diff --git a/Eigen/src/SVD/JacobiSVD.h b/Eigen/src/SVD/JacobiSVD.h
index cb411232b..ebb97344e 100644
--- a/Eigen/src/SVD/JacobiSVD.h
+++ b/Eigen/src/SVD/JacobiSVD.h
@@ -52,7 +52,10 @@ template <typename MatrixType, int Options, int QRPreconditioner, int Case>
 class qr_preconditioner_impl<MatrixType, Options, QRPreconditioner, Case, false> {
  public:
   void allocate(const JacobiSVD<MatrixType, Options>&) {}
-  bool run(JacobiSVD<MatrixType, Options>&, const MatrixType&) { return false; }
+  template <typename Xpr>
+  bool run(JacobiSVD<MatrixType, Options>&, const Xpr&) {
+    return false;
+  }
 };
 
 /*** preconditioner using FullPivHouseholderQR ***/
@@ -75,8 +78,8 @@ class qr_preconditioner_impl<MatrixType, Options, FullPivHouseholderQRPreconditi
     }
     if (svd.m_computeFullU) m_workspace.resize(svd.rows());
   }
-
-  bool run(SVDType& svd, const MatrixType& matrix) {
+  template <typename Xpr>
+  bool run(SVDType& svd, const Xpr& matrix) {
     if (matrix.rows() > matrix.cols()) {
       m_qr.compute(matrix);
       svd.m_workMatrix = m_qr.matrixQR().block(0, 0, matrix.cols(), matrix.cols()).template triangularView<Upper>();
@@ -117,14 +120,12 @@ class qr_preconditioner_impl<MatrixType, Options, FullPivHouseholderQRPreconditi
       internal::destroy_at(&m_qr);
       internal::construct_at(&m_qr, svd.cols(), svd.rows());
     }
-    m_adjoint.resize(svd.cols(), svd.rows());
     if (svd.m_computeFullV) m_workspace.resize(svd.cols());
   }
-
-  bool run(SVDType& svd, const MatrixType& matrix) {
+  template <typename Xpr>
+  bool run(SVDType& svd, const Xpr& matrix) {
     if (matrix.cols() > matrix.rows()) {
-      m_adjoint = matrix.adjoint();
-      m_qr.compute(m_adjoint);
+      m_qr.compute(matrix.adjoint());
       svd.m_workMatrix =
           m_qr.matrixQR().block(0, 0, matrix.rows(), matrix.rows()).template triangularView<Upper>().adjoint();
       if (svd.m_computeFullV) m_qr.matrixQ().evalTo(svd.m_matrixV, m_workspace);
@@ -137,7 +138,6 @@ class qr_preconditioner_impl<MatrixType, Options, FullPivHouseholderQRPreconditi
  private:
   typedef FullPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
   QRType m_qr;
-  TransposeTypeWithSameStorageOrder m_adjoint;
   typename plain_row_type<MatrixType>::type m_workspace;
 };
 
@@ -167,8 +167,8 @@ class qr_preconditioner_impl<MatrixType, Options, ColPivHouseholderQRPreconditio
     else if (svd.m_computeThinU)
       m_workspace.resize(svd.cols());
   }
-
-  bool run(SVDType& svd, const MatrixType& matrix) {
+  template <typename Xpr>
+  bool run(SVDType& svd, const Xpr& matrix) {
     if (matrix.rows() > matrix.cols()) {
       m_qr.compute(matrix);
       svd.m_workMatrix = m_qr.matrixQR().block(0, 0, matrix.cols(), matrix.cols()).template triangularView<Upper>();
@@ -222,13 +222,11 @@ class qr_preconditioner_impl<MatrixType, Options, ColPivHouseholderQRPreconditio
       m_workspace.resize(svd.cols());
     else if (svd.m_computeThinV)
       m_workspace.resize(svd.rows());
-    m_adjoint.resize(svd.cols(), svd.rows());
   }
-
-  bool run(SVDType& svd, const MatrixType& matrix) {
+  template <typename Xpr>
+  bool run(SVDType& svd, const Xpr& matrix) {
     if (matrix.cols() > matrix.rows()) {
-      m_adjoint = matrix.adjoint();
-      m_qr.compute(m_adjoint);
+      m_qr.compute(matrix.adjoint());
 
       svd.m_workMatrix =
           m_qr.matrixQR().block(0, 0, matrix.rows(), matrix.rows()).template triangularView<Upper>().adjoint();
@@ -247,7 +245,6 @@ class qr_preconditioner_impl<MatrixType, Options, ColPivHouseholderQRPreconditio
  private:
   typedef ColPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
   QRType m_qr;
-  TransposeTypeWithSameStorageOrder m_adjoint;
   WorkspaceType m_workspace;
 };
 
@@ -276,8 +273,8 @@ class qr_preconditioner_impl<MatrixType, Options, HouseholderQRPreconditioner, P
     else if (svd.m_computeThinU)
       m_workspace.resize(svd.cols());
   }
-
-  bool run(SVDType& svd, const MatrixType& matrix) {
+  template <typename Xpr>
+  bool run(SVDType& svd, const Xpr& matrix) {
     if (matrix.rows() > matrix.cols()) {
       m_qr.compute(matrix);
       svd.m_workMatrix = m_qr.matrixQR().block(0, 0, matrix.cols(), matrix.cols()).template triangularView<Upper>();
@@ -330,13 +327,12 @@ class qr_preconditioner_impl<MatrixType, Options, HouseholderQRPreconditioner, P
       m_workspace.resize(svd.cols());
     else if (svd.m_computeThinV)
       m_workspace.resize(svd.rows());
-    m_adjoint.resize(svd.cols(), svd.rows());
   }
 
-  bool run(SVDType& svd, const MatrixType& matrix) {
+  template <typename Xpr>
+  bool run(SVDType& svd, const Xpr& matrix) {
     if (matrix.cols() > matrix.rows()) {
-      m_adjoint = matrix.adjoint();
-      m_qr.compute(m_adjoint);
+      m_qr.compute(matrix.adjoint());
 
       svd.m_workMatrix =
           m_qr.matrixQR().block(0, 0, matrix.rows(), matrix.rows()).template triangularView<Upper>().adjoint();
@@ -355,7 +351,6 @@ class qr_preconditioner_impl<MatrixType, Options, HouseholderQRPreconditioner, P
  private:
   typedef HouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
   QRType m_qr;
-  TransposeTypeWithSameStorageOrder m_adjoint;
   WorkspaceType m_workspace;
 };
 
@@ -509,7 +504,6 @@ class JacobiSVD : public SVDBase<JacobiSVD<MatrixType_, Options_> > {
   typedef MatrixType_ MatrixType;
   typedef typename Base::Scalar Scalar;
   typedef typename Base::RealScalar RealScalar;
-  typedef typename Base::Index Index;
   enum : int {
     Options = Options_,
     QRPreconditioner = internal::get_qr_preconditioner(Options),
@@ -654,7 +648,6 @@ class JacobiSVD : public SVDBase<JacobiSVD<MatrixType_, Options_> > {
   internal::qr_preconditioner_impl<MatrixType, Options, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols>
       m_qr_precond_morerows;
   WorkMatrixType m_workMatrix;
-  MatrixType m_scaledMatrix;
 };
 
 template <typename MatrixType, int Options>
@@ -669,7 +662,6 @@ void JacobiSVD<MatrixType, Options>::allocate(Index rows_, Index cols_, unsigned
   m_workMatrix.resize(diagSize(), diagSize());
   if (cols() > rows()) m_qr_precond_morecols.allocate(*this);
   if (rows() > cols()) m_qr_precond_morerows.allocate(*this);
-  if (rows() != cols()) m_scaledMatrix.resize(rows(), cols());
 }
 
 template <typename MatrixType, int Options>
@@ -699,9 +691,8 @@ JacobiSVD<MatrixType, Options>& JacobiSVD<MatrixType, Options>::compute_impl(con
   /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */
 
   if (rows() != cols()) {
-    m_scaledMatrix = matrix / scale;
-    m_qr_precond_morecols.run(*this, m_scaledMatrix);
-    m_qr_precond_morerows.run(*this, m_scaledMatrix);
+    m_qr_precond_morecols.run(*this, matrix / scale);
+    m_qr_precond_morerows.run(*this, matrix / scale);
   } else {
     m_workMatrix =
         matrix.template topLeftCorner<DiagSizeAtCompileTime, DiagSizeAtCompileTime>(diagSize(), diagSize()) / scale;
diff --git a/Eigen/src/SVD/SVDBase.h b/Eigen/src/SVD/SVDBase.h
index 5f04647e9..d1ad63de2 100644
--- a/Eigen/src/SVD/SVDBase.h
+++ b/Eigen/src/SVD/SVDBase.h
@@ -125,7 +125,6 @@ class SVDBase : public SolverBase<SVDBase<Derived> > {
   typedef typename MatrixType::Scalar Scalar;
   typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
   typedef typename Eigen::internal::traits<SVDBase>::StorageIndex StorageIndex;
-  typedef Eigen::Index Index;  ///< \deprecated since Eigen 3.3
 
   static constexpr bool ShouldComputeFullU = internal::traits<Derived>::ShouldComputeFullU;
   static constexpr bool ShouldComputeThinU = internal::traits<Derived>::ShouldComputeThinU;
@@ -355,11 +354,11 @@ class SVDBase : public SolverBase<SVDBase<Derived> > {
         m_isInitialized(false),
         m_isAllocated(false),
         m_usePrescribedThreshold(false),
-        m_computeFullU(false),
-        m_computeThinU(false),
-        m_computeFullV(false),
-        m_computeThinV(false),
-        m_computationOptions(0),
+        m_computeFullU(ShouldComputeFullU),
+        m_computeThinU(ShouldComputeThinU),
+        m_computeFullV(ShouldComputeFullV),
+        m_computeThinV(ShouldComputeThinV),
+        m_computationOptions(internal::traits<Derived>::Options),
         m_nonzeroSingularValues(0),
         m_rows(RowsAtCompileTime),
         m_cols(ColsAtCompileTime),