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Veckstein's method as a modification of the transversal method
VECKSTEIN’S METHOD AS A MODIFICATION OF THE TRANSVERSAL METHOD* V. M. VERBUCK and D. I. MILMAN Sverdlovsk (Received 5 September 1975; revised 9 January 1976)
IT IS shown that Veckstein’s solution of the equation j(x) = 0 is a modification of the transversal method and possesses all its characte~sti~s. In the modern literature devoted to numerical methods of finding the roots of the non-linear equation I=;*$&),
(1)
as well as other methods (bisection, simple iteration, Newton’s method, the transversal method etc.) Veckstein’s method is also encountered [l-3] . Its convergence has been investigated [3] , it is recommended for the mathematical security of computers [2,3], and it is generalized for the solution of systems of non-linear equations in [4]. The selection of Veckstein’s method as an ~de~ndent method is a mis~derstan~g, it is simply a modification of the transversal method. We will demonstrate this.
since
Veckstein’s iterative process has the form
Replacing Eq. (1) by the equivalent equation f(x)=*(z)
-x=0
(3)
and substituting Eq. (3) into Eq. (2), we obtain after transformations
The last expression represents the iterative process of the transversal method with the initial points x0 and a=lp(xo). Consequently, Veckstein’s method is of no independent value. translated by J. Berry
*Zh. vjkhisl. Mat. mat. Fiz., 17,2,507-508,
1977.
21s
B.&f. ~ukhnmed~ev
216
REFERENCES 1.
FIL’CHAKOV, P. F., Numerical and graphical methods of applied mathematics (Chislennye i graficheskie metody prikladnoi matematiki), “Naukova dumka”, Kiev, 1970.
2.
LANCE, G. N., Numerical methods for h&h speed computers (Chislennye metody dlya bystrodeistvuyushchikh vychislitei’nykh mashin), Izd-vo in. lit., Moscow, 1962.
3.
CHEWCHELOV, ~~s~ndental
4.
KHROMOV, L. N., A generalization of Veckstein’s method for solving systems of non-linear equations. Nauchn. tr. Tashkentsk. un-ta, No. 394, 18%192,197O.
A. D., Procedure and standard program for the M-20 computer for finding roots of the equation f(x) = 0 by Veckstein’s method. Dep. in VINITI, No. 6898 - 73 dep.
CONSTRUCTION OF REACHABLE SETS FOR LINEAR DYNAMIC SYSTEMS WITH NOISE* B. M, MUKHAMEDEV Moscow (Received 2 February 1976) DEPENDING on the hypothesis of the information content of the process a numerical method of constructing sets of reachability is proposed or the question of the reachability of some region in the state space is solved. In a number of cases when dynamic systems are studied it may be useful to construct sets of reachab~ty. For linear systems with common constr~ts on the controls and phase variables the numerical method of the const~ction of sets of reachab~ty was proposed in [ 11. In this paper we consider a linear dynamic system with mixed constraints, whose state is determined by controls and the values of undefined factors. Two cases of the informativeness of the course of the process are distinguished, for which either the method of the construction of sets of reachability is proposed, or a method of obtaining an answer to the question of the reachability of some domain in the state space is indicated. A similar problem for a linear dynamic system without constraints on the phase variables was considered in [2] . 1. Let the process be described by the relations r(t+l)=A(t)x(t)+C(t)u(t)+B(t)v(t),
t-0,
1,. . . , T-l,
(1)
where x(t) is the vector of the phase variables, u(t) is the control vector, and v(t) is the vector of the undetermined factors at the step t. The initial state x(0) is specified and the number of steps T is fixed. Here A (I), C(t) and B(t) are matrices of the corresponding dimensions. At each step t we impose on the phase variable vector x(t) and control vector u(t) the common constraints G(t)s(f)-1-Ef1)u(t)+h(t-f10, where G(t) and E(t) are matrices, and h(t) is a column vector. About the vector of the undetermined factors v(t) we know only that it belongs to the set V(t).
Below we will consider two hypotheses about the information on the operating side of the course of the process: 1) at each step t the value of the phase variable vector x(t) and the value *zh. vpchisl Mat. mat. F&z., 17, 2,508-512,
1977.
(2)
IT IS shown that Veckstein’s solution of the equation j(x) = 0 is a modification of the transversal method and possesses all its characte~sti~s. In the modern literature devoted to numerical methods of finding the roots of the non-linear equation I=;*$&),
(1)
as well as other methods (bisection, simple iteration, Newton’s method, the transversal method etc.) Veckstein’s method is also encountered [l-3] . Its convergence has been investigated [3] , it is recommended for the mathematical security of computers [2,3], and it is generalized for the solution of systems of non-linear equations in [4]. The selection of Veckstein’s method as an ~de~ndent method is a mis~derstan~g, it is simply a modification of the transversal method. We will demonstrate this.
since
Veckstein’s iterative process has the form
Replacing Eq. (1) by the equivalent equation f(x)=*(z)
-x=0
(3)
and substituting Eq. (3) into Eq. (2), we obtain after transformations
The last expression represents the iterative process of the transversal method with the initial points x0 and a=lp(xo). Consequently, Veckstein’s method is of no independent value. translated by J. Berry
*Zh. vjkhisl. Mat. mat. Fiz., 17,2,507-508,
1977.
21s
B.&f. ~ukhnmed~ev
216
REFERENCES 1.
FIL’CHAKOV, P. F., Numerical and graphical methods of applied mathematics (Chislennye i graficheskie metody prikladnoi matematiki), “Naukova dumka”, Kiev, 1970.
2.
LANCE, G. N., Numerical methods for h&h speed computers (Chislennye metody dlya bystrodeistvuyushchikh vychislitei’nykh mashin), Izd-vo in. lit., Moscow, 1962.
3.
CHEWCHELOV, ~~s~ndental
4.
KHROMOV, L. N., A generalization of Veckstein’s method for solving systems of non-linear equations. Nauchn. tr. Tashkentsk. un-ta, No. 394, 18%192,197O.
A. D., Procedure and standard program for the M-20 computer for finding roots of the equation f(x) = 0 by Veckstein’s method. Dep. in VINITI, No. 6898 - 73 dep.
CONSTRUCTION OF REACHABLE SETS FOR LINEAR DYNAMIC SYSTEMS WITH NOISE* B. M, MUKHAMEDEV Moscow (Received 2 February 1976) DEPENDING on the hypothesis of the information content of the process a numerical method of constructing sets of reachability is proposed or the question of the reachability of some region in the state space is solved. In a number of cases when dynamic systems are studied it may be useful to construct sets of reachab~ty. For linear systems with common constr~ts on the controls and phase variables the numerical method of the const~ction of sets of reachab~ty was proposed in [ 11. In this paper we consider a linear dynamic system with mixed constraints, whose state is determined by controls and the values of undefined factors. Two cases of the informativeness of the course of the process are distinguished, for which either the method of the construction of sets of reachability is proposed, or a method of obtaining an answer to the question of the reachability of some domain in the state space is indicated. A similar problem for a linear dynamic system without constraints on the phase variables was considered in [2] . 1. Let the process be described by the relations r(t+l)=A(t)x(t)+C(t)u(t)+B(t)v(t),
t-0,
1,. . . , T-l,
(1)
where x(t) is the vector of the phase variables, u(t) is the control vector, and v(t) is the vector of the undetermined factors at the step t. The initial state x(0) is specified and the number of steps T is fixed. Here A (I), C(t) and B(t) are matrices of the corresponding dimensions. At each step t we impose on the phase variable vector x(t) and control vector u(t) the common constraints G(t)s(f)-1-Ef1)u(t)+h(t-f10, where G(t) and E(t) are matrices, and h(t) is a column vector. About the vector of the undetermined factors v(t) we know only that it belongs to the set V(t).
Below we will consider two hypotheses about the information on the operating side of the course of the process: 1) at each step t the value of the phase variable vector x(t) and the value *zh. vpchisl Mat. mat. F&z., 17, 2,508-512,
1977.
(2)