Questions tagged [numerical-linear-algebra]

Question 1

I have been working on an algorithm for efficient matrix–vector multiplication on GPUs in the regime of moderate sparsity (30%–90%), with no structural assumptions on the sparsity pattern. The ...

Question 2

Let $L$ be a lower triangular matrix where all elements on and below the diagonal are positive. The matrix is also highly ill-conditioned. I want to solve $Lx=1$. Additionally, I know that all ...

Question 3

I am studying a generalized eigenvalue problem which can be partitioned as $$ \left[ {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}}\\ {{A_{21}}}&{{A_{22}}} \end{array}} \right]\left[ {\begin{...

Question 4

Let $A$ be $n\times n$ matrix. Can $A$ be written as $A=B_1\cdots B_k$ where $B_i$ are band matrices with constant bandwidth, and $k=O(n)$? Is it always possible? Is there an efficient algorithm for ...

Question 5

Problem: We want to determine whether the following algorithm is stable or not. Data is $x_{1},x_{2} \in \mathbb{C}$, Solution is $x_{1}(x_{2}+1)$, computed as $\text{fl}(x_{1}) \otimes (\text{fl}(x_{...

Question 6

Problem: Let $A \in \mathbb{R}^{m \times n}$ be a nonsingular matrix. Consider the system $Ax = b$. If $b$ is perturbed by $\delta b$, suppose that the solution of the original system is perturbed by $...

Question 7

I'm studying for an exam and I came into the following question regarding the use of eigenvalues to estimate a solution. I'm having trubles understating what the request is, can somebody give me a ...

Question 8

Problem Setup: Let $A \in \mathbb{C}^{m \times m}$ be nonsingular. For each $k$ with $1 \leq k \leq m$, the upper-left $k \times k$ block $A_{1:k,1:k}$ is nonsingular. We want to show that $A$ has an ...

Question 9

I am reading Numerical Linear Algebra by Lloyd N. Trefethen and David Bau, III. On p.16 in this book: Exercise 2.5. Let $S\in\mathbb{C}^{m\times m}$ be skew-hermitian, i.e., $S^*=-S$. (a) Show by ...

Question 10

Let $A = \begin{bmatrix} 1.0000 & 2.0000\\ 1.0001 & 2.0000 \end{bmatrix}$. Suppose we wish to find $Ax = b$, where $b = (3.0000,3.0001)^T$. Instead of $x$, we obtain $x' = ...

Question 11

We have the Householder reflector $F = \begin{bmatrix} -c & s\\ s &c \end{bmatrix}$, where $c$ stands for $\cos{\theta}$, $s$ stands for $\sin{\theta}$ for some $\theta$. We want to know its ...

Question 12

Let $A$ be an $m \times n$ matrix with $m \geq n$. And let $A = \hat{Q}\hat{R}$ be the reduced $QR$ factorization. Suppose that all the diagonal matrix entries $\hat{R}$ are nonzero. We want to show ...

Question 13

The bandwidth of a matrix $M=(m_{ij})$ is defined as the maximum value of $|i-j|$ such that $m_{ij}$ is nonzero. During LU factorization for solving a linear system, reducing the bandwidth of a matrix ...

Question 14

I have a double integral, which I want to solve: $\mathcal{I}_{au}=\int_0^{t'}e^{(a-c)t''}\int_0^{t''}e^{(-b+c)t'''}\text{exp}\big(-\frac{\lambda^2}{2}\left( S_2(t''')^2+S_1(t')^2+S_1(t'')^2+2S_{12}t'...

Question 15

For a least-squares problem find $x$ such that $\|Ax - b\|_2, A \in \mathbb{R}^{m \times n}$ is minimized, the solution of is captured by the pseudo-inverse, $$x = A^\dagger b$$ There exists three ...

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Questions tagged [numerical-linear-algebra]

Efficient sparse matrix vector multiplication at low 30-90% unstructured sparsity. Relevant use cases?

Inversion of an ill-conditioned triangular matrix

Ratio of eigenvector amplitudes

Factorizing a matrix as a product of band matrices

Stability of specific floating point operation algorithm

Inequality of Perturbation of Linear System [duplicate]

Using eigenvalue to estimate the accuracy of the solution of a system

$LU$ decomposition and Leading principal submatrix [duplicate]

How to use Exercise 2.1 to solve Exercise 2.5(a)? (Numerical Linear Algebra by Lloyd N. Trefethen and David Bau, III.)

Condition Number of $A$

Geometric Effect of Householder Reflector

Nozero diagonal entries imply full rank

Bandwidth of a matrix and reducing fill-in during LU factorization

I need help with solving a double integral exp of a quadratic variable

How do you compute the solution to least-squares problem if neither $A^TA$ nor $AA^T$ nor $A$ are invertible?

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