1 files changed, 199 insertions, 7 deletions

diff --git a/eigen/test/qr_colpivoting.cpp b/eigen/test/qr_colpivoting.cpp
index eb3feac..26ed27f 100644
--- a/eigen/test/qr_colpivoting.cpp
+++ b/eigen/test/qr_colpivoting.cpp
@@ -10,21 +10,103 @@
 
 #include "main.h"
 #include <Eigen/QR>
+#include <Eigen/SVD>
+
+template <typename MatrixType>
+void cod() {
+  typedef typename MatrixType::Index Index;
+
+  Index rows = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
+  Index cols = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
+  Index cols2 = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
+  Index rank = internal::random<Index>(1, (std::min)(rows, cols) - 1);
+
+  typedef typename MatrixType::Scalar Scalar;
+  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime,
+                 MatrixType::RowsAtCompileTime>
+      MatrixQType;
+  MatrixType matrix;
+  createRandomPIMatrixOfRank(rank, rows, cols, matrix);
+  CompleteOrthogonalDecomposition<MatrixType> cod(matrix);
+  VERIFY(rank == cod.rank());
+  VERIFY(cols - cod.rank() == cod.dimensionOfKernel());
+  VERIFY(!cod.isInjective());
+  VERIFY(!cod.isInvertible());
+  VERIFY(!cod.isSurjective());
+
+  MatrixQType q = cod.householderQ();
+  VERIFY_IS_UNITARY(q);
+
+  MatrixType z = cod.matrixZ();
+  VERIFY_IS_UNITARY(z);
+
+  MatrixType t;
+  t.setZero(rows, cols);
+  t.topLeftCorner(rank, rank) =
+      cod.matrixT().topLeftCorner(rank, rank).template triangularView<Upper>();
+
+  MatrixType c = q * t * z * cod.colsPermutation().inverse();
+  VERIFY_IS_APPROX(matrix, c);
+
+  MatrixType exact_solution = MatrixType::Random(cols, cols2);
+  MatrixType rhs = matrix * exact_solution;
+  MatrixType cod_solution = cod.solve(rhs);
+  VERIFY_IS_APPROX(rhs, matrix * cod_solution);
+
+  // Verify that we get the same minimum-norm solution as the SVD.
+  JacobiSVD<MatrixType> svd(matrix, ComputeThinU | ComputeThinV);
+  MatrixType svd_solution = svd.solve(rhs);
+  VERIFY_IS_APPROX(cod_solution, svd_solution);
+
+  MatrixType pinv = cod.pseudoInverse();
+  VERIFY_IS_APPROX(cod_solution, pinv * rhs);
+}
+
+template <typename MatrixType, int Cols2>
+void cod_fixedsize() {
+  enum {
+    Rows = MatrixType::RowsAtCompileTime,
+    Cols = MatrixType::ColsAtCompileTime
+  };
+  typedef typename MatrixType::Scalar Scalar;
+  int rank = internal::random<int>(1, (std::min)(int(Rows), int(Cols)) - 1);
+  Matrix<Scalar, Rows, Cols> matrix;
+  createRandomPIMatrixOfRank(rank, Rows, Cols, matrix);
+  CompleteOrthogonalDecomposition<Matrix<Scalar, Rows, Cols> > cod(matrix);
+  VERIFY(rank == cod.rank());
+  VERIFY(Cols - cod.rank() == cod.dimensionOfKernel());
+  VERIFY(cod.isInjective() == (rank == Rows));
+  VERIFY(cod.isSurjective() == (rank == Cols));
+  VERIFY(cod.isInvertible() == (cod.isInjective() && cod.isSurjective()));
+
+  Matrix<Scalar, Cols, Cols2> exact_solution;
+  exact_solution.setRandom(Cols, Cols2);
+  Matrix<Scalar, Rows, Cols2> rhs = matrix * exact_solution;
+  Matrix<Scalar, Cols, Cols2> cod_solution = cod.solve(rhs);
+  VERIFY_IS_APPROX(rhs, matrix * cod_solution);
+
+  // Verify that we get the same minimum-norm solution as the SVD.
+  JacobiSVD<MatrixType> svd(matrix, ComputeFullU | ComputeFullV);
+  Matrix<Scalar, Cols, Cols2> svd_solution = svd.solve(rhs);
+  VERIFY_IS_APPROX(cod_solution, svd_solution);
+}
 
 template<typename MatrixType> void qr()
 {
+  using std::sqrt;
   typedef typename MatrixType::Index Index;
 
   Index rows = internal::random<Index>(2,EIGEN_TEST_MAX_SIZE), cols = internal::random<Index>(2,EIGEN_TEST_MAX_SIZE), cols2 = internal::random<Index>(2,EIGEN_TEST_MAX_SIZE);
   Index rank = internal::random<Index>(1, (std::min)(rows, cols)-1);
 
   typedef typename MatrixType::Scalar Scalar;
+  typedef typename MatrixType::RealScalar RealScalar;
   typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime> MatrixQType;
   MatrixType m1;
   createRandomPIMatrixOfRank(rank,rows,cols,m1);
   ColPivHouseholderQR<MatrixType> qr(m1);
-  VERIFY(rank == qr.rank());
-  VERIFY(cols - qr.rank() == qr.dimensionOfKernel());
+  VERIFY_IS_EQUAL(rank, qr.rank());
+  VERIFY_IS_EQUAL(cols - qr.rank(), qr.dimensionOfKernel());
   VERIFY(!qr.isInjective());
   VERIFY(!qr.isInvertible());
   VERIFY(!qr.isSurjective());
@@ -36,26 +118,59 @@ template<typename MatrixType> void qr()
   MatrixType c = q * r * qr.colsPermutation().inverse();
   VERIFY_IS_APPROX(m1, c);
 
+  // Verify that the absolute value of the diagonal elements in R are
+  // non-increasing until they reach the singularity threshold.
+  RealScalar threshold =
+      sqrt(RealScalar(rows)) * numext::abs(r(0, 0)) * NumTraits<Scalar>::epsilon();
+  for (Index i = 0; i < (std::min)(rows, cols) - 1; ++i) {
+    RealScalar x = numext::abs(r(i, i));
+    RealScalar y = numext::abs(r(i + 1, i + 1));
+    if (x < threshold && y < threshold) continue;
+    if (!test_isApproxOrLessThan(y, x)) {
+      for (Index j = 0; j < (std::min)(rows, cols); ++j) {
+        std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
+      }
+      std::cout << "Failure at i=" << i << ", rank=" << rank
+                << ", threshold=" << threshold << std::endl;
+    }
+    VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
+  }
+
   MatrixType m2 = MatrixType::Random(cols,cols2);
   MatrixType m3 = m1*m2;
   m2 = MatrixType::Random(cols,cols2);
   m2 = qr.solve(m3);
   VERIFY_IS_APPROX(m3, m1*m2);
+
+  {
+    Index size = rows;
+    do {
+      m1 = MatrixType::Random(size,size);
+      qr.compute(m1);
+    } while(!qr.isInvertible());
+    MatrixType m1_inv = qr.inverse();
+    m3 = m1 * MatrixType::Random(size,cols2);
+    m2 = qr.solve(m3);
+    VERIFY_IS_APPROX(m2, m1_inv*m3);
+  }
 }
 
 template<typename MatrixType, int Cols2> void qr_fixedsize()
 {
+  using std::sqrt;
+  using std::abs;
   enum { Rows = MatrixType::RowsAtCompileTime, Cols = MatrixType::ColsAtCompileTime };
   typedef typename MatrixType::Scalar Scalar;
+  typedef typename MatrixType::RealScalar RealScalar;
   int rank = internal::random<int>(1, (std::min)(int(Rows), int(Cols))-1);
   Matrix<Scalar,Rows,Cols> m1;
   createRandomPIMatrixOfRank(rank,Rows,Cols,m1);
   ColPivHouseholderQR<Matrix<Scalar,Rows,Cols> > qr(m1);
-  VERIFY(rank == qr.rank());
-  VERIFY(Cols - qr.rank() == qr.dimensionOfKernel());
-  VERIFY(qr.isInjective() == (rank == Rows));
-  VERIFY(qr.isSurjective() == (rank == Cols));
-  VERIFY(qr.isInvertible() == (qr.isInjective() && qr.isSurjective()));
+  VERIFY_IS_EQUAL(rank, qr.rank());
+  VERIFY_IS_EQUAL(Cols - qr.rank(), qr.dimensionOfKernel());
+  VERIFY_IS_EQUAL(qr.isInjective(), (rank == Rows));
+  VERIFY_IS_EQUAL(qr.isSurjective(), (rank == Cols));
+  VERIFY_IS_EQUAL(qr.isInvertible(), (qr.isInjective() && qr.isSurjective()));
 
   Matrix<Scalar,Rows,Cols> r = qr.matrixQR().template triangularView<Upper>();
   Matrix<Scalar,Rows,Cols> c = qr.householderQ() * r * qr.colsPermutation().inverse();
@@ -66,6 +181,71 @@ template<typename MatrixType, int Cols2> void qr_fixedsize()
   m2 = Matrix<Scalar,Cols,Cols2>::Random(Cols,Cols2);
   m2 = qr.solve(m3);
   VERIFY_IS_APPROX(m3, m1*m2);
+  // Verify that the absolute value of the diagonal elements in R are
+  // non-increasing until they reache the singularity threshold.
+  RealScalar threshold =
+      sqrt(RealScalar(Rows)) * (std::abs)(r(0, 0)) * NumTraits<Scalar>::epsilon();
+  for (Index i = 0; i < (std::min)(int(Rows), int(Cols)) - 1; ++i) {
+    RealScalar x = numext::abs(r(i, i));
+    RealScalar y = numext::abs(r(i + 1, i + 1));
+    if (x < threshold && y < threshold) continue;
+    if (!test_isApproxOrLessThan(y, x)) {
+      for (Index j = 0; j < (std::min)(int(Rows), int(Cols)); ++j) {
+        std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
+      }
+      std::cout << "Failure at i=" << i << ", rank=" << rank
+                << ", threshold=" << threshold << std::endl;
+    }
+    VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
+  }
+}
+
+// This test is meant to verify that pivots are chosen such that
+// even for a graded matrix, the diagonal of R falls of roughly
+// monotonically until it reaches the threshold for singularity.
+// We use the so-called Kahan matrix, which is a famous counter-example
+// for rank-revealing QR. See
+// http://www.netlib.org/lapack/lawnspdf/lawn176.pdf
+// page 3 for more detail.
+template<typename MatrixType> void qr_kahan_matrix()
+{
+  using std::sqrt;
+  using std::abs;
+  typedef typename MatrixType::Index Index;
+  typedef typename MatrixType::Scalar Scalar;
+  typedef typename MatrixType::RealScalar RealScalar;
+
+  Index rows = 300, cols = rows;
+
+  MatrixType m1;
+  m1.setZero(rows,cols);
+  RealScalar s = std::pow(NumTraits<RealScalar>::epsilon(), 1.0 / rows);
+  RealScalar c = std::sqrt(1 - s*s);
+  RealScalar pow_s_i(1.0); // pow(s,i)
+  for (Index i = 0; i < rows; ++i) {
+    m1(i, i) = pow_s_i;
+    m1.row(i).tail(rows - i - 1) = -pow_s_i * c * MatrixType::Ones(1, rows - i - 1);
+    pow_s_i *= s;
+  }
+  m1 = (m1 + m1.transpose()).eval();
+  ColPivHouseholderQR<MatrixType> qr(m1);
+  MatrixType r = qr.matrixQR().template triangularView<Upper>();
+
+  RealScalar threshold =
+      std::sqrt(RealScalar(rows)) * numext::abs(r(0, 0)) * NumTraits<Scalar>::epsilon();
+  for (Index i = 0; i < (std::min)(rows, cols) - 1; ++i) {
+    RealScalar x = numext::abs(r(i, i));
+    RealScalar y = numext::abs(r(i + 1, i + 1));
+    if (x < threshold && y < threshold) continue;
+    if (!test_isApproxOrLessThan(y, x)) {
+      for (Index j = 0; j < (std::min)(rows, cols); ++j) {
+        std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
+      }
+      std::cout << "Failure at i=" << i << ", rank=" << qr.rank()
+                << ", threshold=" << threshold << std::endl;
+    }
+    VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
+  }
 }
 
 template<typename MatrixType> void qr_invertible()
@@ -132,6 +312,15 @@ void test_qr_colpivoting()
   }
 
   for(int i = 0; i < g_repeat; i++) {
+    CALL_SUBTEST_1( cod<MatrixXf>() );
+    CALL_SUBTEST_2( cod<MatrixXd>() );
+    CALL_SUBTEST_3( cod<MatrixXcd>() );
+    CALL_SUBTEST_4(( cod_fixedsize<Matrix<float,3,5>, 4 >() ));
+    CALL_SUBTEST_5(( cod_fixedsize<Matrix<double,6,2>, 3 >() ));
+    CALL_SUBTEST_5(( cod_fixedsize<Matrix<double,1,1>, 1 >() ));
+  }
+
+  for(int i = 0; i < g_repeat; i++) {
     CALL_SUBTEST_1( qr_invertible<MatrixXf>() );
     CALL_SUBTEST_2( qr_invertible<MatrixXd>() );
     CALL_SUBTEST_6( qr_invertible<MatrixXcf>() );
@@ -147,4 +336,7 @@ void test_qr_colpivoting()
 
   // Test problem size constructors
   CALL_SUBTEST_9(ColPivHouseholderQR<MatrixXf>(10, 20));
+
+  CALL_SUBTEST_1( qr_kahan_matrix<MatrixXf>() );
+  CALL_SUBTEST_2( qr_kahan_matrix<MatrixXd>() );
 }