A REMARK ON THE INSTABILITY OF CERTAIN RULES FOR A NUMERICAL SOLUTION OF THE CAUCHY PROBLEM

N. P. FEDENKO

D. W. Ionescu, in the article [1], proposed thirteen new rules for the numerical solution of the problem with an initial condition for the first-order differential equation

\[ y' = f(x,y), \qquad y(x_0)=y_0 . \tag{1} \]

When these rules were tested for stability with respect to an increase in error, it turned out that all these rules are, unfortunately, unstable to one degree or another and, consequently, can be used in computations only for a small number of steps.

It is known [2] that if, for solving the Cauchy problem (1), one adopts a difference rule

\[ y(x_{n+1})=\sum_{i=0}^{p} A_i y(x_{n-i}) + h \sum_{j=-1}^{q} B_j y'(x_{n-j}), \tag{2} \]

then its stability depends on the coefficients \(A_i\); namely, in order to judge stability it is necessary to find the roots of the characteristic equation

\[ \lambda^{p+1}=\sum_{i=0}^{p} A_i \lambda^{p-i}. \tag{3} \]

For the rule (2) to be stable, two conditions must be satisfied: a) all roots \(\lambda_i\) of equation (3) lie in the unit circle \(|\lambda|\leqslant 1\), b) all multiple roots lie strictly inside this circle. It is also well known that if there are roots whose moduli are greater than unity, then the instability of rule (2) will be the stronger, the greater the moduli of such roots.

Below we give the list of D. W. Ionescu’s rules and, correspondingly, write out approximate values of the roots \(\lambda_i\) of the characteristic equation (3):

\[ \begin{aligned} 1.\quad y(x_6)={}& 28.4y(x_0)+426y(x_1)+825y(x_2) \\ &{}-400y(x_3)-750y(x_4)-128.4y(x_5) \\ &{}+6h\,[y'(x_0)+30y'(x_1)+150y'(x_2) \\ &{}\qquad +200y'(x_3)+75y'(x_4)+6y'(x_5)] + R_1; \end{aligned} \]

\[ \lambda_1=1,\quad \lambda_2=-0.0792,\quad \lambda_3=-0.409,\quad \lambda_4=1.365, \]

\[ \lambda_5=-5.253,\quad \lambda_6=-122.294. \]

1. \[ y(x_6)=y(x_0)+101[y(x_1)-y(x_5)]+425[y(x_2)-y(x_4)]+ \]
   \[ +30h\{[y'(x_1)+y'(x_5)]+10[y'(x_2)+ \]
   \[ +y'(x_4)]+20y'(x_3)\}+R_2; \]
   \[ \lambda_1=1,\quad \lambda_2=-1,\quad \lambda_3=-0.0104,\quad \lambda_4=-0.241, \]
   \[ \lambda_5=-4.147,\quad \lambda_6=-96.601. \]

2. \[ y(x_6)=\frac{1}{3}\{3y(x_0)-32[y(x_1)-y(x_5)]-5[y(x_2)-y(x_4)]\}+ \]
   \[ +20h[y'(x_2)+y'(x_4)]+R_3; \]
   \[ \lambda_1=1,\quad \lambda_2=-1,\quad \lambda_3=-0.0924,\quad \lambda_4=-10.818, \]
   \[ \lambda_{5,6}=-0.122\pm0.993i. \]

3. \[ y(x_6)=\frac{1}{6}\{6y(x_0)+31[y(x_1)-y(x_5)]-50[y(x_2)-y(x_4)]\}+ \]
   \[ +5h[y'(x_1)+y'(x_5)]+R_4; \]
   \[ \lambda_1=1,\quad \lambda_2=-1,\quad \lambda_3=-0.290,\quad \lambda_4=-3.451, \]
   \[ \lambda_{5,6}=-0.713\pm0.701i. \]

4. \[ y(x_6)=y(x_0)+2.953125[y(x_1)-y(x_5)]+ \]
   \[ +0.46875h\{9[y'(x_1)+y'(x_5)]+20y'(x_3)\}+R_5; \]
   \[ \lambda_1=1,\quad \lambda_2=-1,\quad \lambda_3=-0.343,\quad \lambda_4=-2.917, \]
   \[ \lambda_{5,6}=-0.153\pm0.988i. \]

5. \[ y(x_6)=y(x_0)+135[y(x_2)-y(x_4)]+ \]
   \[ +6h\{9[y'(x_2)+y'(x_4)]+28y'(x_3)\}+R_6; \]
   \[ \lambda_1=1,\quad \lambda_2=-1,\quad \lambda_{3,4}=\pm0.0858i,\quad \lambda_{5,6}=\pm11.662i. \]

6. \[ y(x_6)=y(x_0)-9[y(x_1)-y(x_5)]+45[y(x_2)-y(x_4)]+ \]
   \[ +60h\,y'(x_3)+R_7; \]
   \[ \lambda_1=1,\quad \lambda_2=-1,\quad \lambda_{3,4}=0.100\pm0.113i,\quad \lambda_{5,6}=4.401\pm4.986i. \]

7. \[ y(x_6)=-y(x_0)+16[y(x_1)+y(x_5)]+ \]
   \[ +65[y(x_2)+y(x_4)]-160y(x_3)+ \]
   \[ +60h[y'(x_2)-y'(x_4)]+R_8; \]
   \[ \lambda_{1,2}=1,\quad \lambda_3=0.0527,\quad \lambda_4=-0.207,\quad \lambda_5=-4.835,\quad \lambda_6=18.989. \]

8. \[ y(x_6)=-y(x_0)-11.5[y(x_1)+y(x_5)]+ \]
   \[ +25[y(x_2)+y(x_4)]-25y(x_3)- \]
   \[ -7.5h[y'(x_1)-y'(x_5)]+R_9; \]
   \[ \lambda_{1,2}=1,\quad \lambda_3=-0.0741,\quad \lambda_4=-13.500,\quad \lambda_{5,6}=0.0368\pm0.999i. \]

9. \[ y(x_6)=\frac{1}{9}\{9y(x_0)+284[y(x_1)-y(x_5)]-175[y(x_2)-y(x_4)]\}+ \]

\[ +\frac{20h}{3}\{2[y'(x_1)+y'(x_5)]+5[y'(x_2)+y'(x_4)]\}+R_{10}; \]

\[ \lambda_1=1,\quad \lambda_2=-1,\quad \lambda_3=-0.0311,\quad \lambda_4=-32.160, \]
\[ \lambda_{5,6}=0.318\pm0.948i. \]

\[ \begin{aligned} 11.\quad y(x_6)=&\ y(x_0)-24[y(x_1)-y(x_5)]-375[y(x_2)-y(x_4)]\\ &-60h\{3[y'(x_2)+y'(x_4)]+8y'(x_3)\}+R_{11}; \end{aligned} \]

\[ \lambda_1=1,\quad \lambda_2=-1,\quad \lambda_3=0.0298,\quad \lambda_4=-0.0934, \]
\[ \lambda_5=-10.710,\quad \lambda_6=34.775. \]

\[ \begin{aligned} 12.\quad y(x_6)=&\ y(x_0)+22.875[y(x_1)-y(x_5)]-75[y(x_2)-y(x_4)]\\ &+3.75h\{3[y'(x_1)+y'(x_5)]-20y'(x_3)\}+R_{12}; \end{aligned} \]

\[ \lambda_1=1,\quad \lambda_2=-1,\quad \lambda_3=-0.0388,\quad \lambda_4=0.392, \]
\[ \lambda_5=2.552,\quad \lambda_6=-25.780. \]

\[ \begin{aligned} 13.\quad y(x_6)=&-\frac{1}{3}\{3y(x_0)+172[y(x_1)+y(x_5)]\\ &+125[y(x_2)+y(x_4)]-600y(x_3)\}\\ &-20h\{[y'(x_1)-y'(x_5)]+5[y'(x_2)-y'(x_4)]\}+R_{13}; \end{aligned} \]

\[ \lambda_{1,2}=1,\quad \lambda_3=-0.0177,\quad \lambda_4=-0.424,\quad \lambda_5=-2.358,\quad \lambda_6=-56.534. \]

As is seen from this list, for all of D. V. Ionescu’s rules the corresponding equations (3) have roots whose moduli exceed 1. Let us note that for rules 1, 2, 10, 11, 13 there exist roots with very large moduli, and the instability of these rules will be especially strong. These rules themselves, apparently, will have no practical application. The error will increase least rapidly for rules 4 and 5, for which the moduli of the roots are not very large, and these rules can probably be used in computations for a small number of steps.

References

1. Ionescu D. V. Rev. roumaine math. pureset appl., 9, No. 3, 1964, 237—243.
2. Berezin I. S. and Zhidkov N. P. Methods of Computation, vol. II, Fizmatgiz, Moscow, 1959.

Received by the editors
November 20, 1964

Institute of Mathematics, Academy of Sciences of the BSSR