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§4.45 Methods of Computation ‣ Computation ‣ Chapter 4 Elementary Functions

1. §4.45(i) Real Variables
2. §4.45(ii) Complex Variables
3. §4.45(iii) Lambert W -Function

The function ln ⁡ x can always be computed from its ascending power series after preliminary scaling. Suppose first 1 / 10 ≤ x ≤ 10 . Then we take square roots repeatedly until | y | is sufficiently small, where

After computing ln ⁡ ( 1 + y ) from (4.6.1)

For other values of x set x = 10m ⁢ ξ , where 1 / 10 ≤ ξ ≤ 10 and m ∈ ℤ . Then

Let x have any real value. First, rescale via

Then

and since | y | ≤ 1 2 ⁢ ln ⁡ 10 = 1.15 ⁢ … , ey can be computed straightforwardly from (4.2.19).

Let x have any real value. We first compute ξ = x / π , followed by

Then

and since | θ | ≤ 1 2 ⁢ π = 1.57 ⁢ … , sin ⁡ θ and cos ⁡ θ can be computed straightforwardly from (4.19.1) and (4.19.2).

The other trigonometric functions can be found from the definitions (4.14.4)–(4.14.7).

The function arctan ⁡ x can always be computed from its ascending power series after preliminary transformations to reduce the size of x . From (4.24.15) with u = v = ( ( 1 + x2 ) 1 / 2 − 1 ) / x , we have

Beginning with x 0 = x , generate the sequence

until x n is sufficiently small. We then compute arctan ⁡ x n from (4.24.3), followed by

Another method, when x is large, is to sum

compare (4.24.4).

As an example, take x = 9.47376 . Then

For the remaining inverse trigonometric functions, we may use the identities provided by the fourth row of Table 4.16.3. For example, arcsin ⁡ x = arctan ⁡ ( x ⁢ ( 1 − x2 ) − 1 / 2 ) .

The hyperbolic functions can be computed directly from the definitions (4.28.1)–(4.28.7). The inverses arcsinh , arccosh , and arctanh can be computed from the logarithmic forms given in §4.37(iv), with real arguments. For arccsch , arcsech , and arccoth we have (4.37.7)–(4.37.9).

See Luther (1995), Ziv (1991), Cody and Waite (1980), Rosenberg and McNamee (1976), Carlson (1972a). For interval-arithmetic algorithms, see Markov (1981). For Shift-and-Add and CORDIC algorithms, see Muller (1997), Merrheim (1994), Schelin (1983). For multiprecision methods, see Smith (1989), Brent (1976).

For ln ⁡ z and ez

See §1.9(i) for the precise relationship of ph ⁡ z to the arctangent function.

The trigonometric functions may be computed from the definitions (4.14.1)–(4.14.7), and their inverses from the logarithmic forms in §4.23(iv), followed by (4.23.7)–(4.23.9). Similarly for the hyperbolic and inverse hyperbolic functions; compare (4.28.1)–(4.28.7), §4.37(iv), and (4.37.7)–(4.37.9).

For other methods see Miel (1981).

For x ∈ [ − 1 / e , ∞ ) the principal branch W 0 ⁡ ( x ) can be computed by solving the defining equation W ⁢ eW = x numerically, for example, by Newton’s rule (§3.8(ii)). Initial approximations are obtainable, for example, from the power series (4.13.6) (with t ≥ 0 ) when x is close to − 1 / e , from the asymptotic expansion (4.13.10) when x is large, and by numerical integration of the differential equation (4.13.4) (§3.7) for other values of x .

Similarly for W − 1 ⁡ ( x ) in the interval [ − 1 / e , 0 ) (with t ≤ 0 in (4.13.6)).

See also Barry et al. (1995b) and Chapeau-Blondeau and Monir (2002).

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