Finding roots. Lecture 4

Transcription

1 Finding roots Lecture 4

2 Finding roots: Find such that 0 or given. Bisection method: The intermediate value theorem states the obvious: i a continuous unction changes sign within a given interval, it has at least one root in such interval. b Suppose the unction has one root in the interval. a Then i I partition the interval into two at the midpoint, the root is either in the right or the let semi-intervals. To see where, I just need to check i at the ends o the semi-intervals the sign o the unction is the same or not. Iterating this procedure I can bracket the root within an interval o ever increasing sharpness. The method is not ast but is very robust: it always works.

3 ewton-raphson method: Tangent First approimation Initial guess Start with a guess. Find the place where the tangent at your initial guess intersects the abscissa. This is your irst approimation. I the unction is continuous and you start reasonably close to the root, you will get a convergent procedure. Second approimation Alternatively, pick a point 0 and, R 0 + R 0 ' 0 0 Thereore R 0 ' 0 ' 0 ' Second derivatives small.

4 The method thereore consists on iterating ' And one needs a stopping criteria, or instance, e < < or e or a given e or < e All o these may have diiculties in practice...

5 Pathologies o ewton-raphson:

6 Another issue is that the method requires computing the derivative o the unction. This might be hard to do analytically, so we might want to choose to do it numerically. Or to have a method that does not require it: ' Combining with ' Secant method, geometrically, - -

7 A variation o this method is the regula alsi method. In the secant method one computes the secant o successive points, whereas in regula alsi one checks that the two points have unctional values that are opposite in sign. I the last two approimations have the same sign, the method digs back into previous estimates to get a dierent sign. This has the advantage o keeping the root bracketed Secant 4 Regula Falsi

8 Rates o convergence: Error in i-th iteration with respect to the correct root. Bisection method: ε i+ ε i / lim Thereore + n n n ε ε Converges linearly ewton s method: ' ' ε ε " ' R R R R R + + since And R ε I we substitute, we get, ' " R R ε ε So convergence is quadratic.

9 unction rtsaeuncd,,,acc parameter mait00 call uncd,l,d call uncd,h,d il*h.ge.0. pause 'root must be bracketed' il.lt.0.then l h Order interval so a<0 else h l swapl lh hswap endi rtsae.5*+ doldabs- ddold call uncdrtsae,,d do j,mait irtsae-h*d-*rtsae-l*d-.ge.0. *.or. abs.*.gt.absdold*d then doldd d0.5*h-l rtsael+d il.eq.rtsaereturn else I ewton-raphson out o range, bisect ewton-raphson doldd d/d temprtsae rtsaertsae-d itemp.eq.rtsaereturn endi iabsd.lt.acc return call uncdrtsae,,d i.lt.0. then lrtsae l Check else hrtsae h endi continue pause 'rtsae eceeding maimum iterations' return end Routine that combines bisection and ewton-raphson to enjoy the advantages o both.

10 Systems o non-linear equations: Are very hard. One is essentially looking in a multidimensional space or intersections o suraces, with a priori no clue as to where they may happen and no guarantee that one ound everything. We can write ormulae very similar to the ones we introduced here or many variables and equations. r r r r r Fi F + d F + J d + O J ik i i r d J F ewton-raphson. It is usually better to combine this method with a minimization idea, since the latter is very eicient to implement see umerical Recipes i F F i i k

11 Chaos and ewton-raphson: Suppose one considers the equation z 3-0. The root is at z. Suppose one now applies ewton-raphson to ind this root and asks the question which points close to z converge to that root And suppose one poses the problem in the comple plane. The answer: How could this happen? The method has an instability i 0! At such points a small deviation in the guess can yield wildly diering net approimations. Chaos!

12 Summary Bisection always works, but may require many tries. ewton-raphson sometimes ails, and possible pathologies are many.