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32 Some Fundamental Numerical Methods
2.13. Write the trapezoidal integration formulas for T o .a, To." and To.,. Then
use extrapolation formula (2.27) to find Simpson rules T,.o and T\,\.
2.14. Calculate
numerically, using:
(a) Simpson's rule with five samples.
(b) The trapezoidal rule with three evenly spaced samples.
(c) The trapezoidal rule with five evenly spaced samples.
(d) The Romberg extrapolation to the limit, using the preceding
results in (b) and (c) above.
2.15. Evaluate Chebyshev polynomials T 4
(0.8) and T.(3.1) by:
(a) Horner's nesting method
T(x) =ao+ x{ a, + x[a,+ x( ... )J}.
(b)
(c)
Trigonometric or hyperbolic identities.
Numerical recursion
T;(x) = 2xT; _ ,(x) - T;_,(x).
2.16. Chebyshev polynomials of the second kind are defined by
Pk(y)=yPk-I-Pk-,·
where PI = I and P,=y. Find P
"
P 4 , P" and Po numerically for y= 1.5.
2.17. Find an expression for x at the n-l extreme values of
To(x) = cos(n cos- IX).
2.18. Write the power series equivalent to
P(x) = TI(x) + 4T,(x) + 2T,(x) - T 4 (x)
= ao+ a1x+ a2x 2 + a3x] + a4x4;
in other words, find the a i
coefficients. See Table 2.2 for T i
.
2.19. Suppose that q(y) is defined on -7