The document discusses various numerical methods for finding the roots or zeros of equations, including closed and open methods. Closed methods like bisection and false position trap the root within a closed interval by repeatedly dividing the interval in half. Open methods like Newton-Raphson and secant methods use information about the nonlinear function to iteratively refine the estimated root without being restricted to an interval. The document also covers methods for equations with multiple roots like Muller's method.
12. CLOSED METHODS Page 2. Error in the bisection method For the bisection method is known that the root is within the range, the result must be within Dx / 2, where Dx = xb - xa. The solution in this case is equal to the midpoint of the interval xr = (xb + xa) / 2 Should be expressed by xr = (xb + xa) / 2 Dx / 2 Approximate Error replacing
19. OPEN METHODS Page Consider the function: x3 + 4x2 -10 = 0 has a root in [1, 2] You can unwind in: a. x = g 1 ( x ) = x – x 3 – 4 x 2 +10 b. x = g 2 ( x ) = ½(10 – x 3 ) ½ c . x = g 3 ( x ) = (10/(4 + x )) ½ d. x = g 4 ( x ) = x – ( x 3 + 4 x 2 – 10)/(3 x 2 + 8 x )
25. OPEN METHODS Page 3. Alternative method to evaluate the derivative (secant method) . The secant method starts at two points (no one like Newton's method) and estimates the tangent by an approach according to the expression: The expression of the secant method gives us the next iteration point:
26. OPEN METHODS Page 3. Alternative method to evaluate the derivative (secant method) . : In the next iteration, we use the points x1 and x2para estimate a new point closer to the root of Eq. The figure represents geometric method.
27. OPEN METHODS Page Multiple roots In the event that a polynomial has multiple roots, the function will have zero slope when crossing the x-axis Such cases can not be detected in the bisection method if the multiplicity is even. In Newton's method the derivative is zero at the root. Usually the function value tends to zero faster than the derivative and can be used Newton's method . :
28. OPEN METHODS Page 1 . Muller Method . This method used to find roots of equations with multiple roots, and is to obtain the coefficients of the parabola passing through three selected points. These coefficients are substituted in the quadratic formula to get the value where the parabola intersects the X axis, the estimated result. The approach can be facilitated if we write the equation of the parabola in a convenient way. One of the biggest advantages of this method is that by working with the quadratic formula is therefore possible to locate real estate, and complex roots. Formula The three initial values are denoted as needed xk, xk-1 xk-2. The parabola passes through the points (xk, f (xk)) (xk-1, f (xk-1)) and (xk-2, f (xk-2)), if written in the form Newton, then: where f [xk, xk-1] f [xk, xk-1, xk-2] denote subtraction divided. This can be written as: where The next iteration is given by the root that gives the equation y = 0
29. OPEN METHODS Page 2 . Lin-Bairstow . The Lin-Bairstow method finds all the roots (real and complex) of a polinomioP (x). Given initial values of r and s, made a synthetic divide P (x) by (x2 - rx - s). Use Newton's method to find r and s values that make the waste is zero, ie, find the roots of the system of equations. bn(r, s) = 0, (55) bn−1(r, s) = 0. (56) Using the recursive rule r ← r+ Δ r (57) s ← s+ Δ s Where Once you find a quadratic factor of P (x) is solved with the formula and work continues to take Q (x) as the new polynomial P (x).
30. Page „ The best thing about the future is that it comes only one day at a time.“ Abraham Lincoln (1809-1865)
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