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Advanced Methods for Nonlinear Equation Solutions

Learn about iterative methods like Bisection's & Newton’s methods for finding roots of nonlinear equations with error estimates.

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Advanced Methods for Nonlinear Equation Solutions

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  1. Mathematics1 Applied Informatics Štefan BEREŽNÝ

  2. 3rd lecture

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

  4. 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 equationf(x) = 0. • initial approximationc0 • iterative sequencec1, c2, c3, … etc • Methods based on the construction of an iterative sequence are called iterative methods MATHEMATICS 1 Applied Informatics

  5. 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 MATHEMATICS 1 Applied Informatics

  6. 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 (–, b0  a1, b1  a2, b2  ...   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. MATHEMATICS 1 Applied Informatics

  7. Darboux theorem If function f is continuous on an interval I = a, b and x1, x2 are any two points from interval I then to any given number  between f(x1) and f(x2) there exists a point  between x1 and x2 such that f() = . MATHEMATICS 1 Applied Informatics

  8. 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). MATHEMATICS 1 Applied Informatics

  9. 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. MATHEMATICS 1 Applied Informatics

  10. Bisection’s method Choice of the initial approximation: Put c0 = (a + b)/2. Calculation of the further approximations: If f(c0) f(b)  0 then c (c0, b. Therefore we change a and we put a = c0. If f(c0) f(b)  0 then ca, c0. We change b and we put b = c0. Further, we put c1 = (a + b)/2. Similarly, we obtain c2, c3, … etc. MATHEMATICS 1 Applied Informatics

  11. 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, c0 – 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 c0 – cd/2n+1. MATHEMATICS 1 Applied Informatics

  12. Newton’s method Suppose that: - function f has a second derivative f ′′(x) ateach 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. MATHEMATICS 1 Applied Informatics

  13. Newton’s method Choice of the initial approximation: The initial approximation c0 can be chosen to be equal an arbitrary point of the interval a, b such that f (c0) f ′′(c0)  0. (Among others, this inequality is satisfied by one of the points a and b.) MATHEMATICS 1 Applied Informatics

  14. Newton’s method Calculation of the further approximations: To approximate the curve y = f (x) in the neighborhood of the point [c0, f (c0)], we use a tangent line to the graph of f at this point. The point where this line crosses the x-axis is called c1. Similarly, the point where the tangent line to the graph of f at point [c1, f(c1)] crosses the x-axis is the next approximation c2, etc. MATHEMATICS 1 Applied Informatics

  15. 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: MATHEMATICS 1 Applied Informatics

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

  17. 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: MATHEMATICS 1 Applied Informatics

  18. Newton’s method MATHEMATICS 1 Applied Informatics

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

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