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CHAPTER 3: Roots of Equations By Erika Villarreal
Roots of Equations The determination of the roots of an equation is one of the oldest problems in mathematics and there have been many efforts in this regard. Its importance is that if we can determine the roots of an equation we can also determine the maximum and minimum values of matrices, solving systems of linear differential equations. Page
CLOSED  METHODS Closed domain methods are methods that start with two values of x between which is the root of the equation, x = α, the interval is reduced consistently keeping the root within the interval. The two methods that fall into this category are:  1. Graphical method  2. Bisection method (assuming the range in two)  3. Method of false position  These methods will be developed in the following sections. Closed methods are robust in the sense that ensure that a solution will be obtained because the root is? Trapped? on a closed interval. The counterpart of them is they have a slow convergence. Page
CLOSED METHODS Graphic methods consist of plotting the function f (x) and observe where the function crosses the x-axis Example 1: Find the following:  Page  1. Graphical Methods
CLOSED METHODS Example 2: Find the following:  Page  1. Graphical Methods f ( x ) = sen 10 x  + cos 3 x
CLOSED METHODS It's about finding the zeros of f (x) = 0 Where f is a continuous function on [a, b] with f (a) f (b) with different signs According to the mean value theorem, there is p    [a, b] such that f (p) = 0. The method involves dividing the interval in half and trace half containing p. The process is repeated to achieve the desired accuracy. Page  2. Bisection method
CLOSED METHODS Half of the interval containing p Page  2. Bisection method First iteration of the algorithm
CLOSED METHODS Half of the interval containing p Page  2. Bisection method First iteration of the algorithm
CLOSED METHODS Accession: ends a, b, number  of iterations or, tolerance tol Page  2. Bisection method Bisection algorithm 1. p=a; i=1; eps=1; 2. mientras f(p)  0 y i   ni eps>tol   2.1. pa = p;   2.2. p = (a+b)/2   2.3. si f(p)*f(a)>0 entonces a=p;   2.4. sino   2.5.  si f(p)*f(b)>0 entonces b=p;   2.6. i = i + 1; eps = |p-pa|/p;
CLOSED METHODS Page  2. Bisection method double biseccion(double a, double b, double error, int ni){ double p,pa,eps; int i; p = a; i = 1; eps = 1; while(f(p) != 0 && i<ni && eps > error){ pa = p; p = (a+b)/2; if(f(p)*f(a)>0) a = p; else if(f(p)*f(b)>0) b = p; i++; eps = fabs(p-pa)/p; } return p; }
Methods closed Page  2. Bisection method
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
CLOSED METHODS This method considers that the range limit is closer to the root. From Figure clearing Page  3. Method of false position
CLOSED METHODS Example 1: Find the following:  Page  3. Method of false position
CLOSED METHODS Page  False position in C
OPEN METHODS Open domain methods are not restricted to an interval root. Consequently, these methods are not as robust as methods dimensional and can diverge. However, these methods use information coming from the nonlinear function to refine the estimated result. Thus, these methods are more efficient than dimensional methods. The most representative methods are: 1. Fixed-point iterative method. 2. Newton-Raphson Method 3. Secant method 4. Muller Method Page
OPEN METHODS A fixed point of a function g (x) is a number p such that g (p) = p. Given a problem f (x) = 0, we can define a function g (x) with a fixed point p in different ways.For example g (x) = x - f (x). Theorem If g    C [a, b] and g (x)    C [a, b] for all x    C [a, b], then g has a fixed point in [a, b]. If in addition g '(x) exists in (a, b) and a positive constant k <1 exists | G '(x) | <= k,  For  all x   (a, b) Then the fixed point in [a, b] is unique Page  1. Fixed-Point Iteration
OPEN METHODS Page  1. Graph of fixed-point algorithm
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 )
OPEN METHODS Page  1.  Fixed point iterations
OPEN METHODS Page  1. Functions plotted in Mathlab
OPEN METHODS Page  1. Cases of non-convergence
OPEN METHODS The equation of the tangent line is y  –  f ( x n ) =  f ’  ( x n )( x  –  x n ) When  y  = 0,  x  =  x n +1  and 0  – f ( x n ) =  f ’  ( x n )( x n +1 –  x n ) Page  2.  Newton Method .
OPEN METHODS Example f(x) = x – cos(x) f’(x) = 1 + sen(x) Page  2.  Newton Method .  p n+1  = p n  – (p n  – cos(p n ))/(1 + sen(p n )) Taking p 0  = 0, finding p n   f(p n ) f’(p n ) p n+1 0 -1 1 1 1 0.459698 1.8414 0.7503639 0.7503639 0.0189 1.6819 0.7391128 0.7391128 0.00005 1.6736  0.7390851  0.7390851 3E-10 1.6736 0.7390851
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:  
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.
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 .  :  
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
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).
Page  „ The best thing about the future is that it comes only one day at a time.“ Abraham Lincoln (1809-1865)
Page  Software and Tools for Microsoft PowerPoint. The website with innovative solutions. Save time and money by automating your presentations. www.presentationpoint. com http://www.uv.es/diaz/mn/node17.html http://www.slideshare.net/nestorbalcazar/mtodos-numricos-03 MÉTODOS NUMÉRICOS Maestría en Ingeniería de Petróleos Escuela de Ingeniería de Petróleos David Fuentes Díaz Escuela de Ingeniería de Mecánica .  Métodos Numéricos (SC–854) Solución de ecuaciones no lineales M. Valenzuela 2007–2008 (5 de mayo de 2008) BIBLIOGRAPY

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  • 1. CHAPTER 3: Roots of Equations By Erika Villarreal
  • 2. Roots of Equations The determination of the roots of an equation is one of the oldest problems in mathematics and there have been many efforts in this regard. Its importance is that if we can determine the roots of an equation we can also determine the maximum and minimum values of matrices, solving systems of linear differential equations. Page
  • 3. CLOSED METHODS Closed domain methods are methods that start with two values of x between which is the root of the equation, x = α, the interval is reduced consistently keeping the root within the interval. The two methods that fall into this category are:  1. Graphical method  2. Bisection method (assuming the range in two)  3. Method of false position  These methods will be developed in the following sections. Closed methods are robust in the sense that ensure that a solution will be obtained because the root is? Trapped? on a closed interval. The counterpart of them is they have a slow convergence. Page
  • 4. CLOSED METHODS Graphic methods consist of plotting the function f (x) and observe where the function crosses the x-axis Example 1: Find the following: Page 1. Graphical Methods
  • 5. CLOSED METHODS Example 2: Find the following: Page 1. Graphical Methods f ( x ) = sen 10 x + cos 3 x
  • 6. CLOSED METHODS It's about finding the zeros of f (x) = 0 Where f is a continuous function on [a, b] with f (a) f (b) with different signs According to the mean value theorem, there is p  [a, b] such that f (p) = 0. The method involves dividing the interval in half and trace half containing p. The process is repeated to achieve the desired accuracy. Page 2. Bisection method
  • 7. CLOSED METHODS Half of the interval containing p Page 2. Bisection method First iteration of the algorithm
  • 8. CLOSED METHODS Half of the interval containing p Page 2. Bisection method First iteration of the algorithm
  • 9. CLOSED METHODS Accession: ends a, b, number of iterations or, tolerance tol Page 2. Bisection method Bisection algorithm 1. p=a; i=1; eps=1; 2. mientras f(p)  0 y i  ni eps>tol 2.1. pa = p; 2.2. p = (a+b)/2 2.3. si f(p)*f(a)>0 entonces a=p; 2.4. sino 2.5. si f(p)*f(b)>0 entonces b=p; 2.6. i = i + 1; eps = |p-pa|/p;
  • 10. CLOSED METHODS Page 2. Bisection method double biseccion(double a, double b, double error, int ni){ double p,pa,eps; int i; p = a; i = 1; eps = 1; while(f(p) != 0 && i<ni && eps > error){ pa = p; p = (a+b)/2; if(f(p)*f(a)>0) a = p; else if(f(p)*f(b)>0) b = p; i++; eps = fabs(p-pa)/p; } return p; }
  • 11. Methods closed Page 2. Bisection 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
  • 13. CLOSED METHODS This method considers that the range limit is closer to the root. From Figure clearing Page 3. Method of false position
  • 14. CLOSED METHODS Example 1: Find the following: Page 3. Method of false position
  • 15. CLOSED METHODS Page False position in C
  • 16. OPEN METHODS Open domain methods are not restricted to an interval root. Consequently, these methods are not as robust as methods dimensional and can diverge. However, these methods use information coming from the nonlinear function to refine the estimated result. Thus, these methods are more efficient than dimensional methods. The most representative methods are: 1. Fixed-point iterative method. 2. Newton-Raphson Method 3. Secant method 4. Muller Method Page
  • 17. OPEN METHODS A fixed point of a function g (x) is a number p such that g (p) = p. Given a problem f (x) = 0, we can define a function g (x) with a fixed point p in different ways.For example g (x) = x - f (x). Theorem If g  C [a, b] and g (x)  C [a, b] for all x  C [a, b], then g has a fixed point in [a, b]. If in addition g '(x) exists in (a, b) and a positive constant k <1 exists | G '(x) | <= k, For all x  (a, b) Then the fixed point in [a, b] is unique Page 1. Fixed-Point Iteration
  • 18. OPEN METHODS Page 1. Graph of fixed-point algorithm
  • 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 )
  • 20. OPEN METHODS Page 1. Fixed point iterations
  • 21. OPEN METHODS Page 1. Functions plotted in Mathlab
  • 22. OPEN METHODS Page 1. Cases of non-convergence
  • 23. OPEN METHODS The equation of the tangent line is y – f ( x n ) = f ’ ( x n )( x – x n ) When y = 0, x = x n +1 and 0 – f ( x n ) = f ’ ( x n )( x n +1 – x n ) Page 2. Newton Method .
  • 24. OPEN METHODS Example f(x) = x – cos(x) f’(x) = 1 + sen(x) Page 2. Newton Method . p n+1 = p n – (p n – cos(p n ))/(1 + sen(p n )) Taking p 0 = 0, finding p n f(p n ) f’(p n ) p n+1 0 -1 1 1 1 0.459698 1.8414 0.7503639 0.7503639 0.0189 1.6819 0.7391128 0.7391128 0.00005 1.6736 0.7390851 0.7390851 3E-10 1.6736 0.7390851
  • 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)
  • 31. Page Software and Tools for Microsoft PowerPoint. The website with innovative solutions. Save time and money by automating your presentations. www.presentationpoint. com http://www.uv.es/diaz/mn/node17.html http://www.slideshare.net/nestorbalcazar/mtodos-numricos-03 MÉTODOS NUMÉRICOS Maestría en Ingeniería de Petróleos Escuela de Ingeniería de Petróleos David Fuentes Díaz Escuela de Ingeniería de Mecánica . Métodos Numéricos (SC–854) Solución de ecuaciones no lineales M. Valenzuela 2007–2008 (5 de mayo de 2008) BIBLIOGRAPY