Reports of the Academy of Sciences of the USSR

1957, Volume 115, No. 3

MATHEMATICS

M. S. GORNSTEIN

NUMERICAL SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS

(Presented by Academician M. V. Keldysh on 6 II 1957)

I. Let it be required to find a simple root \(Z\) of the equation \(f(z)=0\), lying in the domain \(G:\ |z-\zeta|<\rho\), where \(\zeta\) is a fixed number; \(f(z)\) is a function of a complex variable, regular in the domain \(G\). We represent this equation in the form \(z=\varphi_n(z;\zeta)\), where \(\varphi_n(z;\zeta)=z-H_n(z;\zeta)f(z)\); \(H_n(z;\zeta)\) is a polynomial of degree \(n\) in \(z\), whose coefficients depend on \(\zeta\). Expand the functions \(H_n(z;\zeta)\) and \(f(z)\) in powers of \(z-\zeta\):

\[ H_n(z,\zeta)=\sum_{k=0}^{n} h_{nk}(z-\zeta)^k,\quad f(z)=\sum_{s=0}^{\infty} f_s (z-\zeta)^s \quad \left(f_s=\frac{f^{(s)}(\zeta)}{s!}\right). \]

Then

\[ \varphi_n(z;\zeta)=\sum_{k=0}^{\infty}(z-\zeta)^k, \]
where \(\varphi_{n0}=\zeta-f_0h_{n0}\), \(\varphi_{n1}=1-f_1h_{n0}-f_0h_{n1}\),

\[ \varphi_{nk}=-\sum_{s=0}^{k} f_{k-s}h_{ns}\quad (k=2,3,\ldots;\ h_{ns}=0\ \text{for } s>n), \]
and
\[ \varphi_{nk}= \frac{1}{k!}\left.\frac{d^k\varphi_n(z;\zeta)}{dz^k}\right|_{z=\zeta}. \]

Subject the function \(\varphi_n(z;\zeta)\) to the following \(n+1\) conditions:
\[ \left.\frac{d^k\varphi_n(z;\zeta)}{dz^k}\right|_{z=\zeta}=0 \quad\text{for } k=1,2,\ldots,n+1, \]
which gives a system of \(n+1\) linear equations \(\varphi_{nk}=0\) \((k=1,2,\ldots,n+1)\) with unknowns \(h_{n0},h_{n1},\ldots,h_{nn}\). The determinant of this system is

\[ g_n= \left| \begin{array}{ccccc} f_1 & f_0 & 0 & \ldots & 0\ 0\\ f_2 & f_1 & f_0 & \ldots & 0\ 0\\ \cdot & \cdot & \cdot & \cdot & \cdot\ \cdot\\ f_n & f_{n-1} & f_{n-2} & \ldots & f_1\ f_0\\ f_{n+1} & f_n & f_{n-1} & \ldots & f_2\ f_1 \end{array} \right|. \tag{1} \]

The solution of the system (under the condition \(g_n\ne0\)) will be
\(h_{nk}=g_{nk}/g_n\) \((k=0,\ldots,n)\), where \(g_{nk}\) are the algebraic cofactors of the elements of the first row of determinant (1).

The given equation can be written in the form

\[ z=z_n+r_n(z;\zeta), \tag{2} \]

where

\[ z_n=\zeta-f_0g_{n-1}/g_n, \tag{3} \]

\[ r_n(z;\zeta)=\sum_{k=n+2}^{\infty}\varphi_{nk}(z-\zeta)^k. \tag{4} \]

We take \(z_n\) as the approximate value of the root and shall call it the approximation of the \(n\)-th order. The error of the \(n\)-th approximation is equal to \(r_n(z;\zeta)\).

II. Expanding the determinant \(g_n\) by the elements of the first column, we obtain for it the recurrence formula

\[ g_n=\sum_{s=0}^{n}(-f_0)^s f_{s+1}g_{n-1-s}\qquad (n=0,1,\ldots;\ g_{-1}=1). \tag{5} \]

Using the known properties of determinants, we find

\[ g_{nk}=(g_{k-1}g_{n-1}-g_{k-2}g_n)/(-f_0)^k \qquad (k=0,\ldots,n;\ g_{-1}=1,\ g_{-2}=0). \]

One can express \(g_n\) directly in terms of \(f_s\),

\[ g_n=\sum_{s=0}^{n}(-1)^s A_s^{(n)} f_0^s . \]

However, as \(n\) and \(s\) increase, the expressions for \(A_s^{(n)}\) rapidly become complicated, and it is hardly advantageous to use them for large values of \(n\) and \(s\). We write out several of the first values of \(A_s^{(n)}\) (the notation \(m^{(k)}=m(m-1)\ldots(m-k+1)\) has been introduced):

\[ \begin{aligned} A_0^{(n)}&=f_1^{\,n+1};\\ A_1^{(n)}&=n f_1^{\,n-1}f_2;\\ A_2^{(n)}&=(n-1)^{(1)}f_1^{\,n-2}f_3+(n-1)^{(2)}f_1^{\,n-3}f_2^2/2!; \end{aligned} \]

\[ A_3^{(n)}=(n-2)^{(1)}f_1^{\,n-3}f_4+(n-2)^{(2)}f_1^{\,n-4}f_2f_3+(n-2)^{(3)}f_1^{\,n-5}f_2^3/3!; \]

\[ \begin{aligned} A_4^{(n)} &=(n-3)^{(1)}f_1^{\,n-4}f_5 +(n-3)^{(2)}f_1^{\,n-5}(f_2f_4+f_3^2/2!)\\ &\quad +(n-3)^{(3)}f_1^{\,n-6}f_2^2f_3/2! +(n-3)^{(4)}f_1^{\,n-7}f_2^4/4!,\ldots \end{aligned} \]

The first expressions for \(g_n\) (in terms of \(f_s\)) have the form:

\[ g_0=f_1;\qquad g_1=f_1^2-f_0f_2;\qquad g_2=f_1^3-2f_0f_1f_2+f_0^2f_3; \]

\[ g_3=f_1^4-3f_0f_1^2f_2+f_0^2(2f_1f_3+f_2^2)-f_0^3f_4; \]

\[ g_4=f_1^5-4f_0f_1^3f_2+3f_0^2(f_1f_3+f_2^2) -2f_0^3(f_1f_4+f_2f_3)+f_0^4f_5,\ldots \]

III. We introduce the notation

\[ g_n/g_{n-1}=a_n\quad (n=0,1,\ldots);\qquad f_0/a_n=l_n\quad (n=0,1,\ldots); \tag{6} \]

\[ f_n/f_1=m_n\quad (n=2,3,\ldots) \]

(it is assumed that \(f_1\ne 0\)). From (5) we obtain

\[ a_n=f_1+\sum_{s=1}^{n}(-f_0)^s f_{s+1}:\frac{g_{n-1}}{g_{n-1-s}} = \]

\[ =f_1+\sum_{s=1}^{n}\frac{(-f_0)^s f_{s+1}}{a_{n-1}\ldots a_{n-s}}, \]

or

\[ a_n=f_1(1+b_n), \tag{7} \]

where

\[ b_n=\sum_{s=1}^{n}(-1)^s l_{n-1}\ldots l_{n-s}m_{s+1}. \tag{8} \]

Dividing \(f_0\) by both sides of (7), we obtain

\[ l_n=\frac{l_0}{1+b_n}. \tag{9} \]

Lemma. If the conditions are satisfied (capital letters denote moduli of numbers)

\[ L_0=|f_0/f_1|<3-2\sqrt{2}=0.171572\ldots,\qquad M_{s+1}<\bigl((3-2\sqrt{2})/L_0\bigr)^s \tag{10} \]

\[ (s=1,2,\ldots) \]

the sequence \(a_n\) is bounded, and \(a_n\ne 0\) (hence also \(g_n\ne 0\)) for \(n=0,1,\ldots\).

Proof. First suppose that \(L_0<1\). Take a number \(q\) from the interval \((L_0,1)\) and another number
\[ q_1=(q-L_0)/(2q-L_0). \]
Obviously, \(q_1<1\). Require that \(q_1>q\). It is easy to show that, for \(L_0<3-2\sqrt{2}\), one has \(q_1>q\) for all values of \(q\) from the specified interval, and for \(q=L_0(1+1/\sqrt{2})\) the ratio \(q_1/q\) assumes the largest possible value:
\[ (q_1/q)_{\max}=(2-3\sqrt{2})/L_0. \]

By complete induction one can prove that, when conditions (10) are satisfied, \(L_n<q\) for \(n=0,1,2,\ldots\). From (8) we obtain
\[ B_n\le \sum_{s=1}^{n} L_{n-1}\ldots L_{n-s}M_{s+1} <\sum_{s=1}^{n} q^s\left(\frac{q_1}{q}\right)^s <\frac{q_1}{1-q_1} =\sqrt{2}-1<0.5. \]
Hence, taking (7) into account, the assertions of the lemma follow.

IV. Denote the lower and upper bounds of the sequence \(A_n\) respectively by \(a\) and \(A\); \(a\le A_n\le A\). In addition, suppose that the sequence
\[ F_n=|f_n|=|f_n(\zeta)/n!| \]
is bounded, and let
\[ F_n\le F\qquad (n=2,3,\ldots). \tag{11} \]

Theorem 1. If conditions (10), (11) and the condition \(\mu<1\) are satisfied, where
\[ \mu=A\rho\bigl(1+F/(1-L_0)A\bigr)/a, \]
then the sequence (3), which can be written in the form \(z_n=\zeta-l_n\), converges to a root of the given equation.

Proof. It is enough to show that \(\lim_{n\to\infty} R_n=0\). From (4) we obtain
\[ R_n<\sum_{k=n+2}^{\infty}\Phi_{nk}\rho^k,\qquad \Phi_{nk}\le \sum_{s=0}^{k} F_{k-s}H_{ns}\le F\sum_{s=0}^{n} H_{ns} \]
(\(h_{ns}=0\) for \(s>n\)). It can be shown that
\[ H_{ns}<\frac{F A^{n-s}(A+F/(1-L_0))^{s-1}}{(1-L_0)a^{n+1}} \quad (s=1,2,\ldots,n) \]
and
\[ H_{n0}<A^n/a^{n+1}, \]
so that
\[ \Phi_{nk}<F\sum_{s=0}^{n} \frac{F A^{n-s}\bigl(A+F/(1-L_0)\bigr)^{s-1}}{(1-L_0)a^{n+1}} < \frac{F A^n}{a^{n+1}} \left(1+\frac{1}{1-L_0}\frac{F}{A}\right)^n, \]
\[ R_n<K\rho\mu^{n+1}\qquad (n>0), \tag{12} \]
where
\[ K=F/A(1-\rho)\bigl(1+F/(1-L_0)A\bigr),\qquad \mu=A\rho\bigl(1+F/(1-L_0)A\bigr)/a. \]
In view of the condition \(\mu<1\), we shall have \(\lim_{n\to\infty}R_n=0\). The theorem is proved.

The error estimate of the \(n\)-th approximation \((n=1,2,\ldots)\) is given by expression (12). As for the zeroth-order approximation, for it we have
\[ r_0=-\frac{1}{f_1}\sum_{k=2}^{\infty} f_k(z-\zeta)^k. \]
Suppose that in this series the first of the numbers \(f_k\) that is not equal to zero is \(f_r\) \((r\ge 2)\). Then we obtain
\[ R_0<\bigl(F/(1-\rho)F_1\bigr)\rho^r. \]

Remarks 1. In practice, when computing the error, the following simplifications may be allowed: 1) replace \(1-L_0\) and \(1-\rho\) by unity; 2) take \(a=A=F_1=|f_1|\). Then
\[ K=F/(F+F_1),\qquad \mu=(1+F/F_1)\rho. \]

1. In the case of a real root of a real function one obtains
   \[ K=F/(F+F_1)\bigl(1+\rho(1+F_1/F)\bigr). \]

2. It can be shown that: if a sequence \(f_k\) \((k=0,1,\ldots)\) has been found corresponding to some number \(\zeta\) satisfying the conditions of Theorem 1, then in some \(\rho\)-neighborhood of the number \(\zeta\) there lies a root of the equation, and the sequence \(z_n\) converges to the root. One may take \(\rho\approx L_0\).

Example. \(z^3-6z^2+109z-306=0\). Let \(\zeta=3\). Then \(f_0=-6\), \(f_1=100\), \(f_2=3\), \(f_3=1\), \(f_k=0\) for \(k>3\). One may use formu-

by (3), computing \(g_n\) by (5), or by the formula \(z_n=\zeta-l_n\), computing \(b_n\) by (8) and \(l_n\) by (9). Finally, one may compute \(g_n\) (up to \(n=4\)) from the expressions in item II. We obtain
\(z_0=3.060;\quad z_1=3.05989;\quad z_2=3.0598945;\quad z_3=3.0598945279;\quad z_4=3.059894527995;\ldots\)
The errors, computed from the simplified values of \(K\) and \(\mu\), are:
\(R_0<0.11\cdot 10^{-3};\quad R_1<0.23\cdot 10^{-5};\quad R_2<0.14\cdot 10^{-6};\quad R_3<0.87\cdot 10^{-8};\quad R_4<0.53\cdot 10^{-10}\).
The actual errors are:
\(0.11\cdot 10^{-3};\quad 0.23\cdot 10^{-5};\quad 0.2\cdot 10^{-7};\quad 0.2\cdot 10^{-9};\quad 10^{-12}\).

V. Generalization of Newton’s process. Denote by \(z_{n_1n_2}\) the approximation of order \(n_2\) under the condition that \(z_{n_1}\) is taken as the initial value of the root; in general, denote by \(z_{n_1\ldots n_{m-1}n_m}\) the approximation of order \(n_m\) under the condition that the initial value of the root is \(z_{n_1\ldots n_{m-1}}\). It is assumed that \(n_m\geq 0\) \((m=1,2,\ldots)\).

Theorem 2. If the initial value of the root \(\zeta\) satisfies the conditions of Theorem 1, then the sequence \(z_{n_1}, z_{n_1n_2}, z_{n_1n_2n_3},\ldots\) converges to the root of the equation.

Proof. We shall first assume that all \(n_k>0\). We have
\(|z-z_{n_1}|=R_{n_1}<K\rho\mu^{n_1+1}\).
Take \(z_{n_1}=\zeta_1\) and, for the value \(\zeta_1\), find \(z_{n_1n_2}\). We have
\(\rho_1=K\rho\mu^{n_1+1}\),
\(|z-z_{n_1n_2}|<K_1\rho_1\mu_1^{n_2+2}\), or
\(R_{n_1n_2}<K_1K^{n_2+2}\rho\mu^{(n_1+2)(n_2+2)-1}\).
After \(m\) such steps we obtain \(z_{n_1\ldots n_m}\) with error

\[ R_{n_1\ldots n_m}<K_{m-1}K_{m-2}^{n_m+2}\cdots K^{(n_m+2)(n_{m-1}+2)\cdots(n_2+2)} \rho\mu^{[(n_1+2)\cdots(n_m+2)]-1}. \tag{13} \]

Since all \(K<1\) and \(\mu<1\), it follows that
\(\lim_{m\to\infty} R_{n_1\ldots n_m}=0\), and the theorem is proved for this case. It is not difficult to show that the theorem will also be true in the case when zeros occur among the numbers \(n_k\), if \(F_{\rho}/(1-\rho)F_1<1\).

Remarks. 1. For a rough determination of the error, one may put \(K_1=K_2=\cdots=K\) in (13). Then we obtain

\[ R_{n_1,\ldots,n_m}<K^{1+(n_m+2)+\cdots+[(n_m+2)\cdots(n_2+2)]} \rho\mu^{[(n_1+2)\cdots(n_m+2)]-1}. \]

In the case \(n_1=n_2=\cdots=n\), we obtain

\[ R_{n_m}<K^{\frac{(n+2)^m-1}{n+1}}\rho\mu^{(n+2)^m-1}. \]

1. In the case \(n_1=n_2=\cdots=0\) (Newton’s process), the error estimate for the \(m\)-th step will be
   \(R_{0^m}<K^{2^m-1}\rho^{2^m-1}r\), where \(K=F/(1-\rho)F_1\), or, in simplified form, \(K=F/F_1\).

2. Theorem 2 can also be applied in the case of a non-analytic function having derivatives up to order \(n+2\) \((n\geq 0)\), if for the initial value \(\zeta\) the conditions of Theorem 1 are satisfied up to and including order \(n+2\). In this case one must have \(n_m\leq n\) \((m=1,2,\ldots)\).

VI. Remarks. The first approximation was first published in \((^2)\), and then was rediscovered by other authors \((^{3,4})\). Approximations up to third order are found in \((^{5,6})\). Approximations of arbitrary order, expressed by the determinant (1), are given in \((^{7-9})\). In \((^9)\) there is also formula (5) of the present paper. In all the cited works the question of the existence of \(\lim_{n\to\infty} z_n\) is not posed; nor is the question of convergence to the root of the generalized Newton process resolved, except for one case: \(n_1=n_2=\cdots=1\) \((^{10})\). As for the question of convergence of Newton’s process itself, there is an extensive literature, which I shall not touch upon here.

Received
10 XII 1956

CITED LITERATURE

\(^1\) M. S. Gornshtein, DAN, 78, No. 2 (1951).
\(^2\) Halley, Phil. Trans. Roy. Soc., 18 (1694).
\(^3\) J. S. Frame, Am. Math. Monthly, 51 (1944).
\(^4\) H. S. Wall, Am. Math. Monthly, 55 (1948).
\(^5\) E. Shroder, Math. Ann., 2 (1870).
\(^6\) J. Kiss, Acta Techn. Acad. Sci. Hung., 7, No. 3–4 (1954).
\(^7\) H. W. Richmond, Lond. Math. Soc., 19, No. 73 (1944).
\(^8\) A. P. Domoryad, Tr. Sredneaziatsk. univ., nov. ser., 7, No. 36 (1954).
\(^9\) H. J. Hamilton, Am. Math. Monthly, 57, No. 8 (1950).
\(^ {10}\) G. S. Salekhov, DAN, 82, No. 4 (1954).