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.
3
votes
1answer
52 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
0answers
36 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
110 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
168 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
62 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
128 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 ...
10
votes
1answer
199 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 ...
4
votes
1answer
311 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
195 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 ...
7
votes
1answer
116 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
251 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 ?
8
votes
2answers
166 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
1answer
225 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
734 views
What is counterpoise correction?
What is counterpoise correction exactly ? Can you explain when it is needed and why ?
