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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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 ...
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103 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, ...
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1answer
136 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. ...
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1answer
108 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 ...
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201 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 ...
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311 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
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1answer
76 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 ...
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141 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 ...
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295 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 ...
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2answers
671 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
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220 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 ...
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131 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 ...
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473 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 ?
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201 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 ?
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357 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 ?
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2k views

What is counterpoise correction?

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