PDL::Matrix -- a derived matrix class that implements column-major constructors and methods

- NAME
- VERSION
- SYNOPSIS
- DESCRIPTION

- Overview
- Implementation

- FUNCTIONS

- mpdl, PDL::Matrix::pdl
- mzeroes, mones, msequence
- vpdl
- vzeroes, vones, vsequence
- PDL::Matrix::slice, PDL::Matrix::dice
- PDL::Matrix::at
- PDL::Matrix::set
- PDL::Matrix::reshape
- mdims
- kroneckerproduct
- det_general
- trace
- vcrossp, PDL::Matrix::crossp

- BUGS AND PROBLEMS
- TODO
- `AUTHOR(S)`
- COPYRIGHT

# NAME

PDL::Matrix -- a derived matrix class that implements column-major constructors and methods

# VERSION

This document refers to version PDL::Matrix 0.01 of PDL::Matrix

# SYNOPSIS

      use PDL::Matrix;

      $m = mpdl [[1,2,3],[4,5,6]];
      $m = PDL::Matrix->pdl([[1,2,3],[4,5,6]]);
      $m = msequence(4,3);
      @dimsa = $a->mdims; # 'dims' is not overloaded

      $v = vpdl [0,1,2,3]
      $v = vzeroes(4);

# DESCRIPTION

## Overview

This package tries to help people who want to use PDL for 2D matrix computation with lots of indexing involved . It provides a PDL subclass so one- and two-dimensional piddles that are used as vectors resp. matrices can be typed in using traditional matrix convention.

The original pdl class refers to the first index as the first row, the second index as the first column of a matrix. Consider

      print $B = sequence(3,2)
      [
       [0 1 2]
       [3 4 5]
      ]

    which gives a 2x3 matrix in terms of the matrix convention, but the
    constructor used (3,2). This might get more confusing when using
    slices like sequence(3,2)->slice("1:2,(0)") : with traditional
    matrix convention one would expect [2 4] instead of [1 2].

This subclass PDL::Matrix overloads the constructors and indexing functions of pdls so that they are compatible with the usual matrix convention, where the first dimension refers to the row of a matrix. So now, the above example would be written as

      print $B = PDL::Matrix->sequence(3,2) # or $B = msequence(3,2)
      [
       [0 1]
       [2 3]
       [4 5]
      ]

Routines like eigenvalue or matrix inversion can be used without any changes.

Furthermore one can construct and use vectors as n x 1 matrices without mentioning the second index '1'.

## Implementation

`PDL::Matrix` works by overloading a number of PDL constructors and methods such that first and second args (corresponding to first and second dims of corresponding matrices) are effectively swapped. It is not yet clear if PDL::Matrix achieves a consistent column major look-and-feel in this way.

# FUNCTIONS

## mpdl, PDL::Matrix::pdl

constructs an object of class PDL::Matrix which is a piddle child class, where the first index refers to the first column of the two-dimensional piddle.

        $m = mpdl [[1,2,3],[4,5,6]];
        $m = PDL::Matrix->pdl([[1,2,3],[4,5,6]]);

## mzeroes, mones, msequence

constructs a PDL::Matrix object similar to the piddle constructors zeroes, ones, sequence

## vpdl

constructs an object of class PDL::Matrix which is of matrix dimensions (n x 1)

        print $v = vpdl [0,1];
        [
         [0]
         [1]
        ]

## vzeroes, vones, vsequence

constructs a PDL::Matrix object with matrix dimensions (n x 1), therefore only the first scalar argument is used.

        print $v = vsequence(2);
        [
         [0]
         [1]
        ]

## PDL::Matrix::slice, PDL::Matrix::dice

same as slice, dice for normal piddles, but reflecting the matrix convention by swapping the first two arguments.

        print  sequence(3,2)->slice("1:2,(0)") # piddle
        [1 2]
        print msequence(3,2)->slice("1:2,(0)") # PDL::Matrix
        [2 4]

## PDL::Matrix::at

same as at for piddles, but reflecting the matrix convention by swapping the first two arguments

If only one scalar argument is used, we assume the object to be a vector and look only at the first column.

## PDL::Matrix::set

set a particular value in a PDL::Matrix object. Note that this has to be called as an object method rather than a function

print msequence(3,3)->`set(2,0,-1)` # ok with PDL::Matrix convention [ [ 0 1 2] [ 3 4 5] [-1 7 8] ]

print set msequence(3,3), 2,0,-1 # does not conform with PDL::Matrix convention [ [ 0 1 -1] [ 3 4 5] [ 6 7 8] ]

## PDL::Matrix::reshape

same as reshape for piddles, but reflecting the matrix convention by swapping the first two arguments

## mdims

returns the dimensions of the PDL::Matrix object in matrix convention

`dims` is NOT overloaded by PDL::Matrix to make sure that methods like PDL::transpose still work. So use `mdims` to get the dims in the PDL::Matrix notation.

        print msequence(3,2)->mdims
        3 2

## kroneckerproduct

returns kroneckerproduct of two matrices. This is not efficiently implemented.

## det_general

returns a generalized determinant of a matrix. If the matrix is not regular, one can specify the rank of the matrix and the corresponding subdeterminant is returned. This is implemented using the `eigens` function.

## trace

returns the trace of a matrix (sum of diagonals)

## vcrossp, PDL::Matrix::crossp

similar to PDL::crossp, however reflecting PDL::Matrix notations

# BUGS AND PROBLEMS

Because we change the way piddles are constructed, not all pdl operators may be applied to piddle-matrices. The inner product is not redefined. We might have missed some functions/methods. Internal consistency of our approach needs yet to be established.

# TODO

check all PDL functions, benchmarks, optimization, lots of other things ...

# `AUTHOR(S)`

Stephan Heuel (stephan@heuel.org), Christian Soeller (c.soeller@auckland.ac.nz)

# COPYRIGHT

All rights reserved. There is no warranty. You are allowed to redistribute this software / documentation under certain conditions. For details, see the file COPYING in the PDL distribution. If this file is separated from the PDL distribution, the copyright notice should be included in the file.