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:

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 Enumerables 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 1e-8 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

singleton_class.epsilon has first preference over class.epsilon.

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

rubyforge.org/frs/?group_id=273

Repository

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 rank, which is equal to the number of columns only in the case of a nonsingular square matrix.

Details

Author:	James M. Lawrence <quixoticsycophant@gmail.com>
Requires:	Ruby 1.8.1 or later
License:	Copyright (c) 2004-2008 James M. Lawrence. Released under the MIT license.

License

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.