- #1
swevener
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Homework Statement
(a) If [itex]x_{1},\ldots, x_{n}[/itex] are distinct numbers, find a polynomial function [itex]f_{i}[/itex] of degree [itex]n - 1[/itex] which is 1 at [itex]x_{i}[/itex] and 0 at [itex]x_{j}[/itex] for [itex]j \ne i[/itex]. Hint: the product of all [itex](x - x_{j})[/itex] for [itex]j \ne i[/itex] is 0 at [itex]x_{j}[/itex] if [itex]j \ne i[/itex]. This product is usually denoted by
[tex]\prod_{\substack{j = 1 \\ j \ne i}}^{n} (x - x_{j}).[/tex]
(b) Now find a polynomial function [itex]f[/itex] of degree [itex]n - 1[/itex] such that [itex]f(x_{i}) = a_{i}[/itex], where [itex]a_{1},\ldots,a_{n}[/itex] are given numbers. (You should use the functions [itex]f_{i}[/itex] from part (a). The formula you will obtain is called the "Lagrange interpolation formula.")
3. [strike]The attempt at a solution[/strike] Questions
Why are these polynomials of degree [itex]n - 1[/itex]? Because of the [itex]j \ne i[/itex]?
[strike]And the hint in part (a), where does that come from? Why can we say the product is zero if[/strike] [itex]j \ne i[/itex]? Figured this one out. I misread the problem.
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