diff --git a/tutorial/5 - flash attention2.ipynb b/tutorial/5 - flash attention2.ipynb
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+{
+ "cells": [
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Based on the flash attention1, the developer has implemented flash attention2, whose main changes are to put the tiled  Q in the outer loop and the tiled K and V in the inner loop, and to change the way of calculating online softmax and O. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import torch\n",
+    "\n",
+    "torch.manual_seed(456)\n",
+    "\n",
+    "N,d=16,8\n",
+    "Q_mat=torch.rand((N,d))\n",
+    "K_mat=torch.rand((N,d))\n",
+    "V_mat=torch.rand((N,d))\n",
+    "\n",
+    "expected_softmax=torch.softmax(Q_mat@K_mat.T,dim=1)\n",
+    "expected_attention=expected_softmax@V_mat\n",
+    "\n",
+    "# tile size for matmul, no op bigger than this size can be stored in SRAM\n",
+    "Br=4\n",
+    "Bc=d\n",
+    "\n",
+    "# variables outside the for loop represent the global memory\n",
+    "# they are the only ones bigger than what the SRAM can store\n",
+    "O=torch.zeros((N,d))\n",
+    "\n",
+    "# For the 2 variables below, they may be removed in a serially executed code (in particular the outter for loop)\n",
+    "# They are needed in parallelized execution where each thread block need to sync its findings with the others\n",
+    "for block_start_Br in range(0,N,Br):\n",
+    "    block_end_Br=block_start_Br+Br\n",
+    "    # line 4, load a block of Q_mat from HBM\n",
+    "    Qi=Q_mat[block_start_Br:block_end_Br,:]\n",
+    "    # line 5, initialize Oi, li and mi.\n",
+    "    Oi=torch.zeros((Br,d))  # shape Br x d\n",
+    "    li=torch.zeros((Br,1))  # shape Br x 1\n",
+    "    mi=torch.full((Br,1),-torch.inf)  # shape Br x 1\n",
+    "\n",
+    "    for block_start_Bc in range(0,N,Bc):\n",
+    "        block_end_Bc=block_start_Bc+Bc\n",
+    "\n",
+    "        # line 7, load a block from matmul input tensor\n",
+    "        Kj=K_mat[block_start_Bc:block_end_Bc,:]\n",
+    "        Vj=V_mat[block_start_Bc:block_end_Bc,:]\n",
+    "\n",
+    "        # line 8,QKt at the tile level\n",
+    "        Sij=Qi@Kj.T\n",
+    "\n",
+    "        # line 9, find max of each row regarding the current block and the previous ones we have already visited\n",
+    "        mi_new=torch.max(torch.column_stack([mi,torch.max(Sij,dim=1).values[:,None]]),dim=1).values[:,None]\n",
+    "        \n",
+    "        # line 9,compute the softmax numerator like if we only had the data from this block (and nothing before or after)\n",
+    "        Pij_hat=torch.exp(Sij-mi_new)\n",
+    "\n",
+    "        # line 9,adjusting factor (see online softmax computation above) leveraging the rule of exponentiation\n",
+    "        li=torch.exp(mi-mi_new)*li+torch.sum(Pij_hat,dim=1)[:,None]\n",
+    "        \n",
+    "        # line 10\n",
+    "        Oi=Oi*torch.exp(mi-mi_new)+Pij_hat@Vj\n",
+    "\n",
+    "        # update the mi\n",
+    "        mi=mi_new\n",
+    "    \n",
+    "    # line 12\n",
+    "    Oi=Oi/li\n",
+    "\n",
+    "    # line 14\n",
+    "    O[block_start_Br:block_end_Br,:]=Oi\n",
+    "assert torch.allclose(O,expected_attention)"
+   ]
+  }
+ ],
+ "metadata": {
+  "language_info": {
+   "name": "python"
+  },
+  "orig_nbformat": 4
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}