Procedurally generated polynomial functions

[JMS1]

I'm a programmer looking for a way to create polynomial equations from a list of x intercepts and local maxima.

For the sake of discussion we can begin with a function of degree 4. The scale and position of the curve is unimportant so for simplicity's sake the curve can always have x intercepts at 1 and -1. That leaves the declaration of one local maximum to create a unique equation. If I want that maximum to be (0,0) then the equation takes the form (x-1)(x+1)(x)(x). Moving the maximum along the x-axis is simple. For maximum at (a,0) use the formula (x-1)(x+1)(x-a)(x-a) for any 'a' where 1>a>-1. Creating an equation where the local maximum is on the y-axis is also easy where a maximum at (0,-b) can be given by the equation (x-1)(x+1)(x-bi)(x+bi) for any b where 1>b>0

The problems arise when trying to use a maximum not on a zero axis. for example trying to create a local maximum at (0.2,-0.2) using the equation (x-1)(x+1)(x-.2-.2i)(x-.2+.2i) the local maximum is actually at (0.256,-0.190). Not exactly what I was looking for. So what would be a generalized formula which would give me a local maximum at (a,-b)?

 

[fresh_42]

You could operate with ##f(x) = a_n x^n + \dots + a_1x+a_0## instead and formulate your conditions in terms of derivatives of ##f##, i.e. in conditions on the ##a_i##.

 

[JMS1]

You could operate with ##f(x) = a_n x^n + \dots + a_1x+a_0## instead and formulate your conditions in terms of derivatives of ##f##, i.e. in conditions on the ##a_i##.

I could, but how?

Taking the example I gave of the function ##f(x)=(x-1)(x+1)(x-.2-.2i)(x-.2+.2i)##, or otherwise written ##f(x)=x^4-.4x^3-.76x^2+.4x-.24##, I get the derrivitive ##f'(x)=4x^3-1.2x^2-1.52x+.4##. taking the value at ##x=.2## I get ##y=.08##. Subtracting this from the derivative I get ##f'(x)=4x^3-1.2x^2-1.52x+.32##. The integral of this is ##f(x)=x^4-.4x^3-.76x^2+.32x##. Evaluating this equation at ##x=.2## I get ##y=.032##. Now subtracting ##.232## from the function I get ##f(x)=x^4-.4x^3-.76x^2+.32x-.232##. Evaluating this function I do indeed have a local maximum at ##(0.2,-0.2)##. However while I've been changing the function to match the local maximum I've managed to move the ##x## intercepts from my standard ##-1, 1##, to ##-0.972, 1.041##.

In the end I've traded one two-dimensional discrepancy for two one-dimensional discrepancies. Calculus may be the way forward with this, but If it is I still don't know how to apply it.

 

[fresh_42]

Let's gather what you have. You are looking for a polynomial ##f## of ##\deg(f) = 4 \; , \; f(1) = f(-1) = 0 \; , \; f(0.2) = -0.2## and the maximum condition gives ##f'(0.2) = 0## and ##f''(0.2) < 0.##
These conditions on the coefficients ##a_i## of ##f## mean:
##a_0 + a_1 + a_2 + a_3 + a_4 = 0##
##a_0 - a_1 + a_2 - a_3 + a_4 = 0##
##a_0 + a_1 \cdot (0.2)^1 + a_2 \cdot (0.2)^2 + a_3\cdot (0.2)^3 + a_4\cdot (0.2)^4 = -0.2##
##a_1 + 2a_2 \cdot (0.2)^1 + 3a_3\cdot (0.2)^2 + 4a_4\cdot (0.2)^3 = 0##
##a_2 + 3a_3\cdot (0.2)^1 + 6a_4\cdot (0.2)^2 < 0##
Now any solution to these conditions gives you a required polynomial ##f##. In general there can be more than one solution or even none.

 

[chiro]

Hey JMS1.

Following on from fresh_42's advice, I'd list the constraints one by one with whatever information you have and then rectify that system.

If you don't have a full rank system then you will get multiple solutions.

Also - are you trying to do anything besides have a maximum at a point along with the intercepts?

If you get an inconsistency then it means you will probably get a function that is inconsistent with the constraints (like how a linear system when row reduced has a bunch of zero's in a row but a non-zero element in the final column). That's how you tell for linear systems but for non-linear you get a complex solution when a consistent one would result in real numbers.

 

[Stephen Tashi]

I'm a programmer looking for a way to create polynomial equations from a list of x intercepts and local maxima.

For the sake of discussion we can begin with a function of degree 4. The scale and position of the curve is unimportant so for simplicity's sake the curve can always have x intercepts at 1 and -1.

How many points specifying local maxima will you have for each polynomial ? - just one given (x=a, y=b = local max) point per polynomial or several?

 

[JMS1]

Let's gather what you have. You are looking for a polynomial ##f## of ##\deg(f) = 4 \; , \; f(1) = f(-1) = 0 \; , \; f(0.2) = -0.2## and the maximum condition gives ##f'(0.2) = 0## and ##f''(0.2) < 0.##
These conditions on the coefficients ##a_i## of ##f## mean:
##a_0 + a_1 + a_2 + a_3 + a_4 = 0##
##a_0 - a_1 + a_2 - a_3 + a_4 = 0##
##a_0 + a_1 \cdot (0.2)^1 + a_2 \cdot (0.2)^2 + a_3\cdot (0.2)^3 + a_4\cdot (0.2)^4 = -0.2##
##a_1 + 2a_2 \cdot (0.2)^1 + 3a_3\cdot (0.2)^2 + 4a_4\cdot (0.2)^3 = 0##
##a_2 + 3a_3\cdot (0.2)^1 + 6a_4\cdot (0.2)^2 < 0##
Now any solution to these conditions gives you a required polynomial ##f##. In general there can be more than one solution or even none.

It works!

It looks simple and robust enough to be applied to a general case, thanks.

BTW: for future reference or google searching for help, what would you call this system?

 

[JMS1]

How many points specifying local maxima will you have for each polynomial ? - just one given (x=a, y=b = local max) point per polynomial or several?

For even polynomials I have ##x## intercepts at ##-1## and ##1##. For odd polynomials I put a double ##x## intercept at ##1## or in other words it's both an ##x## intercept and a zero slope point. . The scalar for the highest order term or ##a_n## is always ##1##. That leaves ##n-2## degrees of freedom for even degree polynomials and ##n-3## degrees of freedom for odd degree polynomials. Each local maximum fixes ##2## degrees of freedom, so for ##n=4## and ##n=5## I need one local maximum. for ##n=6## and ##n=7## I need 2 local maxima, and so on.