Some Stability Theorems for Ordinary Difference Equations

Some Stability Theorems for Ordinary Difference EquationsAuthor(s): James HurtSource: SIAM Journal on Numerical Analysis, Vol. 4, No. 4 (Dec., 1967), pp. 582-596Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2949723 .

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SIAM J. NUMER. ANAL.

Vol. 4, No. 4, 1967 Printed in U.S.A.

SOME STABILITY THEOREMS FOR ORDINARY DIFFERENCE EQUATIONS*

JAMES HURTt

Summary. LaSalle [5], [6], [7] and others have developed a generalization of the "second method" of Liapunov which utilizes certain invariance properties of solutions of ordinary differential equations. Invariance properties of solutions of ordinary difference equations are utilized here to develop stability theorems similar to those in LaSalle [5]. As illustrations of the application of these theorems, a region of convergence is derived for the Newton-Raphson and secant iteration methods. A modi- fication of one of these theorems is given and applied to study the effect of roundoff errors in the Newton-Raphson and Gauss-Seidel iteration methods.

1. Introduction. An ordinary difference equation is an equation of the type:

(1) x(k + 1) = f(k, x(k)),

where each x and f(k, x) are elements of X, an n-dimensional vector space. Since the notation used in (1) can become very clumsy, the somewhat neater E notation is used. If E is defined as the operator where Ex(k) - x(k + 1), then (1) can be written as

(1') Ex = f(k, x),

where the arguments of x and Ex are understood to be k. A function x(k; ko , x0) is called a solution of the difference equation (1)

if it satisfies the following three conditions:

(a) x(k; ko , x0) is defined for ko < k < ko + K for some integer K > 0; (b) x(ko ; ko , xo) -xo, the initial vector; (c) x(k + 1; ko , xo) = f(k, x(k; ko , xo) ) for ko < k < ko + K-1.

Hereafter, it is assumed that a solution to (1) exists and is unique for all k ? ko and that this solution is continuous in the initial vector x0. More specifically, if {x4) is a sequence of n-vectors with xn -+ x0 as n -o 00,

then the solutions through xn converge to the solution through x0:

x(k; ko, Xn) ->x(k; ko , xo) as n-- o.

* Received by the editors December 13, 1966. t Center for Dynamical Systems, Brown University, Providence, Rhode Island.

Now at Department of Mathematics, University of Iowa, Iowa City, Iowa 52240. The author received support from the National Aeronautics and Space Admin- istration under their Traineeship Program, and the preparation and reproduction of this paper was supported by the same Administration under Contract NAS8-11264.

582



STABILITY THEOREMS FOR DIFFERENCE EQUATIONS 583

For all k on any compact interval, this convergence is assumed to be uniform.

For any n-vector x, let I x I denote any vector norm of x. For any nonempty set of n-vectors A, denote the distance from x to A by d(x, A):

d(x,A) = infI x - yj:y E Al.

Introduce the vector X to X and define d(x, oX) = I x . Let A* = A U I oI and d(x,A*) = min{d(x,A),d(x, oX ).

A point p E X is a positive limit point of x(k) if there is a sequence { k,4 with kn+1 > kn --x oX, and x(kn) -? p as n -- oo. The union of all the positive limit points of x(k) is the positive limit set of x(k).

2. The general stability theorem. Let G be any set in the vector space X. G may be unbounded. Let V(k, x) and W(x) be real-valued functions defined for all k _ ko and all x in G. If V(k, x) and W(x) are continuous in x, V(k, x) is bounded below, and

AV(k, x) = V(k + 1,f(k, x)) -V(k, x) < -W(x) ? 0

for all k > ko and all x in G, then V is called a Liapunov function for (1) on G. Let G be the closure of G, including oX if G is unbounded, and define the set A by

(2) A = {x E G:W(x) = 0l.

The following result is the difference analogue to Theorem 1 in LaSalle [5].

THEOREM 1. If there exists a Liapunov function V for (1) on G, then each solution of (1) which remains in G for all k _ ko approaches the set A* = A U Xo}I ask-+ co.

Proof. Let x(k) be a solution to (1) which remains in G for all k > ko. Then, by assumption, V(k, x(k)) is a monotone nonincreasing function which is bounded from below. Hence, V(k, x(k)) must approach a limit as k -- oo, and W(x(k)) must approach zero as k -+ oo. From the definition of A* and the continuity of W(x), we get d(x(k), A*) O 0 as k -* oo. Note that if G is unbounded and there exists a sequence x,j such that xn E G, I xI -X oo and W(xn) -+0 as n -+ oo, then it is possible to have an unbounded solution under the conditions of the theorem. If G is bounded or if W(x) is bounded away from zero for all sufficiently large x, then all solutions which remain in G are bounded and approach a closed, bounded set contained in A as k -* o.

This theorem can be used to prove easily all of the usual Liapunov stability theorems (see, for example, Hahn [11 and Kalman and Bertram [3]). For example, if G is the entire space X and W(x) is positive definite, then A = 101 and all solutions approach the origin as k -+ c*. However,



584 JAMES HURT

as the following example shows, other considerations can be used to determine if a solution x(k) will remain in G. The difference equation is given by

(3) Ex= x-2 for > 0.

Let the set G be the set of positive numbers. Then, if x > 0, we obtain Ex > 0 from (3), and all solutions which start in G remain in G. The function V(k, x) = V(x), where

X V(x = l+ 2 1 + XI'

is a Liapunov function for (3) on G since V(x) _ 0 and

AV(x)= x -2 - x -x( - x)(x3- 1) w )?

1 + X-4 + X2 (1 + X2)(1 +X4)

We have W(x) = 0 when x=0, x=1, and W(x) - 0 as x -+ . Thus, the set A* is the set {O, 1, oo }. Each solution with xo > 0 approaches A* as k -* oo. A look at the solutions to (3),

x (k) =-xo (2)k

shows that this is exactly the case. If xo = 1, then x(k) = 1 for all k. If x0 < 1, then x(k) -* 0 for even k and x(k) X-+ for odd k.

Quite often the set G can be constructed so that all solutions which start in some smaller set G, remain in G. One such case is covered in the following corollary.

COROLLARY 1. Let u(x) and v(x) be continuous real-valued functions. Let V(k, x) be such that

u(x) _ V(k, x) < v(x)

for all k > ko. For some r, define the sets G = G(,) and G1 = G1(,q) as

G(O) = {x:u(x) < t1},

GI(n) = {x:v(x) < v}.

If V is a Liapunov function for (1) on G(q), then all solutions which start in G1(-) remain in G(67) and approach A as k -> co.

Proof. Let x(k) be a solution of (1) with x(ko) G G1(iq). Then,

u(x(k)) ? V(k,x(k)) < V(ko,x(ko)) < v(x((ko)) < v

for all k > koI, implying that x((k) G G(,) for all k> ko . Theorem 1 and Corollary 1 give sufficient conditions for the positive

limit set of a solution x(k) to be contained in A. There is an art to finding the best V, W, u and v, i.e., the functions V, WV, u and v which give the largest G, the largest G1 and the smallest A. Often more information about




the behavior of the solutions can be obtained by considering several different Liapunov functions and combining the results from each.

The following example is taken from Vidal and Laurent [10]. The sampled control systems covered in that paper are described by the difference equation

(4) Ex = M(k, x)x,

where M(k, x) is a matrix. For any vector norm, I x , define the norm of the matrix M(k, x) by

I M(k, x) I = min {b: I M(k, x)y I _ b I y I for all y 0}.

Then clearly, I M(k, x)x I < I M(k, x) x I . For the difference equation (4), try the Liapunov functioii V(k, x) = x i. Then,

AV(k, x) = I M(k, x)x - x

? (I M(k, x) I-1) x.

Let u(x) = v(x) = V(k, x) = x l; then Gj(t) = G(t) = {x: I x I < i}7. For all x in G(t1) and all k ? ko, let M(k, x) I _ a(x) and W(x)

+(1 -a(x)) I x . Then we have

AV(k, x) < -W(x).

If a(x) < 1 for all x in G(i), then -W(x) ? 0, the set A is the origin and possibly something on the boundary of G( i). Since V(k, x(k)) is a nonincreasing function of k and the boundary of G( ) is a level surface of V(k, x), the solutions cannot approach the boundary of G( i). Hence, all solutions which start in G( i) remain in G( i) and approach the origin as k -* o. The set G(it) is called a domain of stability for the system (4). The best G( i) is chosen by picking v7 as large as possible without violating the inequality a(x) < 1 for all x in G(i).

Various choices for the vector norm will result in various a(x) and various domains of stability. Since each is sufficient, the union of all these domains of stability is also a domain of stability.

If M(k, 0) is a constant matrix, independent of k, and the spectral radius of M(k, 0) is less than one, then there is a vector norm such that a(x) is continuous in x and a(0) < 1, indicating that there is a nonempty domain of stability (see Ostrowski [8] for a proof of this statement).

The following example illustrates that the results obtained in Theorem 1 and Corollary 1 are the best possible without further assumptions. The difference equation is

Ex = y,

Ey = a2x + p(k)y,



586 JAMES HURT

where 0 < a < 1 and O < <5 p(t) < 1 - a2. If p(k) = p, a constant, then the conditions for stability are satisfied and all solutions approach the origin as k -+ oo.

Try the Liapunov function

V(k, x, y) = a2x2 + y2

Then,

AV(k, x, y) = -a 2p(k)(x - y)2 + a 2(p(k) - (1 - a ))x

+ (p(k) + l)(p(k) - (1 - a ))y

< -a2p(k)(x _ y)2 < -a26(X _ -Y)2 _ _V(X, y) ? 0.

From Corollary 1, we see that all solutions are bounded and x(k) -y(k) 0 as k -- X .

If

1 - a 2 p(k) + ak+1

for all k 2 0, then this p(k) satisfies the conditions given above and one solution of the difference equation (5) is

x(k) = 1 + ak *1 as k oo

y(k) = 1+ak+l1 as k-oo.

The results obtained are the best possible. Notice, however, that this p(k) approaches 1- a2 as k *c. If, instead of p(k) < 1-a2, we knew that p(k) < 1 - a2 - e for some e > 0, then we obtain

AV(k, x, y) < -a26(X _ y)2 - a2ex2-(1 + 6)y2 = W (X, Y) < 0,

and the only point where W1(x, y) = 0 is x = y = 0. In this case, all solutions approach the origin as k -- oo.

3. Autonomous difference equations. If the function f(k, x) in (1) is independent of k, then the difference equation is said to be autonomous:

(6) Ex = f(x).

Just as is the case for autonomous differential equations, solutions to (6) are essentially independent of ko; so we assume ko = 0 and write the solution as x(k; xo). A function x*(k) is said to be a solution for (6) on (- , oo) if, for any ko in (- , oc), we have for all k > ko ,

x(k - ko; x*(ko)) = x*(k).




A set B is an invariant set of (6) if xo G B implies that there is a solution x*(k) for (6) on (-oo, o ) such that x*(k) E B for all k and x*(O) = xo .

LEMMA 1. The positive limit set B of any bounded solution of (6) is a nonempty, compact, invariant set of (6).

Proof. Let x(k) be a bounded solution of (6) and B its positive limit set. For each p E B, there is a monotone sequence of integers {kj} such that k- oo and x(kn) -- p as n --* . Then each function yn(k) = x(k + kn) is a solution of (6) with Yn(O) -* p as n -m o. From continuity in the initial conditions, these functions approach the solution x(k; p) as n - oo. By extending each function yn (k) to -kn , we can extend the solution x(k; p) to - o. The simultaneous convergence to x(k; p) and B implies that x(k; p) E B for all k, and so B is an invariant set. The fact that B is nonempty and compact is obtained from the definition of a positive limit set and the boundedness of x(k).

For an autonomous equation, Theorem 1 can be strengthened as follows. THEOREM 2. If there exists a Liapunov function V(x) for (6) on some

set G, then each solution x(k) which remains in G is either unbounded or approaches some invariant set contained in A as k -f oo.

Proof. From Theorem 1, x(k) -* A U {X} as k -* o. If x(k) is unbounded, then Lemma 1 does not hold. If x(k) is bounded, then its positive limit set is an invariant set.

If the set M is defined as the union of all the invariant sets contained in A, then x(k) -+ M as k -* X whenever x(k) remains in G and is bounded. The set M may be considerably smaller than the set A. Under the conditions of Theorem 2, an unbounded solution can exist only if G is unbounded and there is a sequence {xnj, x. E G, x. -* o and AV(xn) -O0 as n -* xo.

Corollary 1 can be restated in a similar manner. COROLLARY 2. If, in Theorem 2, the set G is of the form

G = G(t) = {x:V(x) < j}

for some v > 0, then all solutions which start in G remain in G and approach M as k -+ c.

This corollary can be used to obtain regions of convergence for various iterative methods which can be described by an autonomous difference equation. A region of convergence is a set G C X such that, if x(0) C G, then x(k) C G for all k > 0 and x(k) approaches the desired vector as k -* o. The largest region of convergence is the union of all regions of convergence. The secant and Newton-Raphson methods are treated as ex- amples. (For a derivation and discussion of these methods see, for example, Traub [9] or Ostrowski [8].)

The secant method for finding a root of f(z) = 0 (f(z) and z are complex numbers) is given by assuming values for z1 and Z2, then forming the se-



588 JAMES HURT

quence IZk} by repeated application of the following:

(7) Zk+2 = Zk+1 - (Zk+1 - Zk)f(Zk+l)

f(Zk+l) - f(Zk)

We assume that, for every k, Zk+I $? Zk and f(zk+l) 9 f(zk), so this iteration formula is well defined for all k. Let a be the desired root of f(z) = 0 and let

f(a + e) = f'(a)e + g(a, e)e2.

Then, letting Zk = a + ek for each k, (7) becomes

ek+2 = M(a, ek , ek+1)ekek+l ,

where

M(a, ek , ek+l) = g(a, ek+l)ek+1 - g(a,ek)ek f(a + ek+1) -f(a +ekY

With the assumption that a is a simple root of f(z) = 0 and g(a, e) is continuous and bounded in e, then M(a, ek , ek+1) is continuous and bounded for ek, ek+l small enough.

The difference equation (8) is obtained by letting xl(k) = ek and x2(k) - ek+1

ExI = X2 (8)

EX2 = M(a, x1, X2)XIX2v

Consider the Liapunov function Vq(xl, x2) = j xl Jq + I x2 Iq for some q ?1. Then,

AVq(Xl X2) = -(1 - M M(a, X1 , X2)X2 iq) I X1 q

and AV,(xl, x2) ? 0 if I M(a, xl, x2)x2 ? 1. Let G6(rq) be the set Gq(tq) - {(xl , x2): ( ixl iq + I x2 lq)l/q < q}. Since x2 = 0 implies

I M(a, Xl, X2)X2 I = 0 < 1,

there is some q > 0 such that i M(a, xl, x2)x2 ? 1 for all (xl, x2) in G,(77). From Corollary 2, this G,(77) is a region of convergence for the secant method. If the initial guesses z1 and z2 are such that (xI , x2) E Gq(7) for some q, then (xl, x2) will remain in G,(t ) for all k and approach an invariant set contained in the set A = {(X1, X2) E Gq(t):xj = 0}. The only invariant set of (8) with xi = 0 is the origin xi = x2 = 0, so we obtain (xl , x2) - (0O, 0) as k -* o, and the method converges.

If, for i e i < fo, we obtain

If'(a + e) I > F, i g(a, e) i < G,




then we get that I M(a, X1 , X2)X2 I < 1 if i x2 I < F/G. Thus, 77 can be taken 2 2 as the smaller of 77o and F/G. For the particular equation f(z) _ z - a2

we get I f (a + e) I > 2 1 a - 2-?o and g(a, e) 1 for I e K< ro . IJn this case, we can choose =77o = 2 1 a

It should be noted that the set Gq(77), or even the union of these sets for all q _ 1, is not always the largest region of convergence. For the simple equation f(z) = Z2 - a2, almost any choice of z1, Z2, provided only that Zl $4 Z2 and f(z1) $ f(Z2), will lead to a sequeilce which will converge either to +a or to -a. However, if z1 and Z2 are in the region defined by Gq( t), then not only will the sequence converge to a but this convergence will be uniform in the sense that I Zk- a + I Zk?1 - a I will be a decreasing function of k.

Corollary 2 can also be used to find a region of convergence for the Newton-Raphson method. The Newton-Raphson method for finding a root of f(z) = 0 (f(z) and z are n-vectors) is given by assuming a value for Z , then forming the sequence {Zk} by repeated application of the following:

Fa -1

Zk+1 = Zk a- (Zk) f (Zk)

where [Of(Zk)/OZ] is the matrix of partial derivatives of f. Here, we assume that [tf(Zk)/dzl always has an inverse. If the desired root is a simple root, theni this is the case. By letting a be the desired root, expanding f(a + e) as

f(a + e) -[aLf (a) e +?fo(e)

and letting Zk a + ek , then the difference equation becomes

ek+l = +M1(ek)[M'2(ek)ek -fo(ek)I,

where 311(e) = [cf(a + e)/dz]-' and Mi12(e) = [cf(a + e)/dz - af(a)/az]. Let i e I be some vector norm (see Ostrowski [8]). If a is a simple root of f(z) = 0 and f is twice continuously differentiable at z = a, then for each 7 > 0 there exists a positive constant k(-q) such that, for all e with J e I < q, we have I M(e)(M1/2(e)e - fo(e)) < k(t) I e 12. Then, letting V(e)

e , we obtain

AV(e) < -(1- k(7) Iel ) IeI

and AV(e) ? 0 if k(r) I e J < 1. Using Corollary 2, we obtain a region of convergence G(no) = {z:I z - a I < o, where -0 = min (,, 1/(k(-))). We can choose t7 so as to maximize fo , thus obtaining the best region of convergence obtainable with this Liapunov function.

For the case where z and f(z) are complex numbers, if there is some F > 0 such that Ifo(e) i _ F I e 12 for all z, where i z - a i < -q, some



590 JAMES HURT

2 s f'(a) /F, then ve can obtain

k(q) -3 F

and the best (with this k(,q)) region of convergence is given by G(no), where

77a=0 f()

For the simple case f(z) Z a 2, we get so = | a However, a sharper estimate may be used for k( ) which results in qo = a j . This latter case is the best possible. Any disc centered at a with radius larger than 2 1 a i will have points inside the disc which will map outside the disc on the next iteration, and AV(x) is positive for some values of x.

It should be noted that the region of convergence G(tjo) is not always the largest region of convergence. For the simple equation f(z) = Z2 - a 2, any initial guess z1 4 0 will lead to a sequence {zj} which will converge either to +a or to -a.

4. Periodic difference equations. If, in the difference equation (1), f(k, x) is T-periodic for some integer T _ 1 and fixed x, i.e., f(k + T, x) = f(k, x) for all k, x, then the difference equation is said to be a T-periodic difference equation. A function x*(k) is said to be a solution for (1) on (- oo, o) if, for any ko in (-oc, )oowehaveforallk ? ko,

x(k; ko , x*(ko) ) = x*(k).

A set B is an invariant set of (1) if xo C B implies that there are a ko and a solution x*(k) for (1) on (-X X, ? ) such that x*(ko) = xo and x*(k) E B for all k.

LEIMmA 2. Let x(k) be a solution of (1) that is bounded for all k > ko. Then the positive limit set of x(k) is an invariant set of (1).

Proof. This lemma is proven in a manner very similar to that used in Lemma 1. The ko used in the definition of an invariant set is obtained in the following manner: If {knT is a monotone sequence such that x(kn)

p C B, the positive limit set of x(k), then there is a sequence of integers {Mn} such that kn - M,,7T C [0, T) for all n. The set [0, T) consists of a finite number of integers, so at least one of these integers, ko, must satisfy ko-= kn - MT7 for an infinite number of n's. The solution x(k; koJ, p) is then shown to be the limit of the functions y,,(k) = x(k + kn) and is in B for all k, thus demonstrating that B is an invariant set of (1).

Theorem I can now be restated for T-periodic difference equations. THEOREM 3. Let V(k, x) be a T-periodic, continuous function which is

bounded below for all x in some set G. For 7c _ ko and x in G, let AV(k, x) < 0 and define the set A byA = {(k,x) AV(kI,x) = 0,x C G}.LetM be the




union of all solutions x(k) of (1) such that (k, x(k)) E A for all k. Then each solution of (1) which remains bounded and in Gfor all k >_ ko approaches some invariant set contained in M as k -c oo.

Proof. The function V(k, x(k)) is nonincreasing and bounded below, hence AV(k, x(k))-* 0 as k ct. The continuity of V and AV implies that d(((k, x(k)), A) 0 as k oo. Since x(k) must approach an invariant set as k -a: , it must approach M as k - oo.

An unbounded solution is possible under the conditions of Theorem 3 only if G is unbounded and there exists a sequence { (knI, xn) } with jXn I -* 0, and AV(kn, xn) -+ 0 as n -* o. If G is bounded or if AV(k, x) is bounded away from zero for all sufficiently large x, then all solutions of (1) which remain in G are bounded and approach M as k -* o.

5. Asymptotically autonomous difference equations. If the difference equation (1) can be written in the form:

(10) Ex = H(x) + F(k, x),

where F(k, x) - 0 as k -0> o uniformly for all x in any compact set, the difference equation is said to be an asymptotically autonomous difference equation. With each asymptotically autonomous difference equation (10), there is the associated autonomous difference equation

(11) Ex = H(x).

LEMmA 3. The positive limit set of any bounded solution of the asymptotically autonomous difference equation (10) is an invariant set of the autonomous difference equation (11).

This lemma is proven in the same manner as Lemma 1. Theorem 1 could now be restated in a manner similar to Theorem 2, but the following more general statement has proved to be more useful in its applications.

THEOREM 4. If a solution x(k) of the difference equation (1) approaches a closed, bounded set A as k -+oo, and if x(k) is also a solution of the asymptotically autonomous difference equation (10), then it approaches the largest invariant set of ( 11 ) contained in A as k -*oo .

As an example of the application of Theorem 4, consider the following difference equation:

Ex = cz - s(1 - p(k))y,

Ey = sx + c(l - p(k))y,

where c = cos w, s = sin w, 0 < X < 2r, 0 < a < p(k) ? 2 -e < 2. With the Liapunov function V(x, y) = x2 + y2, we get

AV(x, y) = -p(k)(2 - p(k))y2 < -bsy2 ? 0



592 JAMES HURT

Applying Corollary 1, we obtain that all solutions for (12) are bounded and y(k) -O as k -- o.

Let xi(k), yi(k) be a solution for (12); then yi(k) is bounded and approaches 0 as k -. Also, xl(k), yi(k) is a solution of the difference equation

Ex = cx - sy + p(k)yl(k), (13)

Ey = sx + cy - p(k)yi(k).

This difference equation is asymptotically autonomous to the difference equation

Ex = cx - sy,

Ey = sx + cy.

The only invariant set of (14) with y = 0 is the origin x = y = 0 since 0 < w < 27r. By Theorem 4, all solutions of (12) approach this invariant k and set, the origin, as k m*.

6. Practical stability. For many difference equations a solution is considered a stable solution if it enters and remains in a sufficiently small set. For example, under the proper conditions all solutions of the Newton- Raphson equation (9) approach the desired solution as k - m. But, when the effects of roundoff errors are considered, this is no longer the case. However, if all the solutions become and remain close to the desired solution, then the method is judged to be satisfactory. This type of stability is called practical stability. The following theorem and corollaries are con- cerned with practical stability for the difference equation

(15) Ex = f(k, x).

THEOREM 5. Given is a set G C X, possibly unbounded. Let V(x) and W(x) be continuous, real-valued functions defined on G and such that, for all k and all x in G,

(i) V(x) _ 0, (ii) AV(k, x) = V(f(k, x)) - V(x) < W(x) < a

for some constant a ? 0. Let the set S be the set

S = {x E G:W(x) >_ 0.

Let b = sup I V(x) :x E SI and the set A be the set

A = {xEG:V(x) < b + a}.

Then any solution x(k) which renwins in G and enters A when k = k, remains in A for all k ? k1.




The properties of S, A and V(x) are used to show that, if x(k) is in A, then x(k + 1) is in A. The theorem follows by induction.

COROLLARY 3. If 6 = sup {-W(x) :x C G - A} > 0, then each solution x(k) of (15) which remains in G enters A in a finite number of steps.

If x(k) does not enter A in a finite number of steps, then k-1

V(x(k)) V(x(ko)) + E AV(k, x(n)) n-ko

? V(x(ko)) - (k - ko)

and V(x(kI)) - a- as k -X oc, a contradiction since V(x) ? b + a for all x in G - A.

COROLLARY 4. If G is of the form G = G(,q) = {x: V(x) < t and the conditions of Theorem 5 and Corollary 3 are satisfied, then all solutions which start in G remain in G and enter A in a finite number of steps.

Corollary 4 can be used to study the effects of roundoff errors in the Newton-Raphson method. Without errors, the Newton-Raphson method is given by (9). With errors, this method is given by

(16) Zk+1 = Zk - (Zk)l f(zk) + h(k, Zk),

where all that is known about the error term h(k, Zk) is its upper bound, say ih(k, Zk) ? _ e for some vector norm and some e > 0. A value for e

can be obtained by assuming that Zk is known exactly and studying the steps of the computations in great detail to estimate the error in zk+1 . This error term includes the effects of errors in the functions f(z) and af(z)/3z, errors in evaluating f(z) and [&f(z)/9z] ', and any other errors that may be encountered. Often it is not very difficult to find an estimate for e; the problem is to determine the net effect of the term h((k, z) on the positive limit set of a solution z(k).

With the same assumptions onf(z) and the same expansions used before, the difference equation (16) becomes

ek+1 = Ml(ek)F[M'2(ek) ek - fo(ek)I + h(i(k, ek).

With V(e) = e 1, we obtain

AV(e) < -(1- k(,7) le )el + e = +W(e) _ E.

The set S becomes

S= {e: W(e) _5 = {e: I e ? bl where

b(7) = b _ 1 - - 4 = + 2k(q)E2 +

162k(77)



594 JAMES HURT

provided that 4k(fl)E < 1. If 4k(fl)E > 1, then W(e) > 0 everywhere and the iterations may not converge. The set A is defined by

A = Ie:V(e) < b + el = Ie:I eI < b + el. We note that, for - small enough, we have

W(e) < -(b - k(rI)(b + E)2) =

From Corollary 4, if 6 > 0 we have that all solutions which start in G( lu)

remain in G(-fo), enter A in a finite number of interations, and remain in A thereafter. Here, flu is not quite the same as before. We must choose qo such that k(77o) lo < 1, 4k(77o)e < 1, and b(-lo) - k(lo)(b(flo) + )2 > 0. If fl is the smallest positive solution of flk( fl) = 1, then choosing qo < 77, will satisfy both k(-lo)-lo < 1 and b - k(-lo)(b + E)2 > 0. The condition 4k(fbo) E < 1 becomes a condition on the precision or accuracy required in the computations.

Thus one effect of roundoff errors is to reduce the region of convergence. Another effect of roundoff errors is that the error of each Zk cannot generally be reduced much below the value b + E = 2E + 2k(-q)E2 + - - *, no matter how many iterations are perforned. The value b + E is called the ultimate accuracy obtainable with roundoff errors. Notice that, for small E, the ultimate accuracy is approximately 2E, or about twice the roundoff errors committed at each step.

If the ultimate accuracy is large, then the method is judged to be a poor one since the effect of small roundoff errors is a large error in the computed solution. If the ultimate accuracy is small, then the method is judged to be a good one since small roundoff errors have a small effect on the computed solution. In this sense, the Newton-Raphson method is judged to be a good method.

For a nonsingular matrix A, many iteration methods for solving Ax = b for the vector x are described by the difference equation

(17) Xk+1 = Bxk + C,

where the matrix B and the vector c are determined in some fashion by A and b. For example, if A = Q + R, then B = -Q 'R and c = Q'lb would be a possibility. B and c must have the property that xo = Bxo + c if and only if Axo = b. The iterations Xk will converge to the solution xo if and only if p(B), the spectral radius of B, is less than one. (For a derivation of several of these methods, see, for example, Kunz [4] or Hildebrand [2].) Choose a vector nonn I x I such that I B j = X < 1. Since p(B) < 1, this can always be done.

Let xo be the desired solution and let Xk = xo + ek . Then the ek satisfy the difference equation

ek+1 = Bek + h(k, ek),




where the term h(k, ek) represents the roundoff errors committed at step k. We assume that there exist positive constants v and E such that I h(k, e) < E for all k and all e, Ie < q.

Try the Liapunov function V(e) = e 1. Then,

AV(e) ? -(1 - X) Ie + E= W(e) < E.

Then the set S is given by

S = {e: W(e) > 0} lel < 1 I

and b = e/( -AX). The set A is given by

A = e: V(e) < b+e} = {eI lel < if

If v > (2 - X)e/(1- X), then we can choose

G = {e:V(e) < t} = {e:I e I < ni, and Corollary 4 holds. Thus, if el is in the set G, then the solution will remain in G, will enter A after a finite number of iterations, and will remain in A for all following iterations.

By looking at the set A, we see that the ultimate accuracy is given by

2-)E b + e = 2 xe.

We note that, if X is very nearly one, then this ultimate accuracy may be large even if e is small. For example, if X = 1 - a, then b + E = (a-1 + 1 ) e > e/a, and e/a may be large. This indicatesthat these iteration methods will give acceptable results only if X = B I is considerably less thar one.

Acknowledgments. The author wishes to thank Professor J. K. Hale, Professor J. P. LaSalle and Professor Mr. A. Feldstein of Brown University for their encouragement, advice and criticism in the preparation of this article. Also, the author is grateful for the financial support rendered to him by the National Aeronautics and Space Administration under their Traineeship Program.

REFERENCES

[1] WOLFGANG HAHN, Theory and Application of Liapunov's Direct Method, Prentice- Hall, Englewood Cliffs, New Jersey, 1963.

[2] F. B. HILDEBRAND, Introduction to Numerical Analysis, Prentice-Hall, Engle- wood Cliffs, New Jersey, 1952.



596 JAMES HURT

[3] R. E. KALMAN AND J. E. BERTRAM, Control system analysis and design via the 'second method' of Lyapunov. Part II, Discrete-time systems, Trans. ASME Ser. D. J. Basic Engrg., 82 (1960), pp. 394-400.

[41 K. S. KUNZ, Numerical Analysis, McGraw-Hill, New York, 1957. [5] J. P. LASALLE, An invariance principle in the theory of stability, TR 66-1, Center

for Dynamical Systems, Brown University, Providence, 1966. [61- , Asymptotic stability criteria, Proc. Symposia in Applied Mathematics,

vol. XIII, American Mathematical Society, Providence, 1962, pp. 299- 307.

[71 - , Liapunov's second method, Stability Problems of Solutions of Differential Equations, Proc. NATO Advanced Study Institute, 1965, Edizioni Oderisi, Gubbio, 1966, pp. 95-106.

[8] A. OSTROWSKI, Solution of Equations and Systems of Equations, Academic Press, New York, 1960.

[9] J. F. TRAUB, Iterative Methods for the Solution of Equations, Prentice-Hall, Engle- wood Cliffs, New Jersey, 1964.

[10] P. VIDAL AND F. LAURANT, Asymptotic stability of sampled non-linear systems with variablesampling periods-a bionic application, Session 30C, IFAC, London, 1966.



Some Stability Theorems for Ordinary Difference Equations

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