I have a trivial question about Ralston's book (If after clicking on the link the download doesn't start, copy the link and paste it in the browser's tab.)
I do not follow how can I obtain from $(5.2-5)$ by inserting into it $y_i=x_i^j,j=1$ the second equation in $(5.2-6)$ (with $h$=1):
$$1=-\sum_{i=0}^p ia_i+\sum_{i=-1}^p b_i,j=1$$.
If I take $y_i=y_{n+1}=x$ I obtain $$x=-\sum_{i=0}^p a_i x+\sum_{i=-1}^p b_i\cdot 1,j=1$$ but at the same time if I take $y_i=y_{n+1}=1$ I obtain $$1=-\sum_{i=0}^p a_i \cdot 1+\sum_{i=-1}^p b_i\cdot 0,j=1.$$
None of these two equations resemble the second equation in $(5.2-6),j=1.$
Snippets from the book:
You have to be careful when inserting, you want $$ x_{n+1}^j=\sum_{i=0}^{p}a_i(x_{n-i})^j+h\sum_{i=-1}^pb_i j(x_{n-i})^{j-1} $$ at $n=0$ and with $x_i=ih$.