Abstract
Using a finite number of additions, subtractions, multiplications , and comparisons, the only functions of one variable that one can compute are piecewise polynomials.
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- 1.
This kind of approximation is sometimes called Chebyshev approximation. Throughout this book, Chebyshev approximation means least squares approximation using Chebyshev polynomials. Chebyshev worked on both kinds of approximation.
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Chebfun is available at www.chebfun.org.
- 3.
According to Trefethen [458], that result was known by Chebyshev but the first proof was given by Kirchberger in his Ph.D. dissertation [273].
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An exception is multiple precision computations (see Chapter 7), since it is not possible to precompute and store least squares or minimax approximations for all possible precisions.
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Of course, this is not the right way to implement the square root function: first, it is straightforward to reduce the domain to [1 / 4, 1], second, Newton–Raphson’s iteration for \(\sqrt{a}\):
$$x_{n+1} = \frac{1}{2}\left( x_n + \frac{a}{x_n}\right) ,$$or (to avoid divisions) Newton–Raphson’s iteration for \(1/\sqrt{a}\):
$$ x_{n+1} = \frac{x_n}{2}\left( 3-ax_n^2\right) $$followed by a multiplication by a, or digit recurrence methods [179] are preferable.
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Another solution is to drastically reduce the size of the interval where the function is being approximated. This is studied in Chapter 6.
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And yet, they can have better global behavior than expected. See for instance reference [190].
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Assuming that Horner’s scheme is used, and that we first compute \(x^2\).
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Unless some parallelism is available in the processor being used or the circuit being designed. For instance, as pointed out by Koren and Zinaty [278], if we can perform an addition and a multiplication simultaneously, then we can compute rational functions by performing in parallel an add operation for evaluating the numerator and a multiply operation for evaluating the denominator (and vice versa). If the degrees of the numerator and denominator are large enough, the delay due to the division may become negligible. However, we will see in Chapter 5 that the availability of parallelism also makes it possible to very significantly accelerate the evaluation of polynomials.
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The basic idea behind this is that, although division is less frequently called than multiplication, it is so slow (on most existing computers) that the time spent by some numerical programs in performing divisions is not at all negligible compared to the time spent in performing other arithmetic operations.
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A polynomial equal to a sum of squares of polynomials is nonnegative on all the real line, not only on [a, b]. To show nonnegativity on [a, b] only, a change of variables may be necessary. Nonnegativity of the degree-n polynomial P(x) for \(x \in [a,b]\) is equivalent to nonnegativity of the polynomial \(Q(y) = (1+y^2)^n \cdot {} P( (a+by^2)/(1+y^2))\) for \(y \in \mathbb {R}\). Hence, it is Q, not P that one should try to express as a sum of squares.
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The Sollya package is available at http://sollya.gforge.inria.fr.
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© 2016 Springer Science+Business Media New York
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Muller, JM. (2016). The Classical Theory of Polynomial or Rational Approximations. In: Elementary Functions. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-7983-4_3
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DOI: https://doi.org/10.1007/978-1-4899-7983-4_3
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