{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Symmetric tensors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "using TensorDec"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We consider symmetric tensors or equivalently homogeneous polynomials, in the following variables:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = @ring x0 x1 x2;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A symmetric tensor of order d=4 and of rank 3."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "0.5x0^{4} - 2.0x0^{3}x1 + 11.0x0^{3}x2 + 15.0x0^{2}x1^{2} + 9.0x0^{2}x1x2 - 8.625x0^{2}x2^{2} - 2.0x0x1^{3} + 9.0x0x1^{2}x2 + 6.75x0x1x2^{2} + 9.6875x0x2^{3} + 2.5x1^{4} + 3.0x1^{3}x2 + 3.375x1^{2}x2^{2} + 1.6875x1x2^{3} - 1.68359375x2^{4}"
      ],
      "text/plain": [
       "0.5x0⁴ - 2.0x0³x1 + 11.0x0³x2 + 15.0x0²x1² + 9.0x0²x1x2 - 8.625x0²x2² - 2.0x0x1³ + 9.0x0x1²x2 + 6.75x0x1x2² + 9.6875x0x2³ + 2.5x1⁴ + 3.0x1³x2 + 3.375x1²x2² + 1.6875x1x2³ - 1.68359375x2⁴"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d=4; F = (x0+x1+0.75x2)^d + 1.5*(x0-x1)^d -2.0*(x0-x2)^d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The graph of the homogeneous polynomial $(x_0+x_1+0.75x_2)^4 + 1.5(x_0-x_1)^4 -2(x_0-x_2)^4$ in polar coordinates on the sphere looks like this:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![tensor](tensor.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We associate to $t$, the following (truncated) series in the dual variables, after substituting $x_0$ by 1:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Computing its decomposition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "w, Xi = decompose(F);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "yields the weights `w`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Float64,1}:\n",
       "  1.4999999999999996\n",
       " -1.9999999999999987\n",
       "  0.9999999999999999"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and the corresponding points $\\Xi$, which are the coefficient vectors of $x_0, x_1, x_2$ in the linear forms of the decomposition of the tensor F. They are normalized to have norm 1:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Float64,2}:\n",
       "  1.0           1.0          1.0 \n",
       " -1.0           4.15916e-16  1.0 \n",
       "  1.03483e-16  -1.0          0.75"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Xi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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