NUMERICAL STABILITY OF A CLASS (OF SYSTEMS) OF NONLINEAR EQUATIONS

Transcription

1 MATEMATIQKI VESNIK ), 7 33 UDK originalni nauqni rad research paper NUMERICAL STABILITY OF A CLASS OF SYSTEMS) OF NONLINEAR EQUATIONS Zlatko Udovičić Abstract In this article we consider stability of nonlinear equations which have the following form: Ax + F x) =b, ) where F is any function, A is a linear operator, b is given and x is an unknown vector We give under some assumptions about function F and operator A) a generalization of inequality: X X X A A b b b ) equation ) estimates the relative error of the solution when the linear equation Ax = b becomes the equation Ax = b ) and a generalization of inequality: X X X A b b A + A A b b b A ) A A equation 3) estimates the relative error of the solution when the linear equation A x = b becomes the equation A x = b ) Basic results Teorem Let V be a normed space, let the linear operator A: V V be invertible and bounded, let the inverse operator of the operator A be also bounded, let b,b V and let the functions F,F : V V and the set S V have the following properties: the function F is Lipschitz on S, ie, L >0) x,x S) F x ) F x ) L x x, and the constant L is such that the inequality L A > 0, holds; M >0) x S) F x) M x ; and 3 ε 0) x S) F x) F x) ε AMS Subject Classification: 65J5 Keywords and phrases: Numerical stability, nonlinear equations 7 3)

2 8 Z Udovičić If X S is a solution of the equation Ax + F x) =b and X S is a solution of the equation Ax + F x) =b, then the following inequality holds: X X A A + M) b b X L A + ε ) b b Proof Since AX + F X )=b,wehave b = AX + F X ) AX + F X ) A + M) X and we can conclude that A + M) 4) X b On the other hand, from X X = A b b ) F X ) F X ))) it follows that X X A b b + F X ) F X ) + F X ) F X ) ) A b b + L X X + ε), and that A b b + ε) X X L A 5) Finally, from 4) and 5) we have X X A A + M) b b X L A b + ε ) b which proves the theorem If F 0andF 0 in this case we have L = M = ε = 0), then the proved inequality becomes ) Theorem Let V be a normed space, let the linear operators A,A : V V be invertible and bounded, let their inverse operators be also bounded, let b,b V and let the function F : V V and the set S V have the following properties: the function F is Lipschitz on S, ie, L >0) x,x S) F x ) F x ) L x x, and the constant L is such that the inequality L A > 0 holds; M >0) x S) F x) M x ; 3 the function F is bounded on the set S, ie, B 0) x S) F x) B If X S is a solution of the equation A x + F x) =b and X S is a solution of the equation A x + F x) =b, then the following inequality holds: X X X A A + M) L b b A b b b A A A + A A ) + B I A A b

3 Numerical stability of nonlinear equations 9 Proof Since X = A b F X )), we have A X = A X + b A X F X ) =A A ) X + b F X ) =A A ) A b F X )) + b F X ) =A A ) A b A A ) A F X )+b F X ) =A A ) A b + b A A F X ), and we can apply the previous theorem to the equations A x + F x) =b and A x + A A F x) =A A ) A b + b Condition 3 of the theorem is satisfied since for every x S the inequality F x) A A F x) F x) I A A B I A A holds So, X X A A + M) X L b b A A ) A b A + b + B ) I A A b A A + M) L b b A + A A b b b A A + B ) I A A A b The theorem has been proved If F 0 in this case we have L = M = B = 0), then the inequality just proved becomes 3) From the theorems just proved we can conclude that relatively small changes of operator A, function F or vector b) in the equation ) may cause relatively big changes in the solution if the number A A + M) L A 6) is big enough, so we can take this number as a measure of stability of equation ) It is obvious that the equation ) gets more badly conditioned as the number 6) increases Since the inequality A A > always holds, the number 6) is greater than one whenever inequality L A > 0 holds

4 30 Z Udovičić A note If the normed space X is complete and the subset S X is closed, if the function F satisfies the condition of Theorem Theorem ) and if ϕ S) S where ϕ x) =A b F x)), then the array generated by the recursive formula x n+ = A b F x n )),n N 7) converges to the unique solution of the equation ) for every x 0 S Indeed, the function ϕ is a contraction since for every x, y S we have ϕ x) ϕ y) A b F x) b + F y) L A x y, while from the condition of Theorem Theorem ) we have that L A <, and therefore in accordance with Banach fixed point theorem, the array defined by formula 7) will converge to the unique solution of the equation ) 3 Examples The first example will give under certain assumptions) a sufficient condition for stability of polynomial with real coefficients We thoroughly considered polynomials of the third degree Example Let a polynomial with real coefficients P x) =ax 3 +bx +cx+d, a, c 0) have at least one zero in the segment [α, β] Furthermore, let F x) = ax 3 + bx and let Λ = max { α, β } Then we have that x [α, β]) F x) a Λ + b Λ ) x and max x [α,β] F x) = max { F α), F β), F ) } b 3a and in Theorems and we can put that M = a Λ + b Λ, and that { L = max F α), F β), F b ) } 3a If the condition L c > 0 L< c is satisfied then, in accordance with Theorems and we can say that if the number c +M c L = +M/ c L/ c which is always greater than one) is close enough to one, then relatively small changes in coefficients of the polynomial P will not cause relatively great changes in roots of the polynomial So, if linear term in polynomial P is more dominant c M and c L), the polynomial P is better conditioned We can do the same thing with polynomial of the fourth degree P x) =ax 4 + bx 3 + cx + dx + e, a, d 0) and conclude that the number d +M d L the numbers M and L have the same meaning) can be used as a measure of stability of the polynomial P So, the polynomial P is in this case also better conditioned if the number d +M d L is closer to one Of course, we can use the same technics for the polynomials of higher degrees, but in that case the problem of effective finding of number L is much more complex

5 Numerical stability of nonlinear equations 3 Example Let V be a normed space, let d V be a fixed vector, and let the function F : V V be defined by x V ) F x) = x d We shall consider the relative error of solution when the equation A x + F x) =b becomes the equation A x + F x) =b Since for every x, x,x V inequality F x ) F x ) x x d and equality F x) = x d, hold, we can put L = M = d So, if the condition d A > 0, is A A + d ) satisfied we can take the number d A as a measure of stability for the considered equation The following example is a numerical realization of Example Example 3 The solution of the system max {x, y} +0x 000y = 000 max {x, y} 00x 000y = 000 ) is X =, while the solution of the system 9800 max {x, y} +0x 000y = 000 max {x, y} 00x 000y = 000, ) 985 is the vector X = Stability of the considered ) system can be estimated by using the previous example V = R, d =, and the norm is the uniform norm of the space R ) The relative error of the matrix A = when this matrix becomes the matrix A = is A A %), A while the relative error of the solution when the first system becomes the second one is X X %) So, the relative error of the solution is approximately 500 times bigger then the relative error of the matrix X A According to the proved theorems our system is badly conditioned since A A + d ) d A = 0030

6 3 Z Udovičić It should be noted that the influence of nonlinear term in this example ) is 000 irrelevant The relative error of solution, when linear system A x = b = 000 becomes the system A x = b is approximately 05%, too We would like to point out that this system may also be solved by using the Banach fixed-point theorem see Section ) The first one of the following examples has a theoretical character, while the second one is its numerical realization Example 4 Let V be a normed space, let d V and r>0 be a fixed vector and a real number, let S = {x V x r } and let the function F : V V be defined by x V ) F x) = x d We shall estimate the relative error of the solution when equation A x + F x) =b becomes equation A x + F x) =b Since for every x, x,x S inequalities F x ) F x ) = x d x d = x + x ) x x d r d x x and F x) = x d r d x, hold, we can put M = r d and L =r d So, if the condition r d A > A A + r d ) 0 is satisfied then the number r d A can be taken as a measure of stability of the considered equation Example 5 The solution of the system x + y + 750x +50y = x + y +x 3y = { ) } x which belongs to the set S = R y x + y is X = while the solution of the system ) , x + y + 750x +50y = x + y +x y = { ) } x which belongs) to the set S = R y x + y is the vector X = Stability { of ) the considered system } can be estimated ) by using Example 4 V = x R, S = R y x + y, d =, while the norm is the Euclidean

7 Numerical stability of nonlinear equations norm of the space R ) Relative error of the matrix A = when this matrix becomes the matrix A = is A A %), A while the relative error of the solution when the first system becomes the second one is X X %) So, the relative error of the solution is approximately X 700 times bigger than the relative error of matrix A According to the proved A theorems the system is badly conditioned since A + d ) d A = 57 Contrary to Example 3, the influence of nonlinear term is important now In ) this example, the relative error of solution when linear system A x = b = becomes the system A x = b is approximately 47% REFERENCES [] Bahvalov, N S, Numerical Methods, Science, Moscow, 973 [] Edwards, R E, Functional Analysis, Holt, Rinehart and Winston, Inc, New York, 965 received 05004, in revised form 40005) Faculty of Sciences, Department of Mathematics, Zmaja od Bosne 35, 7000 Sarajevo, Bosnia and Herzegovina zzlatko@pmfunsaba