Questions tagged [eigenvalues]
Questions on symbolically or numerically determining the eigenvalues of matrices (Eigenvalues, Eigensystem) or differential equations (DEigenvalues, DEigensystem, NDEigenvalues, NDEigensystem) in Mathematica. Also includes determining the eigenvalues of differential equations with DSolve or NDSolve.
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What algorithm is Mathematica using to find the smallest eigenvalue so quickly?
My question is what kind of black magic is Mathematica doing to obtain the correct answer so quickly compared to other programming languages?
Details:
I've written a Mathematica notebook to find the ...
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Numeric Solution Hydrogen Atom
I'm trying to solve the Schrödinger equation for the hydrogen atom without made the variable separation of the polar and radial coordinate. It is my test code to extrapolate to another system with its ...
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DEigenvalues and NDEigenvalues return different values
In the following example, DEigenvalues and NDEigenvalues return different results despite having identical arguments. Does anyone know why?
(I use Mathematica 11.3)
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How to compute eigenvalues of linear function (not matrix)?
How to compute eigenvalues of a known linear function?
In Julia, there is a package https://jutho.github.io/LinearMaps.jl/dev/ to compute the matrix representation of given function, then we can ...
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Differing behavior of Eigenvalues and Eigensystem
With the update to v12.0, I seem to be getting different behavior of eigenvalues returned by Eigenvalues and Eigensystem (oddly, ...
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Memory usage for smallest eigenvalues
I have a bunch of hermitian matrices which are huge (of order 2^17 x 2^17) but extremely sparse so that, when I build the matrices, the usage of RAM is low (say of order 1 GB or similar).
The ...
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Solving a matrix pencil (quadratic eigenvalue) problem with Mathematica
According to Wikipedia
The matrix pencil of degree $\ell$ is the matrix-valued function
defined on the complex numbers $L(k) = \sum_{i=0}^{\ell} k^{i} A_{i}$.
Here $A_{\ell}$ are non-zero $n\times n$...
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Does current Mathematica capability allow solving PDEs (e.g. reaction diffusion equation) over surface of a mesh or surfaces
I came across this nice paper (https://arxiv.org/pdf/1605.01583.pdf) where the authors simulated a reaction diffusion system over the surface of a gecko, primarily to understand how various patterns ...
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Where is the mistake in computing the particular eigenvector of the following DFT Matrix?
I have the following matrix (the DFT Matrix for N = 3)
$$W = \frac{1}{\sqrt{3}}\begin{pmatrix} 1 & 1 & 1 \\ 1 & e^{-\frac{i 2 \pi}{3} } & e^{\frac{i 2 \pi}{3} } \\ 1 & e^{\frac{...
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Effectivelly using Compile for calculate a Unitary transformation
I am new to Mathematica, and this is my first post, so if my question is not clear enough, I would be glad to read the comments and edit my question to add more information.
The problem
I need to ...
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Spectral problem for differential vector operator (calculation of EM field in a cavity)
I know that mathematica has a DEigensystem and NDEigensystem which allow one to find eigenfunction and ...
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How to verify the convexity of a function?
I have an optimization problem with the following objective function in $(x,y)$
$$
A\log \left(\sum_{i=1}^n x_i\right)+\log\left(1-\frac{f}{n}\left(\sum_{i=1}^n\frac{x_i}{y_i}\right)\right)
$$
where $...
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Efficiently calculating half of the eigenvectors of a sparse array
Eigenvectors of a sparse array
$\quad$ Problem statement
I want to calculate the eigenvectors corresponding to the negative eigenvalues of an $8L^2 \times 8 L^2$ matrix ($L \sim 30 )$. Most of the ...
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How to use NDEigenvalue to accurately estimate functional determinants?
Goal: Ultimately, I would like to find a trustworthy approximation for the ratio of the functional determinant of two differential operators using the formula
$$ \frac{\text{Det}[\hat{D}_0]}{\text{Det}...
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Finding matrix in Krylov subspace (Lanczos method)
The Lanczos method for finding the smallest eigenvalue of a hermiteian matrix $H$ is based on the construction of a vector subspace (Krylov space) where one can build a matrix $H_{Krylov}$ which is ...
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Difference in computation speed of Eigensystem not to expectation for ParallelTable and Table
I was trying to compute the time it takes for Eigensystem to evaluate while being inside a ParallelTable, as it is well-known LAPACK subroutine has an inbuilt Parallelization to it. And the difference ...
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Orthogonal matrix decomposition of symmetric matrix?
If matrix mat is symmetric, we should be able to decompose it into eigenvalue matrix matJ and orthogonal matrix ...
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What real symmetric matrices of this type can Mathematica find symbolic eigenvalues for?
I'm working on a problem calculating symbolic eigenvalues of matrices that always have a very simple form: they are real and symmetric and usually sparse. They have two distinct symbolic parameters. (...
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First few smallest eigenvalues of a large dense symmetric matrix
I construct a large (say 2000x2000) matrix M whose entries are real random variables drawn from a certain distribution. Most of these values will be nonzero, so <...
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(Possibly bugs? ) Wrong results provided by `DEigensystem`
I was trying DEigensystem with the following code:
...
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Chladni experiment verification
I refurbished my post in order to be more understandable.
After computing simulations of Chladni patterns with Mathematica (see my previous topics), I finally went to practice. I realized my own ...
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Derivative of eigenvalues
I work with 4x4 Hermitian matrices (r). I want to calculate a derivative of a function f[t,r] (ff[t_,r_]=1/2*D[f[t,r],t]), where the function f depends on the absolute value of the eigenvalues of r. ...
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Interface points of NDEigensystem
When solving an eigenvalue problem with "NDEigensystem", e.g. a 1D Eigenvalue problem with the interval composed of different materials, which should be solved by the pure numerical method such as FEM,...
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Antisymmetric Matrix Eigenvector Normalization
So, I have a complex $4n \times 4n$ antisymmetric matrix, $A$ and it has a non-degenerate spectrum. The matrix $A$ then has eigenvalues given by
$$ \beta_{1}, -\beta_{1}, \beta_{2}, -\beta_{2}, ... , ...
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1st-order linear ODE system gives inaccurate/biased solutions
Consider an ODE eigensystem
$$
t(y+\frac{1}{s})a(y)+[(q+\frac{1}{2}+\frac{s}{2}y)+s(y\partial_y+\frac{1}{2})]b(y)=\lambda a(y)\\
t(y+\frac{1}{s})b(y)+[(q+\frac{1}{2}+\frac{s}{2}y)-s(y\partial_y+\frac{...
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Z-eigenvalue problem
I am interested in solving an $N-$dimensional $Z-$eigenvalue problem, which schematically takes the following form
$$
\sum_{b=1,c=1}^{N}T_{abc} X_b X_c=\lambda\,X_a\quad \text{with}\quad \sum_{a=1}^...
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How to improve the efficiency of NIntegrate on complicated functions?
When solving a problem by Matlab and Mathematica both, I find the speed of NIntegrate are far less than dblquad when calling ...
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OpenCL How to read and write to VRAM array
I am quite a newbie to OpenCLLink. My program needs to perform some heavy computations on GPU including solving the eigenvector problems. Since I didn't find a straightforward implementation of this ...
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What is the fastest way to get the lowest eigenvalue and corresponding eigenstate in Mathematica?
I tried to use the Arnoldi method to get the smallest eigenvalue and corresponding eigenstate for large matrix. However, Arnoldi did not give me the desired result:
...
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Determine a negative semidefinite 5*5 matrix
I'd like to find the real parameters $\left\{p,\alpha ,a_{24},c_3,c_7,c_9,c_{10}\right\}$ in $M$, which is a $5\times 5$ real symmetric matrix, such that $M$ is negative semidefinite.
My code for $M$:
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Parallel Matrix Manipulation: find eigenvalues and construct list
I'm having some trouble with the Parallel commands in Mathematica 12.1:
I need to construct a table where its entries are {M, Eigenvalues of X[M]}, where X is a square matrix of dimension N with N big ...
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How can I improve the speed of computing of numerical integration and eigenvalue?
First, let me explain what I am calculating.
I have the following 2D-equation:
$$ w(x,y) = \sum_{m}^{M}\sum_{n}^{N}A_{mn}\sin\left(\frac{2m\pi x}{a}\right)\sin\left(\frac{2n\pi y}{b}\right) $$
...
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Computing the first eigenfunction of the p-Laplacian in a real interval
How can I numerically compute the first (non-negative) eigenfunction $u$ of the $p$-Laplacian ($p>1$) in a bounded interval $(-a,a) \subset \mathbb R$ (up to positive multiplicative constant)?
$$-\...
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What kind of performance should I expect out of Eigensystem using FEAST?
I'm numerically solving a time-independent Schrödinger equation using Eigensystem's FEAST method. It takes a lot longer than I ...
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Block diagonalizing a complex anti-symmetric matrix
I am going to evaluate the block diagonal form of few skew-matrices.
When matrix elements are real I can simply follow the approach suggested in this thread which I have implemented that as
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Boundary value problem, multiple dimensional shooting, coupled eigenvalue problem
Following the one dimensional boundary value problem here, I would like to understand the easiest way to solve a BVP for a coupled system. In the 1D case, BVP can be converted to an initial value ...
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DEigenvalues with Robin B.C. sign problem
To find eigenvalues for $y''=\lambda y$ with robin boundary conditions on one end, and Dirichlet on the other end, I am getting correct value when robin B.C. on the right side, but when I flip things, ...
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Eigenvector matrix of a real positive matrix to be from $SO(n)$. How?
Suppose I have a positive real matrix of dimension $n$ and as such there exists a diagonalizing (rotation) matrix that belongs to $SO(n)$. How do I force Eigensystem...
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Wrong eigenvalues for 2D QHO using DEigensystem[]
I try to solve 2D Quantum Harmonic Oscillator using DEigensystem[] in Mathematica 13.0. Here is my code:
...
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Complex eigenvalue of a symmetric matrix
I am working on a eigenvalue problem. I am using Eigensystem
However, I am facing issues when I change the input parameters:
I get complex eigenvalues. I couldn't understand the reason although my ...
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Finding the (symbolic) eigenvalues of a matrix with some assumption
I have the following 3$\times$3 symmetric matrix with symbolic entries.
Meff =
\begin{array}{ccc}
2 \beta \text{c12}^2 \text{c13}^2 \text{$\eta $11} \text{m1}+\alpha \text{c13}^2 \text{D31} \text{...
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Why does the minimum eigenvalue sharply decrease when the number of basis functions increases from 37 to 38 and more?
I have the following Hamiltonian: H = -1/2 * Laplacian -1/r - 2 * r/5 * (Exp[-r * 1.6] + Exp[-r * 3.1])
I'm trying to find the minimum eigenvalue of this Hamiltonian using the matrix method. As basis ...
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Calculating eigenvalues and solving inequalities with parameters
I would like to calculate b in terms of c and a, in order to satisfy the following conditions:
...
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Why does Sparsearray diagonalization by Krylov technique become slower for very large sizes?
Here is an example sparse matrix, taken from example Hamiltonian, which was a question I asked before in this forum.
...
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Intel MKL ERROR: Parameter 2 was incorrect on entry to ZGEHD2
I have some code which is diagonalising very large random matrices, and storing certain statistical properties of their eigenvalues and eigenvectors.
The code runs fine on my computer. When running it ...
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Eigenvalues broken for nonsymmetric matrices
It looks that eigensystem calculation can easily go broken with real nonsymmetric/general complex matrices. Here are two examples with the possibly complex spectra plotted in the complex plane. I'm ...
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NDEigensystem solutions depend on how many solutions I ask for?
Background
I am using NDEigensystem to solve the following eigenvalue problem:
$$ \left( \begin{matrix} m&-i\partial_x \\ -i\partial_x & -m\end{matrix}\right) \left( \begin{matrix} u_u(x) \\ ...
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Strange behaviour of eigensystem
I encounter with really strange behavior in calculating the eigensystem of a hermitian matrix.
I upload the matrix here. After loading it you can construct the main matrix as follows
...
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