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.

learn more… | top users | synonyms

1
vote
0answers
95 views

Galerkin FEM: Handling Dirichlet boundary condition with quadratic basis function

Consider a simple BVP: $-u_{xx} = f$ with $u(1) = g$ and $-u_x(0) = H$. Following Hughes' notation for Galerkin FEM, the variational function space $V$ is defined first using basis $\{N_A(x)\}$, $A = ...
1
vote
0answers
28 views

Basis for spatially localized function

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, ...
2
votes
0answers
38 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 ...
0
votes
0answers
24 views

Size reduction of matrices in dispersion curve calculation

I have an energy dispersion curve resulted from eigen values of E(k) = eig(T exp(ik) + T' exp(-ik) + H0). Where H0 and T are NxN square matrices and T' is transpose of T and k is wave-vector. So there ...
2
votes
2answers
161 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, ...
3
votes
1answer
250 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
111 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
259 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 ...
0
votes
2answers
431 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
82 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 ...
5
votes
1answer
147 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
404 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
948 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 ...
5
votes
0answers
238 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 ...
8
votes
1answer
138 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
704 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
228 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
457 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 ?
6
votes
2answers
2k views

What is counterpoise correction?

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