...principle
Peebles originally referred to the method as the ``Least Action Principle'', but later realized that Hamilton's principle and the Principle of Least Action do not coincide [for details, see][]Gold. The use of the word ``least'' is neither quite correct, since, as will be seen later, saddle points are also allowed solutions to the action P-95.
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...Mpc
The lower-case ``h'' represent the Hubble parameter, ranging from 0.0 to 1.0, provided 10#10km s11#11 Mpc11#11.
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...103#103.
The mass-to-light ratio M/L is a measure of how much of the matter that is luminous, and is often given in solar values; hence the 104#104.
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...k=0.
The reason we can use the current values of the H and 170#170 is that 43#43 is constant throughout the history of the Universe.
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...148#148.
The dimension of the system is being determined by the number of particles, the number of trial functions, and the three spatial dimensions.
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...procedure.
This procedure is similar to the Newton-Raphson method discussed in Subsection 4.3.3.
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Description of these method can be found in textbooks like KC,Press,NM.
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...efficient.
Gia made use of 189#189. I have not tried this; it is more laborious to apply and my choice seems to work well.
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...AVP.
P-94 referred to the Gaussian quadrature used by Gia as ``very efficient'', but never used it.
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...matrix
A symmetric matrix is said to be positive definite if, and only if, all its eigenvalues are positive DS.
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...matrix
The matrix containing the second partial derivatives of the action at x.
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...eigenvector
The eigenvectors are given by the matrix of the quadratic function, e.g. 1#1 in equation (4.6).
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...method.
This method is also often referred to as ``Quasi-Newton'', but in this thesis the convention of DS is used: since the method in question is a multidimensional generalization of the linear secant method, it can be called a secant method. The term quasi-Newton can be used to describe the Newton-Raphson method with a stepsize determiner.
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...coordinates.
N-body codes can also calculate in comoving coordinates, but the NBODY1 code used here only considers physical coordinates.
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...HREF="node25.html#secopti">4.3.
Two points need to be made: Firstly, due to Peebles not using any conventional name for the method, it is here assumed that what he refers to as ``walking down the gradient'' is in fact the method of Steepest Descent. Secondly, he implements the method somewhat differently from what is done here. The stepsize was multiplied by the mass of the particle under consideration, i.e. the stepsize term of equation (5.1) becomes 289#289, where 268#268 in the case of the method of Steepest Descent is the negative gradient of the action (5.8). This results in the removal of the mass 93#93 in the gradient at each iteration, which in turn simplifies the system, giving it a better scaling. The convergence of the method is then considerably improved. This implementation is not used here though, due to the main point being the comparison of the numerical methods applied on the same system.
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...330#330.
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...338#338
The condition number is the ratio between the maximum and the minimum eigenvalue of the Hessian. More on this in, for example, GMW,Bert.
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...consuming.
The Hessian can be made sparser, and the computation time reduced, by ignoring the cross-terms [see][]P-95. This method was not tested here.
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...convergence.
This method was also attempted in this study, but there were problems with convergence. According to Gia,S-B,Sch, the convergence needs to be aided by, for example, adding some stabilizing terms to equation (5.8). Based on this instability factor, and the fact that this method is not an optimizing method comparable to the ones tested here, the method was not considered any further.
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...Descent.
Assuming that what they refer to as ``walking down the gradient'' is the method of Steepest Descent.
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...implementations
The implementations using the DFP-formula is not considered due to it being deemed not suitable in Section 5.3.
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```

Trond Hjorteland
Mon Jul 5 02:59:28 MET DST 1999