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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Arithmetics\n",
    "\n",
    "## Basic Arithmetics\n",
    "\n",
    "The most basic tensor operations (addition `+`, subtraction `-`, and product `*` with either a scalar or with another tensor) can be accomplished via direct manipulation of tensor cores (see e.g. the [original tensor train paper](https://epubs.siam.org/doi/abs/10.1137/090752286?journalCode=sjoce3))."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4D TT tensor:\n",
      "\n",
      " 32  32  32  32\n",
      "  |   |   |   |\n",
      " (0) (1) (2) (3)\n",
      " / \\ / \\ / \\ / \\\n",
      "1   1   1   1   1\n",
      "\n"
     ]
    }
   ],
   "source": [
    "import tntorch as tn\n",
    "import torch\n",
    "torch.set_default_dtype(torch.float64)\n",
    "import numpy as np\n",
    "\n",
    "t1 = tn.ones([32]*4)\n",
    "t2 = tn.ones([32]*4)\n",
    "\n",
    "t = tn.round((t1+t2)*(t2-2))\n",
    "print(t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can also *assign* values to parts of a tensor:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[6., 6., 2., 2., 2.],\n",
      "        [6., 6., 2., 2., 2.],\n",
      "        [6., 6., 2., 2., 2.],\n",
      "        [3., 3., 1., 1., 1.],\n",
      "        [3., 3., 1., 1., 1.]])\n"
     ]
    }
   ],
   "source": [
    "t = tn.ones(5, 5)\n",
    "t[:3, :] = 2\n",
    "t[:, :2] *= 3\n",
    "print(t.torch())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Advanced Operations\n",
    "\n",
    "Thanks to [cross-approximation](cross.ipynb), *tntorch* supports many other more advanced operations on tensors, including element-wise division `/`, `exp()`, `log()`, `sin()`, etc."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4D TT-Tucker tensor:\n",
      "\n",
      " 32  32  32  32\n",
      "  |   |   |   |\n",
      "  7  13  13   7\n",
      " (0) (1) (2) (3)\n",
      " / \\ / \\ / \\ / \\\n",
      "1   7   7   7   1\n",
      "\n"
     ]
    }
   ],
   "source": [
    "domain = [torch.linspace(0, np.pi, 32)]*4\n",
    "x, y, z, w = tn.meshgrid(domain)\n",
    "\n",
    "t = tn.round(1 / (1+x+y+z+w))\n",
    "print(t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will now try the trigonometric identity $\\sin^2(x) + \\cos^2(x) = 1$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4D TT tensor:\n",
      "\n",
      " 32  32  32  32\n",
      "  |   |   |   |\n",
      " (0) (1) (2) (3)\n",
      " / \\ / \\ / \\ / \\\n",
      "1   13  17  13  1\n",
      "\n"
     ]
    }
   ],
   "source": [
    "t = tn.round(tn.sin(t)**2 + tn.cos(t)**2)\n",
    "print(t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The tensor `t` should be $1$ everywhere. Indeed:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor(1.0000)\n",
      "tensor(1.8159e-15)\n"
     ]
    }
   ],
   "source": [
    "print(tn.mean(t))\n",
    "print(tn.var(t))"
   ]
  }
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