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3. Consider f(x) = x lnx and interpolating points with abscissae xo = 1 and 2₁ = 2. (Keep your calculations accurate to at least 4 decimal places.) (a) Compute the Hermite polynomial H(x) that approximates f(x) using the basis functions Hnj and Hnj in the lecture notes. (b) Compute the divided-difference table and construct the Hermite polynomial G(x) that approximates f(x). (c) The Hermite polynomial can also be obtained using the monomial basis. List the linear system for the coefficients (under the monomial basis) of the Hermite polynomial F(x) that approximates f(x). Use the backslash in matlab to solve this system. Verify that the polynomials obtained in (a), (b), and (c) are actually the same. (d) Suppose we use G(1.5) to approximate f(1.5). Find the error f(1.5) - G(1.5). (e) Use the error formula to find the error bound at x = 1.5, and compare the error bound with the actual error in (d).

3. Consider f(x) = x lnx and interpolating points with abscissae xo = 1 and 2₁ = 2. (Keep your calculations accurate to at least 4 decimal places.) (a) Compute the Hermite polynomial H(x) that approximates f(x) using the basis functions Hnj and Hnj in the lecture notes. (b) Compute the divided-difference table and construct the Hermite polynomial G(x) that approximates f(x). (c) The Hermite polynomial can also be obtained using the monomial basis. List the linear system for the coefficients (under the monomial basis) of the Hermite polynomial F(x) that approximates f(x). Use the backslash in matlab to solve this system. Verify that the polynomials obtained in (a), (b), and (c) are actually the same. (d) Suppose we use G(1.5) to approximate f(1.5). Find the error f(1.5) - G(1.5). (e) Use the error formula to find the error bound at x = 1.5, and compare the error bound with the actual error in (d).

Advanced Engineering Mathematics
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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3. Consider f(x) = x lnx and interpolating points with abscissae xo = 1 and 2₁ = 2. (Keep your calculations accurate to at least 4 decimal places.) (a) Compute the Hermite polynomial H(x) that approximates f(x) using the basis functions Hnj and Hnj in the lecture notes. (b) Compute the divided-difference table and construct the Hermite polynomial G(x) that approximates f(x). (c) The Hermite polynomial can also be obtained using the monomial basis. List the linear system for the coefficients (under the monomial basis) of the Hermite polynomial F(x) that approximates f(x). Use the backslash in matlab to solve this system. Verify that the polynomials obtained in (a), (b), and (c) are actually the same. (d) Suppose we use G(1.5) to approximate f(1.5). Find the error f(1.5) - G(1.5). (e) Use the error formula to find the error bound at x = 1.5, and compare the error bound with the actual error in (d).
Transcribed Image Text:3. Consider f(x) = x lnx and interpolating points with abscissae xo = 1 and 2₁ = 2. (Keep your calculations accurate to at least 4 decimal places.) (a) Compute the Hermite polynomial H(x) that approximates f(x) using the basis functions Hnj and Hnj in the lecture notes. (b) Compute the divided-difference table and construct the Hermite polynomial G(x) that approximates f(x). (c) The Hermite polynomial can also be obtained using the monomial basis. List the linear system for the coefficients (under the monomial basis) of the Hermite polynomial F(x) that approximates f(x). Use the backslash in matlab to solve this system. Verify that the polynomials obtained in (a), (b), and (c) are actually the same. (d) Suppose we use G(1.5) to approximate f(1.5). Find the error f(1.5) - G(1.5). (e) Use the error formula to find the error bound at x = 1.5, and compare the error bound with the actual error in (d).
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