Numerical analysis

What is the condition number $\kappa_\infty(A) = \|A\|_\infty\|A^{-1}\|_\infty$ of $A = \begin{pmatrix} 4 & 0 \\ 0 & 0.004 \end{pmatrix}$?

A solver returns $\hat x$ for $Ax = b$, and you find that $\hat x$ is the exact solution of $(A + \delta A)\hat x = b$ with $\|\delta A\| / \|A\| \approx 10^{-16}$. What can you conclude?

You solve $Ax = b$ in IEEE double precision ($u \approx 10^{-16}$) with a backward-stable method, and $\kappa(A) \approx 10^{10}$. Roughly how many correct significant decimal digits can you expect in the solution?

Which of these invertible matrices has no factorisation $A = LU$ (with $L$ unit lower triangular and $U$ upper triangular) unless rows are swapped?

Why should you solve $Ax = b$ by LU with partial pivoting rather than by computing $A^{-1}$ and multiplying? Choose all correct reasons.

Which Lean conclusion correctly expresses Theorem 1, $\frac{\|\delta x\|}{\|x\|} \le \kappa(A)\frac{\|\delta b\|}{\|b\|}$, without dividing by norms that might be zero?

Explicit Euler is applied to $y' = -200y$ on $[0, 1]$ with $N$ equal steps ($h = 1/N$). What is the smallest $N$ for which the numerical solution decays, i.e. $|1 + h\lambda| < 1$?

Write rk4(f, y0, t_end, n) that applies $n$ steps of the classical fourth-order Runge–Kutta method to $y' = f(t, y)$, $y(0) = y_0$, with step $h = t_{\text{end}}/n$, and returns the approximation to $y(t_{\text{end}})$.

Let $A \in \R^{m \times n}$ ($m > n$) have full column rank and 2-norm condition number $\kappa_2(A) = 10^8$. Which statements are true? Choose all that apply.

Which of these methods are A-stable, i.e. $|R(z)| \le 1$ for every complex $z$ with $\operatorname{Re} z \le 0$? Choose all that apply.