IEEE 754 Single Precision Quiz

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Questions and Answers

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

The exponent of the number

The base of the number

The sign of the number

The significant digits of the number (correct)

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

False (B)

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

127

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

NaN Signup and view all the answers

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

Sign (D) Signup and view all the answers

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

False (B) Signup and view all the answers

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

2 Signup and view all the answers

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

e = 0, m = 0 = Represents zero e = 255, m != 0 = Represents NaN (Not a Number) e = 255, m = 0 = Represents Infinity 0 < e < 255 = Represents a normal number Signup and view all the answers

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

$|x_{k+1} - x_k| < tol$ (A) Signup and view all the answers

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

False (B) Signup and view all the answers

What type of matrix results from Gaussian elimination?

Upper triangular matrix Signup and view all the answers

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

LU Signup and view all the answers

Match the matrix factorization with its corresponding matrix type:

LU factorization = General square matrices Cholesky factorization = Symmetric positive definite matrices Givens rotations = Sparse matrices Signup and view all the answers

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

A specified tolerance (A) Signup and view all the answers

Gaussian elimination is applicable only to singular matrices.

False (B) Signup and view all the answers

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

$f(x_k)$ Signup and view all the answers

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

$A^T = A$ and $x^T Ax > 0$ for all $x ≠ 0$ (C) Signup and view all the answers

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

True (A) Signup and view all the answers

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

A = LU with U = DLT Signup and view all the answers

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

R^T Signup and view all the answers

Match the following matrix properties with their implications:

Symmetric Positive Definite = All eigenvalues are positive Principal Submatrix = Symmetric positive definite Largest Entry in Absolute Value = Located on the diagonal and positive L has rank m, then $LAL^T$ = Symmetric positive definite. Signup and view all the answers

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

Upper triangular (D) Signup and view all the answers

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

False (B) Signup and view all the answers

How many operations are required to perform Cholesky factorization directly?

1/3 n^3 + O(n^2) Signup and view all the answers

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

$c = \frac{a_{11}}{\sqrt{a_{11}^2 + a_{21}^2}}$, $s = \frac{-a_{21}}{\sqrt{a_{11}^2 + a_{21}^2}}$ (A) Signup and view all the answers

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 $c^2 + s^2 = 1$.

True (A) Signup and view all the answers

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

$Q^T Q = I$ Signup and view all the answers

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

orthogonal Signup and view all the answers

Match the following matrix properties with their descriptions:

Orthogonal Matrix = Its transpose multiplied by itself equals the identity matrix. Rotation Matrix = A matrix that represents a rotation in space. Identity Matrix = A square matrix with ones on the main diagonal and zeros elsewhere. Transpose of a Matrix = A matrix formed by interchanging the rows and columns of a given matrix. Signup and view all the answers

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

It implies that the dot product of the second row of Q and the first column of A is zero. (B) Signup and view all the answers

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

False (B) Signup and view all the answers

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

1 Signup and view all the answers

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

4000 (A) Signup and view all the answers

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

False (B) Signup and view all the answers

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

[0.999999999974, 1.999999999951, 3.000000000039, 4.000000000032] Signup and view all the answers

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

1.369 Signup and view all the answers

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

It has an upper bound error of 1.3 x 10^{-10}. (B) Signup and view all the answers

Match the following values with their descriptions:

κ∞(A) = 4000 Actual relative error = 4.002 Upper bound error = 4.004 Condition number context = Stable computation by MATLAB Signup and view all the answers

The solution vector x is defined as [___].

[1, 2, 3, 4] Signup and view all the answers

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

6.01 x 10^5 (A) Signup and view all the answers

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 $A'x = b'$?

$\epsilon_A \cdot K(A) < 1$ (A) Signup and view all the answers

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.

False (B) Signup and view all the answers

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

$K(A) \cdot \epsilon_b$ Signup and view all the answers

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

condition Signup and view all the answers

Match the terms from Theorem 3.7 with their descriptions:

K(A) = Condition number of matrix A \epsilon_A = Bound on relative change in matrix elements \epsilon_b = Bound on relative change in vector b x = Solution vector to the original system Signup and view all the answers

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

Relative errors in A and b, respectively (B) Signup and view all the answers

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.

True (A) Signup and view all the answers

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

sensitivity Signup and view all the answers

Flashcards

Floating Point Number

A representation of real numbers in a way that can support a wide range of values.

Mantissa

The part of a floating point number that contains its significant digits.