How to make sure if a polynomial curve is monotonic under interval [a,b]?

If I got a polynomial curve, and I want to find all monotonic curve segments and corresponding intervals by programming.

What's the best way to do this...

I want to avoid solving equation like f'(x) = 0;

Using some nice numerical ways to do this,like bi-section, is preferred.

f'(x) expression is available.

Thanks.

Add additional details. For example, I get a curve in 2d space, and its polynomial is

x: f(t) y: g(t)

t is [0,1]

So, if I want to get its monotonic curve segment, I must know the position of t where its tangent vector is (1,0).

One direct way to resolve this is to setup an equation "f'(x) = 0".

But I want to use the most efficient way to do this.

For example, I try to use recursive ways to find this. Divide the range [0,1] to four parts, and check whether the four tangents projection on vector (1,0) are in same direction, and two points are close enough. If not, continue to divide the range into 4 parts, until they are in same d开发者_Python百科irection in (1,0) and (0,1), and close enough.

I think you will have to find the roots of f'(x) using a numerical method (feel free to implement any root-seeking algorithm you want, Wikipedia has a list). The roots will be those points where the gradient reaches zero; say x1, x2, x3.

You then have a set of intervals (-inf, x1) (x1, x2) etc, continuity of a polynomial ensures that the gradient will be always positive or always negative between a particular pair of points.

So evaluating the gradient sign at a point within each interval will tell you whether that interval is monotically increasing or not. If you don't care for a "strictly" increasing section, you could patch together adjacent intervals which have positive gradient (as a point of inflection will show up as one of the f'(x)=0 roots).

As an alternative to computing the roots of f', you can also use Sturm Sequences.

They allow counting the number of roots (here, the roots of f') in an interval.

The monotic curve segments are delimited by the roots of f'(x). You can find the roots by using an iterative algorithm like Newton's method.