Merge #471 adding pseudoinverse (#583)

* Adding pseudo inverse
Squashed commit of the following:

commit a773004
Author: Arafatk <arafat@iitkpg.ac.in>
Date:   Wed Mar 23 13:33:47 2016 +0530

    removing loop

commit 9c04728
Author: Arafatk <arafat@iitkpg.ac.in>
Date:   Fri Mar 11 10:32:24 2016 +0530

    Adding Moore pseudo inverse

    Squashed commit of the following:

    commit 2d335f84de81032811b840cbabb1cb58adbbc294
    Author: Arafatk <arafat@iitkpg.ac.in>
    Date:   Fri Mar 11 10:31:07 2016 +0530

        Improving tests

    commit f17061f
    Author: Arafatk <arafat@iitkpg.ac.in>
    Date:   Sat Mar 5 12:25:59 2016 +0530

        Removing un-necessary spaces

    commit fa908ac
    Author: Arafatk <arafat@iitkpg.ac.in>
    Date:   Fri Mar 4 10:33:23 2016 +0530

        Corrections to spec tests

    commit 9d9ab35
    Author: Arafatk <arafat@iitkpg.ac.in>
    Date:   Tue Mar 1 20:40:33 2016 +0530

        Adding Moore-Penrose pseudo-inverse of a matrix
        Squashed commit of the following:

        commit 8c02601d5cd065071c4378c16a5a525477033b87
        Author: Arafatk <arafat@iitkpg.ac.in>
        Date:   Tue Mar 1 20:39:17 2016 +0530

            correcting Docs

        commit b8b7f1f
        Author: Arafatk <arafat@iitkpg.ac.in>
        Date:   Mon Feb 29 23:55:14 2016 +0530

            Squashed commit of the following:

            commit fd2cba0
            Author: Arafatk <arafat@iitkpg.ac.in>
            Date:   Mon Feb 29 23:52:33 2016 +0530

                correcting docs

            commit 87eeaae
            Author: Arafatk <arafat@iitkpg.ac.in>
            Date:   Mon Feb 29 23:26:17 2016 +0530

                Adding complex data type

            commit f3950b1
            Author: Arafatk <arafat@iitkpg.ac.in>
            Date:   Sun Feb 28 09:39:05 2016 +0530

                Adding Moore-Penrose pseudo-inverse of a matrix
                Squashed commit of the following:

                commit dbba0c1
                Author: Arafatk <arafat@iitkpg.ac.in>
                Date:   Sat Feb 27 23:20:31 2016 +0530

                    Adding Pseudo inverse

                commit 32a49e6
                Author: Arafatk <arafat@iitkpg.ac.in>
                Date:   Sat Feb 27 22:24:19 2016 +0530

                    correcting error in docs

                commit 8031cc8
                Author: Arafatk <arafat@iitkpg.ac.in>
                Date:   Sat Feb 27 21:06:55 2016 +0530

                    Correcting load error

                commit 4b0b375
                Author: Arafatk <arafat@iitkpg.ac.in>
                Date:   Sat Feb 27 18:59:25 2016 +0530

                    Correcting Load error in Spec Tests

                commit 2fffafc
                Author: Arafatk <arafat@iitkpg.ac.in>
                Date:   Sat Feb 27 17:22:08 2016 +0530

                    Improving Docs
                    Adding Tests

                commit 6e0ab74
                Author: Arafatk <arafat@iitkpg.ac.in>
                Date:   Wed Feb 24 22:28:21 2016 +0530

                    Adding Moore-Penrose pseudo-inverse of a matrix

* Added pendings for jruby. Also made sure pendings were marked for object dtypes (instead of just using `next`)

Author: translunar

lib/nmatrix/math.rb

@@ -112,6 +112,71 @@ def invert
   end
   alias :inverse :invert
 
+
+  #
+  # call-seq:
+  #     pinv -> NMatrix
+  #
+  # Compute the Moore-Penrose pseudo-inverse of a matrix using its
+  # singular value decomposition (SVD).
+  #
+  # This function requires the nmatrix-atlas gem installed.
+  #
+  # * *Arguments* :
+  #  - +tolerance(optional)+ -> Cutoff for small singular values.
+  #
+  # * *Returns* :
+  #   -  Pseudo-inverse matrix.
+  #
+  # * *Raises* :
+  #   - +NotImplementedError+ -> If called without nmatrix-atlas or nmatrix-lapacke gem.
+  #   - +TypeError+ -> If called without float or complex data type.
+  #
+  # * *Examples* :
+  #
+  #  a = NMatrix.new([2,2],[1,2,
+  #                         3,4], dtype: :float64)
+  #  a.pinv # => [ [-2.0000000000000018, 1.0000000000000007]
+  #                [1.5000000000000016, -0.5000000000000008] ]
+  #
+  #  b = NMatrix.new([4,1],[1,2,3,4], dtype: :float64)
+  #  b.pinv # => [ [ 0.03333333, 0.06666667, 0.99999999, 0.13333333] ]
+  #
+  # == References
+  #
+  # * https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse
+  # * G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press
+  #
+  def pinv(tolerance = 1e-15)
+    raise DataTypeError, "pinv works only with matrices of float or complex data type" unless
+      [:float32, :float64, :complex64, :complex128].include?(dtype)
+    if self.complex_dtype?
+      u, s, vt = self.complex_conjugate.gesvd # singular value decomposition
+    else
+      u, s, vt = self.gesvd
+    end
+    rows = self.shape[0]
+    cols = self.shape[1]
+    if rows < cols
+      u_reduced = u
+      vt_reduced = vt[0..rows - 1, 0..cols - 1].transpose
+    else
+      u_reduced = u[0..rows - 1, 0..cols - 1]
+      vt_reduced = vt.transpose
+    end
+    largest_singular_value = s.max.to_f
+    cutoff = tolerance * largest_singular_value
+    (0...[rows, cols].min).each do |i|
+      s[i] = 1 / s[i] if s[i] > cutoff
+      s[i] = 0        if s[i] <= cutoff
+    end
+    multiplier = u_reduced.dot(NMatrix.diagonal(s.to_a)).transpose
+    vt_reduced.dot(multiplier)
+  end
+  alias :pseudo_inverse :pinv
+  alias :pseudoinverse :pinv
+
+
   #
   # call-seq:
   #     adjugate! -> NMatrix