)). An example illustrates what it
does the best:
In [1]: reverse(collect(2:5))
Out[1]: 4-element Array{Int64,1}:
5
4
3
2
In the code diff the inputs are the x and y-coordinates of the data. The numbering
of the indices starts at 1, not 0.
In [2]: function diff(x::Array,y::Array)
m=length(x) #here m is the number of data points.
#the degree of the polynomial is m-1
a=Array{Float64}(undef,m)
for i in 1:m
a[i]=y[i]
end
for j in 2:m
for i in reverse(collect(j:m))
a[i]=(a[i]-a[i-1])/(x[i]-x[i-(j-1)])