How many decimal places is determined by the precision we set. The initial estimates of Newton’s method are consistent with that of bisection method, which is designed for comparison between these two methods.Ī reference local minimum for the studied function is x=2.356194490…. Initial estimates are boundary values of each range for bisection method. The function used here is f(x)=exp(-x)cos(x). So the following content will study how the initial guess changes convergence properties of these two methods. An initial estimate is needed to use these two methods and no guidance is provided by the methods themselves. These two methods are iterative and can only provide approximations, which is controlled by the convergence criterion. Here, only optimization in one dimension is studied and methods adopted are bisection method and Newton’s method. Normally, the later one is more difficult. This locally lowest value search could be called numerical optimization, which could happen in one dimension or more than one dimension.

There are various methods that could search local minimums for a function. How initial estimates change convergence properties of bisection method and Newton’s method

This entry was posted in 2018, 2nd Post 2018, Hafnium, Student Post on Apby Jonathan Schirmer. et al. First principles methods using CASTEP. In this work, the packing ratio was found to be 1.574, approximately a 4% difference. Furthermore, it is known that the ideal packing ratio for spheres arranged in an hcp lattice is c/a≈1.633. This is found to be well in agreement with a result obtained in a previous post. The unit cell volume was found to be 45.163544 Å^3. The minimized energy was found to be -15.73336 keV and the corresponding lattice constants were found to be a=b=3.2117 Å and c=5.0557 Å. The values above gave a well-converged energy with a reasonable runtime. K-points: 9x9圆 (36 points in the irreducible part of the Brillouin zone (IBZ))īoth the number of k-points and the energy cutoff were varied to ensure reasonable convergence. The following settings were used for Geometry Optimization in CASTEP:ġs2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 4f14 5s2 5p6 5d2 6s2 We maintain Throughout the calculation, symmetry demands that a=b, but a and c are allowed to vary. These values are provided by examples within Materials Studio and are accepted experimental values. The initial values of the lattice parameters are a=b=3.1956 Å and c=5.0511 Å. Initial cell provided for CASTEP calculation.

Materials studio change bond angle