Suppose p(x) is a polynomial of degree n (n ≥ 0), and z; (0 ≤ i ≤n) are a set of distinct. points on the real axis. Is the following relation true? Briefly explain why. p(x) = p[ro] + p[ro, 21] (x − x0) + p[ro, x1, x2] (x − xo)(x − x₁) + ... + p[ro, 1,...,n] (x − xo)(x − x₁)(x - xn-1)
Suppose p(x) is a polynomial of degree n (n ≥ 0), and z; (0 ≤ i ≤n) are a set of distinct. points on the real axis. Is the following relation true? Briefly explain why. p(x) = p[ro] + p[ro, 21] (x − x0) + p[ro, x1, x2] (x − xo)(x − x₁) + ... + p[ro, 1,...,n] (x − xo)(x − x₁)(x - xn-1)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Suppose \( p(x) \) is a polynomial of degree \( n \) (\( n \geq 0 \)), and \( x_i \) (\( 0 \leq i \leq n \)) are a set of distinct points on the real axis. Is the following relation true? Briefly explain why.
\[
p(x) = p[x_0] + p[x_0, x_1](x - x_0) + p[x_0, x_1, x_2](x - x_0)(x - x_1) + \cdots
+ p[x_0, x_1, \ldots, x_n](x - x_0)(x - x_1) \cdots (x - x_{n-1})
\]
This expression represents the Newton form of the interpolating polynomial. It is true if \( p(x) \) is indeed the polynomial interpolating the points \( x_0, x_1, \ldots, x_n \) with their corresponding function values, using Newton's divided differences. The terms \( p[x_0], p[x_0, x_1], \ldots \) are the divided differences, which are coefficients in the polynomial expansion.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2F6d27e681-a76c-4c36-91d2-b1c1181ce837%2Fqn4q0bd_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose \( p(x) \) is a polynomial of degree \( n \) (\( n \geq 0 \)), and \( x_i \) (\( 0 \leq i \leq n \)) are a set of distinct points on the real axis. Is the following relation true? Briefly explain why.
\[
p(x) = p[x_0] + p[x_0, x_1](x - x_0) + p[x_0, x_1, x_2](x - x_0)(x - x_1) + \cdots
+ p[x_0, x_1, \ldots, x_n](x - x_0)(x - x_1) \cdots (x - x_{n-1})
\]
This expression represents the Newton form of the interpolating polynomial. It is true if \( p(x) \) is indeed the polynomial interpolating the points \( x_0, x_1, \ldots, x_n \) with their corresponding function values, using Newton's divided differences. The terms \( p[x_0], p[x_0, x_1], \ldots \) are the divided differences, which are coefficients in the polynomial expansion.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
