Block transposeInPlace() when the matrix is real and square. This yields a large speedup because we transpose in registers (or L1 if we spill), instead of one packet at a time, which in the worst case makes the code write to the same cache line PacketSize times instead of once.

rmlarsen@rmlarsen4:.../eigen_bench/google3$ benchy --benchmarks=.*TransposeInPlace.*float.* --reference=srcfs experimental/users/rmlarsen/bench:matmul_bench
 10 / 10 [====================================================================================================================================================================================================================] 100.00% 2m50s
(Generated by http://go/benchy. Settings: --runs 5 --benchtime 1s --reference "srcfs" --benchmarks ".*TransposeInPlace.*float.*" experimental/users/rmlarsen/bench:matmul_bench)

name                                       old time/op             new time/op             delta
BM_TransposeInPlace<float>/4               9.84ns ± 0%             6.51ns ± 0%  -33.80%          (p=0.008 n=5+5)
BM_TransposeInPlace<float>/8               23.6ns ± 1%             17.6ns ± 0%  -25.26%          (p=0.016 n=5+4)
BM_TransposeInPlace<float>/16              78.8ns ± 0%             60.3ns ± 0%  -23.50%          (p=0.029 n=4+4)
BM_TransposeInPlace<float>/32               302ns ± 0%              229ns ± 0%  -24.40%          (p=0.008 n=5+5)
BM_TransposeInPlace<float>/59              1.03µs ± 0%             0.84µs ± 1%  -17.87%          (p=0.016 n=5+4)
BM_TransposeInPlace<float>/64              1.20µs ± 0%             0.89µs ± 1%  -25.81%          (p=0.008 n=5+5)
BM_TransposeInPlace<float>/128             8.96µs ± 0%             3.82µs ± 2%  -57.33%          (p=0.008 n=5+5)
BM_TransposeInPlace<float>/256              152µs ± 3%               17µs ± 2%  -89.06%          (p=0.008 n=5+5)
BM_TransposeInPlace<float>/512              837µs ± 1%              208µs ± 0%  -75.15%          (p=0.008 n=5+5)
BM_TransposeInPlace<float>/1k              4.28ms ± 2%             1.08ms ± 2%  -74.72%          (p=0.008 n=5+5)

Eigen/src/Core/Transpose.h

@@ -243,7 +243,6 @@ struct inplace_transpose_selector<MatrixType,true,false> { // square matrix
   }
 };
 
-// TODO: vectorized path is currently limited to LargestPacketSize x LargestPacketSize cases only.
 template<typename MatrixType>
 struct inplace_transpose_selector<MatrixType,true,true> { // PacketSize x PacketSize
   static void run(MatrixType& m) {
@@ -260,16 +259,65 @@ struct inplace_transpose_selector<MatrixType,true,true> { // PacketSize x Packet
   }
 };
 
+
+template <typename MatrixType, Index Alignment>
+void BlockedInPlaceTranspose(MatrixType& m) {
+  typedef typename MatrixType::Scalar Scalar;
+  typedef typename internal::packet_traits<typename MatrixType::Scalar>::type Packet;
+  const Index PacketSize = internal::packet_traits<Scalar>::size;
+  eigen_assert(m.rows() == m.cols());
+  int row_start = 0;
+  for (; row_start + PacketSize <= m.rows(); row_start += PacketSize) {
+    for (int col_start = row_start; col_start + PacketSize <= m.cols(); col_start += PacketSize) {
+      PacketBlock<Packet> A;
+      if (row_start == col_start) {
+        for (Index i=0; i<PacketSize; ++i)
+          A.packet[i] = m.template packetByOuterInner<Alignment>(row_start + i,col_start);
+        internal::ptranspose(A);
+        for (Index i=0; i<PacketSize; ++i)
+          m.template writePacket<Alignment>(m.rowIndexByOuterInner(row_start + i, col_start), m.colIndexByOuterInner(row_start + i,col_start), A.packet[i]);
+      } else {
+        PacketBlock<Packet> B;
+        for (Index i=0; i<PacketSize; ++i) {
+          A.packet[i] = m.template packetByOuterInner<Alignment>(row_start + i,col_start);
+          B.packet[i] = m.template packetByOuterInner<Alignment>(col_start + i, row_start);
+        }
+        internal::ptranspose(A);
+        internal::ptranspose(B);
+        for (Index i=0; i<PacketSize; ++i) {
+          m.template writePacket<Alignment>(m.rowIndexByOuterInner(row_start + i, col_start), m.colIndexByOuterInner(row_start + i,col_start), A.packet[i]);
+          m.template writePacket<Alignment>(m.rowIndexByOuterInner(col_start + i, row_start), m.colIndexByOuterInner(col_start + i,row_start), B.packet[i]);
+        }
+      }
+    }
+  }
+  for (Index row = row_start; row < m.rows(); ++row) {
+    m.matrix().row(row).swap(m.matrix().col(row));
+  }
+}
+
 template<typename MatrixType,bool MatchPacketSize>
-struct inplace_transpose_selector<MatrixType,false,MatchPacketSize> { // non square matrix
+struct inplace_transpose_selector<MatrixType,false,MatchPacketSize> { // non square or dynamic matrix
   static void run(MatrixType& m) {
-    if (m.rows()==m.cols())
-      m.matrix().template triangularView<StrictlyUpper>().swap(m.matrix().transpose().template triangularView<StrictlyUpper>());
-    else
+    typedef typename MatrixType::Scalar Scalar;
+    if (m.rows() == m.cols()) {
+      const Index PacketSize = internal::packet_traits<Scalar>::size;
+      if (!NumTraits<Scalar>::IsComplex && m.rows() >= PacketSize) {
+        if ((m.rows() % PacketSize) == 0)
+          BlockedInPlaceTranspose<MatrixType,internal::evaluator<MatrixType>::Alignment>(m);
+        else
+          BlockedInPlaceTranspose<MatrixType,Unaligned>(m);
+      }
+      else {
+        m.matrix().template triangularView<StrictlyUpper>().swap(m.matrix().transpose().template triangularView<StrictlyUpper>());
+      }
+    } else {
       m = m.transpose().eval();
+    }
   }
 };
 
+
 } // end namespace internal
 
 /** This is the "in place" version of transpose(): it replaces \c *this by its own transpose.

test/array_cwise.cpp

@@ -495,7 +495,11 @@ template<typename ArrayType> void array_complex(const ArrayType& m)
   VERIFY_IS_APPROX(m2, m1.transpose());
   m2.transposeInPlace();
   VERIFY_IS_APPROX(m2, m1);
-
+  // Check vectorized inplace transpose.
+  ArrayType m5 = ArrayType::Random(130, 130);
+  ArrayType m6 = m5;
+  m6.transposeInPlace();
+  VERIFY_IS_APPROX(m6, m5.transpose());
 }
 
 template<typename ArrayType> void min_max(const ArrayType& m)