Abstract
Full Text
MATHEMATICS
Yu. Ya. Kaazik
ON A CLASS OF ITERATIVE PROCESSES FOR THE APPROXIMATE SOLUTION OF OPERATOR EQUATIONS
(Presented by Academician A. N. Kolmogorov, 14 IX 1956)
For the approximate solution of the equation
[
P(x)=0,
\tag{1}
]
where (P) is a twice differentiable operator from a Banach space (X) into a normed space (Y), several iterative processes are known which can be represented by the general formula
[
\Delta x_{n+1}=x_{n+1}-x_n
=-(E+\alpha R_n)^{-1}[E+(\alpha+1)R_n]\Gamma_nP(x_n),
\tag{2}
]
where (E) is the identity operator, (\Gamma_n=[P'(x_n)]^{-1}), (R_n=\frac12\Gamma_nP''(x_n)\Gamma_nP(x_n)), and (\alpha) is a real number.
For example, for (\alpha=0) we obtain the “Chebyshev process” (see ((^1))); for (\alpha=-1), the “process of tangent hyperbolas” (see ((^2))); for (\alpha=-2), the process proposed in ((^3)).
Concerning the convergence of the processes (2) to the solution of equation (1) in the case where the operator (P) is analytic, the following theorem holds.
Theorem 1. If the following conditions are satisfied:
1) there exists the inverse operator (\Gamma_0=[P'(x_0)]^{-1}), and (|\Gamma_0P(x)|\le \eta_0);
2) the operator (P) is analytic in the sphere
[
|x-x_0|\le
\frac{\delta_0\eta_0}{1-s_0^2h_0^2(1-h_0)},
\tag{3}
]
and
[
\left|\frac1{j!}\Gamma_0P^{(j)}(x_0)\right|\le A_0H_0^{j-1}
\quad (j=2,3,\ldots);
]
3) the quantities (\eta_0, A_0, H_0) satisfy the inequalities:
[
|\alpha|A_0H_0\eta_0<1,\qquad
|\alpha+1|A_0H_0\eta_0<1,\qquad
h_0=H_0\delta_0\eta_0<1,
]
where
[
\delta_0=
\frac{1-(|\alpha|-1)A_0H_0\eta_0}
{1-|\alpha|A_0H_0\eta_0},
\qquad
q_0=1-A_0\frac{h_0(2-h_0)}{(1-h_0)^2}>0,
]
[
p_0^2=\frac{s_0^2h_0^2}{q_0(1-h_0)^2}\le 1,
]
[
s_0^2=
\frac{\bigl|2+\alpha|-(|2\alpha+\alpha^2|-1)A_0H_0\eta_0\bigr^2(1-h_0)
+A_0\delta_0(1-|\alpha|A_0H_0\eta_0)^2}
{q_0(1-h_0)^2(1-|\alpha|A_0H_0\eta_0)^2},
]
then equation (1) has in the sphere (3) a solution (x^), to which the process (2) converges with rate*
[
|x^*-x_n|\le
\frac{a_0^n p_0^{3^n-1}}{1-a_0p_0^6}\,\delta_0\eta_0
\quad
(a_0=q_0(1-h_0)^3).
\tag{4}
]
Proof. We shall show that, when (x_0) is replaced by (x_1), conditions 1)—3) are not violated. Since (\Delta x_1) can be written in the form
[
\Delta x_1=-\Gamma_0P(x_0)-(E+\alpha R_0)^{-1}R_0\Gamma_0P(x_0),
]
we have
[
|\Delta x_1|\leq \eta_0+\frac{A_0H_0\eta_0^2}{1-|\alpha|A_0H_0\eta_0}
=\delta_0\eta_0.
]
In view of the fact that (x_1) belongs to the sphere (3), we obtain
[
|\Gamma_0(P'(x_0)-P'(x_1))|
=
\left|
\sum_{j=1}^{\infty}\frac{1}{j!}\Gamma_0P^{(1+j)}(x_0)\Delta x_1^j
\right|
\leq 1-q_0<1,
]
and, therefore, there exists
[
Q^{-1}=[E-\Gamma_0(P'(x_0)-P'(x_1))]^{-1},
]
with (|Q^{-1}|\leq 1/q_0). But then (\Gamma_1=Q^{-1}\Gamma_0) also exists and, consequently, condition 1) is satisfied when (x_0) is replaced by (x_1).
The estimates (A_1), (H_1), and (\eta_1) are found, for example, as follows:
[
\left|\frac{1}{j!}\Gamma_1P^{(j)}(x_1)\right|
\leq
\frac{1}{q_0}
\left|\frac{1}{j!}\Gamma_0P^{(j)}(x_0)\right|
+
\frac{1}{q_0}
\left|\frac{1}{j!}\Gamma(P^{(j)}(x_1)-P^{(j)}(x_0))\right|
\leq
]
[
\leq
\frac{1}{q_0}
\left{
A_0H_0^{j-1}
+
A_0H_0^{j-1}
\sum_{i=1}^{\infty}
\frac{(j+i)!}{j!\,i!}h_0^i
\right}
=
\frac{A_0H_0^{j-1}}{q_0(1-h_0)^{j+1}},
]
whence
[
A_1=\frac{A_0}{q_0(1-h_0)^2},\qquad
H_1=\frac{H_0}{1-h_0},
]
[
\Gamma_1P(x_1)
=
\frac{1}{2}Q^{-1}\Gamma_0P''(x_0)(E+\alpha R_0)^{-1}R_0\Gamma_0P(x_0)
{(2+\alpha)E+
]
[
+(E+\alpha R_0)^{-1}R_0]\Gamma_0P(x_0)
+
Q^{-1}\sum_{j=3}^{\infty}\frac{1}{j!}\Gamma_0P^{(j)}(x_0)\Delta x_1^j,
]
[
|\Gamma_1P(x_1)|
\leq
(1-h_0)s_0^2h_0^2\eta_0=\eta_1.
]
The fact that condition 3) is satisfied for (x_1) is now verified directly:
[
A_1H_1\eta_1=p_0^2A_0H_0\eta_0\leq A_0H_0\eta_0,
]
[
\delta_1=
\frac{1-(|\alpha|-1)A_1H_1\eta_1}
{1-|\alpha|A_1H_1\eta_1}
\leq
\frac{1-(|\alpha|-1)A_0H_0\eta_0}
{1-|\alpha|A_0H_0\eta_0}
=\delta_0,
]
[
h_1=H_1\delta_1\eta_1
=
\frac{\delta_1}{\delta_0}s_0^2h_0^3
\leq s_0^2h_0^3\leq h_0<1,
]
[
A_1\frac{h_1(2-h_1)}{(1-h_1)^2}
=
\frac{\delta_1}{\delta_0}p_0^2A_0h_0
\frac{2-h_1}{(1-h_1)^2}
\leq
A_0\frac{h_0(2-h_0)}{(1-h_0)^2}
]
and, thus, (q_1\geq q_0>0).
Finally, it is not difficult to prove that (p_1^2\leq p_0^6) and, quite analogously,
[
(1-h_1)s_1^2h_1^2\leq p_0^4(1-h_0)s_0^2h_0^2.
\tag{5}
]
The analyticity of the operator (P) in the sphere
[
|x-x_1|\leq
\frac{\delta_1\eta_1}{1-s_1^2h_1^2(1-h_1)}
\tag{6}
]
follows from the fact that the sphere (6) is contained in (3). Indeed, if (x) belongs to (6), then
[
|x-x_0|
\leq
|x-x_1|+|\Delta x_1|
\leq
\frac{\delta_1\eta_1}{1-s_1^2h_1^2(1-h_1)}
+\delta_0\eta_0
\leq
]
[
\leq
\frac{\delta_0\eta_1}{1-s_0^2h_0^2(1-h_0)}
+\delta_0\eta_0
=
\frac{\delta_0\eta_0}{1-s_0^2h_0^2(1-h_0)}.
]
Thus, all the conditions of the theorem are satisfied for (x_1), and we can continue the definition of the elements (x_n) and the computation of the quantities associated with them, (\eta_n,\ A_n,\ H_n), etc., by the formulas
[
\eta_{n+1}=(1-h_n)s_n^2h_n^2\eta_n,\quad
A_{n+1}=\frac{A_n}{q_n(1-h_n)^2},\quad
H_{n+1}=\frac{H_n}{1-h_n},
]
[
h_{n+1}\leqslant s_n^2h_n^3,\quad
\delta_{n-1}\leqslant \delta_n,\quad
\rho_{n+1}^2\leqslant \rho_n^6,
]
to which is added the inequality, also obtained from (5),
[
(1-h_{n+1})s_{n+1}^2h_{n+1}^2\leqslant \rho_n^4(1-h_n)s_n^2h_n^2.
]
Repeated application of the last inequalities gives:
[
\rho_n^2\leqslant \rho_0^{2\cdot 3^n},\quad
(1-h_n)s_n^2h_n^2\leqslant a_0\rho_0^{2\cdot 3^n},\quad
\eta_n\leqslant a_0^n\rho_0^{3^n-1}\eta_0.
]
By virtue of this and of the obvious inequality (3^{n+p}-1\geqslant 3^n+6(p-1)) ((p,n=1,2,\ldots)), we obtain
[
|x_{n+p}-x_n|\leqslant
a_0^n\rho_0^{3^n-1}\eta_0\delta_0
\frac{1-a_0^p\rho_0^{6p}}{1-a_0\rho_0^6}.
]
Thus the sequence ({x_n}) converges, i.e., there exists an element (x^=\lim\limits_{n\to\infty}x_n). Passing in the last inequality to the limit as (p\to\infty), we obtain (4). Taking there (n=0) and taking into account the inequality (a_0\rho_0^6\leqslant (1-h_0)s_0^2h_0^2), we see that all (x_p), and also (x^), belong to (3).
Finally, we shall also prove that the element (x^*) obtained is a solution of equation (1). This is obtained by passing to the limit in the equality
[
P(x_n)+P'(x_n)[E+(\alpha+1)R_n]^{-1}(E+\alpha R_n)(x_{n+1}-x_n)=0.
]
Indeed, (|x_{n+1}-x_n|\to 0), while
(|P'(x_n)[E+(\alpha+1)R_n]^{-1}(E+\alpha R_n)|) is bounded. The theorem is proved.
When only the differentiability (up to and including the third order) of the operator (P) is known, the following theorem holds.
Theorem 2. If the following conditions are satisfied:
1) there exists (\Gamma_0), and
[
|\Gamma_0|\leqslant B_0,\quad |\Gamma_0P(x_0)|\leqslant \eta_0;
]
2) in the sphere
[
|x-x_0|\leqslant \frac{\varepsilon_0\eta_0}{1-g_0r_0^2k_0^2}
\tag{7}
]
there exist (P''(x)) and (P'''(x)), and
[
\frac12|P''(x)|\leqslant K,\quad
\frac16|P'''(x)|\leqslant L;
]
3) the numbers (\eta_0,\ B_0,\ K) and (L) satisfy the inequalities
[
|\alpha|k_0=|\alpha|B_0K\eta_0<1,\quad
|\alpha+1|k_0<1,
]
[
2k_0\varepsilon_0
=
2k_0\frac{1-(|\alpha|-1)k_0}{1-|\alpha|k_0}
<1,\quad
l_0\varepsilon_0=B_0L\eta_0^2\varepsilon_0\leqslant Ak_0^2,\quad
\nu_0k_0\leqslant 1,
]
where
[
\nu_0^2=
\frac{|2+\alpha|+A-(|2\alpha+\alpha^2|-1)k_0}
{g_0^2(1-|\alpha|k_0)^2},
\quad
g_0=1-2k_0\varepsilon_0,
]
then equation (1) has in the sphere (7) a solution (x^), to which process (2) converges with the rate*
[
|x^*-x_n|\leqslant
\frac{g_0^n(\nu_0k_0)^{3^n-1}\varepsilon_0\eta_0}
{1-g_0(\nu_0k_0)^6}.
]
The proof differs only slightly from that given in (4) for Newton’s method.
Despite the great generality of Theorems 1 and 2, the conditions and convergence estimates for concrete processes obtained from them are no worse than the conditions and estimates in the previously known theorems for the same processes.
Tartu
State University
Received
29 VI 1956
REFERENCES
- M. I. Nechepurenko, Uspekhi Mat. Nauk 9, no. 2 (60) (1954).
- M. A. Mertvetsova, DAN, 88, No. 4 (1953).
- L. K. Vykhandu, “On iterative methods in the solution of equations,” Author’s abstract of dissertation, Tartu State Univ., 1955.
- L. V. Kantorovich, Uspekhi Mat. Nauk 3, no. 6 (28) (1943).