Download presentation

Presentation is loading. Please wait.

Published byKennedy Fenning Modified over 2 years ago

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.

Similar presentations

Presentation is loading. Please wait....

OK

Applied Informatics Štefan BEREŽNÝ

Applied Informatics Štefan BEREŽNÝ

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on cross site scripting virus Ppt on yamuna action plan Ppt online shopping project in vb Ppt on construction for class 10 cbse Ppt on suspension type insulators for sale Ppt on duty roster for boy Ppt on marie curie inventions Ppt on different solid figures powerpoint Free download ppt on roman numerals Training ppt on msds