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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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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9
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1
answer
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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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2
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Solving eigenvalue BVP with an interface
I have a boundary-value problem, that is defined over two adjacent regions with an interface in the middle, that contains an eigenvalue $\lambda$. The boundary conditions and the equations are ...
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4
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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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7
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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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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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1
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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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2
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Efficiently find all values of parameter such that any of the eigenvalues of a matrix is equal to 1
I wish to find all values for a parameter such that my matrix has an eigenvalue of 1.
Here is an example 16-by-16 matrix with elements depending on the parameter x ...
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3
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1
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The algebraic solution and numerical solutions for eigenvectors are different. Why?
I find that the algebraic solution for the Eigenvectors of a 3x3 matrix is not correct when compared to the numerical solution. I don't see why ?
...
3
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1
answer
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Unbounded solution at boundaries with few combination of values in this BVP solution
I had asked a question here regarding solution to a BVP problem. bbgodfrey provided an excellent answer using the method of integrated least squares.
However, for a few specific set of values of ...
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3
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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 ...
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9
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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.
$$
For this particular ...
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votes
2
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Calculating smallest eigenvalues by real part using Arnoldi method
Bug introduced in 8.0 or earlier and persisting through 11.3
According to the documentation,
Eigenvalues[m,k] gives the first ...
- 1,766
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votes
2
answers
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Wrong eigenvalues from a sparse matrix
Bug introduced after 5.0, in or before 8.0 and persisting through 13.3.
I notice in the following example that wrong smallest 2 eigenvalues are resulted if calculating from a sparse matrix. But it ...
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9
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2
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How do you find the eigenvalues of a PDE (Dynamic Euler-Bernoulli beam)?
I am continuing to work on the vibration of a beam modeled by the Euler-Bernoulli equation. I have had some good answers to simulating the motion which may be found here. Now I wish to calculate the ...
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8
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2
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Solving a biharmonic eigenvalue Problem
I am interested in solving the following biharmonic eigenvalue problem.
$$\begin{array}{cccc}
& \Delta ^2 \Psi (x,y) = \lambda \Psi (x,y), & - \frac{\pi}{2} \le x \le \frac{\pi}{2} & -\...
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Eigenvalue dependent boundary conditions- mathematica
I am dealing with an eigenvalue problem whose boundary conditions are also eigenvalue dependent.
Could anyone please comment whether Mathematica can numerically solve such a problem? For boundary ...
3
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1
answer
407
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Discontinuities in eigenvalues plotting with Plot3D
I have a 6x6 matrix, depending on 2 variables kx and ky and defined as the sum of the two following matrices (...
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3
votes
2
answers
552
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Why ODE's naive finite difference matrix works well for different boundary conditions
We know finite difference method (FDM) can replace $y''(x)$ as $\frac{1}{h^2}[y(x+h)+y(x-h)-2y(x)]$ or so. The naive way to write down the matrix of the differential operator is like the following, ...
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2
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1
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Small positive eigenvalues found for a negative definite matrix
I have encountered a problem with Mathematica that I really don't know how to solve. I'm sorry if I am going to be very verbose but I need it to explain the problem properly (and also make the code I'...
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1
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Eigenvalue problem
my question is about solving an eigenvalue problem of the Helmholtz equation using sinc approximation $\nabla^2E + V (x) = \lambda E$ and $V(x)= X^2 / 2$
I have a problem in calculating the ...
1
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0
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Finding eigenvalues of a differential operator
I am trying to get the eigenvalues of the following differential operator
$$L\psi(r) = -f \partial_r (f \partial_r \psi(r)) + V \psi(r)$$
which must satisfy (obviously)
$$L \psi(r) = \omega^2 \psi(...
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votes
1
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Filter out eigenvalues in NDEigenfunction [duplicate]
I am going to solve TISE for logarithmic potential in two dimensions. For bound state solution, the energy eigenvalues are to come negative. This is my code:
...
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2
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Library for FEAST method is missing
Mathematica (V 12.3.1, Native Mac M1 version) is not letting me use the FEAST method for solving eigenvalue problems. For example,
...
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6
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Solving this challenging ODE
Consider the ODE:
$$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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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 ...
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votes
2
answers
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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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10
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2
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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 ...
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10
votes
1
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290
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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 ...
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9
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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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7
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5
answers
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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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2
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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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2
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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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votes
2
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Noise in Eigenvalues plot
I am trying to Plot Eigenvalues of a Hamiltonian, but I am getting noisy plot, which is incorrect. Here is the code.
...
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1
answer
886
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Finding the eigenvalues (diagonalizing) of a block-diagonal matrix
I have a large $2^N \times 2^N$ matrix. It is the exact Hamiltonian of a spin chain model which I have generated with code I wrote in Fortran. The code block diagonalizes the Hamiltonian into constant ...
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Using/ Extracting data from Interpolating Functions returned from NDEigensystem
I am relatively new to Mathematica and have been trying to use the NDEigensystem command to work with some quantum systems. I am able to get the accurate energy eigenvalues but am having problems with ...
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2
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371
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5
votes
2
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NDEigensystem and radial function equation for Hydrogen atom
I'm trying to numerically solve the radial equation for the 3D hydrogen atom problem, i.e., to find $R(r)$ which satisfies:
$$
-\frac{\hbar^2}{2m}\left[\frac{1}{r}\frac{d}{dr}\left(r^2\frac{dR(r)}{dr}\...
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4
votes
2
answers
618
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Using NDEigensystem to solve coupled eigenvalue problem
I want to find the Eigenvalues and Eigenfunctions for the following Eigenvalue problem
I tried to solve this numerically
...
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4
votes
2
answers
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How to generate the 8^th order symmetric binary matrices whose sum of absolute eigenvalues is 8?
It is needed to generate all 8th order(8 by 8) symmetric binary matrices(of 0's and 1's) such that the sum of the absolute eigenvalues is 8. Listing all the 8th order symmetric binary matrices and ...
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4
votes
2
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How to return multiplicity of each eigenvalue?
I could not find the information so maybe someone know if it possible.
I have a matrix which has several degenerated eigenvalues and I would like Mathematica to return the multiplicity of each ...
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votes
2
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How to collect eigenvectors corresponding to only positive eigenvalues?
Let us consider a matrix of order $n \times n$ with $n/2$ positive and $n/2$ negative eigenvalues. How to collect $n/2$ eigenvectors corresponding to positive eigenvalues in a matrix of order $n \...
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4
votes
1
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Numerically computing the eigenvalues of an infinite-dimensional tridiagonal matrix
I have one infinite dimensional tridiagonal matrix whose eigenvalues I have to compute. How can that be done numerically using Mathematica?
Let me expose the concrete case I want to do it. I shall ...
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3
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1
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Evaluating Hough functions by using NDEigensystem on the Laplace tidal equation
Currently I am looking into the use of Mathematica to solve the classical tidal equation of M. Laplace:
$$\mathcal{F}\Theta+\gamma\Theta=0$$
whose eigenfunctions $\Theta$ are the Hough functions. ...
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3
votes
0
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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,...
3
votes
3
answers
609
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Solving a Sturm-Liouville problem
Ello,
I would like to reproduce the analytical solution of the following eigenvalue problem, or at the least confirm them numerically (especially the eigenvalues):
$$ - \frac{1}{2} y^{\prime \prime} ...
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3
votes
2
answers
186
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3
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2
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267
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Checking NDEigensystem Results
I'm looking to verify the output of a call to NDEigensystem. I'm doing this by plotting the operator acting on the Interpolating Function outputs versus the ...
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2
votes
1
answer
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How can I tell if a matrix is ill-conditioned or Singular by using the Eigensystem function(or LUDecomposition)?
I'm using the Eigensystem function, and I'm trying figure out whether or not it is singular or ill-conditioned. I'm using the function as so:
...
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