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Problem 3. Let pEP2. Consider the interpolation problem with the following degrees of freedom: p(0) = Po, p'(1) = P₁, and f₁ p(x) dx = (p). 1. Write the Vandermonde matrix for this interpolation problem. 2. Write down the coefficients of the cardinal basis for this interpolation problem in the monomial basis. 3. Graph the cardinal basis functions. 4. "Taking the average of a polynomial is the same as computing a dot product with its monomial basis coefficient vector." Explain what this means. 5. Let po = 0, p₁ = 1, and (p) = µ. Let pμ(x) be the solution of the interpolation problem with this data. Graph pu from x = 0 to x = 1 for μ = 0, 1, 2.

Problem 3. Let pEP2. Consider the interpolation problem with the following degrees of freedom: p(0) = Po, p'(1) = P₁, and f₁ p(x) dx = (p). 1. Write the Vandermonde matrix for this interpolation problem. 2. Write down the coefficients of the cardinal basis for this interpolation problem in the monomial basis. 3. Graph the cardinal basis functions. 4. "Taking the average of a polynomial is the same as computing a dot product with its monomial basis coefficient vector." Explain what this means. 5. Let po = 0, p₁ = 1, and (p) = µ. Let pμ(x) be the solution of the interpolation problem with this data. Graph pu from x = 0 to x = 1 for μ = 0, 1, 2.

Advanced Engineering Mathematics
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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Problem 3. Let p = P². Consider the interpolation problem with the following degrees of freedom: p(0) = po, p′(1) = p₁, and ſ₁ p(x)dx = (p). 1. Write the Vandermonde matrix for this interpolation problem. 2. Write down the coefficients of the cardinal basis for this interpolation problem in the monomial basis. 3. Graph the cardinal basis functions. 4. "Taking the average of a polynomial is the same as computing a dot product with its monomial basis coefficient vector." Explain what this means. 5. Let po = 0, p₁ = 1, and (p) = µ. Let pμ(x) be the solution of the interpolation problem with this data. Graph pu from x = 0 to x = 1 for µ = 0, 1, 2.
Transcribed Image Text:Problem 3. Let p = P². Consider the interpolation problem with the following degrees of freedom: p(0) = po, p′(1) = p₁, and ſ₁ p(x)dx = (p). 1. Write the Vandermonde matrix for this interpolation problem. 2. Write down the coefficients of the cardinal basis for this interpolation problem in the monomial basis. 3. Graph the cardinal basis functions. 4. "Taking the average of a polynomial is the same as computing a dot product with its monomial basis coefficient vector." Explain what this means. 5. Let po = 0, p₁ = 1, and (p) = µ. Let pμ(x) be the solution of the interpolation problem with this data. Graph pu from x = 0 to x = 1 for µ = 0, 1, 2.
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