I am looking for a 5 parameter family of analytic functions $f:[0,1]\to \mathbb{R}$ such that

(0) $f$ has zeros at $0,p,1$.

(1) $f$ is convex in $[0,p]$ and concave in $[p,1]$.

(2) The five parameters, $p$ with $0<p<1$ and arbitrary positive values of $-f'(0)$, $-f'(1)$, $f''(0)$, and $-f''(1)$, can be independently prescribed, and $f$ depends continuously on these parameters.

A closed form solution in terms of rational operations and elementary functions is preferred. (I know a solution using a cubic spline with 7 knots, but due to its piecewise construction it is nonanalytic.)

I would already be happy with a 4 parameter family where $p$ is determined implicitly or explicitly by the other 4 parameters, since in spite of many attempts, I could not even satisfy this weaker goal.

  • 1
    $\begingroup$ Sorry to sound naive, but can't you do that with a polynomial ? $\endgroup$ – Loïc Teyssier Apr 24 '20 at 8:07
  • $\begingroup$ @LoïcTeyssier: I don't know. In any case, it seems to be very difficult to give a 5-parameter family of such polynomials. $\endgroup$ – Arnold Neumaier Apr 24 '20 at 8:20
  • $\begingroup$ I think that for polynomials, the degree would have to depend on the data pescribed in (2) and could not be bounded. $\endgroup$ – Arnold Neumaier Apr 24 '20 at 8:27
  • $\begingroup$ I agree, with bounded degree you couldn't get all freedom degree without introducing a root in $[0,1]$ for some values of the parameters. $\endgroup$ – Loïc Teyssier Apr 24 '20 at 8:57
  • $\begingroup$ You should specify more precisely what is a 5-parametric family: a) parametrized by what ($R^5$?) and b) how should a function depend on parameters. $\endgroup$ – Alexandre Eremenko Apr 24 '20 at 12:39

This is a comment to show the narrow range of polynomials that might work. We let $a=-f'(0),\ b=-f'(1),\ c=f''(0),\ d=-f''(1)$.

If $f$ is a polynomial, then $f''(x)$ is of the form $$f''(x)=(p-x)\left(\frac{c(1-x)}{p}+\frac{dx}{1-p}+x(1-x)q(x)\right)$$ for some polynomial $q(x)$ such that $f''(x)/(p-x)$ is positive on $[0,1]$. In particular, the factor of $p-x$ comes from the requirement that $f''(p)=0$, which in turn is the consequence of $f$ being convex on $[0,p]$ and concave on $[p,1]$.

Then there are only four more remaining conditions on $f''$ to solve the problem: \begin{align} -f'(0)=a&:\ \ \ \ \ \ -f'(p) + \int_0^p f''(x)\, dx = a\\ -f'(1)=b&:\ \ \ \ \ \ \ \ \ \ f'(p) + \int_p^1 f''(x)\, dx = -b\\ f(0)=f(p)&:\ \int_p^1 \left(f'(p) + \int_p^u f''(x)\, dx\right)\, du = 0\\ f(p)=f(1)&:\ \int_0^p \left(f'(p) - \int_u^p f''(x)\, dx\right)\, du = 0 \end{align}

So we could try $q(x)$ of the form $$q(x)=r x^i + s (1-x)^j + tx(1-x)$$ and solve the four equations above for $r,s,t$ and $f'(p)$.

The integrals are easy. Solving the equations is easy. Ensuring the positivity of $f''(x)/(p-x)$ is difficult. But if we can choose $i$ and $j$ to make $r,s,t$ positive, that will be enough. The trick is finding how $i$ and $j$ can depend on $a,b,c,d$ to get this positivity, or getting something similar for another form of $q(x)$.

  • $\begingroup$ Please tell me how to do the trick, or is it just a hope than one might exist? $\endgroup$ – Arnold Neumaier Sep 25 '20 at 14:22
  • $\begingroup$ It's just a hope for now... $\endgroup$ – Matt F. Sep 25 '20 at 14:22

This is a very nice question - an innocent-looking problem, easily described - basically construct a horizontal s-shaped analytical function under some well-defined conditions. But I must admit I had changed my mind several times, bouncing back and forth, in trying to decide (and prove) whether it can or cannot be done.

I now believe that the problem can be solved with polynomials as @MattF wrote in his answer, and as you and @LoïcTeyssier suggested in the comments, the degree $n$ of the polynomial depends on the constraints.

The problem is very suited to be represented with polynomials in Bernstein/Bezier form (see also this comprehensive survey). The Bernstein polynomial $P(x)$ is defined as $P(x) = \sum_{i=0}^{n} c_i B_{i}^{n}(x)$, where $B_i^{n}$ are the Bernstein basis polynomials $B_i^{n}(x) = \binom{n}{i} (1-x)^{n-i} x^{i}$.

Below, I will sometimes use freely the term "curve" (or "polynomial curve" or "Bezier curve") for the graph of the polynomial. Also, the points $p_i = (i/n, c_i)$, denoted the control points or control polygon of the curve, have a geometric meaning, which I will be using.

The first thing to notice is that the conditions on the first and second derivatives totally define the first and last three control points. Since the Bezier curves interpolate their endpoints, we have from condition (0): $$c_0 = c_n = 0$$

The first derivative of a Bernstein polynomial of degree $n$ is itself a Bernstein polynomial of degree $n-1$ of the form: $P'(x) = n \sum_{i=0}^{n-1} (c_{i+1} - c_i) B_{i}^{n-1}(x)$ Thus $P'(0) = n (c_1 - c_0)$ and from condition (2) we get: $n (c_1 - c_0) = f'(0)$, and so: $$c_1 = f'(0)/n$$ and in a similar manner: $$c_{n-1} = -f'(1)/n$$.

The second derivative is a Bernstein polynomial of degree $n-2$ of the form: $$P''(x) = n(n-1) \sum_{i=0}^{n-2} (c_{i+2} - 2 c_{i+1} + c_i) B_{i}^{n-2}(x)$$ Therefore, $$P''(0) = n(n-1)(c_2 - 2 c_1 + c_0)$$ and from condition (2) we get: $$n(n-1) (c_2 - 2 c_1) = f''(0)$$ and (with some manipulation): $$c_2 = \frac{f''(0)}{n(n-1)} + 2 c_1 = \frac{f''(0)}{n(n-1)} + 2 \frac{f'(0)}{n}$$.

and in a similar manner: $$c_{n-2} = \frac{f''(1)}{n(n-1)} - 2 c_{n-1} = \frac{f''(1)}{n(n-1)} - 2 \frac{f'(1)}{n}$$.

Furthermore, if the control polygon intersects the $x$-axis only once in $(0, 1)$ then, by the Bezier variation diminishing property, there is only one inner root. This is useful since we want (from condition (0)) only a single inner root (at $x=p$), and the roots at $0$ and $1$ are immediate since we set $c_0 = c_n = 0$.

Another important concept is what I call the "convex-concave property" of the control polygon. I say the control polygon is "convex-concave" if every consecutive control point tuple $(p_i, p_{i+1}, p_{i+2})$ make a left-turn up to some index $i=k$ and from then on all consecutive tuples make right-turns.

The important insight here is that if the control polygon is convex-concave, then so is its Bezier curve. The reason is that, as we saw above, the second derivative of the polynomial is itself a Bezier curve (of degree $n-2$), where its coefficients are $d_i = c_{i+2} - 2c_{i+1} + c_i$. A left-turn tuple gives a positive control point $d_i$, whereas a right-turn tuple gives a negative one. Thus, up to $d_k$ all $d_i$ are positive, and afterwards all are negative. Therefore, by the variation diminishing property, there is only one zero-crossing of the second-derivative control polygon, which means that there is just a single inflection point in $[0, 1]$.

So a convex-concave control polygon certifies that the curve is convex up to some point and concave afterwards.

One thing to notice is that if $c_2 > c_1$ then it may be impossible to satisfy the convex-concave condition. To see this, take for example $c_2 = 0$. Then $c_3$ must be positive or the tuple $(c_1, c_2, c_3)$ will be concave and violate the convex-concave property. If this happens on both sides (e.g., $c_2 = 0$ and $c_{n-2} = 0$) then the convex-concave property cannot be satisfied.

If, however, $c_2 < c_1$ and $c_{n-2} > c_{n-1}$, then you can always construct a convex-concave polygon. For example, we can construct such a polygon by taking all $c_i$ to be on the line between $c_1$ and $c_2$ for $i < k$, and all $c_i$ on the line between $c_{n-1}$ and $c_{n-2}$ for $i \geq k$. We can derive a lower bound on $n$ for which this will be the case. Since $c_2 = \frac{f''(0)}{n(n-1)} + 2 c_1 = \frac{f''(0)}{n(n-1)} + 2 \frac{f'(0)}{n}$, then if we set $n > |f''(0)/f'(0)| + 1$, we will have $c_2 < c_1$. Similarly, for $n > |f''(1)/f'(1)| + 1$, we will have $c_{n-2} > c_{n-1}$

So setting the lower bound $n > \max(|f''(0)/f'(0)| + 1, |f''(1)/f'(1)| + 1)$ guarantees that we can construct convex-concave control polygons with the required end conditions.

This already takes us a long way. Condition (2) is satisfied by the first and last three control points. Condition (1) is satisfied in the weak sense (i.e., with an implicit $p$) by the convex-concave property. And condition (0) is also partially satisified by the first and last control points and the fact that there is only one inner root.

What is missing to fully satisfy all the conditions (at least in the weak sense) is to find a way to force the root and the inflection point (where the curve turns from convex to concave) to be the same point $p$.

We can now think of an algorithm to construct the polynomial, which starts with the first and last three control points, and continues by trying to build a convex-concave control polygon, which satisfies the additional constraints. Apart from the constraints on the first and last three coefficients described above, there are the following constraints.

The polynomial zero-value at $p$: $$P(p) = \sum_{i=0}^{n} c_i B_{i}^{n}(p) = 0$$

The zero-value of the second derivative at $p$: $$P''(p) = \sum_{i=0}^{n-2} (c_{i+2} - 2 c_{i+1} + c_i) B_{i}^{n-2}(p) = 0$$

The inequality constraints to satisfy the convex-concave property: $$(c_{i+2} - 2 c_{i+1} + c_i) > 0 \text{, for } 0 < i < k$$ $$(c_{i+2} - 2 c_{i+1} + c_i) < 0 \text{, for } k \leq i < n-2$$

These constraints are linear equalities and inequalities in the variables $c_i$ and can therefore be solved using a Linear Programming solver (with varying k-values).

Note that the above lower bound on $n$ may not be sufficient. For example, if $p$ is very close to $0$ or $1$, e.g., smaller than $1/n$, then we will need to increase $n$. However, the LP-algorithm also reports if a feasible solution is not found and one can then increase $n$ and try again.

Even though I believe this algorithm is theoretically correct (after all, there are so many degrees of freedom..) I haven't succeeded in proving it. I wrote an initial implementation of the algorithm, and while it works fairly well on some inputs, on others the LP solver fails due to numerical errors (I used scipy's LP solver). This looks to me like an implementation issue (possibly a bug in my code), but there might be something fundamental hiding there, which I'm missing. Below is a figure of polynomials and their second derivatives for $p=0.5$ and varying end conditions, computed by the algorithm.

enter image description here

For the weaker version of the problem (i.e., with an implicit $p$), I believe I can show that a solution exists.

Given the first and second derivative constraints on the endpoints, I'll show that there exists a convex-concave polynomial $P(x)$ (of a sufficiently large degree $n$) that satisfies the end constraints, and for which there exists some value $p$ that satisfies $P(p)=0$ and $P''(p)=0$.

This means that $x=p$ is both the root and the inflection point and with the convex-concave property satisfies the weaker conditions of the problem. Note, I prove the existence of $p$ for a sufficiently large $n$, but don't explicitly compute $p$ nor $n$.

I will use the Weierstrass approximation theorem, which states that any continuous function on a closed interval can be approximated with a polynomial. More specifically, I will use Bernstein's constructive proof of the Wierstrass theorem, which gives the following theorem:

For a continuous function $f$ over $[0, 1]$, for any $\epsilon$, for a sufficiently large $n$, $|f(x) - \sum_{i=0}^n f(i/n) B_{i}^{n}(x)| < \epsilon$.

From Bernstein's theorem, by setting the control polygon to be the function $f$, the following corollary follows:

For any $\epsilon$, for a sufficiently large $n$, the distance between the Bernstein polynomial and its control polygon is smaller than $\epsilon$.

From the corollary it follows that (for a suitable $n$) the root $p$ of $P(x)$ is within a $\delta$-environment of the intersection point of the control polygon and the $x$-axis. Similarly, the inflection point (the root of $P''(x)$) is within a $\delta$-environment of the intersection of the control polygon of $P''(x)$ and the $x$-axis.

For any $n$, I assume we can construct a convex-concave control polygon, which satisfies the end conditions, and has an inflection point above the $x$-axis, i.e., to the right of the root. For a sufficiently large $n$, by the corollary, the curve inflection point will also be to the right of the curve root. We can similarly construct such a configuration that has an inflection point to the left of the root. Below is are example figures of two such corresponding configurations, where the inflection point is the blue dot, and the root is black (the control points are the blue crosses).

inflection point to right of root

inflection point to left of root

Moving continuously from the first configuration to the second (while maintaining the convex-concave property), we will encounter (at least one) value $x=p$ where the inflection point and the root coincide, which satisfies our request.

So, to summarize, I assume we can draw for any $n$ a pair of convex-concave control polygons that satisfy the end conditions and for which one has an inflection point above the $x$-axis and the other has it below (I think this is a reasonable assumption). Then, for a sufficiently large $n$, the polynomial curves will be close enough to their control polygon so that their inflection point is also to the right/left of the polynomial root. Then there should be at least one configuration where the root and inflection point coincide.

Some open issues:

  • Try and convert the weak-constraint proof to an algorithm. Basically, if you have the two configurations with their roots on opposite sides of the inflection point, then a bisection algorithm might be used.

  • Prove that there is always a solution for the problem with the strong constraints (for a given $p$ value). The consequence is that the LP algorithm will eventually give the correct solution.

  • Find a (hopefully analytical/concise/easily computable) upper bound on the polynomial degree $n$.

  • $\begingroup$ Thanks for your efforts. But this does not quite answer my question, as it is a many-parameter family containing some function for each case of my 5 parameters. $\endgroup$ – Arnold Neumaier Oct 16 '20 at 11:40
  • $\begingroup$ I'm not sure I understand what you mean by a many-parameter family. Given the five parameters (or four in the weak version) there are indeed many polynomials that can satisfy the constraints. $\endgroup$ – Iddo Hanniel Oct 18 '20 at 15:42
  • $\begingroup$ I'd like to have a closed form 5-parameter family of solutions, not a recipe for constructing one solution (or many) for each set of parameters that may jump discontinuously when the parameters are varied. $\endgroup$ – Arnold Neumaier Oct 18 '20 at 18:00

Too long to comment.

Here is a suggestion. Consider transforming the question to finding an analytic non-positive convex function $f$ over [0,1] with (i) zeros at 0 and 1, (ii) the required first and second order derivatives and (iii unimodal at $p$. In that case, one can then multiply by $g(x) = \frac{2}{1+e^{K(x-p)}}-1$, $K$ sufficient large and positive, and see that for $h(x)=f(x)g(x)$, $h(p)=0$ and $$ h'' = f''g + 2f'g' + g''f $$ is non-negative (and hence conves) in $[0,p]$ and non-positive in $[p,1]$. Reason for signs of curvature: (i) for $x \in [0,p]$, $f''g$ is non-negative since $f''$ and $g$ are positive, $f'g'$ is non-negative since $f$ reaches minimum at $p$ and $g'$ is non-positive, and $g''f$ is non-negative since $g''$ is non-positive and $f$ is non-positive. The same can be reasoned out for concavity in $[p,1]$.

  • $\begingroup$ Could you please explain how to find $f$? This is still a nontrivial problem. $\endgroup$ – Arnold Neumaier Apr 25 '20 at 9:22
  • $\begingroup$ I wish I had an answer for that. I should have put it as a comment, but was just too long. $\endgroup$ – DSM Apr 25 '20 at 9:34
  • $\begingroup$ $C^\infty$ is not analytic. $\endgroup$ – Hans Apr 25 '20 at 17:54
  • $\begingroup$ @Hans, thanks for catching that. My bad. Have made the change. $\endgroup$ – DSM Apr 26 '20 at 3:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.