IEEE 754 Single Precision Quiz

In the context of floating-point numbers, what does the mantissa represent?

In the IEEE standard for single precision floating-point numbers, the mantissa's first digit is explicitly stored.

What is the bias (K) used in the IEEE single precision floating-point representation?

In the IEEE single precision format, an exponent of 255 with a non-zero mantissa represents ______.

What does S represent in the IEEE single precision floating-point format?

Subnormal numbers have an exponent e = 255 in the IEEE single precision format.

What base (B) is assumed in the IEEE standard as described?

Match the following floating-point representations with their meanings in the IEEE 754 single precision standard:

Which of the following stopping criteria uses the absolute difference of successive approximations?

A small residual $r_k = b - Ax_k$ guarantees a small error $x_k - x$ in solving a linear system $Ax = b$.

What type of matrix results from Gaussian elimination?

Gaussian elimination can be written in compact from as PA = ______ where P denotes a permutation matrix

Match the matrix factorization with its corresponding matrix type:

What does 'tol' represent in the stopping criteria formulas?

Gaussian elimination is applicable only to singular matrices.

In the context of solving $f(x) = 0$, what is the residual $r_k$?

Which of the following is a necessary condition for a matrix $A ∈ R^{n×n}$ to be symmetric positive definite?

If a matrix $A$ is symmetric positive definite, all its diagonal entries must be positive.

What is the result of Gaussian elimination on a symmetric positive definite matrix A without pivoting?

The Cholesky factorization of a symmetric positive definite matrix A is given by $A = ______ · R$.

Match the following matrix properties with their implications:

Given the Cholesky factorization $A = R^T R$, what type of matrix is R?

Cholesky factorization requires pivoting for symmetric positive definite matrices in exact arithmetic.

How many operations are required to perform Cholesky factorization directly?

What are the values of 'c' and 's' in the rotation matrix given that $Q = \begin{bmatrix} c & -s \ s & c \end{bmatrix}$?

The matrix $Q$ defined as $Q = \begin{bmatrix} c & -s \ s & c \end{bmatrix}$, where $s = \sin(\alpha)$ and $c = \cos(\alpha)$ for some angle $\alpha$, is an orthogonal matrix if and only if $c^2 + s^2 = 1$.

What condition must be satisfied for the product of a matrix $Q$ and its transpose $Q^T$ to equal the identity matrix $I$?

A matrix Q is considered __________ if its transpose multiplied by itself equals the identity matrix.

Match the following matrix properties with their descriptions:

In the context of rotation matrices, what does the condition $(QA)_{21} = 0$ imply?

The $Q_{ij,k}$ matrix is always a $2 \times 2$ matrix.

In the $n \times n$ matrix $Q_{ij,k}$, what are the diagonal elements other than $c$ and $s$?

What is the condition number of matrix A, denoted as κ∞(A)?

The actual relative error is larger than the upper bound provided by Theorem 3.7.

What is the numerical solution vector obtained by using MATLAB's backslash on the matrix A?

The relative error from the numerical solution amounts to approximately ___ x 10^{-11}.

Which of the following statements is true regarding the matrix A?

Match the following values with their descriptions:

The solution vector x is defined as [___].

What is the value of K(A) described in the document?

According to Theorem 3.7, what condition must be met to ensure a certain bound when comparing the solutions of two linear systems $Ax = b$ and $A'x = b'$?

For all induced norms, the values $\epsilon_A$ and $\epsilon_b$ are equal to $\bar{
ewline \\epsilon}_A$ and $\bar{
ewline \\epsilon}_b$ respectively, as defined in Theorem 3.7.

In the special case where $A' = A$, according to Theorem 3.7, what is the upper bound for the relative error $\frac{||x' - x||}{||x||}$?

According to Theorem 3.7, if $A$ is invertible and $\epsilon_A \cdot K(A) < 1$, then there exists a relation with the ______ number $K(A)$.

Match the terms from Theorem 3.7 with their descriptions:

In the context of perturbed linear systems (Theorem 3.7), what do $\epsilon_A$ and $\epsilon_b$ represent?

Theorem 3.7 provides a way to quantify the sensitivity of the solution of a linear system to perturbations in the matrix A and the vector b.

According to Theorem 3.7, the condition number $K(A)$ relates to the [blank] of a matrix.