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-1

For a 3x3 matrix, the analytical solution can easily be computed with some symbolic math software like wmaxima.


1

The problem on which I originally made that comment is a linear algebra problem: consider the linear matrix equation $$ \sum_{i=1}^k A_i X B_i = C, $$ where $A_i,B_i,C \in \mathbb{R}^{n\times n}$ are given, and $X\in \mathbb{R}^{n\times n}$ is the unknown. For $k=2$ this is a generalized Sylvester equation, and can be solved using a Bartels-Stewart-type ...


2

The infinite square well potential problem in non-relativistic quantum mechanics has energy eigenvalues $E_n=n^2\hbar^2\pi^2/2mL^2$,where $n^2=\sum_{k=1}^Nn_k^2$($N$=number of dimensions). A problem of interest is, given an energy eigenvalue, the degeneracy(i.e,number of eigenstates with same eigenvalue) are to be found.This amounts to counting the number ...


1

The boundary conditions don't depend on the choice of your basis but on the formulation you have for your problem. If you have a "standard" finite element formulation, you don't need to do anything to apply (homogeneous) Neumann boundary conditions, they are already satisfied by your system. In the most common formulation, Neumann boundary conditions are ...


8

I think the method has too much implementation complexity and too narrow applicability to be worth it. Though the paper is correct to point out the importance of solving the tridiagonal-symmetric eigenproblem in the course of solving the general-symmetric eigenproblem, it neglects to mention that the "frontend" procedure between these two scenarios (...


1

1) Is there a mathematical trick to simplify the above matrix equations? As in, only having to do one inverse operation instead of two. Yes: Schur complement formulation. Your system is equivalent to the larger one $$ \begin{bmatrix} 0 & B\\ B^T & A^T \end{bmatrix} \begin{bmatrix} y\\ z \end{bmatrix} = \begin{bmatrix} -b\\0 \end{bmatrix} $$ with $b=...


0

For stability analysis, we can assume $f = 0$. First we write $$ M \left( u_{k+1} - u_{k} \right) = \Delta t K u_{k+1} $$ or $$ \left( M - \Delta t K \right) u_{k+1} = M u_{k} $$ Take the eigendecomposition of $(K, M)$ (there is an explicit expression in 1D), such that $$ w_i^T K w_j = \delta_{ij} \lambda_i \quad \mbox{and} \quad w_i^T M w_j = \delta_{ij} $$ ...


0

If $X$ is a $(n \times m)$-matrix whose entries are independently generated values from the standard normal distribution, then $X(X^{\top}X)^{-\frac12}$ is a uniformly generated random orthogonal matrix. Source. Here is an implementation with Eigen: #include<iostream> #include<random> #include<Eigen/Eigen> #include<ctime> using ...


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