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°.

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 91E
icon
Related questions
Question
100%
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°.
Theorem (3.3): (Error Bounds for Lagrange Interpolation, Equally Spaced Nodes)
Assume that f(x) is defined on [a,b], which contains equally spaced nodes x=Xo+hk.
Additionally, assume that f(x) and derivatives of f (x), up to order N+1, are continuous and
bounded on the special subintervals [xo,X1], [Xo,X2], and [xo,X3], respectively; that is:
(3.13)
|F(N+1)(x)| < Mn+1 for xo sxs xN
for N=1,2,3. The error terms (3.13) corresponding to the cases N=1,2, and 3 have the following
useful bounds on their magnitude:
|E,(x)| <
h²M2
valid for x e [xo, X1],
(3.15)
|E2(x)| <
h³ M2
valid for x € [X,, Xz],
(3.16)
|E,(x)| < valid for x e [xo,X3],
24
(3.17)
Transcribed Image Text: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°. Theorem (3.3): (Error Bounds for Lagrange Interpolation, Equally Spaced Nodes) Assume that f(x) is defined on [a,b], which contains equally spaced nodes x=Xo+hk. Additionally, assume that f(x) and derivatives of f (x), up to order N+1, are continuous and bounded on the special subintervals [xo,X1], [Xo,X2], and [xo,X3], respectively; that is: (3.13) |F(N+1)(x)| < Mn+1 for xo sxs xN for N=1,2,3. The error terms (3.13) corresponding to the cases N=1,2, and 3 have the following useful bounds on their magnitude: |E,(x)| < h²M2 valid for x e [xo, X1], (3.15) |E2(x)| < h³ M2 valid for x € [X,, Xz], (3.16) |E,(x)| < valid for x e [xo,X3], 24 (3.17)
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps

Blurred answer
Knowledge Booster
Single Variable
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning