naginterfaces.library.specfun.bessel_i0_real¶

naginterfaces.library.specfun.bessel_i0_real(x)[source]¶

bessel_i0_real returns the value of the modified Bessel function $I_{0} (x)$ .

For full information please refer to the NAG Library document for s18ae

Parameters

Returns

Raises

NagValueError

(errno $1$ )

On entry, $x = ⟨ v a l u e ⟩$ .

Constraint: $| x | \leq ⟨ v a l u e ⟩$ .

$| x |$ is too large and the function returns the approximate value of $I_{0} (x)$ at the nearest valid argument.

Notes

bessel_i0_real evaluates an approximation to the modified Bessel function of the first kind $I_{0} (x)$ .

Note: $I_{0} (- x) = I_{0} (x)$ , so the approximation need only consider $x \geq 0$ .

The function is based on three Chebyshev expansions:

For $0 < x \leq 4$ ,

I_{0} (x) = e^{x} {' \sum}_{r = 0} a_{r} T_{r} (t), where t = 2 (\frac{x}{4}) - 1 .

For $4 < x \leq 12$ ,

I_{0} (x) = e^{x} {' \sum}_{r = 0} b_{r} T_{r} (t), where t = \frac{x - 8}{4} .

For $x > 12$ ,

I_{0} (x) = \frac{e^{x}}{\sqrt{x}} {' \sum}_{r = 0} c_{r} T_{r} (t), where t = 2 (\frac{12}{x}) - 1 .

For small $x$ , $I_{0} (x) ≃ 1$ . This approximation is used when $x$ is sufficiently small for the result to be correct to machine precision.

For large $x$ , the function must fail because of the danger of overflow in calculating $e^{x}$ .

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