Local polynomial structure

Remember that a subdivision limit curve consists of pieces, each of which depends only on a finite number, w, of original control points. If those control points he on a polynomial, then that piece of the limit curve will be polynomial or not, depending on the degree, irrespective of the other control points. 

However, a polynomial of degree w— 1 can always be fitted, however many such control points there are, and if w — 1 is less than or equal to the spanning degree, the limit curve will have polynomial pieces. 

Consider first the binary case. Let the scheme be of the form anK where K is the kernel of k entries. 

The total width w of the mask is k + n because each convolution with a adds one to the width. 

The number of control points influencing one span is w — 1. We can always find a polynomial of degree at most w — 2 interpolating all of these points. That polynomial will be spanned if its degree is less than or equal to n — 1 and so we have the condition 

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Independently of the scaling of the computational needs as a function of the size of the system, another, complementary, problem causes additional complications. This problem is related to the fact that the number of inequivalent minima on the total-energy surface as a function of structure grows very fast with the size of the system. In fact, it has been shown [4] that the number of local total-energy minima grows faster than any polynomial in N or, alternatively expressed, that the determination of the global total-energy minimum is a so-called NP-hard problem. 

Cortona developed a method to calculate the electronic structure of solids by calculating individually the electron density of atoms in a unit cell with a spherically averaged Hamiltonian as the local Hamiltonian. The tests of the method have been successful for alkali halides where the density around each nucleus can be well approximated by a spherical description. Goedecker proposed a scheme closely related to the divide-and-conquer approach. The local Hamiltonian is also constructed by truncation in the atomic orbital space. Instead of the matrix diagonalization for the local Hamiltonian described in equation (34) in the divide-and-conquer approach, Goedecker used an iterative diagonalization based on the Chebyshev polynomial approximation for the density matrix. Voter, Kress, and Silver s method is related to that of Goedecker with the use of a kernel polynomial method. 

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