# Part 2 Roots of Equations Why But All Iterative

## Part 2 Roots of Equations Why But All Iterative

McDonald, Ashley, Meteorologist has reference to this Academic Journal, PHwiki organized this Journal Part 2 Roots of Equations Why But All Iterative

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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 Newtons 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 Newtons 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 Bairstows 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

## McDonald, Ashley Meteorologist

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