1. Mathematics1 Mathematics 1 Applied Informatics Štefan BEREŽNÝ

2. 3 rd lecture

3. MATHEMATICS 1 Applied Informatics 3 Štefan BEREŽNÝ Contents Approximate Solution of a Nonlinear Equation Separation of a Root Darboux Theorem Bisection’s method Newton’s method

4. MATHEMATICS 1 Applied Informatics 4 Štefan BEREŽNÝ Approximate Solution of a Nonlinear Equation Definition: Let f ( x ) be a function. Every point c  D( f ) such that f ( c ) = 0 is called the root of the equation f ( x ) = 0. initial approximation c 0 iterative sequence c 1, c 2, c 3, … etc Methods based on the construction of an iterative sequence are called iterative methods

5. MATHEMATICS 1 Applied Informatics 5 Štefan BEREŽNÝ Approximate Solution of a Nonlinear Equation If iterative sequence converges to the root c of the equation f ( x ) = 0 then Error estimates:  c n – c   n, where  n  0 for n  

6. MATHEMATICS 1 Applied Informatics 6 Štefan BEREŽNÝ Separation of a Root By the separation of a root we understand the specification of an interval  a, b  such that the equation f ( x ) = 0 has a unique root c in  a, b . Intervals (– , b 0    a 1, b 1    a 2, b 2  ...    a n–1, b n–1    a n, b n    a n+1,  ) = D( f ) separate the roots of the equation f ( x ) = 0, if each of the intervals includes at most one root.

7. MATHEMATICS 1 Applied Informatics 7 Štefan BEREŽNÝ Darboux theorem If function f is continuous on an interval I =  a, b  and x 1, x 2 are any two points from interval I then to any given number  between f ( x 1 ) and f ( x 2 ) there exists a point  between x 1 and x 2 such that f (  ) = .

8. MATHEMATICS 1 Applied Informatics 8 Štefan BEREŽNÝ Darboux theorem Corollary: If function f is continuous on an interval I =  a, b  and f ( a )  f ( b )  0 then exists a point c in ( a, b ) such that f ( c ) = 0 (the root of the equation f ( x ) = 0).

9. MATHEMATICS 1 Applied Informatics 9 Štefan BEREŽNÝ Bisection’s method Suppose that function f is continuous and strictly monotonic in the interval I =  a, b  and f ( a )  f ( b )  0. These assumptions guarantee the existence of a unique root c of the equation f ( x ) = 0 in interval I =  a, b .

10. MATHEMATICS 1 Applied Informatics 10 Štefan BEREŽNÝ Bisection’s method Choice of the initial approximation: Put c 0 = ( a + b )/2. Calculation of the further approximations: If f ( c 0 )  f ( b )  0 then c  ( c 0, b . Therefore we change a and we put a = c 0. If f ( c 0 )  f ( b )  0 then c   a, c 0 . We change b and we put b = c 0. Further, we put c 1 = ( a + b )/2. Similarly, we obtain c 2, c 3, … etc.

11. MATHEMATICS 1 Applied Informatics 11 Štefan BEREŽNÝ Bisection’s method The error estimate: Denote by d the length of the interval I =  a, b  at the beginning of the calculation. Since c   a, b ,  c 0 – c  d/2. The length of the “variable” interval I =  a, b  (where the root c is separated) decreases by one half at each step. Hence  c 0 – c  d/2 n+1.

12. MATHEMATICS 1 Applied Informatics 12 Štefan BEREŽNÝ Newton’s method Suppose that: - function f has a second derivative f ′′( x ) at each point x   a, b  and f ′′( x ) does not change its sign in I =  a, b . - f ′( x ) ≠ 0 for all x   a, b , - f ( a )  f ( b )  0.

13. MATHEMATICS 1 Applied Informatics 13 Štefan BEREŽNÝ Newton’s method Choice of the initial approximation: The initial approximation c 0 can be chosen to be equal an arbitrary point of the interval  a, b  such that f ( c 0 )  f ′′( c 0 )  0. (Among others, this inequality is satisfied by one of the points a and b.)

14. MATHEMATICS 1 Applied Informatics 14 Štefan BEREŽNÝ Newton’s method Calculation of the further approximations: To approximate the curve y = f ( x ) in the neighborhood of the point [ c 0, f ( c 0 )], we use a tangent line to the graph of f at this point. The point where this line crosses the x -axis is called c 1. Similarly, the point where the tangent line to the graph of f at point [ c 1, f ( c 1 )] crosses the x -axis is the next approximation c 2, etc.

15. MATHEMATICS 1 Applied Informatics 15 Štefan BEREŽNÝ Newton’s method This procedure can easily be expressed computatively. Suppose that you already know the approximation c n and you wish to find the next approximation c n+1. The equation for the tangent line to the graph of f at the point [ c n, f ( c n )] is:

16. MATHEMATICS 1 Applied Informatics 16 Štefan BEREŽNÝ Newton’s method y = 0 corresponds to x = x n+1. So we get the equation, which yields:

17. MATHEMATICS 1 Applied Informatics 17 Štefan BEREŽNÝ Newton’s method The error estimate: It follows from the Mean Value Theorem, applied on the interval with end points c n and c, that exists  between c n and c such that:

18. MATHEMATICS 1 Applied Informatics 18 Štefan BEREŽNÝ Newton’s method

19. MATHEMATICS 1 Applied Informatics 19 Štefan BEREŽNÝ Thank you for your attention.