This leads to the possibility of using very efficient numerical optimization techniques. We demonstrate in this communication that the Hessian of the GRAPE fidelity functional is unusually cheap, having the same asymptotic complexity scaling as the functional itself. Quadratic convergence throughout the active space is achieved for the gradient ascent pulse engineering (GRAPE) family of quantum optimal control algorithms. International Nuclear Information System (INIS) Modified Newton-Raphson GRAPE methods for optimal control of spin systems In this research, we report the effectiveness of the truncated Newton-Raphson method and quasi- Newton method with low-rank Hessian update strategy that are evaluated against the full Newton-Raphson. Liang, Yu Kanapady, Ramdev Chung, Peter W National Research Council Canada - National Science Library In this section, we will look at the secant method, which is another method for identifying the roots of non-linear equations.Truncated Newton-Raphson Methods for Quasicontinuum Simulations Now that we have seen the robustness of the Newton-Raphson method, let’s take a look at another similar numerical method that uses backward divided difference for derivative approximation. Notice that the result is extremely close to zero, suggesting that we have found the correct root. I added this try except block to take into account the fact that the derive() method and other approximate derivative calculation methods such as center_divided_difference() have differing numbers of parameters. One peculiarity that deserves attention is the TypeError exception, which occurs in this case if the number of arguments passed into the function does not match. Lastly, return_history is a flag that determines whether we return the full update history or simply the last value in the iteration as a single value. If the algorithm is unable to find the root within max_iter iterations, it likely means that the function provided does not have a root, or at the very least, the root is not discoverable via the algorithm. max_iter determines how many iterations we want to continue. epsilon is simply some small value we use to decide when to stop the update if the change in the value of the root is so small that it is not worth the extra compute, we should stop. I’ve added some parameters to the function for functionality and customization. Python Implementationīelow is an implementation of the Newton-Raphson method in Python. \[f'(x) = \lim_$, and now we have the update rule as delineated in (4). If you probe the deepest depths of your memory, somewhere you will recall the following equation, which I’m sure all of us saw in some high school calculus class: Hence the motivation for approximation methods, outlined in the section below. For these reasons, we will need some other methods of calculating derivatives as well. Moreover, the list index representation is unable to represent polynomials that include terms whose powers are not positive integers. While it’s great that we can calculate derivatives and integrals, one very obvious drawback of this direct approach is that we cannot deal with non-polynomial functions, such as exponentials or logarithms. For example, we can express $f(x) = x^3 - 20$ as The most obvious, simplest way of representing polynomials in Python is to simply use functions. For the sake of simplicity, let’s first just consider polynomials. Equation Representationīefore we move on, it’s first necessary to come up with a way of representing equations in Python. Specifically, this post will deal with mainly two methods of solving non-linear equations: the Newton-Raphson method and the secant method. After watching a few of his videos, I decided to implement some numerical methods algorithms in Python. His videos did not seem to assume much mathematical knowledge beyond basic high school calculus. While the videos themselves were recorded a while back in 2009 at just 240p, I found the contents of the video to be very intriguing and easily digestable. It was a channel called numericalmethodsguy, run by a professor of mechanical engineering at the University of Florida. Recently, I ran into an interesting video on YouTube on numerical methods (at this pont, I can’t help but wonder if YouTube can read my mind, but now I digress).
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |