add an optimized "apply in place a rotation in the plane",

and make Jacobi and SelfAdjointEigenSolver use it
=> ~ x1.75 speedup for JacobiSVD and x2 for SelfAdjointEigenSolver

Eigen/Core

@@ -194,6 +194,7 @@ namespace Eigen {
 #include "src/Core/products/TriangularMatrixVector.h"
 #include "src/Core/products/TriangularMatrixMatrix.h"
 #include "src/Core/products/TriangularSolverMatrix.h"
+#include "src/Core/products/RotationInThePlane.h"
 #include "src/Core/BandMatrix.h"
 
 } // namespace Eigen

Eigen/src/Core/products/RotationInThePlane.h

@@ -0,0 +1,127 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2009 Gael Guennebaud <g.gael@free.fr>
+//
+// Eigen is free software; you can redistribute it and/or
+// modify it under the terms of the GNU Lesser General Public
+// License as published by the Free Software Foundation; either
+// version 3 of the License, or (at your option) any later version.
+//
+// Alternatively, you can redistribute it and/or
+// modify it under the terms of the GNU General Public License as
+// published by the Free Software Foundation; either version 2 of
+// the License, or (at your option) any later version.
+//
+// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
+// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
+// GNU General Public License for more details.
+//
+// You should have received a copy of the GNU Lesser General Public
+// License and a copy of the GNU General Public License along with
+// Eigen. If not, see <http://www.gnu.org/licenses/>.
+
+#ifndef EIGEN_ROTATION_IN_THE_PLANE_H
+#define EIGEN_ROTATION_IN_THE_PLANE_H
+
+/**********************************************************************
+* This file implement ...
+**********************************************************************/
+
+template<typename Scalar, int Incr>
+struct ei_apply_rotation_in_the_plane_selector;
+
+template<typename VectorX, typename VectorY>
+void ei_apply_rotation_in_the_plane(VectorX& x, VectorY& y, typename VectorX::Scalar c, typename VectorY::Scalar s)
+{  
+  ei_assert(x.size() == y.size());
+  int size = x.size();
+  int incrx = size ==1 ? 1 : &x.coeffRef(1) - &x.coeffRef(0);
+  int incry = size ==1 ? 1 : &y.coeffRef(1) - &y.coeffRef(0);
+  if (incrx==1 && incry==1)
+    ei_apply_rotation_in_the_plane_selector<typename VectorX::Scalar,1>
+      ::run(&x.coeffRef(0), &y.coeffRef(0), x.size(), c, s, 1, 1);
+  else
+    ei_apply_rotation_in_the_plane_selector<typename VectorX::Scalar,Dynamic>
+      ::run(&x.coeffRef(0), &y.coeffRef(0), x.size(), c, s, incrx, incry);
+}
+
+template<typename Scalar>
+struct ei_apply_rotation_in_the_plane_selector<Scalar,Dynamic>
+{
+  static EIGEN_DONT_INLINE void run(Scalar* x, Scalar* y, int size, Scalar c, Scalar s, int incrx, int incry)
+  {
+    for(int i=0; i<size; ++i)
+    {
+      Scalar xi = *x;
+      Scalar yi = *y;
+      *x = c * xi - s * yi;
+      *y = s * xi + c * yi;
+      x += incrx;
+      y += incry;
+    }
+  }
+};
+
+// both vectors are sequentially stored in memory => vectorization
+template<typename Scalar>
+struct ei_apply_rotation_in_the_plane_selector<Scalar,1>
+{
+  static EIGEN_DONT_INLINE void run(Scalar* x, Scalar* y, int size, Scalar c, Scalar s, int, int)
+  {
+    typedef typename ei_packet_traits<Scalar>::type Packet;
+    enum { PacketSize = ei_packet_traits<Scalar>::size, Peeling = 2 };
+    int alignedStart = ei_alignmentOffset(y, size);
+    int alignedEnd = alignedStart + ((size-alignedStart)/(Peeling*PacketSize))*(Peeling*PacketSize);
+
+    const Packet pc = ei_pset1(c);
+    const Packet ps = ei_pset1(s);
+    
+    for(int i=0; i<alignedStart; ++i)
+    {
+      Scalar xi = x[i];
+      Scalar yi = y[i];
+      x[i] = c * xi - s * yi;
+      y[i] = s * xi + c * yi;
+    }
+
+    Scalar* px = x + alignedStart;
+    Scalar* py = y + alignedStart;
+
+    if(ei_alignmentOffset(x, size)==alignedStart)
+      for(int i=alignedStart; i<alignedEnd; i+=PacketSize)
+      {
+        Packet xi = ei_pload(px);
+        Packet yi = ei_pload(py);
+        ei_pstore(px, ei_psub(ei_pmul(pc,xi),ei_pmul(ps,yi)));
+        ei_pstore(py, ei_padd(ei_pmul(ps,xi),ei_pmul(pc,yi)));
+        px += PacketSize;
+        py += PacketSize;
+      }
+    else
+      for(int i=alignedStart; i<alignedEnd; i+=Peeling*PacketSize)
+      {
+        Packet xi   = ei_ploadu(px);
+        Packet xi1  = ei_ploadu(px+PacketSize);
+        Packet yi   = ei_pload (py);
+        Packet yi1  = ei_pload (py+PacketSize);
+        ei_pstoreu(px, ei_psub(ei_pmul(pc,xi),ei_pmul(ps,yi)));
+        ei_pstoreu(px+PacketSize, ei_psub(ei_pmul(pc,xi1),ei_pmul(ps,yi1)));
+        ei_pstore (py, ei_padd(ei_pmul(ps,xi),ei_pmul(pc,yi)));
+        ei_pstore (py+PacketSize, ei_padd(ei_pmul(ps,xi1),ei_pmul(pc,yi1)));
+        px += Peeling*PacketSize;
+        py += Peeling*PacketSize;
+      }
+
+    for(int i=alignedEnd; i<size; ++i)
+    {
+      Scalar xi = x[i];
+      Scalar yi = y[i];
+      x[i] = c * xi - s * yi;
+      y[i] = s * xi + c * yi;
+    }
+  }
+};
+
+#endif // EIGEN_ROTATION_IN_THE_PLANE_H

Eigen/src/Jacobi/Jacobi.h

@@ -28,23 +28,17 @@
 template<typename Derived>
 void MatrixBase<Derived>::applyJacobiOnTheLeft(int p, int q, Scalar c, Scalar s)
 {
-  for(int i = 0; i < cols(); ++i)
+  RowXpr x(row(p));
-  {
+  RowXpr y(row(q));
-    Scalar tmp = coeff(p,i);
+  ei_apply_rotation_in_the_plane(x, y, c, s);
-    coeffRef(p,i) = c * tmp - s * coeff(q,i);
-    coeffRef(q,i) = s * tmp + c * coeff(q,i);
-  }
 }
 
 template<typename Derived>
 void MatrixBase<Derived>::applyJacobiOnTheRight(int p, int q, Scalar c, Scalar s)
 {
-  for(int i = 0; i < rows(); ++i)
+  ColXpr x(col(p));
-  {
+  ColXpr y(col(q));
-    Scalar tmp = coeff(i,p);
+  ei_apply_rotation_in_the_plane(x, y, c, s);
-    coeffRef(i,p) = c * tmp - s * coeff(i,q);
-    coeffRef(i,q) = s * tmp + c * coeff(i,q);
-  }
 }
 
 template<typename Scalar>

Eigen/src/QR/SelfAdjointEigenSolver.h

@@ -378,23 +378,10 @@ static void ei_tridiagonal_qr_step(RealScalar* diag, RealScalar* subdiag, int st
     if (matrixQ)
     {
       #ifdef EIGEN_DEFAULT_TO_ROW_MAJOR
+      ei_apply_rotation_in_the_plane_selector<Scalar,Dynamic>::run(matrixQ+k, matrixQ+k+1, n, c, s, n, n);
       #else
-      int kn = k*n;
+      ei_apply_rotation_in_the_plane_selector<Scalar,1>::run(matrixQ+k*n, matrixQ+k*n+n, n, c, s, 1, 1);
-      int kn1 = (k+1)*n;
       #endif
-      // let's do the product manually to avoid the need of temporaries...
-      for (int i=0; i<n; ++i)
-      {
-        #ifdef EIGEN_DEFAULT_TO_ROW_MAJOR
-        Scalar matrixQ_i_k = matrixQ[i*n+k];
-        matrixQ[i*n+k]   = c * matrixQ_i_k - s * matrixQ[i*n+k+1];
-        matrixQ[i*n+k+1] = s * matrixQ_i_k + c * matrixQ[i*n+k+1];
-        #else
-        Scalar matrixQ_i_k = matrixQ[i+kn];
-        matrixQ[i+kn]  = c * matrixQ_i_k - s * matrixQ[i+kn1];
-        matrixQ[i+kn1] = s * matrixQ_i_k + c * matrixQ[i+kn1];
-        #endif
-      }
     }
   }
 }