Category Archives: Features

ILNumerics for Scientists – Going 3D

Recap

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!

Particle2DBoxThis 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.

Fun with HDF5, ILNumerics and Excel

It is amazing how many complex business processes in major industries today are supported by a tool that shines by its simplicity: Microsoft Excel. ‘Recently’ (with Visual Studio 2010) Microsoft managed to polish the development tools for all Office applications significantly. The whole Office product line is now ready to serve as a convenient, flexible base framework for stunning custom business logic, custom computations and visualizations – with just a little help of tools like ILNumerics.

ExcelWorkbook1In this blog post I am going to show how easy it is to extend the common functionality of Excel. We will enable an Excel Workbook to load arbitrary HDF5 data files, inspect the content of such files and show the data as interactive 2D or 3D plots. Continue reading Fun with HDF5, ILNumerics and Excel

High Performance Fast Fourier Transformation in .NET

„I started using ILNumerics for the FFT routines. The quality and speed are excellent in a .NET environment.“

The Fourier Transform (named after French mathematician and physicist Joseph Fourier) allows scientists to transform signals between time domain and frequency domain. This way, an arbitrary periodic function can be expressed as a sum of cosine terms. Think of the equalizer of your mp3-player: It expresses your music’s signal in terms of the frequencies it is composed of.

The Fast Fourier Transform (FFT) is an algorithm for the rapid computation of discrete Fourier Transforms’ values. Being one of the most popular numerical algorithms, it is used in physics, engineering, math and many other domains.

In terms of software engineering, the Fast Fourier Transform is a very demanding algorithm: In the .NET-framework, a naive approach would cause very low execution speeds. That’s the reason why many .NET-developers have to implement native C-libraries when it comes to FFTs.

ILNumerics uses Intel’s® MKL for Fast Fourier Transforms: That’s why our users don’t have to implement native library’s themselves for high performance FFTs. No matter if they have a scientific or an industrial background, many developers rely on ILNumerics because of its implementation of the Fast Fourier Transform. It’s the fastest you can get today – even for big amounts of data.

ILNumerics provides interfaces to forward and backward Fourier Transformations, for real and complex floating point data, in single and double precision, in one, two or n dimensions. In addition to the MKL’s FFTs, prepared interfaces for FFTW and for AMDs ACML exist.

Learn more about the ILNumerics library and its implementation of Fast Fourier Transformation in C#/.NET in the online documentation!