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Chebyshev polynomials are commonly found in mathematical libraries for calculators/computers. For example, the widely used GNU Compiler Collection uses Chebyshev polynomials to evaluate trigonometric functions. The Chebyshev polynomial approximation of degree 7 for the sine function is s,(x) = 0.9999966013x – 0.1666482357x³ + 0.008306286146x5 - 0.1836274858 x 10-3x The Chebyshev polynomials are designed to remain close to a function across an entire closed interval. They seek to keep the approximation within a specified distance of the function being approximated at every point of that interval. If Sn is a Chebyshev approximation of degree n to the sine function on -, then the error estimate in using Sn(x) is given by n+1 maxIsin x – S, (x)| S -1/25xsn/2" 2" (n + 1)! Like most error estimates of this type, it gives an upper bound to the error. 9. Use the Chebyshev polynomial approximation for x = % and x = 3/2. Compare the results with the values your calculator/computer supplies for sin (1/2) and sin (3/2), as well as with the results from the Maclaurin approximations and the Taylor series approximations obtained in Problems 2, 3, 7, and 8. 10. Define Er(x) = | sin x- S-(x) |. Use technology to graph E, on - 11. Find the local maximum and local minimum values of E, on - Compare these numbers with the error estimate above, and discuss the characteristics of this error. 12. Which approximation for the sine function would be preferable: the Maclaurin approximation, the Taylor approximation, or the Chebyshev approximation? Why?

Chebyshev polynomials are commonly found in mathematical libraries for calculators/computers. For example, the widely used GNU Compiler Collection uses Chebyshev polynomials to evaluate trigonometric functions. The Chebyshev polynomial approximation of degree 7 for the sine function is s,(x) = 0.9999966013x – 0.1666482357x³ + 0.008306286146x5 - 0.1836274858 x 10-3x The Chebyshev polynomials are designed to remain close to a function across an entire closed interval. They seek to keep the approximation within a specified distance of the function being approximated at every point of that interval. If Sn is a Chebyshev approximation of degree n to the sine function on -, then the error estimate in using Sn(x) is given by n+1 maxIsin x – S, (x)| S -1/25xsn/2" 2" (n + 1)! Like most error estimates of this type, it gives an upper bound to the error. 9. Use the Chebyshev polynomial approximation for x = % and x = 3/2. Compare the results with the values your calculator/computer supplies for sin (1/2) and sin (3/2), as well as with the results from the Maclaurin approximations and the Taylor series approximations obtained in Problems 2, 3, 7, and 8. 10. Define Er(x) = | sin x- S-(x) |. Use technology to graph E, on - 11. Find the local maximum and local minimum values of E, on - Compare these numbers with the error estimate above, and discuss the characteristics of this error. 12. Which approximation for the sine function would be preferable: the Maclaurin approximation, the Taylor approximation, or the Chebyshev approximation? Why?

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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Chebyshev polynomials are commonly found in mathematical libraries for calculators/computers. For example, the widely used GNU Compiler Collection uses Chebyshev polynomials to evaluate trigonometric functions. The Chebyshev polynomial approximation of degree 7 for the sine function is s,(x) = 0.9999966013x – 0.1666482357x³ + 0.008306286146x5 - 0.1836274858 x 10-3x7 The Chebyshev polynomials are designed to remain close to a function across an entire closed interval. They seek to keep the approximation within a specified distance of the function being approximated at every point of that interval. If Sn is a Chebyshev approximation of degree n to the sine function on - then the error estimate in using Sn(x) is given by (n+1 maxIsin x – S(x)|s -1/25xSR/2 - 2" (n + 1)! Like most error estimates of this type, it gives an upper bound to the error. 9. Use the Chebyshev polynomial approximation for x = % and x = 3/2. Compare the results with the values your calculator/computer supplies for sin (1/2) and sin (3/2), as well as with the results from the Maclaurin approximations and the Taylor series approximations obtained in Problems 2, 3, 7, and 8. 10. Define Er(x) = | sin x-S-(x) |. Use technology to graph E, on -. 11. Find the local maximum and local minimum values of E, on -. Compare these numbers with the error estimate above, and discuss the characteristics of this error. 12. Which approximation for the sine function would be preferable: the Maclaurin approximation, the Taylor approximation, or the Chebyshev approximation? Why?
Transcribed Image Text:Chebyshev polynomials are commonly found in mathematical libraries for calculators/computers. For example, the widely used GNU Compiler Collection uses Chebyshev polynomials to evaluate trigonometric functions. The Chebyshev polynomial approximation of degree 7 for the sine function is s,(x) = 0.9999966013x – 0.1666482357x³ + 0.008306286146x5 - 0.1836274858 x 10-3x7 The Chebyshev polynomials are designed to remain close to a function across an entire closed interval. They seek to keep the approximation within a specified distance of the function being approximated at every point of that interval. If Sn is a Chebyshev approximation of degree n to the sine function on - then the error estimate in using Sn(x) is given by (n+1 maxIsin x – S(x)|s -1/25xSR/2 - 2" (n + 1)! Like most error estimates of this type, it gives an upper bound to the error. 9. Use the Chebyshev polynomial approximation for x = % and x = 3/2. Compare the results with the values your calculator/computer supplies for sin (1/2) and sin (3/2), as well as with the results from the Maclaurin approximations and the Taylor series approximations obtained in Problems 2, 3, 7, and 8. 10. Define Er(x) = | sin x-S-(x) |. Use technology to graph E, on -. 11. Find the local maximum and local minimum values of E, on -. Compare these numbers with the error estimate above, and discuss the characteristics of this error. 12. Which approximation for the sine function would be preferable: the Maclaurin approximation, the Taylor approximation, or the Chebyshev approximation? Why?
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