Hint: Use Hermite interpolationP00(x), P01(x), P10(x), P11(x) are not related Consider an interpolating polynomial p(x) of degree ≤ 3 satisfying p(xo) = a, p′ (xo) = b, p(x₁) = cand p'(x₁) = d with xo < x₁.(a) For xo0 and ₁=1, find the interpolating polynomials poo (x) and po₁ (x) with (a, b, c, d) =(1,0,0,0) and (a, b, c, d) = (0, 1, 0, 0), respectively.-(b) For the same xo and x₁ as in (a), find the interpolating polynomials p₁0(x) and p₁1(x) with(a, b, c, d) = (0, 0, 1, 0) and (a, b, c, d) = (0, 0, 0, 1), respectively.== 0 and ₁ = 1 given (a, b, c, d).(c) Use (a) and (b) to find the interpolating polynomial for xo(d) Find the interpolating polynomial for the general data (xo, x₁, a, b, c, d).

Answered: Consider an interpolating polynomial…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

Hint: Use Hermite interpolation

P₀₀(x), P₀₁(x), P₁₀(x), P₁₁(x) are not related

Consider an interpolating polynomial p(x) of degree ≤ 3 satisfying p(xo) = a, p′ (xo) = b, p(x₁) = c
and p'(x₁) = d with xo < x₁.
(a) For xo
0 and ₁=
1, find the interpolating polynomials poo (x) and po₁ (x) with (a, b, c, d) =
(1,0,0,0) and (a, b, c, d) = (0, 1, 0, 0), respectively.
-
(b) For the same xo and x₁ as in (a), find the interpolating polynomials p₁0(x) and p₁1(x) with
(a, b, c, d) = (0, 0, 1, 0) and (a, b, c, d) = (0, 0, 0, 1), respectively.
=
= 0 and ₁ = 1 given (a, b, c, d).
(c) Use (a) and (b) to find the interpolating polynomial for xo
(d) Find the interpolating polynomial for the general data (xo, x₁, a, b, c, d).

Expert Solution

Step 1: Introduction

Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts for you. To get remaining sub-parts solved please repost the complete question and mention the sub-parts.

Given that an interpolating polynomial $p open parentheses x close parentheses$ of degree $less or equal than 3$ satisfying $p open parentheses x subscript 0 close parentheses equals a comma space p apostrophe open parentheses x subscript 0 close parentheses equals b comma space p open parentheses x subscript 1 close parentheses equals c$ and $p apostrophe open parentheses x subscript 1 close parentheses equals d$ with $x subscript 0 less than x subscript 1$ .

We need to find the interpolating polynomial with given different values of $a comma b comma c comma d$ .

We know that Hermite interpolating polynomial $H open parentheses x close parentheses$ of a function $y equals p open parentheses x close parentheses$ is calculated as $H open parentheses x close parentheses equals sum u subscript i open parentheses x close parentheses. y subscript i plus sum v subscript i open parentheses x close parentheses. y subscript i apostrophe$ , where:

$table row cell u subscript i open parentheses x close parentheses end cell equals cell open square brackets 1 minus 2 open parentheses x minus x subscript i close parentheses I subscript i apostrophe open parentheses x subscript i close parentheses close square brackets open square brackets I subscript i open parentheses x close parentheses close square brackets squared end cell row cell v subscript i open parentheses x close parentheses end cell equals cell open parentheses x minus x subscript i close parentheses open square brackets I subscript i open parentheses x close parentheses close square brackets squared end cell end table$

$I subscript i open parentheses x close parentheses$ is Langrange interpolating polynomial.