From: Benjamin Brands <benjamin.brands@fau.de>
Date: Fri, 6 Apr 2018 08:13:06 +0000 (+0200)
Subject: add ScaLAPACKMatrix::pseudoinverse()
X-Git-Tag: v9.0.0-rc1~214^2~1
X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=b390e421541fc8545676368448da826696eba220;p=dealii.git

add ScaLAPACKMatrix::pseudoinverse()
---

diff --git a/include/deal.II/lac/scalapack.h b/include/deal.II/lac/scalapack.h
index ae0c0f9b4b..c5f8b5bb9b 100644
--- a/include/deal.II/lac/scalapack.h
+++ b/include/deal.II/lac/scalapack.h
@@ -512,6 +512,26 @@ public:
   void least_squares(ScaLAPACKMatrix<NumberType> &B,
                      const bool transpose=false);
 
+  /**
+   * Compute the pseudoinverse $\mathbf{A}^+ \in \mathbb{R}^{N \times M}$ (Moore-Penrose inverse)
+   * of a real matrix $\mathbf{A} \in \mathbb{R}^{M \times N}$ using the singular value decomposition
+   * $\mathbf{A} = \mathbf{U} \cdot \mathbf{\Sigma} \cdot \mathbf{V}^T$.
+   *
+   * Unlike the inverse, the pseudoinverse $\mathbf{A}^+ = \mathbf{V} \cdot \mathbf{\Sigma}^+ \cdot \mathbf{U}^T$
+   * exists for both rectangular as well as singular matrices $\mathbf{A}$.
+   *
+   * For a rectangular $\mathbf{\Sigma}$ the pseudoinverse is computed by taking the reciprocal
+   * of each non-zero element on the diagonal, leaving the zeros in place, and then transposing $\mathbf{\Sigma}$.
+   * For the numerical computation only the singular values $\sigma_i > \sigma_{\text{max}} \, \text{ratio}$
+   * are taken into account. Upon successful exit, the function returns the number of singular values
+   * fulfilling that condition. That value can be interpreted as the rank of $\mathbf{A}$.
+   *
+   * Upon return this object contains the pseudoinverse $\mathbf{A}^+ \in \mathbb{R}^{N \times M}$.
+   *
+   * The following alignment conditions have to be fulfilled: $MB_A = NB_A$.
+   */
+  unsigned int pseudoinverse(const NumberType ratio);
+
   /**
    * Estimate the condition number of a SPD matrix in the $l_1$-norm.
    * The matrix has to be in the Cholesky state (see compute_cholesky_factorization()).
diff --git a/source/lac/scalapack.cc b/source/lac/scalapack.cc
index 671b5bd6be..05466e26cf 100644
--- a/source/lac/scalapack.cc
+++ b/source/lac/scalapack.cc
@@ -1122,6 +1122,54 @@ void ScaLAPACKMatrix<NumberType>::least_squares(ScaLAPACKMatrix<NumberType> &B,
 
 
 
+template <typename NumberType>
+unsigned int ScaLAPACKMatrix<NumberType>::pseudoinverse(const NumberType ratio)
+{
+  Assert(state == LAPACKSupport::matrix,
+         ExcMessage("Matrix has to be in Matrix state before calling this function."));
+  Assert(row_block_size==column_block_size,
+         ExcMessage("Use identical block sizes for rows and columns of matrix A"));
+  Assert(ratio>0. && ratio<1.,
+         ExcMessage("input parameter ratio has to be larger than zero and smaller than 1"));
+
+  ScaLAPACKMatrix<NumberType> U(n_rows,n_rows,grid,row_block_size,row_block_size,LAPACKSupport::Property::general);
+  ScaLAPACKMatrix<NumberType> VT(n_columns,n_columns,grid,row_block_size,row_block_size,LAPACKSupport::Property::general);
+  std::vector<NumberType> sv = this->compute_SVD(&U,&VT);
+  AssertThrow(sv[0]>std::numeric_limits<NumberType>::min(),ExcMessage("Matrix has rank 0"));
+
+  // Get number of singular values fulfilling the following: sv[i] > sv[0] * ratio
+  // Obviously, 0-th element already satisfies sv[0] > sv[0] * ratio
+  // The singular values in sv are ordered by descending value so we break out of the loop
+  // if a singular value is smaller than sv[0] * ratio.
+  unsigned int n_sv=1;
+  std::vector<NumberType> inv_sigma;
+  inv_sigma.push_back(1/sv[0]);
+
+  for (unsigned int i=1; i<sv.size(); ++i)
+    if (sv[i] > sv[0] * ratio)
+      {
+        ++n_sv;
+        inv_sigma.push_back(1/sv[i]);
+      }
+    else break;
+
+  // For the matrix multiplication we use only the columns of U and rows of VT which are associated
+  // with singular values larger than the limit.
+  // That saves computational time for matrices with rank significantly smaller than min(n_rows,n_columns)
+  ScaLAPACKMatrix<NumberType> U_R(n_rows,n_sv,grid,row_block_size,row_block_size,LAPACKSupport::Property::general);
+  ScaLAPACKMatrix<NumberType> VT_R(n_sv,n_columns,grid,row_block_size,row_block_size,LAPACKSupport::Property::general);
+  U.copy_to(U_R,std::make_pair(0,0),std::make_pair(0,0),std::make_pair(n_rows,n_sv));
+  VT.copy_to(VT_R,std::make_pair(0,0),std::make_pair(0,0),std::make_pair(n_sv,n_columns));
+
+  VT_R.scale_rows(inv_sigma);
+  this->reinit(n_columns,n_rows,this->grid,row_block_size,column_block_size,LAPACKSupport::Property::general);
+  VT_R.mult(1,U_R,0,*this,true,true);
+  state = LAPACKSupport::State::inverse_matrix;
+  return n_sv;
+}
+
+
+
 template <typename NumberType>
 NumberType ScaLAPACKMatrix<NumberType>::reciprocal_condition_number(const NumberType a_norm) const
 {