Bracketing algorithm when root finding. Single root in "quadratic" function

Question

I am trying to implement a root finding algorithm. I am using the hybrid Newton-Raphson algorithm found in numerical recipes that works pretty nicely. But I have a problem in bracketing the root.

While implementing the root finding algorithm I realised that in several cases my functions have 1 real root and all the other imaginary (several of them, usually 6 or 9). The only root I am interested is in the real one so the problem is not there. The thing is that the function approaches the root like a cubic function, touching with the point the y=0 axis...

Newton-Rapson method needs some brackets of different sign and all the bracketing methods I found don't work for this specific case.

What can I do? It is pretty important to find that root in my program...

EDIT: more problems: sometimes due to reaaaaaally small numerical errors, say a variation of 1e-6 in some value the "cubic" function does NOT have that real root, it is just imaginary with a neglectable imaginary part... (checked with matlab)

EDIT 2: Much more information about the problem.

Ok, I need root finding algorithm.

Info I have:

• The root I need to find is between [0-1] , if there are more roots outside that part I am not interested in them.
• The root is real, there may be imaginary roots, but I don't want them.
• Probably all the rest of the roots will be imaginary
• The root may be double in that point, but I think that actually doesn't mater in numerical analysis problems
• I need to use the root finding algorithm several times during the overall calculations, but the function will always be a polynomial
• In one of the particular cases of the root finding, my polynomial will be similar to a quadratic function that touches Y=0 with the point. Example of a real case:
• The coefficient may not be 100% precise and that really slight imprecision may make the function not to touch the Y=0 axis.
• I cannot solve for this specific case because in other cases it may be that the polynomial is pretty normal and doesn't make any "strange" thing.
• The method I am actually using is NewtonRaphson hybrid, where if the derivative is really small it makes a bisection instead of NewRaph (found in numerical recipes).

Matlab's answer to the function on the image: roots:

0.853553390593276 + 0.353553390593278i
0.853553390593276 - 0.353553390593278i
0.146446609406726 + 0.353553390593273i
0.146446609406726 - 0.353553390593273i
0.499999999999996 + 0.000000040142134i
0.499999999999996 - 0.000000040142134i

The function is a real example I prepared where I know that the answer I want is 0.5

Note: I still haven't check completely some of the answers I you people have give me (Thank you!), I am just trying to give al the information I already have to complete the question.

Answer 1

Assuming you have a one-dimensional polynomial problem (which I assume from the imaginary solutions) you can use Sturm sequences to bracket all real roots. See Sturm's theorem.

Answer 2

Welcome to the wonderful world of numerical methods. Watch your hairline; it might start receding as you pull your hair out in frustration.

First off, with numerical root finding, you are toast if you can't bracket the problem. Newton Raphson is nice for polishing off a solution once you get close, and it only works if the derivative near the root is well away from zero. You always need to have some slower technique at hand as a backup because Newton Raphson can send you off to never-never land (i.e., somewhere well outside the bracket). If your function is not a polynomial, the first thing to try is Brent's method. If your function is a polynomial, try Laguerre's method or Jenkins-Traub.

BTW, it sounds like you have a pathological problem. You shouldn't expect particularly good performance. Pathological problems are, well, pathological.

Addendum
If you are having problems with things that appear to be roots, but aren't, you need to take care how you evaluate your function. If you do have a polynomial, form each term of the polynomial, sort by absolute value, and add smallest to largest. This produces better accuracy most of the time, but fails if you have large terms whose sum is nearly zero. If that's the case, you might want to add those canceling terms separately, add the rest smallest to largest, and then compute a grand total -- and your still kinda screwed. That big addition that nearly cancels loses a lot of precision. There's no escape other than extended precision arithmetic.

Answer 3

Well, if your function touches zero but never crosses it, you seem to be looking for a minimum (or a maximum). In which case, you're better off telling computer to do exactly that --- either find the root of a derivative (if you can calculate it analytically), or use a minimization routine. Then check that the function value at the minimum is 'close enough' to zero.

Just to reiterate what was already said by other people:

• don't start with Newton-Raphson method; it's almost always better to start with Brent or even a straightforward bisection (provided you can bracket the root).
• An instability where 'small numerical errors' of the order of 1e-6 have bad effects is worth investigating. Immediate suspects: mixing floats and doubles, loss of precision somewhere etc.

EDIT: So, depending on some parameters, your function has either a zero crossing, or a minimum with zero value, is this correct? In this case, what I'd do is this: use a simple and robust bracketing strategy (e.g. start from [-1, 1], multiply the endpoints by 1.1, check the signs, keep multiplying, something like this). If that succeeds, there's a zero crossing, use a root finding routine. If bracketing fails, use minimization.

Answer 4

Using Newton-Raphson is an act of desperation. You are much better off finding the continued fraction that represents your function and calculating that. A CF will converge much faster and will produce the real root(s). Also, because the CF produces a ratio of two integers you have tight control over numeric precision and don't have to worry about accumulation of rounding errors and other similar hair-pulling-out problems.

To find the real roots of any polynomial function refer to "A Continued Fraction Algorithm for Approximating All Real Polynomial Roots" by David Rosen (1978).

------------ ADDENDUM 1 --- 11 OCT-----------------

Ok, you are solving a sextic. You have several options. The simplest is to use a Taylor approximation (say to the 3rd degree) in conjunction with Halley's method. This is much superior to Newton because it has cubic convergence and you can detect imaginary solutions. The disadvantage is that you will have rounding problems which may result in an incorrect answer.

The ideal option is to find the continued fraction that represents the monic root, because this CF will be computable as an integer ratio of any desired precision, thus elminating the problem of rounding.

One approach to computing this CF is via the Jacobi-Perron algorithm. See the paper Hendy and Jeans: http://www.ams.org/mcom/1981-36-154/S0025-5718-1981-0606514-X/S0025-5718-1981-0606514-X.pdf. This paper shows the exact algorithm for computing cubic and quartic roots via CF approximation.

Note that if the sextic is reducible then it can converted into a quartic and quadratic: http://elib.mi.sanu.ac.rs/files/journals/tm/21/tm1124.pdf. The quartic is then solvable by the algorithm in the Hendy paper.

The general solution to generate a CF for a sextic can be done via the Rogers-Ramunajan CF. See the following paper for the method: http://arxiv.org/pdf/1111.6023v2. This will generate the CF for any sextic.

Answer 5

Ander, thanks for responding to my question (about the interval); sorry for the delay in following up - I have very busy work. Also - before I found the additional information you've provided - I had in mind to explain quite a few things how to handle this and was contemplating how to present that. However, I now believe your case is not too difficult and we can get at it without too much additional stuff, since you apparently have an explicit polynomial expression (coefficients to the various powers).

Let's start with a simple case, to pinpoint the approach.

Step 1. If you have a 2nd degree polynomial, its derivative is first order and has a simple zero (which you can find by bracketing or simply by explicitly solving the equation). (Yes, I know there's a closed formula for the roots of a 2nd degree polynomial also, but for the sake of the current argument, let us forget that). The zero's of the 2nd degree polynomial are then located one at the left side and one at the right side of the zero of the derivative. So, if you also have the interval where the roots of the original function (the 2nd degree polynomial) are to be found, you now have two intervals - left and right of the derivative-zero, each with one zero.

It is important to realize that the original function is MONOTONIC on each subinterval (decreasing on one of them, increasing on the other). Therefore, simply by checking the function values at the ends of the (sub)interval you can determine whether or not they actually bracket a zero. If not, there's a multiple zero (double, in this case) exactly at the zero of the derivative IF the function is zero there (otherwise, it is a double imaginary root of which you've now found the real part).

In case the zero of the derivative lies OUTSIDE the total interval, you will have at most one root inside your interval and you need to check only that particular (sub)interval.

Step 2. Consider now a 3rd order polynomial. Its derivative is 2nd order. The derivative of THAT 2nd order polynomial is again 1st order and you proceed as before to get two subintervals to find the roots of the derivative of the original function. These two roots give you THREE (at most) intervals where you will find the 3 roots of the original (3rd order) function. And also here, you will have intervals (3) where the original function is monotonic (alternatingly increasing/decreasing), making the analysis per subinterval quite easy.

Again, zeros may coincide (2 or even all 3) and may in addition turn out to be complex-valued (i.e. having non-zero imaginary parts). The analysis of the cases is straightforward: check function values at the borders of the intervals to assess whether not there's a sign-change (function is monotonic on each subinterval) and/or whether the function is zero at one of the subinterval-borders.

Step 4. Generalize this with the known polynomial. Let's say - your example - it is 6th order:

a) construct the 5th derivative (i.e. reducing the original to a 1st order polynomial). Compute it's zero (it is at precisely 0.5 in your example). In this case you're already done, but suppose you don't realize that. So you have now 2 intervals 0..0.5 and 0.5..1

b) construct the 4th derivative. Inspect its values at the subinterval-boundaries (0, 0.5, 1) For each subinterval determine if it has a real zero inside. If so, you re-partition your original interval in 3 subintervals, using the two found zeros (you forget about the zero of the 5th derivative). If they coincide (at the previous cut, 0.5) you stick with that 0.5 (don't care whether you've found a true double zero of your 4th derivative there or a "double imaginary") and still have only 2 intervals, but for the sake of the argument let's say you now have 3.

c) construct the 3rd derivative and do likewise as before. You will then have 4 (at most) intervals.

d) And so on. After having processed the 2nd derivative in this fashion you have 5 (at most) intervals, and after processing the 1st derivative you have 6 intervals (or less...) and knowing the function is monotonic on each subinterval, you'll quickly determine in each of them if there's a real root, as always using the know monotonicity of the function in each of the final subintervals.

Adding a note on numerical accuracy at evaluating a function: A first (probably sufficient, in this case) method to reduce noise is NOT to evaluate your function in the way suggested by the original form (i.e. a6 x*6 + a5 x*5 +..), but to rewrite it as:

a0 + x*(a1 + x*(a2 + x*(a3 + x*(a4 + x*(a5 + x*a6)))))

So, in evaluating you proceed:

tmp = a6

tmp = x*tmp + a5

tmp = x*tmp + a4

etcetera.

In case this little rewriting is not sufficient for numerical stability, you should rewrite your polynomial in (for instance) a chebyshev-polynomial expansion and evaluate that one with its recurrence relations. Both (getting the expansion and applying the recurrence relations for evaluation) are rather simple. I can explain, if you need help, but I guess it won't be necessary here.

In all cases, you HAVE to allow for some inaccuracy, i.e. accept that a computation will, generally speaking, NEVER give you the mathematically exact function value. So the assessment whether the function is presumably zero at some point must include some "tolerance", there's no way around this, unfortunately; the best you can aim for is to minimize the noise.

Answer 6

As in your case, you are interested in the real factorization of a real polynomial. One may see that all complex roots come in conjugate pairs which correspond to a real quadratic factor. By finding this real quadratic and completing the square to get the form (x-r)^2 + s you will be able to see the "real" even order root r with an "error" given by s. If s > 0 is too large, you may discard it as probably being complex. If s < 0 is also large, then you have two faraway real roots given by x = r ± √(-s). If s is very small then you might suspect r is a real double root and keep it.

Finding such a quadratic factor may be done using Bairstow's method, which actually applies a two-dimensional Newton method. This gives x^2 + ux + v and r = -u/2; s = v - r^2.