Part 2 Roots of Equations Why But All Iterative
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Chapter 5 Bracketing Methods (Or, two point methods as long as finding roots) Two initial guesses as long as the root are required. These guesses must bracket or be on either side of the root. == > Fig. 5.1 If one root of a real in addition to continuous function, f(x)=0, is bounded by values x=xl, x =xu then f(xl) . f(xu) <0. (The function changes sign on opposite sides of the root) Figure 5.2 Figure 5.3 The Bisection Method For the arbitrary equation of one variable, f(x)=0 Pick xl in addition to xu such that they bound the root of interest, check if f(xl).f(xu) <0. Estimate the root by evaluating f[(xl+xu)/2]. Find the pair If f(xl). f[(xl+xu)/2]<0, root lies in the lower interval, then xu=(xl+xu)/2 in addition to go to step 2. If f(xl). f[(xl+xu)/2]>0, root lies in the upper interval, then xl= [(xl+xu)/2, go to step 2. If f(xl). f[(xl+xu)/2]=0, then root is (xl+xu)/2 in addition to terminate. Compare es with ea (5.2, p. 118) If ea< es, stop. Otherwise repeat the process. Evaluation of Method Pros Easy Always find root Number of iterations required to attain an absolute error can be computed a priori. Cons Slow Know a in addition to b that bound root Multiple roots No account is taken of f(xl) in addition to f(xu), if f(xl) is closer to zero, it is likely that root is closer to xl . How Many Iterations will It Take Length of the first Interval Lo=b-a After 1 iteration L1=Lo/2 After 2 iterations L2=Lo/4 After k iterations Lk=Lo/2k If the absolute magnitude of the error is in addition to Lo=2, how many iterations will you have to do to get the required accuracy = 10-4 in the solution The False-Position Method (Regula-Falsi) If a real root is bounded by xl in addition to xu of f(x)=0, then we can approximate the solution by doing a linear interpolation between the points [xl, f(xl)] in addition to [xu, f(xu)] to find the xr value such that l(xr)=0, l(x) is the linear approximation of f(x). == > Fig. 5.12 Procedure Find a pair of values of x, xl in addition to xu such that fl=f(xl) <0 in addition to fu=f(xu) >0. Estimate the value of the root from the following as long as mula (Refer to Box 5.1) in addition to evaluate f(xr). Use the new point to replace one of the original points, keeping the two points on opposite sides of the x axis. Use the same selecting rules as the bisection method. If f(xr)=0 then you have found the root in addition to need go no further.
See if the new xl in addition to xu are close enough as long as convergence to be declared. If they are not go back to step 2. Why this method Usually faster Always converges as long as a single root. See Sec.5.3.1, Pitfalls of the False-Position Method Note: Always check by substituting estimated root in the original equation to determine whether f(xr) 0. One-sidedness: Watch out (p. 128) Chapter 6 Open Methods Open methods are based on as long as mulas that require only a single starting value of x or two starting values that do not necessarily bracket the root. Figure 6.1
Simple Fixed-point Iteration Bracketing methods are convergent. Fixed-point methods may sometime diverge, depending on the stating point (initial guess) in addition to how the function behaves. Good as long as calculators. Rearrange the function so that x is on the left side of the equation: Example:
Convergence x=g(x) can be expressed as a pair of equations: y1=x y2=g(x) (component equations) Plot them separately. Figure 6.2 Conclusion Fixed-point iteration converges if When the method converges, the error is roughly proportional to or less than the error of the previous step, there as long as e it is called linearly convergent.
Newton-Raphson Method Most widely used method. Based on Taylor series expansion: Newton-Raphson as long as mula Solve as long as A convenient method as long as functions whose derivatives can be evaluated analytically. It may not be convenient as long as functions whose derivatives cannot be evaluated analytically. Fig. 6.5 Fig. 6.6
The Secant Method A slight variation of Newtons method as long as functions whose derivatives are difficult to evaluate. For these cases the derivative can be approximated by a backward finite divided difference. Requires two initial estimates of x , e.g, xo, x1. However, because f(x) is not required to change signs between estimates, it is not classified as a bracketing method. The scant method has the same properties as Newtons method. Convergence is not guaranteed as long as all xo, f(x). Fig. 6.7 Fig. 6.8
The values of the roots are determined by At this point three possibilities exist: The quotient is a third-order polynomial or greater. The previous values of r in addition to s serve as initial guesses in addition to Bairstows method is applied to the quotient to evaluate new r in addition to s values. The quotient is quadratic. The remaining two roots are evaluated directly, using the above eqn. The quotient is a 1st order polynomial. The remaining single root can be evaluated simply as x=-s/r. Refer to Tables pt2.3 in addition to pt2.4
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