Homework 1 : Linear Algebra and Floating Point Arithmetic

Correspondent lab lectures: 1, 2, 3

Direct Methods for the Solution of Linear Systems

1. Given a matrix $A\in {\mathbb{R}}^{n\times n}$ and the vector $x_{\text{true}}=(1,1,\dots ,1{)}^T\in {\mathbb{R}}^n$, write a script that:
   • Computes the right-hand side of the linear system $y=A{x}_{\text{true}}$.
   • Computes the condition number in 2-norm of the matrix $A$. Is it ill-conditioned? What if we use the $\infty$-norm instead of the 2-norm?
   • Solves the linear system $Ax=y$ with the function np.linalg.solve().
   • Computes the relative error between the solution computed before and the true solution $x_{\text{true}}$. Remember that the relative error between $x_{\text{true}}$ and $x$ in ${\mathbb{R}}^n$ can be computed as

$$E(x_{\text{true}},x)=\frac{\Vert x-x_{\text{true}}{\Vert }_2}{\Vert x_{\text{true}}{\Vert }_2}$$

• Plot a graph (using matplotlib.pyplot) with the relative errors as a function of $n$ and (in a new window) the condition number in 2-norm $K_2(A)$ and in $\infty$-norm, as a function of $n$.

2. Test the program above with the following choices of $A\in {\mathbb{R}}^{n\times n}$:
   • A random matrix (created with the function np.random.rand()) with size varying with $n=\{10,20,30,\dots ,100\}$.
   • The Vandermonde matrix (np.vander) of dimension $n=\{5,10,15,20,25,30\}$ with respect to the vector $x=\{1,2,3,\dots ,n\}$.
   • The Hilbert matrix (scipy.linalg.hilbert) of dimension $n=\{4,5,6,\dots ,12\}$.

Floating Point Arithmetic

1. The Machine epsilon $ϵ$ is the distance between 1 and the next floating point number. Compute $ϵ$, which is defined as the smallest floating point number such that it holds: $$\text{fl}(1+ϵ)>1$$

Tips: use a while structure.

2. Let’s consider the sequence $a_n={\left(1+\frac{1}{n}\right)}^n$. It is well known that:

$$\underset{n\to \infty }{\lim }{a}_n=e$$

where $e$ is the Euler constant. Choose different values for $n$, compute $a_n$ and compare it to the real value of the Euler constant. What happens if you choose a large value of $n$? Guess the reason.

3. Let’s consider the matrices:

$$A=\left[\begin{array}{cc}4& 2\\ 1& 3\end{array}\right],B=\left[\begin{array}{cc}4& 2\\ 2& 1\end{array}\right]$$

Compute the rank of $A$ and $B$ and their eigenvalues. Are $A$ and $B$ full-rank matrices? Can you infer some relationship between the values of the eigenvalues and the full-rank condition? Please, corroborate your deduction with other examples. Tips: Please, have a look at np.linalg

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Other assignments:

• Homework 2
• Homework 3
• Homework 4