- ...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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Mpc
- The lower-case ``
*h*'' represent the Hubble parameter, ranging from 0.0 to 1.0, provided 10#10km s11#11 Mpc11#11.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...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.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
*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.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...procedure.
- This procedure is similar to the Newton-Raphson method discussed in
Subsection 4.3.3.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...quadrature.
- Description of these method can be found in textbooks like
KC,Press,NM.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...AVP.
- P-94 referred to the
Gaussian quadrature used by Gia as ``very efficient'', but never used it.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...matrix
- A symmetric matrix is
said to be positive definite if, and only if, all its eigenvalues are positive
DS.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...matrix
- The matrix containing the second partial derivatives of the
action at
**x**.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...eigenvector
- The
eigenvectors are given by the matrix of the quadratic function, e.g. 1#1 in
equation (4.6).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...coordinates.
- N-body codes can also calculate in comoving coordinates, but the
`NBODY1`code used here only considers physical coordinates.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...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.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...330#330.
- For more information on the Divided Differences method,
see [Chapter 6.2]KC.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...Descent.
- Assuming that what they refer to as ``walking down the
gradient'' is the method of Steepest Descent.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...implementations
- The implementations using the
DFP-formula is not considered due to it being deemed not suitable in Section
5.3.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Mon Jul 5 02:59:28 MET DST 1999