# Numerical recipes

About: This article is published in Computers in Physics.The article was published on 1990-10-01. It has received 17845 citations till now.

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TL;DR: A detailed description and comparison of algorithms for performing ab-initio quantum-mechanical calculations using pseudopotentials and a plane-wave basis set is presented in this article. But this is not a comparison of our algorithm with the one presented in this paper.

47,666 citations

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TL;DR: The likelihood function for macromolecular structures is extended to include prior phase information and experimental standard uncertainties and the results derived are consistently better than those obtained from least-squares refinement.

Abstract: This paper reviews the mathematical basis of maximum likelihood The likelihood function for macromolecular structures is extended to include prior phase information and experimental standard uncertainties The assumption that different parts of a structure might have different errors is considered A method for estimating σA using `free' reflections is described and its effects analysed The derived equations have been implemented in the program REFMAC This has been tested on several proteins at different stages of refinement (bacterial α-amylase, cytochrome c′, cross-linked insulin and oligopeptide binding protein) The results derived using the maximum-likelihood residual are consistently better than those obtained from least-squares refinement

14,622 citations

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TL;DR: In this paper, a selfconsistent density functional method using standard norm-conserving pseudopotentials and a flexible, numerical linear combination of atomic orbitals basis set, which includes multiple-zeta and polarization orbitals, was developed and implemented.

Abstract: We have developed and implemented a selfconsistent density functional method using standard norm-conserving pseudopotentials and a flexible, numerical linear combination of atomic orbitals basis set, which includes multiple-zeta and polarization orbitals. Exchange and correlation are treated with the local spin density or generalized gradient approximations. The basis functions and the electron density are projected on a real-space grid, in order to calculate the Hartree and exchange-correlation potentials and matrix elements, with a number of operations that scales linearly with the size of the system. We use a modified energy functional, whose minimization produces orthogonal wavefunctions and the same energy and density as the Kohn-Sham energy functional, without the need for an explicit orthogonalization. Additionally, using localized Wannier-like electron wavefunctions allows the computation time and memory required to minimize the energy to also scale linearly with the size of the system. Forces and stresses are also calculated efficiently and accurately, thus allowing structural relaxation and molecular dynamics simulations.

8,723 citations

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TL;DR: In this paper, the uncertainty inherent in any observational estimate of the IMF is investigated by studying the scatter introduced by Poisson noise and the dynamical evolution of star clusters, and it is found that this apparent scatter reproduces quite well the observed scatter in power-law index determinations, thus defining the fundamental limit within which any true variation becomes undetectable.

Abstract: A universal initial mass function (IMF) is not intuitive, but so far no convincing evidence for a variable IMF exists. The detection of systematic variations of the IMF with star-forming conditions would be the Rosetta Stone for star formation.
In this contribution an average or Galactic-field IMF is defined, stressing that there is evidence for a change in the power-law index at only two masses: near 0.5 M⊙ and near 0.08 M⊙. Using this supposed universal IMF, the uncertainty inherent in any observational estimate of the IMF is investigated by studying the scatter introduced by Poisson noise and the dynamical evolution of star clusters. It is found that this apparent scatter reproduces quite well the observed scatter in power-law index determinations, thus defining the fundamental limit within which any true variation becomes undetectable. The absence of evidence for a variable IMF means that any true variation of the IMF in well-studied populations must be smaller than this scatter.
Determinations of the power-law indices α are subject to systematic errors arising mostly from unresolved binaries. The systematic bias is quantified here, with the result that the single-star IMFs for young star clusters are systematically steeper by Δα≈0.5 between 0.1 and 1 M⊙ than the Galactic-field IMF, which is populated by, on average, about 5-Gyr-old stars. The MFs in globular clusters appear to be, on average, systematically flatter than the Galactic-field IMF (Piotto & Zoccali; Paresce & De Marchi), and the recent detection of ancient white-dwarf candidates in the Galactic halo and the absence of associated low-mass stars (Ibata et al.; Mendez & Minniti) suggest a radically different IMF for this ancient population. Star formation in higher metallicity environments thus appears to produce relatively more low-mass stars. While still tentative, this is an interesting trend, being consistent with a systematic variation of the IMF as expected from theoretical arguments.

6,784 citations

### Cites methods from "Numerical recipes"

...…the masses, m, into 30 log10 m bins which subdivide the range 22:1 # log10 m # 12:1: Power laws are fitted using weighted linear regression (e.g. Press et al. 1994) to subranges that are defined as follows: b1 6; log10 0:01 , lm # log10 0:08 ; b2 6; log10 0:08 , lm # log10 0:50 ; b3 4; log10…...

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TL;DR: The Weighted Histogram Analysis Method (WHAM) as mentioned in this paper is an extension of Ferrenberg and Swendsen's multiple histogram technique for complex biomolecular Hamiltonians.

Abstract: The Weighted Histogram Analysis Method (WHAM), an extension of Ferrenberg and Swendsen's Multiple Histogram Technique, has been applied for the first time on complex biomolecular Hamiltonians. The method is presented here as an extension of the Umbrella Sampling method for free-energy and Potential of Mean Force calculations. This algorithm possesses the following advantages over methods that are currently employed: (1) It provides a built-in estimate of sampling errors thereby yielding objective estimates of the optimal location and length of additional simulations needed to achieve a desired level of precision; (2) it yields the “best” value of free energies by taking into account all the simulations so as to minimize the statistical errors; (3) in addition to optimizing the links between simulations, it also allows multiple overlaps of probability distributions for obtaining better estimates of the free-energy differences. By recasting the Ferrenberg–Swendsen Multiple Histogram equations in a form suitable for molecular mechanics type Hamiltonians, we have demonstrated the feasibility and robustness of this method by applying it to a test problem of the generation of the Potential of Mean Force profile of the pseudorotation phase angle of the sugar ring in deoxyadenosine. © 1992 by John Wiley & Sons, Inc.

5,784 citations