1. Numerical Linear Algebra

Iterative Solvers — Quiz

Test your understanding of iterative solvers with 5 practice questions.

Practice Questions

Question 1

What primary feature distinguishes the Flexible GMRES (FGMRES) algorithm from classical GMRES when solving preconditioned systems?

Question 2

In the Conjugate Gradient method applied to the system $A x=b$ with $A=\begin{pmatrix}4&1\\\\1&3\end{pmatrix},\;b=\begin{pmatrix}1\\\\2\end{pmatrix},$ and initial guess $x_0=\begin{pmatrix}0\\\\0\end{pmatrix},$ what is the approximate solution $x_1$ after one iteration?

Question 3

If a real symmetric positive-definite matrix $A$ has only $m$ distinct eigenvalues, in exact arithmetic the Conjugate Gradient method will produce the exact solution in at most how many iterations?

Question 4

In the Bi-Conjugate Gradient (BiCG) algorithm, breakdown occurs when the scalar $\rho_k=\hat r_k^T r_k$ becomes zero. What is the significance of this breakdown?

Question 5

Which step in the GMRES algorithm leads to an $\mathcal O(k^2)$ computational cost and growing storage as the iteration count $k$ increases?