Problems "multiroot"

Find the minimum of the Rosenbrock function,
```
f(x,y) = (1-x)² + 100(y-x²)² ,
```
by finding the root of it's gradient (that is, the point where the gradient is zero).
(Non-relativistic) hydrogen atom with shooting method in boundary value problems.
Introduction:
The s-wave radial Schrödinger equation for the Hydrogen atom reads (in units "Bohr radius" and "Hartree"),
```
-(1/2)f'' -(1/r)f = εf ,
```
where f(r) is the radial wave-function, ε is the energy, and primes denote the derivative over r.
The bound s-state wave-function satisfies this equation and the two boundary conditions,
```
f(r → 0) = r-r², (prove this)

f(r → ∞) = 0 .
```
These two boundary conditions can only be satisfied for certain discrete (negative) values of the energy.
Since one cannot integrate numerically to ∞ one substitutes ∞ with a reasonably large number, rmax, such that it is much larger than the typical size of the hydrogen atom but still managable for the numerical inregrator (say, rmax = 10 Bohr radii),
```
f(rmax)=0 .
```
Let F_ε(r) be the solution (to be found numericall via gsl_odeiv) to our differential equation with energy ε and initial condition F_ε(r → 0)=r-r². Generally, for a random negative ε, this solution will not satisfy the boundary condition at rmax. It will only be satisfied when ε is equal one of the bound state energies of the system.
Now define an auxiliary function
```
M(ε) ≡ F_ε(rmax) .
```
The shooting method is then equivalent to finding the root of the equation
```
M(ε) = 0 .
```
Exercises:
1. Find the lowest root, ε₀, of the equation M(ε) = 0 for, say, rmax=8. Plot the resulting function and compare with the exact result (ε₀=-1/2, f₀(r)=re^-r – check this).
2. (Optional) Investigate the convergence of the solution towards the exact result as function of rmax.
3. (Optional) Try also to use a more precise boundary condition for bound states (which have ε<0),
  f(r → ∞) = r e^-kr , (prove this)
  where k=√(-2ε). This should allow you to use a smaller rmax.