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.
454
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Eigenvalues broken in Version 12.0
Bug introduced in 12.0 and fixed in 12.1
The following code calculates the eigenvalues of a certain complex matrix, which come in pairs of opposite complex numbers. Therefore one can check whether ...
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1
answer
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Complex eigenvalues from a sparse Hermitian matrix
Bug introduced in 9.0 or earlier and persisting through 13.2.0.
I notice in the following example that wrong complex eigenvalues are resulted if calculating from a Hermitian sparse matrix, which ...
- 4,594
28
votes
2
answers
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Wrong eigenvalues from a sparse matrix
Bug introduced after 5.0, in or before 8.0 and persisting through 12.2.
I notice in the following example that wrong smallest 2 eigenvalues are resulted if calculating from a sparse matrix. But it ...
- 4,594
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8
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How to check if a vector is an eigenvector of a matrix using mathematica?
Here is a vector
$$\begin{pmatrix}i\\7i\\-2\end{pmatrix}$$
Here is a matrix
$$\begin{pmatrix}2& i&0\\-i&1&1\\0 &1&0\end{pmatrix}$$
Is there a simple way to determine ...
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20
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Tracking Eigenvalues Through a Crossing
Suppose I have a matrix which depends on some parameter. I want to compute the eigenvalues as a function of this parameter, and then plot them. For example, I may have a matrix representing the ...
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votes
2
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Library for FEAST method is missing
Mathematica (V 12.3.1, Native Mac M1 version) is not letting me use the FEAST method for solving eigenvalue problems. For example,
...
- 331
14
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1
answer
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Specifying initial vector for finding Eigenvectors using Arnoldi method
I am trying to speedup the calculation of eigenvalues, given that I have good guesses for the eigenvectors. From what I know of Arnoldi/Lanczos, my good guesses should be helpful. Unfortunately, I am ...
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Solving this challenging ODE
Consider the ODE:
$$w^{(4)}(x) + (L-x)w''(x) - w'(x) = 0 $$
with some of the following boundary conditions:
free: $w'' = 0$, $w'''=0$,
clamped: $w = 0$, $w'=0$,
pivot: $w = 0$, $w''=0$.
Two such ...
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13
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3
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An ODE system easily polluted with spurious eigenvalues
I tried solving the eigenvalue problem of a 1st-order ODE system (see the code below)
with NDEigenvalue.
(One option I found in it seems to be ...
- 4,594
13
votes
3
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Routh-Hurwitz criterion not giving correct answer when done manually?
Consider the system:
\begin{align}
\frac{dS}{dt} &= \nu N -\frac{\beta S I}{N} + \xi R - \nu S\\
\frac{dE}{dt} &= \frac{\beta S I}{N}- \sigma E -\nu E \\[2ex]
...
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0
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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 ...
- 131
12
votes
3
answers
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How to set interface conditions for optical waveguide in NDEigensystem?
I have been working on waveguide mode analysis using FEM in Mathematica for a week, but I haven't succeeded until now.
The optical fiber-like waveguide is featured with different refractive index in ...
- 4,785
12
votes
3
answers
297
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Efficient eigendecomposition of DPR1 matrices
I'm finding that the following bit is the bottleneck in my code
...
- 7,558
11
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2
answers
912
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Making an interactive visualization of the eigenvectors of two-dimensional matrices
I've recently stumbled upon this very nice interactive visualization of eigenvectors of two-dimensional matrices, and how powers $A^k$ act on various vectors.
How can this sort of visualization be ...
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1
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Mathematica won't give eigenvectors but Wolfram Alpha will? What am I doing wrong?
If I ask Mathematica to find the eigenvectors and eigenvalues of the matrix:
...
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10
votes
5
answers
740
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NDEigenvalues vs. FindRoot for finding the eigenvalues of Airy equation?
Say I am trying to find the first 5 eigenvalues of the differential equation $f''(x)=\lambda x f(x)$, on the interval [-1,0], with boundary conditions $f(-1)=f(0)=0$.
I will try to do this 3 ways, ...
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Nonlinear ODE eigenvalue problem
How does one find eigenvalues $\lambda$ of the following problem?
$$ \frac{\mathrm{d}^2 u}{\mathrm{d}x^2} = \lambda \left( -u + u^2 \right),$$
$$ u(0) = u(1) = 0. $$
Can this be tackled by ...
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2
answers
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NDEigensystem cannot solve numerically the 3D Coulomb problem, while DSolve returns the right answer
After having derived by hand the eigenvalues and eigenfunctions for the 3D and 2D hydrogen atom, I want to solve the systems numerically using Mathematica. I need to do this because my next step is to ...
- 633
10
votes
1
answer
279
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Inconsistent behaviour between Active and Inactive forms of PDEs (finite element method)
Bug introduced in 13.0 or earlier and fixed in 13.1.0
I have been using the finite element tools in Mathematica version 13.0 to solve eigenvalue problems in physics (Schrödinger equation) using ...
- 145
9
votes
1
answer
970
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Finding Eigenvalues for a boundary value problem
I have a 10x10 linearized BVP which I can write as $$\mathbf{y}'(x) = \mathbf{A}(\omega) \mathbf{y}(x)$$ subject to boundary conditions $$\mathbf{B} \cdot \mathbf{y} = \mathbf{0}, \quad x=0 \\ \mathbf{...
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votes
1
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Eigendecomposition of a matrix with a variable
I have an issue with a decomposition of a matrix $B$ that is positive semidefinite and that depends on a parameter $x$. Writing $\lambda_i\geq0$ the eigenvalues and $\psi_i$ the corresponding ...
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2
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How do you find the eigenvalues of a PDE (Dynamic Euler-Bernoulli beam)?
I am continuing to work on the vibration of a beam modeled by the Euler-Bernoulli equation. I have had some good answers to simulating the motion which may be found here. Now I wish to calculate the ...
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Analytic solution to Orr-Sommerfeld-Squire equations for a special case
Hello everybody in Mathematica SE. Although my question is related to flow stability analysis, this should be a general application of MMA to solve a system of ODEs. Thank you for your suggestion!
...
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9
votes
1
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How to solve this 2nd-order ODE with singularity?
I tried solving the eigenvalue problem of a 2nd-order ODE $$[b^2(k-2)^2y^2-2b(k-2)(1+2ky)+4k^2+b^2(k-2)3y]f(y) \\- 3b(3by-2)f'(y)\\-(3by-2)^2f''(y)=\lambda f(y)$$ with ...
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9
votes
1
answer
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NDEigenvalues complains not Hermitian with large dimension differential operator
The following snippet calculates eigenvalues and eigenfunctions of a null operator (just as an example):
...
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9
votes
1
answer
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Gaining precision/accuracy with NDEigenvalues
See further down for an important note
Background
I study (one component of) the semi-classical Pauli operator,
$$
P_h=-h^2\Delta+ih(-y,x)\cdot\nabla+\frac{x^2+y^2}{4}-h.
$$
For this particular ...
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2
answers
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Lowest Magnitude Eigenvalues of Large Sparse Matrices
I am trying to find the first three lowest eigenvalues of large sparse matrices of size range $10^3 - 10^5$. The matrices depend on some parameter $x$, so I first construct the matrices and then use ...
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Solving the Schrödinger Equation by exact diagonalization
I am solving the Schrödinger equation via finite difference, via the substitution
where we are assuming $V_1 = V_N = \infty$. I solved this using Mathematica for the case that $V(x) = 0$ and get the ...
- 2,079
8
votes
2
answers
466
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Solving a biharmonic eigenvalue Problem
I am interested in solving the following biharmonic eigenvalue problem.
$$\begin{array}{cccc}
& \Delta ^2 \Psi (x,y) = \lambda \Psi (x,y), & - \frac{\pi}{2} \le x \le \frac{\pi}{2} & -\...
- 1,697
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1
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Fast method to calculating signature of a matrix
For calculation of the topological properties of a hamiltonian, sometimes we need the signature of that matrix. This means we only need number of positive eigenvalues. One simple way is to first ...
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2
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Calculating smallest eigenvalues by real part using Arnoldi method
Bug introduced in 8.0 or earlier and persisting through 11.3
According to the documentation,
Eigenvalues[m,k] gives the first ...
- 1,746
8
votes
2
answers
271
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Why Mathematica gives wrong eigenvalues for this equation?
Here is an eigenvalue problem in cylindrical coordinate: $$\mu(r)\frac{\partial}{\partial r} \left( \frac{1}{\mu(r)}\frac{1}{r}\frac{\partial (ru)}{\partial r} \right)=-p^2u$$ where p is the required ...
8
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1
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How to solve this 1st-order linear ODE system with a few discrete eigenvalues?
I am trying to solve the eigensystem of a 1st-order linear ODE system in the region $(-\infty,\infty)$ and with Dirichlet boundary condition at the infinities
$$
-i\partial_xu(x)+f^*(x)v(x)=\lambda u(...
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0
answers
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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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0
answers
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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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7
votes
5
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Spectral problem for differential operator
I want to find numerically eigenvalue and eigenfunctions some nontrivial differential operator. But I can not find, how to do it in Mathematica.
For the sake of simplicity let us discuss the trivial ...
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votes
1
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685
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Is there a bug in Eigensystem[]?
Does Eigensystem[] produce incorrect output for symmetric matrices with integer components?
The following eigensystem decomposition of a 12x12 matrix and its ...
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7
votes
1
answer
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Sensitivity analysis of parameter on eigenvalues of predator-prey model
I am trying to do a sensitivity analysis of the parameter g on the eigenvalues of this simple predator-prey Lotka Volterra model. I know that this code is entirely ...
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7
votes
1
answer
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Which methods are available for NDEigensystem?
I am trying to find out what methods/options are available for NDEigensystem and descriptions of their use. Perusing the help, online Q&As, Mathematica's in-...
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votes
1
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785
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NDSolve eigenvalue problem of bound state
I am trying to solve this eigenvalue problem:
\begin{align}
\mu \Psi(r) & =
-\frac{1}{2}\left ( \Psi^{\prime \prime}(r) + \frac{2}{r} \Psi' (r)\right )
-4\pi \Psi(r) \int _0^\infty dr' r'^2 \frac{...
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votes
1
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NDEigensystem for structural vibration
Having installed version 11 I thought I would check an old Stack Exchange vibration problem using NDEigensytem. The old problem was Test a wooden board's vibration mode and I think this was before ...
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2
answers
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Unable to evaluate Eigenvalues and Eigenvectors for a matrix
I have the following 3X3 matrix M and I wish to find its eigenvectors and eigenvalues
...
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votes
0
answers
246
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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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votes
0
answers
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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, ...
- 1,069
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votes
0
answers
155
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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 ...
- 355
6
votes
3
answers
414
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Linear ODEs with NDSolve
I am trying to solve some first order linear odes. My code is below:
...
6
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2
answers
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How to normalize a list of eigenvectors?
Here is a simple eigenvector problem solution
m = {{2, Sqrt[15]}, {Sqrt[15], 4}};
v = Eigenvectors[m]
However, the list of vectors v is not normalized. The ...
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6
votes
4
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FindInstance won't compute this simple expression
I want to find instances where this standard 3x3 symmetric matrix has only positive eigenvalues.
So I run ...
6
votes
3
answers
282
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