Polynomial Evaluation

We discuss now the computational cost for evaluating the Taylor polynomial value at some point

When evaluating the compiler (usually) translates it to multiplications. Thus, the naive algorithm

p = b(0)
do i=1,n
  p = p + b(i)*x**i
end do

will require additions and , i.e. multiplications.

A better alternative is to save the computed power of from one iteration to the next. Since , iteration will need just two multiplications (not ) to compute .

p = b(0)
powx = 1.0
do i=1,n
  powx = powx*x     ! powx = x**i
  p = p + b(i)*powx
end do

This second algorithm needs additions and multiplications.

We can do even better than this, by rewriting the polynomial in nested form

We have to start with the last term and loop back to the first. The algorithm goes as follows

p = b(n)
do i=n-1,0,-1
  p = b(i) + x*p
end do

and requires additions and only multiplications.