Approximating tanh by a rational function

Question

I want to approximate $\tanh$ by a low-degree rational function of form $$ r(x) = \frac{p(x)}{q(x)} = \frac{p_2 x^2 + p_1 x + p_0}{q_1 x + q_0} $$ such that the $L^2$ norm over the fixed interval $[x_0, x_1]$ is small: $$ I(p, q) = \frac{1}{2} \int_{x_0} ^{x_1} \left(r(x) - \tanh(x) \right)^2 \mathrm{d}x. $$ That is, I would like to solve for the coefficients $p_2, p_1, p_0, q_1, q_0$. Note that $\mathrm{deg}(p) \leq 2$ and $\mathrm{deg}(q) \leq 1$.

However, I don't know where to begin, so I'm looking for suggestions on how to approach this problem; pointers to relevant numerical methods would also be greatly appreciated.

Answer 1

Not an answer but too long for a comment.

An analytical expression of the $L^2$ norm seems very difficult (not to say impossible).

What I would suggest is to built the $[2,1]$ Padé approximant of $\tanh(x)$ around $a=\frac{x_0+x_1}2$. This would give $$\tanh(x)=\frac{3 \tanh (a)+2 (x-a)+ (\tanh (a)-\coth (a))(x-a)^2}{3+ (3 \tanh (a)-\coth (a))(x-a)}+O\left((x-a)^4\right)$$

Trying for $x_0=1$ and $x_1=2$

$$\int_1^2 \Big[\tanh(x)- \text{Padé}\Big]^2\,dx=3.90537\times10^{-7}$$ seems to be already decent. For the same problem, a full optimization would give $1.15644\times10^{-8}$

$$\int_2^3 \Big[\tanh(x)- \text{Padé}\Big]^2\,dx=8.80730\times10^{-9}$$ $$\int_3^4 \Big[\tanh(x)- \text{Padé}\Big]^2\,dx=1.66518\times10^{-10}$$

$$\log\Bigg[\int_n^{n+1} \Big[\tanh(x)- \text{Padé}\Big]^2\,dx\Bigg]=-3.87288\, n-11.2126$$

May be, you could use the sum of these Padé approximants each of them being built for a short interval.

Edit

The key problem is that we cannot have even an explicit solution for $$\int \left(\tanh (x)-\frac{a x}{b x+1}\right)^2 \, dx$$ There is one thing you could try taking into account the fact that the $[2,1]$ Padé approximant is $O\left((x-a)^4\right)$.

Let $$\tanh(x)=\sum_{n=0}^3 b_n \,(x-a)^n$$ and define the $L^2$ norm $$\Phi(b_0,b_1,b_2,b_3)=\int_{x_0}^{x_1} \Bigg[\tanh(x)-\sum_{n=0}^3 b_n \,(x-a)^n\Bigg]^2\,dx$$ developing the integrand, we have terms in $x^k$ $(k=0,\cdots,6)$ (no problem with these) and a series of integrals $$I_n=\int x^n \, \tanh(x)\,dx \qquad \qquad k=0,1,2,3$$

They do not make much problems $$I_0=\log (\cosh (x))$$ $$I_1=\frac{1}{2} \left(x \left(x+2 \log \left(e^{-2 x}+1\right)\right)-\text{Li}_2\left(-e^{-2 x}\right)\right)$$ $$I_2=-x \text{Li}_2\left(-e^{-2 x}\right)-\frac{1}{2} \text{Li}_3\left(-e^{-2 x}\right)+\frac{x^3}{3}+x^2 \log \left(e^{-2 x}+1\right)$$ $$I_3=\frac{1}{4} \left(-6 x^2 \text{Li}_2\left(-e^{-2 x}\right)-6 x \text{Li}_3\left(-e^{-2 x}\right)-3 \text{Li}_4\left(-e^{-2 x}\right)+x^4+4 x^3 \log \left(e^{-2 x}+1\right)\right)$$ and $$\int \tanh^2(x)\,dx=x-\tanh(x)$$ So, we have all the elements. Now solve $$\frac{\partial \Phi}{\partial b_0}=\frac{\partial \Phi}{\partial b_1}=\frac{\partial \Phi}{\partial b_2}=\frac{\partial \Phi}{\partial b_3}=0$$ which makes a linear system of four equations for four unknowns.

So, we have the series expansion that we could transform as a $[2,1]$ Padé approximant around the midpoint $a=\frac {x_0+x_1}2$ $$\sum_{n=0}^3 b_n \,(x-a)^n\sim \frac{p_2 (x-a)^2 + p_1 (x-a) + p_0}{q_1 (x-a) + q_0}$$ with $$p_2=b_2^2-b_1 b_3 \qquad p_1=b_1 b_2-b_0 b_3 \qquad p_0=b_0 b_2\qquad q_1=-b_3\qquad q_0=b_2$$

This procedure has been tried with $x_0=1$, $x_1=3$, $a=2$.

Using the original Padé approximant leads, for the $L^2$ norm, to a value equal to $1.67\times 10^{-4}$. The optimization of the series gives $3.57\times 10^{-6}$. The new Padé approximant leads to $3.33\times 10^{-4}$.

Much work for a more than bad result.

Answer 2

Here is a (bad!) attempt. First, suppose without loss of generality that $q_1 = 1$, and define $$ Q^{(k)}(q_0) = \int_{x_0} ^{x_1} \frac{x^k \tanh(x)}{x + q_0} \mathrm{d}x, $$ then expanding the optimality conditions $\partial_{p_{(\cdot)}} I = 0$ we obtain $$ \begin{align} \partial_{p_0} I &= \int_{x_0} ^{x_1} (r(x) - \tanh(x)) \frac{1}{x + q_0} \mathrm{d}x = \int_{x_0} ^{x_1} \frac{r(x)}{x + q_0} \mathrm{d}x - Q^{(0)} (q_0) = 0 \\ \partial_{p_1} I &= \int_{x_0} ^{x_1} (r(x) - \tanh(x)) \frac{x}{x + q_0} \mathrm{d}x = \int_{x_0} ^{x_1} \frac{x r(x)}{x + q_0} \mathrm{d}x - Q^{(1)} (q_0) = 0 \\ \partial_{p_2} I &= \int_{x_0} ^{x_1} (r(x) - \tanh(x)) \frac{x^2}{x + q_0} \mathrm{d}x = \int_{x_0} ^{x_1} \frac{x^2 r(x)}{x + q_0} \mathrm{d}x - Q^{(2)} (q_0) = 0 \end{align} $$ where for a fixed value of $q_0$ such that $-q_0 \not\in [x_0, x_1]$, the above becomes a system of 3 equations in $p_2, p_1, p_0$. If we can precompute a discretization of $Q^{(k)} (q_0)$ for different values of $q_0$, then we may search over the combination of coefficients that approximately minimizes the norm.