Last time I started with one of the easiest problems in quantum mechanics: the particle in a box. This time I’ll add 1 dimension and we’ll see a particle in a 2D box. To visualize its wave function and density we need 3D surface plots.

2D Box

This time we have a particle that is confined in a 2D box. The potential within the box is zero and outside the box infinity. Again the solution is well-known and can be found on Wikipedia. This time the state of the wave function is determined by two numbers. These are typically called quantum numbers and refer to the X and the Y direction, respectively.

The absolute size of the box doesn’t really matter and we didn’t worry about it in the 1D case. However, the relative size of the length and the width make a difference. The solution to our problem reads

$\Psi_{n,k}(x,y) = \sqrt{\frac{4}{L_x L_y}} \cdot \sin(n \cdot \pi \cdot x / L_x) \cdot \sin(k \cdot \pi \cdot y / L_y)$

The Math

Very similar to the 1D case I quickly coded the wave function and the density for further plotting. I had to make sure that the arrays are fit for 3D plotting, so the code looks a little bit different compared to last post’s

     public static ILArray<double> CalcWF(int EVXID, int EVYID, double LX, double LY, int MeshSize)
     {
        ILArray<double> X = linspace<double>(0, LX, MeshSize);
        ILArray<double> Y = linspace<double>(0, LY, MeshSize);

        ILArray<double> Y2d = 1;
        ILArray<double> X2d = meshgrid(X, Y, Y2d);

        ILArray<double> Z = sqrt(4.0 / LX / LY) * sin(EVXID * pi * X2d / LX) * sin(EVYID * pi * Y2d / LY);

        return Z.Concat(X2d,2).Concat(Y2d,2);
     }

Again, this took me like 10 minutes and I was done.

The Visualization

This time the user can choose the quantum numbers for X and Y direction, the ratio between the length and the width of the box and also the number of mesh points along each axis for plotting. This makes the visualization panel a little bit more involved. Nevertheless, it’s still rather simple and easy to use. This time it took me only 45 minutes – I guess I learned a lot from last time.

The result

Here is the result of my little program. You can click and play with it. If you’re interested, you can download the Particle2DBox source code. Have fun!

This is a screenshot of the application. I chose the second quantum number along the x axis and the fourth quantum number along the y axis. The box is twice as long in y direction as it is in x direction. The mesh size is 100 in each direction. On the left hand side you see the wave function and on the right hand side the probability density.

The post ILNumerics for Scientists – Going 3D appeared first on The ILNumerics Blog.

During a lengthy beta phase we received a whole bunch of precise and enormous helpful feedback from you. We really appreciate that and again would like to say thanks!

ILNumerics Optimization Toolbox adds a number of functions to ILNumerics, useful to find solutions of common optimization problems. Since everything is nicely integrated into ILNumerics it helps you solve the problem easily and very efficiently. Optimization applications like the one shown as screenshot above can now get realized in a couple of minutes!

This blog post sheds some light on the very first steps for using the new Optimization Toolbox. It helps you to start quickly and lists some common documentation sources.

Obtaining the Optimization Toolbox

The optimization toolbox is available as a dedicated package and so must be individually purchased and installed. ILNumerics 4.6 or above is needed for that to work. Existing customers can evaluate the new toolbox by installing an extended trial on top of your existing ILNumerics Ultimate VS installation. Just let us know and we will lead you the way to the download. Another option is the trial: it includes all toolboxes, hence you can start right away by downloading and installing a trial of ILNumerics Ultimate VS and get familiar with all optimization methods easily.

Optimization Toolbox Setup

The optimization toolbox obviously depends on the ILNumerics Computation Engine. It installs the managed assemblies ILNumerics.Optimization.dll into the GAC and also makes them available inside Visual Studio as reference to your application projects:

Afterwards, the ILNumerics.Optimization class is found in the ILNumerics namespace which commonly is included in your source files anyway. A super-mini but complete starting example in a fresh new C# console application could look as follows:

using System;
using ILNumerics; 

namespace ConsoleApplication1 {
    class Program : ILMath {

        static ILRetArray<double> myObjFunc(ILInArray<double> A) {
            using (ILScope.Enter(A)) {
                return sum((A - 1) * A);
            }
        }
        static void Main(string[] args) {
            ILArray<double> M = Optimization.fmin(myObjFunc, ones(1,4));
            Console.WriteLine(M);
        }
    }
}

Note how we derived our console class from ILNumerics.ILMath! This allows us to omit the namespace.class identifier and ‘ILMath.sum’ and ‘ILMath.ones’ become just: ‘sum’ and ‘ones’.

If you are lucky enough to use Visual Studio 2015, it will allow you the inclusion of the static ILNumerics.Optimization class into your ‘using’ directives on top of your source code files also. This will inject all public functions of the Optimization class into your scope also which gives an even shorter syntax. ‘fmin’ and all other optimization functions are now readily available right next to the common ILMath functions:

That’s it already! Happy optimizing!

Optimization Examples

A number of example applications have been added to the examples section. Basically, they accompany the corresponding online documentation and let you follow the tutorials step by step on your own machine.

In order to start with an example application, you download the example as a zip package and extract it into a new folder on your local machine. Open the ‘*.csproj’ contained in the example package with Visual Studio. You will notice missing references in the References node of your project in the Solution Explorer:

Just remove these old references and replace them with references to the files actually installed on your system – using the common method described above.

Hit F5 and run the example!

Documentation

The ILNumerics Optimization Toolbox online documentation is available here:

http://ilnumerics.net/ilnumerics-optimization-toolbox.html

The API /class documentation for all functions of ILNumerics.Optimization is available here:

http://ilnumerics.net/apidoc/?topic=html/T_ILNumerics_Optimization.htm

Wrap up

We gave a short introduction in how to obtain and install ILNumerics Optimization Toolbox for ILNumerics Computing Engine. We pointed you to the common places of examples and documentation.

If you run into trouble or have useful suggestions and feedback, let us know! Contact sales@ilnumerics.net for any licensing questions.

The post Directions to the ILNumerics Optimization Toolbox appeared first on The ILNumerics Blog.

I’ve been working as a scientist at universities for 10 years before deciding to go into industry. The one thing I hated most was coding. At the end of the day coding for scientists is like running for a football player. Obviously, you need it but it’s not what you’re here for.

I really dreaded the coding and the debugging. So much precious time for something that was so clear on paper and I just wanted the solution of my equations to see whether my idea made sense or not. More often than not scientists find that their idea was not so great and now they had spent so much time coding just to find out that the idea didn’t work.

To make things worse even in the case that the idea was good the code was typically in a scripted language like Python or Matlab and now needed to be rewritten in Fortran or C++. Isn’t there a better way of doing things? Isn’t there a way where the idea creation and the formulation is the bottle neck and not the coding? I could’ve spent so much more time on valuable thinking instead of senseless coding.

Now, obviously, I’m going to tell you how ILNumerics does exactly that. And, obviously, I do that because I work for them. Still, this is my experience and this is how easy it was. And at the end of the day, you’re not going to buy our product because of this blog post, rather because you downloaded our free trial and convinced yourself.

The Problem

Let’s start with one of the simplest problems in quantum mechanics – the particle in a box. This is a quantum mechanical particle in a potential-free box with an infinite potential outside the box. In other words, the particle is forced to stay within the box. Anybody unfamiliar with this problem can read the Wikipedia article about this.

The solution to this problem is well-known and can be found analytically. As always in quantum mechanics, the solution is quantized with infinitely many solutions – also known as states. Each of the solutions consists of a wave function associated with a corresponding energy.

$\Psi_n(x) = \sqrt{\frac{2}{L}} \sin(n \cdot \pi \cdot  x)$

In the equation above $n$ designates the $n^{th}$ state and $L$ is the length of the box, set to 1 in our case.

The Goal

My goal is to display the wave function and the probability density of a user-chosen state. I choose a simple problem and a simple goal because I’m just getting started. Obviously, this is not one of the problems I faced as a scientist in my earlier life, but it is a nice example.

The Solution

Let’s start with the mathematical part. First I implemented the wave function for a given state. As the wave function is supposed to be plotted it is discretized over a mesh. Here is my body of my function:

     ILArray<double> X = linspace<double>(0,1,MeshSize);
     ILArray<double> Y = sqrt(2.0)*sin(EVID*pi*X);
     return vertcat(X,Y);

The parameter EVID is the eigenvector ID, i.e. an integer greater than 0 to identify which state should be calculated. I coded the same for the density, which basically means squaring the wave function stored in vector Y.

Done in less than 10 minutes!!!

The Graphs

The rest is all about the presentation to the user. So I created a simple Windows Form with a numeric up/down counter (to allow the user to choose the state) and two panels for simple line plots. I just needed to wire all the components to make sure that the plots were rescaled and updated after the user changes the state. This took me some more time, because it was the first time for me. Still it took me only 60 minutes and done.

The Result

Here you can find the result of my simple little program. You can download the source code and play with it. Have fun!

The post ILNumerics for Scientists – An easy start appeared first on The ILNumerics Blog.

Lets say, you went to that conference and want to give your new friends some insight into your brand new simulation results. The PC in the internet cafe enables you to fetch the data from your NAT storage at home. But will you be able to do anything with it on that plain Windows PC?

Or you want to localize a certain test data set but cannot remember its rather cryptic name. Or you might want to manage the latest measurement results from todays atmospheric observation satellite scans. The data are huge but often require some sort of preprocessing. There should be some easy way to filter them by the meta data within the files, right?

Other than getting the data from some application layer, we now want to interface plain old file objects. Of course, you store your data in HDF5 format, right? You do so, because HDF5 is portable, very efficient, flexible and you are in good company.

Let’s see. We have a fresh Windows PC and we know every Windows installation nowadays comes with Powershell. Powershell itself is based on the .NET framework and hence efficiently handles any .NET assembly. It should be easy to use ILNumerics with Powershell! All we still need is some way to access the HDF5 files. ILNumerics, natively is able to read and write Matlab mat files up to version 6. It currently lags on native HDF5 support.

Luckily, the HDF Group provides a large collection of high quality tools for HDF support. Among them you’ll find a .NET wrapper and … a brand new Powershell module: PSH5X! Together with Gerd Heber, the leading inventor of PSH5X, we did a feasibility study with the goal to investigate the options of utilizing HDF5 and ILNumerics together in Powershell. It can be downloaded here. We were quite impressed by the options this brings.

This blog post will describe the necessary steps to setup Powershell for ILNumerics and HDF5.

Getting Started

Basically, the installation process for any Powershell module consists of

1. Getting the module files and its dependencies from somewhere,
2. Deploying the module files into a special folder on your machine, and
3. Importing the module in your session.

The PSH5X homepage gives all information on how to get ready using the HDF5 Powershell module. Just download the package and follow the three steps on the page. At the end, HDF5 signals you a successful installation by displaying its version numbers.

Since ILNumerics depends on several other modules, we provide a small bootstrapper script. Just open up your favorite Powershell IDE (PowerShell_ISE.exe comes with any recent Windows) and copy/paste the following line:

(new-object Net.WebClient).DownloadString('http://ilnumerics.net/media/InstallILNumericsPSM.ps1') | iex

If you are curious, what this does – just ommit the trailing | iex and the script is not executed but displayed for your inspection.

The installer will ask for the installation folder (global under System32/ or local in your user profile), fetches the latest ILNumerics package for the current platform from the official nuget repository and install it into the selected module folder. In addition it loads the TypeAccelerator Powershell module and installs it into the same module directory. Note, the accelerators have been slightly modified in order to make them work with Powershell 3 and hence are fetched from our ILNumerics server. However, credits fully belong to poshoholic for his great work.

Note, the installation has to be done only once. Afterwards, on the next Powershell session, simply re-import needed modules by typing – lets say:

PS> Import-Module ILNumerics 

Go!

If everything was setup correctly, we can now use the full spectrum of the involved modules:

PS> [ilmath]::rand(4,5).ToString()
<Double> [5,4]
   0,72918    0,87547    0,43167    0,94942
   0,58024    0,75562    0,96125    0,83148
   0,22454    0,20583    0,82285    0,83144
   0,13300    0,40047    0,58829    0,87012
   0,50751    0,05496    0,02814    0,48764 

Nice. But what about the MKL? Are the correct binaries really installed as well?

PS> [ilf64] $A = [ilmath]::rand(1000,1000)
PS> Measure-Command { [ilf64]$C = [ilmath]::rank($A) }
Days              : 0
Hours             : 0
Minutes           : 0
Seconds           : 0
Milliseconds      : 920
Ticks             : 9202311
TotalDays         : 1,06508229166667E-05
TotalHours        : 0,00025561975
TotalMinutes      : 0,015337185
TotalSeconds      : 0,9202311
TotalMilliseconds : 920,2311

PS> $C.ToString()
1000

We have almost all options from C#:

PS> [ilf64] $part = $A['10:15;993:end']
PS> $part.ToString()
<Double> [11,7]
   0,08522    0,87217    0,59997    0,57363    0,22956    0,02006    0,02359
   0,33479    0,49003    0,65269    0,97772    0,28322    0,69505    0,70372
   0,30072    0,68705    0,47112    0,68627    0,65030    0,40454    0,63026
   0,15639    0,30391    0,22992    0,69310    0,65716    0,51797    0,68110
   0,72854    0,60188    0,50740    0,74499    0,13459    0,88481    0,12445
   0,80525    0,60180    0,69256    0,74825    0,64388    0,16792    0,45266 

Lets sort the first row of $part, keeping track of original positions:

PS> [ilf64] $indices = 0.0
PS> [ilf64] $sorted = [ilmath]::sort($part['0,1;:'],$indices,0,$false)
PS> $sorted.ToString()
<Double> [2,7]
   0,02006    0,02359    0,08522    0,22956    0,57363    0,59997    0,87217
   0,28322    0,33479    0,49003    0,65269    0,69505    0,70372    0,97772
PS> $indices.ToString()
<Double> [2,7]
         5          6          0          4          3          2          1
         4          0          1          2          5          6          3 

This is all interactive. Of course, we can write complete functions and even complex algorithms that way.
One of the best things: Even in Powershell ILNumerics saves your memory and meets all expectations regarding execution speed. Powershell allows you to consequently use ILNumerics’ typing and scoping rules.

In our feasibility study with Gerd Heber, we show how easy it gets to access an HDF5 file, to convert its data to ILNumerics arrays (implicitly), filter and manipulate a little and even create a full interactive 3D surface graph from it. We demonstrate how to use the type accelerators and to mimic the using statement for artificial scoping. Take a look and let us know, what you think!

The post Putting on a Good Show with HDF5, ILNumerics, and PowerShell appeared first on The ILNumerics Blog.