Consider the positive floating point number system $s\times b^e$, where $b=10$, $s$ is a 15-digit significant $1\leq s\leq (10 - 1\times 10^{-14})$ and $-300\leq e \leq 300$. This is close to the double-precision standard. What is the machine epsilon in this system (i.e.\ the smallest number which, when added to 1, gives a result distinct from 1)?

Consider the floating-point system from the previous question. Newton's method is applied (in exact arithmetic) to find a root of a function $f(x)$. Let the initial error be $\epsilon_0=0.1$, and assume Newton's method converges quadratically from here. Approximately how many iterations are required before the error is smaller than the machine epsilon of the floating-point system?

The small-angle approximation for trigonometric functions is based on a Taylor expansion about $x=0$, up to quadratic terms.  For which angle does the approximation of $\sin(x)$ have a relative error exceeding approximately $1.0\%$? [Note: Relative error: $\frac{f_{true} - f_{approx}}{ f_{true} } \times 100\%$]

Consider an twice-continuously differentiable periodic function $f(x)$, with

with at least one root $\tilde x$ in the interval $[0,2\pi]$.  A fixed-point iteration (FPI) $x_{i+1} = \phi(x_i)$ is applied with a particular initial guess $\hat x_0\in\mathbb{R}$ and converges to $\tilde x$.  What can be said about the behaviour of this FPI for other initial guesses $x_0\neq \hat x_0$?

Using the property of fixed-point iterations, that they are guaranteed to converge if $|\varphi'(x)| < 1$ for $x \in [\tilde x, x_0]$, and noting that Newton is a fixed-point iteration with

in what starting interval is Newton's iteration guaranteed to converge to the exact root $\tilde x = \pi/2$ of $f(x) = \cos x = 0$?

In the course ``Introduction to Aerospace Engineering II'' you might have encountered Kepler's equation for elliptical orbital properties:

where $M$ is the mean anomaly, $E$ the eccentric anomaly and $0 \leq e < 1$ the eccentricity. Unfortunately given $M$ and $e$ the value of $E$ can not be found analytically. The iteration

is used to approximate a solution, with an appropriate starting guess $E_0$. For which values of $M$ is the iteration guaranteed to converge to a root, given any value of $0 \leq e < 1$ and any $E_0 \in \mathbb{R}$?

Consider the two-variable problem consisting of two scalar equations:

Using Newton's method starting with $x_0=(1.5,1.0)^T$, what is the estimate of the root after single iteration? [Hint: For multiple equations and variables the derivatives form a matrix.]

Newton's method is used to find the root of $f(x) = x^4$. Given that $e_{i+1}=K e_{i}^n$, where $e_i$ is the error at iteration $i$, what are $K$ and $n$? [Hint: Starting from the definition of the Newton iteration, derive an expression for the error.]

Consider the nodes $[x_0,\dots,x_4] = [-2,-1,0,1,2,3]$, and the Lagrange basis polynomial $\ell_0(x)$ with the usual property $\ell_0(x_i) = \delta_{0i}$. The function $\ell_0(x)$ is itself approximated by a monomial basis

with $p(x)$ interpolating $\ell_0(x)$ at $6$ distinct unspecified nodes. What is the value of $a_5$? [Hint: You only need to find the coefficient of the highest-degree term in the polynomial, which simplifies the calculation.]

Consider the interpolant

where $i = \sqrt{-1}$.  Note that the coefficients $c_k\in\mathbb{C}$ and basis functions $\varphi_k(x):\mathbb{R}\rightarrow\mathbb{C}$ are complex-valued, while $x\in\mathbb{R}$.  You are given a set of $N+1$ distinct nodes $0 \leq x_0< x_1 < \dots < x_N < 2\pi$ where $N+1$ is odd.  For what relation between $K$ and $N$ does there exist a unique solution to the interpolation problem for any complex-valued samples $f(x_0),\ldots f(x_N) \in \mathbb{C}$?