Mathematics 1 Applied Informatics tefan BEREN MATHEMATICS 1
Mathematics 1 Applied Informatics Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics
3 rd lecture Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics
Contents • Approximate Solution of a Nonlinear Equation • Separation of a Root • Darboux Theorem • Bisection’s method • Newton’s method Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 3
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 Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 4
Approximate Solution of a Nonlinear Equation If iterative sequence converges to the root c of the equation f(x) = 0 then Error estimates: cn – c n, where n 0 for n Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 5
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 . . . an– 1, bn– 1 an, bn an+1, ) = D(f) separate the roots of the equation f(x) = 0, if each of the intervals includes at most one root. Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 6
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( ) = . Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 7
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). Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 8
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. Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 9
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. Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 10
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. Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 11
Newton’s method Suppose that: - function f has a second derivative f ′′(x) at each point x a, b and f Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 12
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. ) Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 13
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. Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 14
Newton’s method This procedure can easily be expressed computatively. Suppose that you already know the approximation cn and you wish to find the next approximation cn+1. The equation for the tangent line to the graph of f at the point [cn, f (cn)] is: Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 15
Newton’s method y = 0 corresponds to x = xn+1. So we get the equation , which yields: Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 16
Newton’s method The error estimate: It follows from the Mean Value Theorem, applied on the interval with end points cn and c, that exists between cn and c such that: Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 17
Newton’s method Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 18
Thank you for your attention. Štefan BEREŽNÝ MATHEMATICS 1 Applied Informatics 19
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