Suppose that ƒ € C²([x0, x1]) for xo < x₁, and let P(x) be the linear interpolant for ƒ at x and x₁. Using the theorem on the error in polynomial interpolation, derive the following bound: 1 where h = x1 — Xo. - |ƒ(x) − P(x)| ≤ =h²_max¸|ƒ"(x)\, €[20,21] 8
Suppose that ƒ € C²([x0, x1]) for xo < x₁, and let P(x) be the linear interpolant for ƒ at x and x₁. Using the theorem on the error in polynomial interpolation, derive the following bound: 1 where h = x1 — Xo. - |ƒ(x) − P(x)| ≤ =h²_max¸|ƒ"(x)\, €[20,21] 8
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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Error in polynomial interpo
![**Polynomial Interpolation Error Bound**
Suppose that \( f \in C^2([x_0, x_1]) \) for \( x_0 < x_1 \), and let \( P(x) \) be the linear interpolant for \( f \) at \( x_0 \) and \( x_1 \). Using the theorem on the error in polynomial interpolation, derive the following bound:
\[
|f(x) - P(x)| \leq \frac{1}{8} h^2 \max_{x \in [x_0, x_1]} |f''(x)|,
\]
where \( h = x_1 - x_0 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2Fe61b0a77-17b6-4c6d-a781-ce74ca899251%2Fkejgyq_processed.png&w=3840&q=75)
Transcribed Image Text:**Polynomial Interpolation Error Bound**
Suppose that \( f \in C^2([x_0, x_1]) \) for \( x_0 < x_1 \), and let \( P(x) \) be the linear interpolant for \( f \) at \( x_0 \) and \( x_1 \). Using the theorem on the error in polynomial interpolation, derive the following bound:
\[
|f(x) - P(x)| \leq \frac{1}{8} h^2 \max_{x \in [x_0, x_1]} |f''(x)|,
\]
where \( h = x_1 - x_0 \).
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