(c) Determine the natural cubic spline that interpolates the data f(0) = 1, f(3) = 2, f(8) = 3 and find the approximate value of f(3.2) correct to four decimal places.
(c) Determine the natural cubic spline that interpolates the data f(0) = 1, f(3) = 2, f(8) = 3 and find the approximate value of f(3.2) correct to four decimal places.
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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3c
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![Let f(x) = x cos (2x) – a, xo = 0, 1 = 0.3,
X2 = 0.7.
(a) Find Lagrange interpolating polynomial for f(x) using the three given nodes
leaving all coefficients correct to 5 decimal points.
(b) Using the nodes r, and a1, construct the Hermite interpolating polynomial H3(r)
for f(x) using the Lagrange coefficient polynomials expressing all coefficients
correct to 5 decimal places.
(c) Determine the natural cubic spline that interpolates the data
f(0) = 1, f(3) = 2, f(8) = 3
and find the approximate value of f(3.2) correct to four decimal places.
(d) The cubic Legendre polynomial is P2(x) = (5x³ – 3r). Prove that it is orthog-
onal (over [-1, 1]) to all polynomials of degree 2.
%3D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38d1926f-bca1-4711-adbc-8837cf013bc2%2Fae73041f-8b9f-4f9e-b623-60806342834c%2Fidnsjc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let f(x) = x cos (2x) – a, xo = 0, 1 = 0.3,
X2 = 0.7.
(a) Find Lagrange interpolating polynomial for f(x) using the three given nodes
leaving all coefficients correct to 5 decimal points.
(b) Using the nodes r, and a1, construct the Hermite interpolating polynomial H3(r)
for f(x) using the Lagrange coefficient polynomials expressing all coefficients
correct to 5 decimal places.
(c) Determine the natural cubic spline that interpolates the data
f(0) = 1, f(3) = 2, f(8) = 3
and find the approximate value of f(3.2) correct to four decimal places.
(d) The cubic Legendre polynomial is P2(x) = (5x³ – 3r). Prove that it is orthog-
onal (over [-1, 1]) to all polynomials of degree 2.
%3D
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