All Questions
1,732 questions
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What is the theoretical interest of finding closed-form solutions of infinite series?
I was reading this when I came across Gourevitch's conjecture.
My understanding is that solutions to these series are of practical interest. If one encounters such a series, being able to solve it ...
- 217
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4
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2k
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When Have Numerology and Computational Experimentation Been Successful?
When has numerology been successfully used in math and science? The Monstrous Moonshine conjecture led to a Fields medal for Borcherds. Balmer's formula for hydrogen spectra led to the Bohr model of ...
10
votes
5
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8k
views
Shifted QR algorithm—why does the shift help?
I read that a way to speed up the convergence rate of the QR algorithm is to shift the target
matrix. It is not so clear to me why this helps. The convergence rate depends on the
minimum gap between ...
- 353
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votes
2
answers
3k
views
How do you tell if a system of linear inequalities has a solution?
A naive solution would be to optimize a dummy variable via linear programming and see if a result is returned. I imagine there must be a more direct way.
- 693
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4
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2k
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How to solve Ax=b incrementally ?
Hi, everyone.
What I am struggling is the following problem. I have a linear matrix equation $Ax=b$, where $A$ is a known $n \times n$ large sparse real matrix, $x$ and $b$ are known $n \times 1$ ...
- 101
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3
answers
12k
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What's an efficient way to calculate covariance for a large data set?
What is the best algorithm for computing covariance that would be accurate for a large number of values like 100,000 or more?
- 3,613
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4
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2k
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Reference request for conceptual numerical analysis
I am interested in clean algorithms for approximating solutions and so I am interested in numerical analysis, but most of the books I have seen get bogged down in error analysis or they spend a lot of ...
- 4,351
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2
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2k
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Is there a standard name for (non-square) matrices with orthonormal columns?
One encounters often in numerics non-square matrices with orthonormal columns, i.e., $U\in\mathbb{R}^{m\times n}$, with $m > n$, such that $U^TU=I$ (but, clearly, $UU^T \neq I$).
Is there a name ...
- 20.2k
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1
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449
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How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ efficiently
Let $M$ be a real symmetric integer valued positive definite matrix with $\det(M) \geq 1$. I would like write code to compute
$$S_M= \sum_{x \in \mathbb{Z}^n} e^{-x^TMx}.$$
One option is to simply ...
- 3,377
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2
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4k
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How to implement Horner’s scheme for multivariate polynomials?
Background
I need to solve polynomials in multiple variables using Horner's scheme in Fortran90/95. The main reason for doing this is the increased efficiency and accuracy that occurs when using ...
- 203
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1
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1k
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Use of games to approximate solutions to Partial Differential Equations
Hi there,
Hopefully the mathematics community can help me out this one, I'm currently studying my senior capstone at my college, and decided to do some research on a chapter in Stanley Farlow's book "...
- 203
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3
answers
1k
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Geodesic triangles in finite element method
I've been working on a new method for 2-dimensional finite element method (FEM) on Riemannian manifolds that involves using geodesic triangles instead of approximating them in an embedded form using &...
- 500
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2
answers
1k
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Machin-like formulas for logarithms
I found this math puzzle blog post
http://fredrikj.net/blog/2013/03/machin-like-formulas-for-logarithms/
which I'm reposting here with permission. I'm setting this to community wiki to minimize the ...
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3
answers
6k
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Solving a system of linear inequalities -- what is the dimension of the solution set?
It is well known how to solve a system of linear equations $A{\bf x} = {\bf b}$, but how do we solve a system of linear inequalities $A{\bf x} \leq {\bf b}$?
For the applications I have in mind the ...
- 7,857
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3
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1k
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Rapid evaluation of multivariate normal integral
I'm implementing a model that requires me to numerically evaluate a multivariate normal integral of the following form
$$\int_{-\infty}^\infty \phi(z)\displaystyle\prod_{i=1}^N \Phi(a_iz+b_i) \, dz,$...
- 101
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1
answer
595
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Fast checking that overdetermined polynomial system does not have a solution
As a result of some inductive procedure for each $n$ I have a system of about $n^2$ polynomial equations with $n$ variables with integer coefficients, which can be precisely computed. As the system is ...
- 728
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1
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2k
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Area of filled Julia sets
The recent question Area of the boundary of the Mandelbrot set ? prompted me to ask this question.
There has been some work on estimates for the area of the Mandelbrot set, e.g., a paper by John H. ...
- 3,022
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1
answer
2k
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Sum of difference moduli vs. sum of modulus differences
This is a failed attempt of mine at creating a contest problem; the failure is in the fact that I wasn't able to solve it myself.
Let $x_1$, $x_2$, ..., $x_n$ be $n$ reals. For any integer $k$, ...
- 33.8k
10
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1
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411
views
Network flows with capacities on pairs of edges
Take a standard network flow problem: a directed graph with nonnegative capacities on each edge, a source $s$, a sink $t$. We all know how to find the maximum flow from $s$ to $t$.
Now add edge-pair ...
- 37.7k
10
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1
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297
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Confusion with practically implementing rational approximations
Writing a program visualizing Ford circles I've encountered a seemingly purely programmatic puzzle but then gradually realized there are some mathematical aspects of it which I don't understand. Let ...
- 18.4k
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1
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4k
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Special considerations when using the Woodbury matrix identity numerically
Are there any special considerations when using the Woodbury matrix identity numerically? What is the best metric for numerical stability in this case? Can anyone point me to a good reference?
The ...
- 211
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0
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722
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Fractional Matching version of Hall's Marriage theorem
Let $G=(S,T,E)$ be a bipartite graph, $|S|=|T|$. Then the following are equivalent:
1) there exist a perfect matching in $G$;
2) there exist non-negative weights on edges such that the sum of ...
- 109k
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439
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Evaluating Shintani cone zeta functions
Hi everyone
I am trying the evaluate sums of the form
$$ \sum_{n_1>0,n_2>0,\ldots,n_m>0} \frac{1}{\big((a_{1,1}n_1 +\ldots +a_{1,m}n_m)^k \ldots (a_{m,1}n_1+ \ldots +a_{m,m}n_m)^k\big)}$$
...
- 265
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3
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6k
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Eigenvalues of non-symmetric matrix and its transpose
What more can be said about the eigenvalues (especially the spectrum) of the $N \times N$ matrix ${\bf M} = {\bf A} + {\bf A}^T$ in terms of $\bf A$ if $\bf A$ is not symmetric and its eigenvalues are ...
- 385
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2
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775
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Is there a systematic theory for Gibbs measures (better if on Hilbert spaces)?
During these first months in my PhD, I realized how my computational problems can be drastically reduced to one single problem:
Find an efficient way to sample from a Gibbs measure.
Let me ...
- 211
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2
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1k
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Why are there so few zero-dimensional polynomial system solvers and is this because there is no real market for them?
My questions involve the quotes below from wikipedia regarding solving polynomial systems, which given the size of the market for Big Data & Predictive Analysis applications I find puzzling:
"...
- 159
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1
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6k
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Proving that a binary matrix is totally unimodular
I'm working on a set of problems for which I can formulate binary integer programs. When I solve the linear relaxations of these problems, I always get integer solutions. I would like to prove that ...
- 213
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3k
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Inverse of a totally unimodular matrix
A unimodular matrix $M$ is a square integer matrix having determinant $+1$ or $−1$.
A totally unimodular matrix (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. A ...
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2
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843
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How did they come up with the MRRW bound?
Among the good asymptotic bounds in coding theory in the MRRW bound. It is obtained by using the linear programming problem of Delsarte's and providing a solution. The LP problem is
Suppose $C \...
- 885
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3
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657
views
Degree necessary of a polynomial?
Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...
- 13.9k
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4
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529
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Software tools for medium-scale systems of polynomial equations
I am attempting to find all real solutions of a system of 12 polynomial equations in 12 unknowns. The equations each have total degree 6 and contain up to 1700 terms. I am only interested in real ...
- 561
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3
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871
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Multiprecision numerical evaluation of integral: Sage vs. PARI/GP vs. mpmath
I am trying to compute thousands of integrals of the below type, that comes up in a conformal mapping problem, to as many accurate digits as possible (preferably 50+):
$$
\int_{-1}^1\textrm{d}t \frac{...
- 193
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1
answer
531
views
Approximating power series coefficients --- Why does a clearly illegitimate method (sometimes) work so well?
For reasons that don't matter here,
I want to estimate the power series coefficients
$t_{ij}$ for the rational function
$$T(x,y)= {(1+x)(1+y)\over 1- x y(2+x+y+x y)}=\sum_{i,j} t_{ij}x^iy^j$$
Using a ...
- 23k
9
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2
answers
9k
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Conditions for convergence of Euler's method
It is known that a sufficient and necessary condition for
$$\dot y(t) = f(y(t), t), \quad t > 0, \quad y(0) = y_0$$
to have a unique solution is $f$ Lipschitz in $y$ and continuous in $t$. However, ...
- 773
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1
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295
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Definition of packing property
Definition 1:
A clutter $C$ is said to have the packing property if $C$ and all of its minors satisfy the König property.
where,
vertex cover of $C$ is a set of vertices that have non-empty ...
- 381
9
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2
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568
views
Some Mathematical Questions on Gravitational Waves and Numerical Relativity
Due to the recent spate of detections of gravitational waves by LIGO, my amateurish interest in the mathematics of general relativity has been revived.
The wave-forms of the detected gravitational ...
- 1,396
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2
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3k
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An optimization problem for points on the sphere (master's dissertation)
First, by means of a disclaimer, some background. I am entering the fourth and final year of an undergraduate master's degree in maths, and a quarter of the maximum credit for this year will be for a ...
- 3,612
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310
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An extrapolation method
I've stumbled upon a method of extrapolation that I haven't seen before.
We are trying to approximate $f(0)$ for a certain function $f$, which we have only measured
at points $x_0, \ldots, x_N$ in ...
- 54.2k
9
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2
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929
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Numerical approximation to the Wasserstein metric?
Are there numerical methods for approximating/calculating the Wasserstein metric in particular cases?
Suppose that $f$ and $g$ are two density functions with the same support. How can I calculate the ...
- 91
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2
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818
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Approximation theory on the disc
Chebyshev theory provides a very effective method for approximating continuous real valued functions on the unit interval. Is there something similar for continuous real valued functions on the ...
- 56.9k
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1
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693
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What happens to continuous spectrum upon discretization?
Excuse me for a bit of an vague question, but I haven't been able to find a definite answer for this for quite some time. My question is regarding (mostly non-normal )linear operators and their ...
- 91
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1
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385
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Adding a multiple of the Identity to a LU factorized matrix
Suppose a square, dense, symmetric matrix $A$ has been factorized into $L$ and $U$ components by performing a LU decomposition. Now let $B = A+\lambda I$. Is there any way to efficiently compute the ...
- 339
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816
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A Closed Form for the Diagonal Matrix Nearest an Arbitrary Square Matrix
If I have a square matrix in $\mathbf{A} \in \mathbb{R}^{n \times n}$, I want to find another diagonal matrix $\mathbf{D} \in \mathbb{R}^{n \times n}$ that minimizes the residual $ \min_\mathbf{D} || \...
- 193
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1
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Uniform sampling from general simplex with a twist
This is part of a question I had asked elsewhere, and then some of the links redirected me to CS stack exchange.
Given $0\leq a_1\leq\dots\leq a_D\leq1$ (all strictly positive), I want to draw points ...
- 221
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561
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Padé approximations of $e$
The following question came up in the analysis of some algorithm.
Let $R_{s,t}(z)$ be the Padé approximants of $e^z$, and define $r_{s,t} = R_{s,t}(1)$. Using the explicit expression for the error ...
- 1,906
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3
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2k
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Are there ill-conditioned problems in infinite precision arithmetric?
It is hoped that in the future with the advent of quantum computing that fundamental operations on a computer will have arbitrarily high precision. Moreover, that even with such high precision, ...
- 3,217
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2
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1k
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Mathematics of sustainable development and energy sobriety in the classroom
Faculty members are encouraged to highlight the connection between the courses we teach and climate change, and raise awareness of the issue in our lectures, across subjects in my university. I am ...
8
votes
5
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14k
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Eigenvalues of A+B where A is symmetric positive definite and B is diagonal
If I have a symmetric positive definite matrix A and a diagonal matrix B, and I know the eigenvalues of both A and B (by iterative numerical computation in A's case and trivially for B), is there any ...
- 135
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2
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1k
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Is quadrature still considered part of numerical analysis?
This question may admittedly sound strange, but having received several desk-rejects (all of them being based on being "out of scope" for the journal in question) from numerical analysis ...
- 3,065
8
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2
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583
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Efficiently computing a few localized eigenvectors
Let $H = \triangle + V(x) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. I am interested in domain decomposition for an eigenproblem involving $H$.
The lowest 1000 eigenfunctions of $H$, $ \psi_i $, can ...
- 817