# Questions tagged [basis-set]

A linearly independent set of elements of a vector space such that any element in that vector space can be expressed as a linear combination of the elements of the basis.

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### What is counterpoise correction?

What is counterpoise correction exactly ? Can you explain when it is needed and why ?
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### Which libraries have good implementations of Basis splines?

I'm looking to use the finite element method with B-splines as my function basis. Which C/C++ libraries have good B-spline support? Specifically, I'm looking for an implementation of a stable ...
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### How to do FEM in sector elements?

Suppose we have an element in the polar coordinate as $0\leq r\leq 1$, $0\leq\theta\leq \alpha$. What are the correct basis functions in this element? For this kind of mesh, there are a lot of ...
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### Multi-matrix orthogonal basis problem

Suppose we are given a set of symmetric, positive definite matrices $A_1,A_2,\ldots,A_k\in\mathbb{R}^{n\times n}$. Is there any numerical method or reduction to a known problem (e.g. eigenvalue ...
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### Orthonormalized Bernstein polynomials using Gram-Schmidt

I was wondering, before trying to do that myself, has anyone attempted to do orthonormalization of Bernstein polynomials using Gram-Schmidt? I discussed this with several people and have been told ...
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### Definition of Lagrange nodes in Gmsh

When gmsh uses higher-order tetrahedral elements, there is an underlying Lagrange basis used to specify the map from reference space to the element. I'm trying to load a gmsh mesh of 3rd degree ...
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### Efficient computation of the extension of a linear basis to completion when the basis is almost complete (ideally using LAPACK routines)

I have a $p \times n$ matrix $B$ (where $n < p$) with orthonormal columns and would like to find a numerically efficient way to extend this matrix to get a complete $p$-dimensional orthonormal ...
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### Using SVD to biorthogonalize left and right eigenvectors?

I have a set of left and right eigenvectors from an nonsymmetric eigenproblem, and I'd like to biorthogonalize them. I tried Gram-Schmidt, but this fails for most cases. I then read that the SVD is ...
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### Computing size of N-Dimensional Polynomial Basis and Efficient Representation of Basis

A problem I have had on my mind recently has been a compact way to compute the size of an $N$-Dimensional Polynomial basis of some order $p$, where a linear basis is $p=1$. I have attempted searching ...
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### How should I choose the knot sequence when using B-splines as a basis for solving a PDE?

I'm looking to solve the Schrödinger equation with a basis made of a tensor product of basis splines. A number of papers describe calculations made with a program designed this way, but they never ...
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### Differences between Gaussian and Slater functions on the quality of the results?

Given two computational programs, one using a Gaussian basis, and the other using Slater basis, what are the practical differences, advantages and disadvantages for each choice ?
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### Does energy decrease with basis set size in density functional theory?

Based on the variational principle, one might expect that the ground state energy of a density functional theory (DFT) calculation will decrease as the basis set size increases. (As I understand it, ...
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### Trying to generate a wave function basis set

For a little project I'm working on, I am trying to generate a wavefunction basis set I can use in Quantum Monte Carlo (DMC to be specific). Preferably, it would be a linear combination of Slater ...
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### Forming Basis Functions from 6-31G Basis Set for Carbon Atom

I am a computer science grad and I am working to write an electronic structure calculation program and I am stuck at forming basis functions using 6-31G Basis set for atoms having higher atomic ...
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### Gram-Schmidt method to identify linearly dependent vectors

A method to orthogonalize a set of vectors (vectors of unit length that are mutually orthogonal) is the Gram-Schmidt process: http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process Note that the ...
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### Orthonormal basis for hexahedron

Orthogonal polynomials are often preferred as basis functions. Recently I learned selecting orthonormal basis further simplifies the mass matrix from diagonal to simply the identity matrix when used ...
I have an energy dispersion curve obtained from the eigenvalues of $$E(k) = \text{eig}(T e^{ik} + T^H e^{-ik} + H_0),$$ where $H_0$ and $T$ are $N\times N$ square matrices, $T^H$ is the Hermitian ...