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Mathematics1 Mathematics 1 Applied Informatics Štefan BEREŽNÝ

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3 rd lecture

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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

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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

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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

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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.

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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 ( ) = .

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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).

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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 .

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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.

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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.

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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.

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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.)

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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.

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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:

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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:

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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:

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MATHEMATICS 1 Applied Informatics 18 Štefan BEREŽNÝ Newton’s method

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

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