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Let's put the origin in your corner of ${\mathbb R}^3$, and say you have measurements of the field $\vec{F}$ at lattice points $\epsilon [i,j,k]$ for integers $0 \le i \le m$, $0 \le j \le n$, $0 \le …

If $H$ is continuous, $T(x)$ defined by $\int_0^{T(x)} H(s)\; ds = x$ satisfies the differential equation $$ \dfrac{dT}{dx} = \dfrac{1}{H(T)}$$ and standard numerical methods for differential equatio …

Saddle-point approximation should work fine, if $x_0 > 0$. If the singularity at 0 bothers you, integrate instead from $x_0/2$ to 1: the integral from $0$ to $x_0/2$ is $O(exp(-a x_0^2/4))$.

For convenience let the interval be $[0,1]$, and let's look at the case $f(t) = t^p$. We have $E_n(t^p) = 0$ for $p \le 1$, while by Faulhaber's formula we have $$\eqalign{E_n(t^p) &= \dfrac{1}{p+1} …

It's easy to see why this is: $\cos(\alpha) \sim 1 - \alpha^2/2$ for $\alpha$ near $0$, so an error of $\delta$ in $\cos(\alpha)$ can produce an error of about $\sqrt{2\delta}$ in $\alpha$ as computed …

It's easy to come up with a differential equation that has a known general solution, expressed in the form $F(t,y) = constant$ where $F$ is differentiable: $$ \dfrac{\partial F}{\partial t} + \dfrac{ …

Here is a first-order approximation. I'll write $\tilde{v} = v + \rho w + O(\rho^2)$ and $\tilde{\lambda} = \lambda + \rho \mu + O(\rho^2)$. We may assume $\|v\| = \|\tilde{v}\| = 1$, so $v^T w = 0$. …

My interpretation of the problem: given $n$ pairs $(x_j, f(x_j)$ with $a \le x_1 < x_2 < \ldots < x_n \le b$, you want an approximation to $\int_a^b f(x)\, dx$. One way, that would give the correc …

Why not use the standard numerical methods for solving a system of equations, available in Maple, Matlab, Mathematica etc?

If $x$ is represented in floating-point as $y \times 10^{-d}$, $0.1 < y \le 1$, $d \in \mathbb N$, note that $$ x \ln(x) = y (\ln(y) - d \ln(10)) \times 10^{-d} $$ which shouldn't be a problem to eval …

To solve $f_n(x)=x$ (where $f_n$ is the $n$-fold composition of $f$), write $f_n(x)-x$ as a rational function, take the numerator (which is a polynomial, I think of degree $2^n-2$), and find its roots …

If you mean $f'' + 2 f (1-f^2)$, that can be solved either numerically (by standard techniques, available e.g. using Maple's dsolve(..., numeric)) or symbolically. The solutions of the differential e …

Not an answer, but I'll just note that there are some special cases that reduce to ODE's. With $u(x,t) = v(x+at)$, the differential equation becomes $$ (1+a) v' - a^2 (v'')^2 = v \tag{1}$$ In part …

Not an answer, but this may help with asymptotics: According to Maple the o.g.f. for $H^s_n$ is $$ \sum_{j=1}^\infty j^{-s} (-1)^{j-1} \sum_{n=j}^\infty {n \choose j} x^n = {\frac {1}{-1+x}{\it …

What you're saying, I think, is that (for a particular $t_1$) the map $\Psi$ from $x(0)$ to $x(t_1)$ is affine: $\Psi(x) = \Phi x + c$ where $\Phi$ is linear. If this was true for all $t_1$, the diff …