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Math – a namespace of mathematical functions and libraries for Dyalog APL

This repository contains a Math namespace for Dyalog APL with functions for finding eigenvalues, eigenvectors and discrete Fourier transforms.

These functions are implemented as calls into shared libraries. Source code and build scripts for the libraries are included.

Using the namespace

Download an appropriate Math-*.zip from the releases page and extract the files somewhere on your computer.

In order for Windows to find the DLLs you'll have to put them:

  • in the current directory (this might be determined by the properties of the shortcut you use to start Dyalog), or
  • in the directory containing the Dyalog executable dyalog.exe, or
  • somewhere on your %PATH%.

The functions in this namespace include complex arithmetic. Dyalog represents complex numbers a+bi as aJb.

Eigen

Monadic function Eigen takes an n×n real or complex matrix and returns an (n+1)×n result of Eigen: Values⍪⍉↑Vectors:

┌───┬───┬───┬───┐
│  v a l u e s  │  ── Eigen values
├───┼───┼───┼───┤
│ v │ v │ v │ v │  ┐
├ e ┼ e ┼ e ┼ e ┤  │
│ c │ c │ c │ c │  │
├ t ┼ t ┼ t ┼ t ┤  ├─ Eigen vectors.
│ o │ o │ o │ o │  │
├ r ┼ r ┼ r ┼ r ┤  │
│   │   │   │   │  ┘
└───┴───┴───┴───┘

Eigen has been constructed from LAPACK (Linear Algebra Package) double- precision C functions which are available as source code from www.netlib.org/lapack

LAPACK.DLL contains these C functions. They can all be called individually through ⎕NA. You need to examining each function's parameters in the corresponding *.C file (downloaded from the internet) in order to correctly specify their result and argument types.

For example, look at the ⎕NA call for dgeev_ in Eigen. Compare this with the parameters specified in file DGEEV.C . All the other double-precision real and complex LAPACK functions can be called in this way using ⎕NA.

Trace the following line in order to see Eigen in action:

  test.eigen    ⍝ run and trace 10 times

Domino

Domino is a cover function for APL's primitive ⌹ function. It has been included for backwards compatibility with the Math workspace which was supplied with previous versions of Dyalog APL.

Trace the following line in order to see Domino in action:

  test.domino   ⍝ run and trace

Fourier

Fourier takes a real or complex array right argument. The left argument signifies:

1: Fourier Transform (default).
¯1: Inverse Fourier Transform.

Fourier has been constructed from FFTW (the Fastest Fourier Transform in the World). The FFTW source code of C functions is available from www.fftw.org

The FFTW.DLL contains all these functions. They can be called individually through ⎕NA after examining each function's parameters in the FFTW documentation.

Check that

  {⍵=¯1 Fourier Fourier ⍵}↓?((5?5),2)⍴100

Trace the following line in order to see Fourier in action:

  test.fourier   ⍝ run and trace

Note that C cover functions dft and idft have been added to the DLL.

These functions are:

// dft.c discrete fourier transform

#include <fftw.h>

void dft(int *rank, const int *shape, double *data)
{
   fftwnd_plan plan;
   plan = fftwnd_create_plan(*rank, shape, FFTW_FORWARD, FFTW_IN_PLACE);
   fftwnd_one(plan, (void*)data, 0);
   fftwnd_destroy_plan(plan);
}

// idft.c inverse discrete fourier transform

#include <fftw.h>

void idft(int *rank, const int *shape, double *data)
{
   fftwnd_plan plan;
   plan = fftwnd_create_plan(*rank, shape, FFTW_BACKWARD, FFTW_IN_PLACE);
   fftwnd_one(plan, (void*)data, 0);
   fftwnd_destroy_plan(plan);
}

To see an example of these functions, type:

      test.eigen ⋄ test.domino ⋄ test.fourier

Building the libraries

The supported build configurations are:

  • Linux (x86 and x86-64)
  • Cross-compiling from Linux to Windows (x86 and x86-64)

To build natively:

  • install some 32- and 64-bit Fortran compilers with sudo apt-get install gfortran-multilib
  • cd into the root of the repository
  • type make -f Makefile.linux to make Math-linux.zip

To cross-compile (tested on Ubuntu 17.04):

  • install some cross-compilers with sudo apt-get install gfortran-mingw-w64
  • cd into the root of the repository
  • type make -f Makefile.windows to make Math-windows.zip