Theory.

1. How do you pass your integrand to gsl_integration_qags rutine?
2. What are the arguments epsabs, epsrel and abserr (correspondingly the 4th, 5th and last argument) of gsl_integration_qags rutine?
3. What does gsl_integration_qags return?
4. How does gsl_integration_qags return the calculated result?
5. Does gsl_integration_qags estimates the error of the calculated result?
6. Suppose you integrand does not take any extra parameters: what du you do then with the .params member of the corresponding gsl_function structure?
7. What is the limit argument (the 4th argument) of the gsl_integration_qagi rutine?
8. What is the argument of the gsl_integration_workspace_alloc function?
9. Do you need to free any memory having finished with your gsl_integration_qag* integrations?
10. Suppose your C-function returns an integer value, like gsl_integration_qags function, which you would normally call as

    int flag = gsl_integration_qags(arguments);

    can you call the function simply as

    gsl_integration_qags(arguments);

    ?
11. Suppose you need to calculate an integral with a given requirement to absolute error. How do you then set up your parameters of gsl_integration_qags routine?

Practice: numerical integration with Gauss-Kronrod algorithms.

1. In the variational method in quantum mechanics one calculates the expectation value E[ψ] of a Hamiltonian H,

   E[ψ] ≡ ⟨ψ|H|ψ⟩/⟨ψ|ψ⟩ ,

   with some trial function ψ. The trial function is then optimized to provide the minimum of the expectation value.

   Consider the Hamiltonian operator for one-dimensional oscillator,

   Ho = - (1/2) (d²/dx²) + (1/2) x² ,

   and the trial function in the form of a Gaussian,

   ψα(x) = exp(-αx²/2) ,

   where α is the variational parameter. Calculate numerically

   E(α) ≡ ⟨ψα|Ho|ψα⟩/⟨ψα|ψα⟩ .

   and plot the result – you should see a minimum of E=0.5 at α=1 (why?).

   Hints:

   1. The norm-integral, ⟨ψ_α|ψ_α⟩, has the form (check it)

      ⟨ψα|ψα⟩ = ∫-∞∞ exp(-αx²) dx .

      It is actually analytic, but you have to calculate it numerically.

   2. The Hamiltonian-integral, ⟨ψ_α|H_o|ψ_α⟩, has the form (check it also)

      ⟨ψα|Ho|ψα⟩ = ∫-∞∞ (-α²x²/2 + α/2 + x²/2)*exp(-αx²) dx .

      It is actually also analytic, but you again have to calculate it numerically.

   3. Since the integration limits are infinite, you migh want to use gsl_integration_qagi function.