= Linalg - Ruby Linear Algebra Library

A Fortran-based linear algebra package.

=== Features

Major features:
* Cholesky decomposition
* LU decomposition
* QR decomposition
* Schur decomposition
* Singular value decomposition
* Eigenvalues and eigenvectors of a general matrix
* Minimization by least squares
* Linear equation solving
* Stand-alone LAPACK bindings: call any LAPACK routine from directly from ruby.

Minor features:
* Convenient iterators
* Condition numbers and condition number estimates
* Nullspace, rank, nullity
* Inverse
* Pseudo-inverse
* Determinant
* 2-norm, 1-norm, infinity-norm, Frobenius norm

=== Getting Started

Everything you need to know is in:
* Linalg::DMatrix  
* Linalg::Iterators

and this README.

=== Tutorial

  $ irb
  irb(main):000:0> require 'linalg'
  => true
  irb(main):000:0> include Linalg
  => Object

==== Construction

  irb(main):000:0> DMatrix[[1,2,3], [4,5,6]]
  => 
    1.000000  2.000000  3.000000
    4.000000  5.000000  6.000000
  
  irb(main):000:0> DMatrix.rows [[1,2,3], [4,5,6]]
  => 
    1.000000  2.000000  3.000000
    4.000000  5.000000  6.000000
  
  irb(main):000:0> DMatrix.columns [[1,2,3], [4,5,6]]
  => 
    1.000000  4.000000
    2.000000  5.000000
    3.000000  6.000000

  irb(main):000:0> a = DMatrix.new(3, 3) { |i, j| 10*i + j }
  => 
    0.000000  1.000000  2.000000
   10.000000 11.000000 12.000000
   20.000000 21.000000 22.000000

  irb(main):000:0> DMatrix.new(3, 3, 99)          
  => 
   99.000000 99.000000 99.000000
   99.000000 99.000000 99.000000
   99.000000 99.000000 99.000000

  irb(main):000:0> DMatrix.diagonal [3,4,5]
  => 
    3.000000  0.000000  0.000000
    0.000000  4.000000  0.000000
    0.000000  0.000000  5.000000

  irb(main):000:0> DMatrix.diagonal(4) { |i| i*i }
  => 
    0.000000  0.000000  0.000000  0.000000
    0.000000  1.000000  0.000000  0.000000
    0.000000  0.000000  4.000000  0.000000
    0.000000  0.000000  0.000000  9.000000

  irb(main):000:0> DMatrix.diagonal(4, 99)
  => 
   99.000000  0.000000  0.000000  0.000000
    0.000000 99.000000  0.000000  0.000000
    0.000000  0.000000 99.000000  0.000000
    0.000000  0.000000  0.000000 99.000000

==== Indexing

Indexing is in (row, column) order.  This is the convention for
Mathematics and Fortran.  It is opposite from C convention.

  irb(main):000:0> a
  => 
    0.000000  1.000000  2.000000
   10.000000 11.000000 12.000000
   20.000000 21.000000 22.000000
  
  irb(main):000:0> a[1,0]
  => 10.0
  irb(main):000:0> a[2,0]
  => 20.0
  irb(main):000:0> a[0,1]
  => 1.0
  irb(main):000:0> a[0,2]
  => 2.0

Index boundaries are strongly enforced
  irb(main):000:0> a[-1,0]   
  IndexError: out of range
          from (irb):27:in `[]'
          from (irb):27

==== Enumerables

There are several abstract <tt>Enumerable</tt>s which you may obtain
from a matrix: columns, rows, elements, and diagonal elements.
  
  irb(main):000:0> a
  => 
    0.000000  1.000000  2.000000
   10.000000 11.000000 12.000000
   20.000000 21.000000 22.000000
  
  irb(main):000:0> a.columns.class
  => Linalg::Iterators::ColumnEnum
  irb(main):000:0> cols = a.columns.map { |x| x }
  => [
    0.000000
   10.000000
   20.000000
  , 
    1.000000
   11.000000
   21.000000
  , 
    2.000000
   12.000000
   22.000000
  ]
  irb(main):000:0> rows = a.rows.map { |x| x }
  => [
    0.000000  1.000000  2.000000
  , 
   10.000000 11.000000 12.000000
  , 
   20.000000 21.000000 22.000000
  ]
  irb(main):000:0> a.elems.map { |x| x }
  => [0.0, 10.0, 20.0, 1.0, 11.0, 21.0, 2.0, 12.0, 22.0]
  irb(main):003:0> a.elems.find_all { |x| x > 10 }
  => [20.0, 11.0, 21.0, 12.0, 22.0]
  irb(main):008:0> a.diags.map { |x| x }
  => [0.0, 11.0, 22.0]

Another method of constructing a matrix is to join rows or columns,

  irb(main):000:0> DMatrix.join_columns [cols[0], cols[2]]
  => 
    0.000000  2.000000
   10.000000 12.000000
   20.000000 22.000000

  irb(main):000:0> DMatrix.join_rows [rows[0], rows[2]]
  => 
    0.000000  1.000000  2.000000
   20.000000 21.000000 22.000000

==== Enumerable-like Iterators with Index Pairs

A matrix itself is not +Enumerable+, but a select number of
+Enumerable+-like methods are provided.

  irb(main):000:0> a
  => 
    0.000000  1.000000  2.000000
   10.000000 11.000000 12.000000
   20.000000 21.000000 22.000000
  
  irb(main):000:0> a.each_with_index { |e, i, j| puts "row #{i} column #{j} : #{e}" } ; nil
  row 0 column 0 : 0.0
  row 1 column 0 : 10.0
  row 2 column 0 : 20.0
  row 0 column 1 : 1.0
  row 1 column 1 : 11.0
  row 2 column 1 : 21.0
  row 0 column 2 : 2.0
  row 1 column 2 : 12.0
  row 2 column 2 : 22.0
  => nil
  irb(main):000:0> a.map_with_index { |e, i, j| e*i*j }
  => 
    0.000000  0.000000  0.000000
    0.000000 11.000000 24.000000
    0.000000 42.000000 88.000000

  irb(main):000:0> a.each_upper_with_index { |e, i, j| puts "a[#{i}, #{j}] : #{e}" } ; nil
  a[0, 1] : 1.0
  a[0, 2] : 2.0
  a[1, 2] : 12.0
  => nil
  irb(main):000:0> a.each_lower_with_index { |e, i, j| puts "a[#{i}, #{j}] : #{e}" } ; nil
  a[1, 0] : 10.0
  a[2, 0] : 20.0
  a[2, 1] : 21.0
  => nil

==== Epsilon Comparison

For good and bad, a default epsilon of <tt>1e-8</tt> is provided for
comparison, nullspace identification, and symmetric testing.

You can change +default_epsilon+ class-wide or on a per-object basis,
or simply pass an explicit epsilon to any of these methods.

  irb(main):000:0> a = DMatrix.rand(3, 3)
  => 
   0.824730  0.305527  0.044433 
  -0.582865 -0.351364 -0.752941 
   0.103417 -0.254290  0.216312 
  
  irb(main):000:0> b = a.map { |e| e + 0.000001 }
  => 
   0.824731  0.305528  0.044434 
  -0.582864 -0.351363 -0.752940 
   0.103418 -0.254289  0.216313 
  
  irb(main):000:0> a.within(1e-4, b)
  => true
  irb(main):000:0> a.class.default_epsilon
  => 1.0e-08
  irb(main):000:0> a =~ b
  => false
  irb(main):000:0> a.singleton_class.default_epsilon
  => nil
  irb(main):000:0> a.singleton_class.default_epsilon = 0.0001
  => 0.0001
  irb(main):000:0> a =~ b
  => true
  irb(main):000:0> b =~ a
  => false
  
<tt>singleton_class.epsilon</tt> has first preference over
<tt>class.epsilon</tt>.

==== Singular Value Decomposition

  irb(main):000:0> a = DMatrix.rand(4, 7) ;
  irb(main):000:0* u, s, vt = a.singular_value_decomposition
  => [
  -0.747003  0.304315 -0.144972 -0.573029 
  -0.435034 -0.814506  0.381951  0.037926 
   0.207010 -0.490811 -0.777727 -0.333753 
  -0.458125  0.055467 -0.477741  0.747535 
  , 
   2.186983  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000 
   0.000000  1.719562  0.000000  0.000000  0.000000  0.000000  0.000000 
   0.000000  0.000000  1.474243  0.000000  0.000000  0.000000  0.000000 
   0.000000  0.000000  0.000000  0.676138  0.000000  0.000000  0.000000 
  , 
   0.276463 -0.345917  0.573929 -0.416910 -0.139648  0.526564  0.062694 
  -0.838456 -0.430716  0.199266  0.170144  0.087954  0.077829  0.170372 
  -0.022382 -0.403705 -0.054483 -0.077530 -0.042506 -0.160232 -0.894461 
  -0.079171  0.476099  0.396614  0.253131  0.573392  0.315335 -0.342737 
   0.034285 -0.212892 -0.265934 -0.537325  0.749204 -0.111692  0.142407 
  -0.377150  0.344083 -0.443538 -0.422834 -0.235075  0.530130 -0.165989 
  -0.265280  0.376113  0.450755 -0.509553 -0.160031 -0.545940 -0.041002 
  ]
  irb(main):000:0> u*s*vt
  => 
  -0.854950  0.241548 -0.975366  0.688628  0.061092 -0.907441  0.310691 
   0.896671  0.717254 -0.845643  0.121186  0.000445 -0.692125 -0.810721 
   0.876331  0.562343  0.064623 -0.300574 -0.218113  0.285261  0.987489 
  -0.381214  0.830467 -0.317184  0.616481  0.468055 -0.247913  0.410178 
  
  irb(main):000:0> a
  => 
  -0.854950  0.241548 -0.975366  0.688628  0.061092 -0.907441  0.310691 
   0.896671  0.717254 -0.845643  0.121186  0.000445 -0.692125 -0.810721 
   0.876331  0.562343  0.064623 -0.300574 -0.218113  0.285261  0.987489 
  -0.381214  0.830467 -0.317184  0.616481  0.468055 -0.247913  0.410178 
  
  irb(main):000:0> u*u.t
  => 
   1.000000  0.000000 -0.000000  0.000000 
   0.000000  1.000000  0.000000  0.000000 
  -0.000000  0.000000  1.000000  0.000000 
   0.000000  0.000000  0.000000  1.000000 
  
  irb(main):000:0> vt.t*vt
  => 
   1.000000  0.000000  0.000000  0.000000  0.000000  0.000000 -0.000000 
   0.000000  1.000000 -0.000000 -0.000000  0.000000 -0.000000  0.000000 
   0.000000 -0.000000  1.000000 -0.000000 -0.000000  0.000000  0.000000 
   0.000000 -0.000000 -0.000000  1.000000 -0.000000 -0.000000  0.000000 
   0.000000  0.000000 -0.000000 -0.000000  1.000000 -0.000000  0.000000 
   0.000000 -0.000000  0.000000 -0.000000 -0.000000  1.000000 -0.000000 
  -0.000000  0.000000  0.000000  0.000000  0.000000 -0.000000  1.000000 
  
==== Eigenvectors and Eigenvalues

  irb(main):000:0> a = DMatrix.rand(5, 5)
  => 
  -0.319566  0.633985  0.335298 -0.150403  0.758559 
  -0.633389  0.444269  0.375873  0.521107  0.247966 
   0.757654  0.504831  0.160970 -0.241885 -0.949746 
  -0.174517  0.351239 -0.600079 -0.533921  0.851118 
  -0.736717  0.006612 -0.941311 -0.417801  0.555841 
  
  irb(main):000:0> eigs, re, im = a.eigensystem   
  => [
  -0.232169 -0.540550 -0.215201 -0.216959  0.036677 
  -0.449048 -0.031420 -0.114366 -0.228818  0.062841 
   0.682632  0.283575  0.019432  0.540907  0.073854 
   0.364843 -0.597905  0.000000 -0.592036  0.000000 
   0.381257 -0.207285 -0.407649 -0.490737 -0.076896 
  , 
  -1.088525 
  -0.093563 
  -0.093563 
   0.791621 
   0.791621 
  , 
   0.000000 
   0.604163 
  -0.604163 
   0.158934 
  -0.158934 
  ]
  irb(main):000:0> a*eigs.column(0)
  => 
   0.252722 
   0.488799 
  -0.743062 
  -0.397141 
  -0.415008 
  
  irb(main):000:0> re[0]*eigs.column(0)
  => 
   0.252722 
   0.488799 
  -0.743062 
  -0.397141 
  -0.415008 

==== QR factorization

  irb(main):000:0> a = DMatrix.rand(4, 7) ;
  irb(main):000:0* q, r = a.qr
  => [
   -0.593983  0.263367 -0.572195 -0.500414
   -0.486715 -0.780258  0.331047 -0.211458
   -0.184137 -0.328403 -0.586889  0.716803
   -0.613503  0.462588  0.467507  0.437109
  , 
   -1.180464 -0.295897 -0.478813 -0.300434  0.360094  0.738799 -0.571687
    0.000000  0.873800 -0.167873  0.612545  0.462280  0.097813 -0.733398
    0.000000  0.000000  1.659791 -0.439355 -0.234388 -0.063566  0.149709
    0.000000  0.000000  0.000000 -0.047795  0.805122 -0.158261  0.754104
  ]
  irb(main):000:0> q*r
  => 
    0.701176  0.405888 -0.709530  0.615091 -0.360919 -0.297505 -0.316607
    0.574550 -0.537772  0.913498 -0.467057 -0.783803 -0.423482  0.740588
    0.217367 -0.232473 -0.830816  0.077752  0.496553 -0.244298  0.798800
    0.724218  0.585743  0.992061  0.241379  0.235274 -0.506903  0.411086
  
  irb(main):000:0> a
  => 
    0.701176  0.405888 -0.709530  0.615091 -0.360919 -0.297505 -0.316607
    0.574550 -0.537772  0.913498 -0.467057 -0.783803 -0.423482  0.740588
    0.217367 -0.232473 -0.830816  0.077752  0.496553 -0.244298  0.798800
    0.724218  0.585743  0.992061  0.241379  0.235274 -0.506903  0.411086
  
  irb(main):000:0> q.t*q
  => 
    1.000000  0.000000 -0.000000 -0.000000
    0.000000  1.000000  0.000000 -0.000000
   -0.000000  0.000000  1.000000  0.000000
   -0.000000 -0.000000  0.000000  1.000000

See the Linalg::DMatrix documentation for more info.  The various
tests in test/ are also instructive.

=== Download

* http://rubyforge.org/frs/?group_id=273

=== Repository

* https://github.com/quix/linalg

=== Notes

There are four matrix types: +SMatrix+, +DMatrix+, +CMatrix+, and
+ZMatrix+ -- single precision, double precision, single precision
complex, and double precision complex, respectively.  They are all
available with basic functionality, however the more complex routines
you see here currently lie only in +DMatrix+.

If you have used +narray+, note that +linalg+ uses the mathematical
definition of <em>rank</em>, which is equal to the number of columns
only in the case of a nonsingular square matrix.

=== Details

Author::   James M. Lawrence <[email protected]>
Requires:: Ruby 1.8.1 or later
License::  Copyright (c) 2004-2008 James M. Lawrence.
           Released under the MIT license.

=== License

Copyright (c) 2004-2008 James M. Lawrence

If +linalg+ begins to smoke, get away immediately.  Seek shelter and
cover head.

Permission is hereby granted, free of charge, to any person obtaining
a copy of this software and associated documentation files (the
"Software"), to deal in the Software without restriction, including
without limitation the rights to use, copy, modify, merge, publish,
distribute, sublicense, and/or sell copies of the Software, and to
permit persons to whom the Software is furnished to do so, subject to
the following conditions:

The above copyright notice and this permission notice shall be
included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.