Numerical Methods Quiz

Test your understanding — Weeks 5–8

These quizzes help you distinguish between numerical approaches covered in Weeks 5–8 of Programming & Numerical Methods. Each section tests a different topic. Select your answer, then click Check to see whether you’re right — and why.

Section 1 — Root-Finding

1. You need to find the steady-state capital stocks of a 3-country Solow model — a system of three non-linear equations in three unknowns. Which approach is most appropriate?
   Brute-force grid search over a 3-D grid scipy.optimize.fsolve with an initial guess scipy.optimize.minimize with BFGS Lagrange interpolation of the solution

2. You run fsolve on a non-linear system with initial guess x₀ = [0.5, 0.5, 0.5] and get a solution. You change to x₀ = [5, 5, 5] and get a different answer. What is the most likely explanation?
   The system has multiple roots and each initial guess converges to a different one fsolve has a bug in its implementation The function is not continuous You need to use brute-force search instead

3. Your non-linear equation has a parameter that must satisfy 0 < s < 1 (e.g., a saving rate). You use fsolve and it returns s = 1.3. What is the best response?
   Accept the result — the algorithm always finds the correct solution Use a bounded solver or reformulate the problem to respect the constraint Increase the tolerance of fsolve Switch from fsolve to Nelder–Mead

Section 2 — Optimisation

4. You minimise the 5-dimensional Rosenbrock function with brute-force grid search and with BFGS. Brute-force takes 17 seconds; BFGS takes 0.01 seconds with comparable accuracy. Why is BFGS so much faster?
   BFGS uses gradient information to take intelligent steps, while brute force evaluates the function at every point on a grid that grows exponentially in dimension BFGS uses a coarser grid Brute force uses more memory BFGS only works in low dimensions

5. You want to minimise a function but you cannot compute or approximate its gradient. Which method is most appropriate?
   BFGS L-BFGS-B Nelder–Mead Sequential Quadratic Programming

6. You are solving a consumer’s constrained utility maximisation problem: max u(x) subject to p·x ≤ I and x ≥ 0. You have coded three approaches from the lecture: penalty method, augmented Lagrangian, and SQP (Newton on the Lagrangian). Which statement is correct?
   The penalty method always converges faster because it is simpler If you set the penalty parameter P too large from the start, the penalty method can give completely wrong answers due to ill-conditioning The augmented Lagrangian and the penalty method are the same thing SQP cannot handle equality constraints

7. You need to minimise a function subject to the constraint x₁² + x₂² ≤ 1. The feasible set is bounded. Which of the following methods can directly handle this inequality constraint?
   Brute-force search Nelder–Mead COBYLA BFGS

Section 3 — Function Approximation

8. You interpolate f(x) = 1/(1+x²) on [−5, 5] with uniformly spaced nodes and a degree-11 polynomial. The approximation oscillates wildly near the boundaries. What is this phenomenon called, and how do you fix it?
   Overfitting — reduce the polynomial degree Runge’s phenomenon — switch to Chebyshev nodes Numerical instability — use higher floating-point precision Underfitting — increase the number of nodes

9. You need to approximate a value function V(k) that has a kink (is continuous but not differentiable at a point). Which approximation method handles this best?
   High-degree Lagrange interpolation Chebyshev polynomials Cubic splines A single global polynomial of degree 30

10. When constructing a Chebyshev approximation of f on [a, b], why do we map the interval to [−1, 1] before computing the nodes?
    It makes the function easier to differentiate Chebyshev polynomials Tₙ(x) = cos(n·arccos(x)) are defined on [−1, 1], so we transform the domain to use the optimal node placement It reduces the number of function evaluations needed It converts the interpolation into a root-finding problem

Section 4 — Choosing the Right Method

11. Match the problem to the best numerical approach: “Find the saving rate s ∈ (0,1) such that the Solow steady-state condition k* = (sA/δ)^(1/(1−α)) matches a target capital stock.”
    Chebyshev approximation Constrained optimisation with COBYLA Root-finding with fsolve (rewrite as F(s) = 0) Cubic spline interpolation

12. Match the problem to the best numerical approach: “Represent the value function V(k) on a grid of 50 capital values so that you can evaluate V at any k between grid points during value function iteration.”
    Root-finding with fsolve Penalty-method optimisation Cubic spline or Chebyshev interpolation Brute-force grid search

13. Match the problem to the best numerical approach: “Maximise a consumer’s Cobb-Douglas utility u(x) = Σ αᵢ log(xᵢ) subject to a budget constraint p·x ≤ I and non-negativity xᵢ ≥ 0.”
    Root-finding with fsolve Constrained optimisation (SQP, penalty method, or augmented Lagrangian) Chebyshev regression Lagrange interpolation

14. You are comparing BFGS, Nelder–Mead, and L-BFGS-B for minimising a function on a box [0.5, 2]⁵. Which statement is true?
    Nelder–Mead respects bounds natively; BFGS and L-BFGS-B do not L-BFGS-B can handle box bounds; BFGS and Nelder–Mead cannot All three methods respect box constraints None of the three can handle bounds