Questions tagged [numerical-linear-algebra]

Questions on the various algorithms used in linear algebra computations (matrix computations).

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estimating the determinant of a large PSD matrix

Is the any property of a potentially large positive semi-definite matrix that can be leveraged on in estimating its determinant (at least in estimating if the determinant is large or small)? The ...
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Is matrix inversion stable?

As the title suggests, if I perturb the entries of an arbitrary matrix by a very small value (say increase every entry by 0.01), how different will the inversion of this perturbed matrix compared to ...
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Condition number of a matrix with factorization $A=QR$

Let $A=QR$ when $Q$ is orthogonal and R is triangular superior proof $\dfrac{1}{n}k_1(A)\leq k_1(R) \leq n k_1(A)$ and $k_2(A)=k_2(R)$ So we have for the second part $k_2(A)=||A||_2|||A^{-1}||_2$ and ...
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Example of matrices where computing the inverse is the most efficient method

I know there exists matrices where for example LU-factorization is not the most efficient way of solving the linear system of equations: $$Ax=b \tag{1}$$ Examples of such matrices are triangular or ...
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Find a matrix A with Col(A) = 1 and Ax = y and try to minimize the L0_norm of A

I need to develop a one-time trading system to match the buyers and sellers and try to minimize the number of trades. (The price is fixed so we only consider demand and supply). Suppose the total ...
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Transform an equation to get a non-negative root

I have the following equation $$f(x) = \frac{1}{\|h(x)\|_2} - \frac{1}{x} =0 \tag{1}$$ where $$h(x) = -\left[A+ x I\right]^{-1}g,\tag{2}$$ $A$ a symmetric matrix and $g$ a column vector. The initial ...
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System of equations with plus function

Let $A\in \mathbb{R}^{m\times n}$ and $b\in \mathbb{R}^m$ and $\lambda >0$. how can I solve the following system of equations? \begin{equation} A^T(Ax-b)_++\lambda x=0 \end{equation} where the plus ...
34 views

How to calculate this product without inverting T?

Let $T\in \mathbb{R}^{n\times n},$ which is an upper triangular matrix,and $x,y\in \mathbb R^n$. What is the most efficient way to evaluate $$x^\top T^{-2}y,$$ and what is the cost in flops? ...
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SVD of product of three matrices

Are there any library implementation tor numerical routine for SVD of product of three general matrices? I know the three matrices separately and want to avoid multiplying them directly before SVD. ...
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Simultaneous orthogonalization in Arnoldi iteration

Recently, I was studying Stewart's A Krylov-Schur Algorithm for Large Eigenproblems. After a survey, the so called expansion phase is implemented, with the description given below: For the sake of ...
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How to solver the least square problem involves with the variable is the product of Hadamard Product?

I encounted an least square problem involves hadamard product of two $100\times 1$ matrix: X, Y: $$A_{N\times100}*(X_{100\times 1}∘Y_{100\times 1})_{100\times 1} = B_{N\times 1}$$ In above equation,...
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Expressing the spectral radius as a function of the matrix size

Currently, I am working with a (non-symmetric) discretization matrix $A$ coming from a convection-diffusion problem without elimination of boundary conditions. Regarding the SOR algorithm, I have to ...
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Show that spectral radius is $\rho(M^{-1}K)\ge1$

Given a splitting of symmetric singular matrix by A=M-K where M is nonsingular. Show that spectral radius of $\rho(M^{-1}K)\ge1$. Recall spectral radius is $\rho(A)=\max_i|\lambda_i|$ Hint: any zero ...
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Are Python/MATLAB/Mathematica numerical eigenvectors affected by eigenvalue degeneracies outside region of calculation?

I have a discretized 2D mesh over which I calculate eigenvalues and eigenvectors of some Hermitian 2 x 2 matrix at each point along a closed loop parameterized by parameter t. The eigenvectors are ...
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How to determine the eigenvalues and eigenvector of the Householder matrix

Here is the specification again: Let $w \in \mathbb{R}^{n}$ and $||w||=1$. The $n \times n$- matrix. $$P:=I-2ww^t$$ is called the Householder transform and $w$ is called the Householder vector. ...
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Finding the projection of a cycle polygon with maximum area and no intersections

Given a polygon (forming a loop) embedded in euclidean space of dimension 3 or higher, how to find the 2d projection maximizing its area and such that it doesn't intersect ? My initial idea was to ...
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Some questions about well condition of a problem

Considerer the matrix $A \in \mathbb{R}^{n\times n}$ and the system $Ax=b$ with $b=(1,1)$ find an upper bound of $||\delta_x||_\infty/||x||_\infty$ in terms of $||\delta_b||_\infty/||b||_\infty$, with ...
61 views

For what $x$ does the function $g(x) = \frac{x^2+a}{2x}$ converge to fixed point $\sqrt{a}$

What is the interval where the function converges to this fixed point? So we have a function $$g(x) = \frac{x^2+a}{2x}$$ We want to know the interval where this function for sure converges to fixed ...
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Find a basis for $S(1,0)h[a,b]=\{p∈C0[a,b],p|Ii∈P1(Ii)\}$
On the intervall $[a,b]$ let $a = x_o< x_1 < x_2 < ... < x_n = b$ be a decomposition. Consider the Vectorspace of the piecewise linear functions S_h^{(1,0)}[a,b] = \{p \in C^0[a,b], p|_{{...
Uniquness of a QR-Decomposition: Show that there exists an orthogonal diagonal matrix $S \in \mathbb{R}^{n \times n}$
Let $A=Q_{1} R_{1}=Q_{2} R_{2}$ be two $Q R$-decompositions of a quadratic matrix $A \in \mathbb{R}^{n \times n}$ with full rank, i.e., $\operatorname{rank}(A)=n$. This means \$ Q_{1}, Q_{2} \in ...