Chapter 2 roots of equations

Chapter 2 roots of equations

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  ROOTS OF EQUATIONS Author Lizeth Paola Barrero Riaño Industrial University of Santander 2010 A root or solution of equation f(x)=0 are 2. CLOSED METHODS the values of x for which the equation These are called closed methods because holds true. The numerical methods are are necessary two initial values to the used for finding roots of equations, some root, which should “enclose” or to be to of them are: the both root sides. The key feature of 1. GRAPHICAL METHOD these methods is that we evaluate a do- main or range in which values are close to It is a simple method to obtain an ap- the function root; these methods are proximation to the equation root f(x) =0. known as convergent. Within the closed It consists of to plot the function and de- methods are the following methods: termine where it crosses the x-axis. At this point, which represents the x value 2.1 BISECTION (Also called Bol- where f(x) =0, offer an initial approxima- zano method) tion of the root. The method feature lie in look for an in- terval where the function changes its sign The graphical method is necessary to use when is analyzed. The location of the sign any method to find roots, due to it allows change gets more accurately by dividing us to have a value or a domain values in the interval in a defined amount of sub- which the function will be evaluated, due intervals. Each of this sub intervals are to these will be next to the root. Like- evaluated to find the sign change. The wise, with this method we can indentify if approximation to the root improves ac- the function has several roots. cording to the sub-intervals are getting smaller. The following is the procedure: Step 1: Choose lower, xl, and upper, xu, values, which enclose the root, so that the function changes sign in the interval. This is verified by checking that: f  xl  f  xu   0 1
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  Step 2: Anapproximation of the xr root, relatively inefficient. Therefore this is determined by: method is an improved alternative based on an idea for a more efficient approach xl  xu to the root. xr  2 This method raises draw a straight line Step 3: Realize the following evaluations joining the two interval points (x, y) and to determine in what subinterval the root (x1, y1), the cut generated by the x-axis is: allows greater approximation to the root. f  xl  f  xr   0 If , then the root is Using similar triangles, the intersection within the lower or left subinterval, can be calculated as follows: so, do xu=xr and return to step 2. f  xl  f  xr   0 If , then the root is within the top or right subinterval, so, do xl=xr and return to step 2. f  xl  f  xr   0 If , the root is equal to xr; the calculations ends. The final equation for False position The maximum number of iterations to ob- method is: tain the root value is given by the follow- ing equation: The calcula- tion of the root xr requires replacing one of the other two values so that they al- TOL = Tolerance ways have opposite signs, what leads these two points always enclose the root. 2.2 METHOD OF FALSE POSITION Sometimes, depending on the function, this method works poorly, while the bi- Although the bisection method is techni- section method leads better approxima- cally valid to determine roots its focus is tions. section method leads better ap- 2
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  section method leadsbetter approxima- cate the root of f(x) = e-x – x tions. Solution: Like f(x)=0 ó e-x – x=0 3. OPEN METHODS Expressing of the form x=g(x) result us: x= e-x These methods are based on formulas Beginning with an initial value of xo=0, we that require a single initial value x, or a can apply the iterative equation xi+1=g(xi) couple of them but do not necessarily that and calculate: contain the root. Because of this feature, sometimes these methods diverge or move away from the root, according to i xi Ea(%) Et(%) grows the number of iterations. It is im- 0 0 100.0 portant to know that when the open 1 1.000000 100.0 76.3 methods converge, these are more effi- 2 0.367879 171.8 35.1 cient than methods that use intervals. 3 0.692201 46.9 22.1 Open methods are: 4 0.500473 38.3 11.8 5 0.606244 17.4 6.89 3.1 SIMPLE FIXED POINT ITE- RATION 6 0.545396 11.2 3.83 7 0.579612 5.90 2.20 Open methods employ a formula that pre- 8 0.560115 3.48 1.24 dicts the root, this formula can be devel- oped for a single iteration of a fixed point 9 0.571143 1.93 0.705 (also called point iteration or successive 10 0.564879 1.11 0.399 substitution) to change the equation f (x) = 0 so that is: Thus, each iteration bring near increas- x=g(x) ingly to the estimated value with the true This formula is employ to predict a new x value of the root, that is to say value in function of the previous x valor, 0.56714329 through: 3.2 NEWTON RAPHSON xi+1=g(xi) METHOD Subsequently, these iterations are used to The most widely used formula to find calculate the approximate error so that roots, is the Newton Raphson, argues that the least error indicates the root of the if it indicates the initial value of x1 as the function in matter. value of the root, then it is possible to ex- xi 1  xi tend a tangent line from the point (x1,f a= x100 (x1)). Where this straight line crosses the xi 1 x-axis will be the point of improve ap- proach. For instance: Use simple iteration of a fixed point to lo- 3
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  This method couldderived in a graphically The approximation of the derivative is ob- or using Taylor’s series. tained as follows: Newton Raphson formula Substituting this equation in the Newton From its reorder is obtained the value of Raphson formula we get: the desired root. The above equation is the secant formula. 4. MULTIPLES ROOTS A multiple root corresponds to a point where a function is tangential to the x axis For instance, double root results of: f(x)=(x – 3)(x – 1)(x – 1) ó f(x)=x3 – 5x2 – 7x - 3 Because a value of x makes two terms in the previous equation are zero. Graphi- cally this means that the curve tangen- tially touches the x axis in the double root. The function touches the axis but not cross in the root. Note: The Newton Raphson method has a strong issue for its implementation, this is due to the derivative, as in some functions is extremely difficult to evaluate the de- rivative. 3.3 Secant Method The secant method allows an approxima- tion of the derivative by means of a di- vided difference; this method avoids fal- ling into the Newton Raphson problem, as it applies for all functions regardless of whether they have difficulty in evaluating A triple root corresponds to the case its derivative. where a value of x makes three terms in an equation equal to zero, as: 4
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  an equation equalto zero, as: Where m is the root multiplicity (ie, m=2 for a double root, m=3 for a triple root, f(x)=(x – 3)(x – 1)(x – 1)(x – 1) etc.). This formulation may be unsatisfac- ó f(x)=x4 – 6x3 + 12x2– 10x + 3 tory because it presumes the root multi- plicity knowledge. In this case the function is tangent to the axis at the root and crosses the axis. b) New function definition The new function is replaced in the Newton-Raphson’s method equation, so you get an alternative: c) Secant Method Modification Substituting the new function (explained In general, odd multiplicity of roots previously) in the equation of the secant crosses the axis, while the pair multiplicity method, we have does not cross. The multiple roots offer some difficulties to numerical methods: 1. The fact that the function does not change sign on multiple pairs roots pre- 5. POLYNOMIALS ROOTS vents the use of reliable methods that Below are described the methods to find use intervals. polynomial equations roots of the general 2. Not only f(x) but also f'(x) approach form: to zero. These problems affect the New- ton-Raphson and the secant methods, which contain derivatives in the denomi- nator of their respective formulas. Where n is the polynomial order and the a Some modifications have been proposed are constant coefficients. The roots of to alleviate these problems, Ralston and such polynomials have the following rules: Rabinowitz (1978) proposed the following a. For equation of order n, there are n formulas: real or complex roots. It should be a) Root Multiplicity noted that these roots are not necessar- ily different. b. If n is odd, there is at least one real root. 5
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  c. If the roots are complex, there is a Thus, this parable is used to intersect the conjugate pair (i.e., λ + µi y λ - µi) where three points [x0, f(x0)], [x1, f(x1)] and [x2, f (x2)]. The coefficients of the previous equa- tion are evaluated by substituting one of these three points to make: A predecessor of Muller method is the se- cant method, which obtain root, estimating f ( x0 )  a( x0  x2 )2  b( x0  x2 )  c a projection of a straight line on the x-axis through two function values (Shape f ( x1 )  a( x1  x2 ) 2  b( x1  x2 )  c 1). Muller's method takes a similar view, but f ( x2 )  a( x2  x2 ) 2  b( x2  x2 )  c projected a parabola through three points (Shape 2). f ( x2 )  c The last equation generated that , The method consist in to obtain the coeffi- in this way, we can have a system of two cients of the three points, replace them in equations with two unknowns: the quadratic formula and get the point where the parabola intersects the x-axis The f ( x0 )  f ( x2 )  a( x0  x2 )2  b( x0  x2 ) approach is easy to write, as appropriate this would be: f ( x1 )  f ( x2 )  a( x1  x2 ) 2  b( x1  x2 ) f 2 ( x)  a( x  x2 ) 2  b( x  x2 )  c Defining this way: h0  x1  x0 h1  x2  x1 f ( x 2 )  f ( x1 ) f ( x1 )  f ( x 2 ) 1  0  x 2  x1 x1  x0 Substituting in the system: (h0  h1 )b  (h0  h1 ) 2 a  h0 0  h1 1 h1b  h1 a  h1 1 2 The coefficients are: Shape 1 1   0 a b  ah1   1 h1  h0 c  f ( x2 ) Finding the root, the conventional solution is implemented, but due to rounding error po- tential, we use an alternative formulation:  2c x3  x 2  b  b 2  4ac Shape 2 6
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  Solving: use the synthetic division. So given  2c x3  x 2  fn(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0 b  b  4ac 2 By dividing between f2(x) = x2 – rx – s, we The great advantage of this method is that have as a result the following polynomial we find real and imaginary roots. Finding the error this will be: fn-2(x) = bnxn-2 + bn-1xn-3 + … + b3x + b2 x3  x 2 Ea   100% with a residue R = b1(x-r) + b0, the residue x3 will be zero only if b1 and b0 are. To be an approximation method, this is per- formed sequentially and iteratively, where The terms b and c, are calculated using the x1, x2, x3replace the points x0, x1, x2 carrying following recurrence relation: the error to a value close to zero. bn = an 5. 1 Bairstow’s Method bn-1 = an-1 + rbn bi = ai + rbi+1 + sbi+2 Bairstow's method is an iterative process related approximately with Muller and New- cn = bn ton-Raphson methods. The mathematical cn-1 = bn-1 + rcn process depends of dividing the polynomial ci = bi + rci+1 + sci+2 between a factor. Finally, the approximate error in r and s can Given a polynomial fn(x) find two factors, a be estimated as: quadratic polynomial f2(x) = x2 – rx – s y fn-2 (x). The general procedure for the Bairstow method is: 1. Given fn(x) y r0 y s0 When both estimated errors failed, the roots 2. Using Newton-Raphson method calcu- values can be determined as: late f2(x) = x2 – r0x – s0 y fn-2(x), such as, the residue of fn(x)/ f2(x) be equal to zero. 3. The roots f2(x) are determined, using the general formula. 4. Calculate fn-2(x)= fn(x)/ f2(x). 5. Do fn(x)= fn-2(x) 6. If the polynomial degree is greater BIBLIOGRAPHY than three, back to step 2 7. If not, we finish CHAPRA, Steven C. y CANALE, Raymond P.: Métodos Numéricos para Ingenieros. The main difference of this method in rela- McGraw Hill 2007. 5th edition. tion to others, allows calculating all the poly- nomial roots (real and imaginaries). To calculate the polynomials division, we 7