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.

Filter by
Sorted by
Tagged with
2
votes
0answers
23 views

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 ...
4
votes
1answer
48 views

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 ...
5
votes
0answers
150 views

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 ...
1
vote
1answer
183 views

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 ...
6
votes
2answers
1k views

Finding Shape Functions for a Triangle in 3D coordinate space

I have a Triangular Plane T given by three points in 3D space $$P_1 = (x_1,y_1,z_1)\\ P_2 = (x_2,y_2,z_2) \\ P_3 = (x_3,y_3,z_3)$$ I want to find the Shape Functions on this plane as $N_1(x,y,z),N_2(...
4
votes
1answer
168 views

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 ...
8
votes
5answers
729 views

Choice of basis in FEM

Briefly, what are the different types of basis used in FEM and why is the nodal basis so popular and advantageous in the finite element context?
4
votes
1answer
315 views

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 ...
1
vote
0answers
58 views

Basis for spatially localized function [closed]

I'm solving an ODE, where currently the dependent variables are the (time dependent) spatial Fourier coefficients. It turns out that the phenomena I'm interested in describing is spatially localized, ...
3
votes
0answers
80 views

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 ...
1
vote
0answers
51 views

Size reduction of matrices in dispersion curve calculation

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 ...
2
votes
2answers
2k views

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, ...
8
votes
1answer
5k views

Fitting one set of points to another by a rigid motion

I'm not really sure how to explain this problem clearly, so please bear with me. I have a basis of 3 orthonormal unit vectors and a position, a standard 4x4 transform matrix in computer graphics. ...
2
votes
1answer
126 views

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 ...
4
votes
2answers
628 views

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 ...
1
vote
2answers
2k views

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 ...
4
votes
1answer
115 views

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 ...
7
votes
1answer
174 views

Conjugate Gradient with Hierarchical Basis Functions: How can the hierarchical base be decomposed?

I'm trying to implement a Conjugate Gradient solver using Hierarchical Basis Functions, following this paper. In section 3 the paper says that the hierarchical basis matrix $S$ can be decomposed into ...
11
votes
1answer
1k views

How can I compute a basis for a matrix Lie algebra given a finite set of generators?

Given an arbitrary set of (numerical) square complex matrices $\mathcal{A}=\{A_1,A_2,\cdots,A_m\}$, I am interested in computing the real matrix Lie algebra generated by $\mathcal{A}$, call it $\...
6
votes
2answers
2k views

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 ...
6
votes
0answers
419 views

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 ...
10
votes
1answer
167 views

Polynomials that are orthogonal over curves in the complex plane

Various important sets of polynomials (Legendre, Chebyshev, etc.) are orthogonal over some real interval with some weighting. Are there known families of polynomials that are orthogonal over other ...
2
votes
3answers
4k views

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 ?
9
votes
2answers
577 views

What does it mean for a basis set to be “correlation consistent” ?

Some basis sets are said to be "correlation consistent". What does it mean in practice ?
7
votes
2answers
1k views

How to choose a basis set for ab-initio evaluations ?

How do I pick a basis set for an ab-initio Hartree-Fock evaluation ? In other words, what are the important characteristics of a basis set so that a proper choice can be made ?
15
votes
2answers
7k views

What is counterpoise correction?

What is counterpoise correction exactly ? Can you explain when it is needed and why ?