{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Multivariate exponential decompositon"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "using TensorDec"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We consider the following function, which is a sum of 6 complex exponentials "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "nbpresent": {
     "id": "8c3a870c-7722-416e-9162-bec1f05a0a1f"
    }
   },
   "outputs": [],
   "source": [
    "f = (u,v) -> 0.5*cos(0.7*pi*(u+v))+0.6*sin(4*pi*u)-0.2*cos(pi*v);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![waves](waves.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to recover the frequencies or exponents of these exponential terms and their coefficients, we sample the function on a grid $({\\alpha_1 \\over T}, {\\alpha_2 \\over T}), \\alpha=(\\alpha_1,\\alpha_2)\\in A\\subset \\mathbb{N}^2$. This defines a sequence of moments $\\sigma_{\\alpha}=f({\\alpha_1 \\over T}, {\\alpha_2 \\over T})$. We compute its generating series truncated in degree $\\leq 5$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "nbpresent": {
     "id": "e1b828a2-c7e6-444f-bef3-e471bafb192f"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2569085959993554dx2^4 - 0.22417045976016967dx1^3dx2 + 0.7358257607718759dx1*dx2^4 + 0.8585922907464658dx1 + 0.5575373543042985dx1^2dx2 + 0.46210935078676274dx1^2dx2^3 + 0.6050846776084937dx1^2 + 0.02699524986977328dx1^5 - 0.15759364518763863dx1^3 + 0.2775204557293506dx2^3 + 0.29774707771034303dx2 - 0.2874793003806999dx1^3dx2^2 + 0.5095797473748392dx1^2dx2^2 + 0.2906101273580203dx2^2 + 0.7717888541929423dx1*dx2^3 + 0.22699524986977343dx2^5 - 0.5338499631663495dx1^4dx2 + 0.8328361327510712dx1*dx2 + 0.8039080170899477dx1*dx2^2 - 0.4519219149027473dx1^4 + 0.3"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = @ring x1 x2\n",
    "L = monoms(X,5)\n",
    "T = 10\n",
    "mnt = (V->f(V[1]/T,V[2]/T))\n",
    "sigma = series(mnt, L)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Computing its decomposition using svd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "nbpresent": {
     "id": "20ac1305-b35b-4eb7-9784-c2a1aadd277a"
    }
   },
   "outputs": [],
   "source": [
    "w, Xi = decompose(sigma);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "yields the weights $\\omega$ of the exponential terms in f and the exponentials $\\Xi$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "nbpresent": {
     "id": "15e6313c-37a5-4cc3-9854-daabb827029a"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6×2 Array{Complex{Float64},2}:\n",
       " 6.84243e-12+1.89601e-12im  -1.46851e-12-1.0im        \n",
       " 6.84243e-12-1.89601e-12im  -1.46851e-12+1.0im        \n",
       " 1.88616e-12-0.7im           4.84085e-13-0.7im        \n",
       " 1.88616e-12+0.7im           4.84085e-13+0.7im        \n",
       " 8.94708e-13-4.0im          -2.79182e-13+8.46664e-15im\n",
       " 8.94708e-13+4.0im          -2.79182e-13-8.46664e-15im"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log.(Xi')*T/pi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By taking the log and scaling by ${T\\over \\pi}$, we recover the frequency vectors within precision $1O^{-11}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "nbpresent": {
     "id": "7348f10f-4db3-4f43-a2d0-228e05052ff6"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6-element Array{Complex{Float64},1}:\n",
       "  0.08650714237109196 + 0.05016487136224489im\n",
       "  0.08650714237109196 - 0.05016487136224486im\n",
       " -0.17203277153436108 - 0.18139659731722088im\n",
       "  -0.1720327715343611 + 0.18139659731722083im\n",
       " -0.16999220553356456 - 0.24718950232123527im\n",
       " -0.16999220553356453 + 0.2471895023212353im "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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