Consider the Lagrange coefficient polynomial L2x(x) that are used for quadratic interpolation at the nodes xo,X1, and x2. Define g(x)=L20(x)+L2,1(x)+L22(x)-1. a. Show that g is a polynomial of degree <2. b. Show that g(xµ)=0 for k=0,1,2.
Consider the Lagrange coefficient polynomial L2x(x) that are used for quadratic interpolation at the nodes xo,X1, and x2. Define g(x)=L20(x)+L2,1(x)+L22(x)-1. a. Show that g is a polynomial of degree <2. b. Show that g(xµ)=0 for k=0,1,2.
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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![Exercises:
1. Consider the Lagrange coefficient polynomial L2k(x) that are used for quadratic
interpolation at the nodes x,X1, and x2. Define g(x)=L2,0(x)+L2,1(x)+L22(x)-1.
a. Show that g is a polynomial of degree < 2.
b. Show that g(Xg)=0 for k=0,1,2.
2. Consider the function f(x)=sin(x) on the interval [0,1]. Use theorem(3.3) to determine the
step size h so that:
a. linear Lagrange interpolation has an accuracy of 10°.
b. quadratic Lagrange interpolation has an accuracy of 10°.
c. cubic Lagrange interpolation has an accuracy of 10°.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbf628e11-bec3-480d-8958-9d4dfe2689c6%2F4970a0ea-f206-4115-9cd4-bbaf43f04c33%2Fhjaaphw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercises:
1. Consider the Lagrange coefficient polynomial L2k(x) that are used for quadratic
interpolation at the nodes x,X1, and x2. Define g(x)=L2,0(x)+L2,1(x)+L22(x)-1.
a. Show that g is a polynomial of degree < 2.
b. Show that g(Xg)=0 for k=0,1,2.
2. Consider the function f(x)=sin(x) on the interval [0,1]. Use theorem(3.3) to determine the
step size h so that:
a. linear Lagrange interpolation has an accuracy of 10°.
b. quadratic Lagrange interpolation has an accuracy of 10°.
c. cubic Lagrange interpolation has an accuracy of 10°.
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