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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 < 1. (a) For xo = 0 and 1= 1, find the interpolating polynomials poo (x) and poi (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 ₁ 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. (c) Use (a) and (b) to find the interpolating polynomial for xo = 0 and 1= 1 given (a, b, c, d).

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 < 1. (a) For xo = 0 and 1= 1, find the interpolating polynomials poo (x) and poi (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 ₁ 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. (c) Use (a) and (b) to find the interpolating polynomial for xo = 0 and 1= 1 given (a, b, c, d).

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.3: Change Of Basis
Problem 17EQ
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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).
Transcribed Image Text: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).
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