Computation of Real Roots of Functions
Abstract Generated abstract
The paper studies the computation of positive real roots of functions represented by entire power series with positive coefficients, focusing on the relation between root locations and continued fractions for the logarithmic derivative. It proves positivity of the coefficients in the corresponding continued fraction under stated assumptions and uses denominators of its convergents to approximate the smallest roots. The method is applied to Bessel functions, deriving explicit polynomial approximations and formulas for the first two roots of \(I_\nu(x)\), together with stated error estimates. It also establishes a series identity for a product of Bessel functions and uses it to obtain sharper approximations for the first roots in selected cases.
Full Text
Computing Real Roots of Functions
V. E. Kornilov
(Presented by the Department of Higher Mathematics)
1. Let an analytic function \(f(x)\) be represented by the power series
\[ f(x)=\sum_{k=0}^{\infty} c_k x^k,\quad c_0=1,\quad c_k>0,\quad |x|<\infty, \]
and suppose that the equation \(f(-x)=0\) has real positive roots
\(0<a_1<a_2<\cdots\).
For the partial function \(f'(x):f(x)\) the following theorem holds.
Theorem. The continued fraction for the partial function
\[ \frac{f'(x)}{f(x)} = \frac{a_1}{1} + \frac{a_2x}{1} +\cdots+ \frac{a_nx}{1} +\cdots \tag{1} \]
has positive coefficients of the links, \(a_n>0\).
Proof. By the method of mathematical induction it is not difficult to prove that the coefficients \(s_i\) \((i=1,2,\ldots)\) in the equality
\[ f'(x):f(x)=\sum_{k=1}^{\infty} s_k(-x)^k \]
are power sums ([1], p. 245) of the reciprocal values \(1:a_k\) of the roots \(a_k\) of the function
\[ f(-x)=\prod_{k=1}^{\infty}\left(1-\frac{x}{a_k}\right), \qquad s_i=\sum_{m=1}^{\infty}\frac{1}{a_m^i}\quad (i=1,2,\ldots). \]
The sequence \(s_1,s_2,\ldots\) is linearly independent; therefore the determinants
\[ \Delta_k= \begin{vmatrix} s_1 & \cdots & s_k\\ \cdot & \cdot & \cdot\\ s_k & \cdots & s_{2k-1} \end{vmatrix} \ne 0, \qquad \delta_k= \begin{vmatrix} s_2 & \cdots & s_{k+1}\\ \cdot & \cdot & \cdot\\ s_{k+1} & \cdots & s_{2k} \end{vmatrix} \ne 0. \]
The determinants \(\Delta_k,\delta_k>0\), because for any finite sum
\[ \sum_{k=0}^{2n} c_k x^k \]
the quadratic form
\[ \sum_{i,k=1} s_{i+k-2}x_i x_k \qquad (s_0=2n) \]
is positive
defined ([2], problems 949, 943). Since the determinants \(\Delta_k, \delta_k>0\), the coefficients \(a_k>0\) ([3], p. 23), as was required to prove.
- The smallest real roots of the function \(f(x)\) are approximately equal to the roots of the polynomial \(q_{2n+1}(x)\) \((n=1,2,\ldots)\), which is the denominator of a convergent of the equality (1). Thus, the smallest roots of the Bessel function \(I_\nu(x)\) are approximately equal to the smallest roots of the following polynomial
\[ q_{2n+1}(x)=\sum_{m=0}^{n} \frac{C_{2n+1-m}^{m}\left(-\dfrac{x^2}{4}\right)^m} {(\nu+1)_m(\nu+2n+2-m)_m}. \tag{2} \]
For example, for the first two roots we have
\[ a_{1,2}= \sqrt{ -\frac{8}{3}M \pm \frac{4}{3}\sqrt{M(\nu^2+6\nu+17)} } +\Delta \quad (M=\nu^2+6\nu+8) \]
and the relative error \(\delta_1<0.0001\) \((\nu=0)\),
\[ \delta_2=\left|\frac{\Delta_2}{a_2}\right|<0.082\;(\nu=0),\qquad \delta_1<0.0081\;(\nu=5). \]
- Theorem. For the product of two Bessel functions we have the equality
\[ \frac{I_\nu(x)\cdot I_\nu(x)} {[\Gamma(\nu+1)]^{-1}} \sum_{k=0}^{\infty} \frac{(-1)^k\left(\dfrac{x}{2}\right)^{4k+2\nu}} {k!(1+\nu)_k(1+\nu)_{2k}}. \tag{3} \]
Proof. For the coefficient \((-1)^k c_k\) \((k=1,2,\ldots)\) we obtain ([4], p. 191, [4])
\[ (-1)^k c_k= \frac{(-1)^k}{k!\,k!\,(1+\nu)_k(1+\nu)_k} \times \]
\[ \times \lim\,{}_4F_3 \left[ \begin{array}{c} a,\;1+\dfrac{a}{2},\;-k,\;-\nu-k;\;-1\\[4pt] \dfrac{a}{2},\;1+a+k,\;1+a+\nu+k \end{array} \right] = \frac{(-1)^k}{k!(1+\nu)_k(1+\nu)_{2k}}, \]
as was required to prove.
The following equalities are special cases of (3):
\[ \operatorname{sh}x\,\sin x = \sum_{k=0}^{\infty} \frac{(-1)^k\left(\dfrac{x}{2}\right)^{4k+2}} {k!\left(\dfrac12\right)_{k+1}\left(\dfrac12\right)_{2k+1}}; \qquad \frac{\operatorname{ch}x}{\sec x} = \sum_{k=0}^{\infty} \frac{(-1)^k\left(\dfrac{x}{2}\right)^{4k}} {k!\left(\dfrac12\right)_k\left(\dfrac12\right)_{2k}}. \tag{4} \]
- We multiply the series (3) with argument \(-x^4\) by the same series with argument \(x^4\). In this way we obtain an alternating series with respect to the argument \(-x^8\)
\[ f(-x^8)=1-c_1x^8+c_2x^{16}-c_3x^{24}+\cdots \tag{5} \]
The minimal positive roots of the Bessel function \(I_\nu(x)\) are approximately equal to the minimal real roots of the function \(f(-x^8)\). By virtue of equality (5), the first two roots of the Bessel function \(\alpha_1\) and \(\alpha_2\) are equal to
\[ \alpha_{1,2}=2\sqrt{ 2(\nu+2)^2(\nu+3)_2 \left[ M \pm \sqrt{M^2-N(\nu+1)^4(\nu+5)_4} \right] }+\Delta, \tag{6} \]
where
\[ M=(5\nu+11)(\nu+5)_4;\qquad N=50\nu^2+462\nu+1084. \]
Moreover, the remainder term \(\Delta\) is, in absolute value, the following:
\[ |\Delta|<\frac{c_3}{x^{-24}}= \frac{ 5(25\nu^4+628\nu^3+5873\nu^2+24098\nu+36384)x_4^2 }{ 2^{24}\cdot 6(\nu+1)_3(\nu+1)_4(\nu+1)_6(\nu+1)_{12} }. \tag{7} \]
For \(\alpha_1,\alpha_2\) of the function \(I_\nu(x)\) we have the relative error
\[ \delta_1<2.12\cdot 10^{-8}\;(\nu=0),\qquad \delta_2<0.004\;(\nu=0),\qquad \delta_1<9.71\cdot 10^{-5}\;(\nu=5) \]
(\(\delta_2\) was obtained by direct computation of the root and comparison with the tabulated value).
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