OrthoPoly2D.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra\n",
    "using Jacobi\n",
    "using Plots\n",
    "using Test"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Orthogonal Polynomials on Genral Quadrilateral"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Construct, Display and Evaluating\n",
    "\n",
    "All the concepts and algorithms developed for orthogonal polynomial in 1D can be extended to polynomial in 2 variables $(x,y)$.\n",
    "\n",
    "Lets start with constructing a polynomial in 2D. Suppose we have 1D polynomial in $x$ and $y$ directions described by coefficients $p_x$ and $p_y$. We can construct a 2D polynomial using the tensor product 1D polynomial basis.\n",
    "\n",
    "For example, consider $p_x = [1, 2, 3],\\quad (p(x)=1 + 2x + 3x^2)$ and $p_y=[4,5], \\quad (p(y)=4+5y)$, then\n",
    "$$\n",
    "P = px \\otimes py = \\begin{bmatrix}4 & 5\\\\8 & 10\\\\12 & 15 \\end{bmatrix}, \\quad P(x,y) = 4+8x+12x^2+5y+10xy+15x^2y\n",
    "$$\n",
    "You notice that we can also represent $P(x,y)$ as\n",
    "$$\n",
    "P(x,y) = y^0\\Big( 4+8x+12x^2 \\Big) + y^1\\Big( 5+10x+15x^2 \\Big) = x^0\\Big(4 +5y \\Big)+x^1\\Big(8 +10y\\Big)+x^2\\Big(12 +15y\\Big)\n",
    "$$\n",
    "which shows for $P$ of size $(n,m)$ column $j$ is a polynomial of form $y^{j-1} p_{n-1}(x)$ and row $i$ is $x^{i-1} q_{m-1}(y)$.\n",
    "\n",
    "The following functions construct, display, and evaluate polynomial in 2D from $p_x, p_y$. Note that in polyval function we use Horner's rule in 1D since we can represnet column $j$ of $P_{n\\times m}$ by $y^{j-1} p_{n-1}(x)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "polyval2d (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function poly2d(px, py)\n",
    "    \"\"\"\n",
    "    Construct a 2D polynomial based on 1D basis in x and y directions.\n",
    "    Input:\n",
    "    px: polynomial in x direction\n",
    "    py: polynomial in y direction\n",
    "    Return:\n",
    "    Matrix p, which is the coefficients of a 2D polynomial p(x,y)\n",
    "    \"\"\"\n",
    "    px = reshape(px, :)\n",
    "    py = reshape(py, :)\n",
    "    # degree of poly in x\n",
    "    n = length(px)\n",
    "    # degree of poly in y\n",
    "    m = length(py)\n",
    "    # construct poly coefficients in 2D and store it in p\n",
    "    p = zeros(n, m)\n",
    "    for i=1:length(px)\n",
    "        for j=1:length(py)\n",
    "            p[i,j] = px[i] * py[j]\n",
    "        end\n",
    "    end\n",
    "    \n",
    "    return p\n",
    "end\n",
    "\n",
    "function polystring(p)\n",
    "    \"\"\"Construct a string representation of the polynomial p(x,y) in 2D\n",
    "    p: polynomial coefficients in x and y directions created by poly2d function \n",
    "    \"\"\"\n",
    "    n,m = size(p)\n",
    "    k = n * m\n",
    "    pol = Array{String}(undef, k)\n",
    "    kk = 1\n",
    "    for j=1:m\n",
    "        for i=1:n\n",
    "            a = p[i,j] \n",
    "            pol[kk] = \"$a x^$(i-1) y^$(j-1)\"\n",
    "            kk += 1 \n",
    "        end\n",
    "    end\n",
    "            \n",
    "    join((pol[i] for i=1:k), \" + \")\n",
    "end\n",
    "\n",
    "function polydisp(name, p)\n",
    "    \"\"\"Display the polynomial with friendly math formatting.\n",
    "    p: polynomial coefficients in x and y directions created by poly2d function\n",
    "    \"\"\"\n",
    "    \n",
    "    s = join(('$', \"$name(x,y) = \", polystring(p), '$'))\n",
    "    # If you download this notebook, you can uncomment the following\n",
    "    # line to display rendered math.\n",
    "    display(\"text/latex\", s)\n",
    "    # This feature is broken in Ed's \"Saturn\" so we'll just display\n",
    "    # plain text.\n",
    "    #display(\"text/plain\", s)\n",
    "end\n",
    "\n",
    "function polyval2d(p, x, y)\n",
    "    \"\"\"Evaluate function described by coefficients p at point (x,y).\n",
    "    Input:\n",
    "    p: polynomial coefficients in 2D build from poly2D(px,py) function \n",
    "    Return:\n",
    "    The value of p(x,y) at (x,y). Note (x,y) can be provided as a vector\n",
    "    \"\"\"\n",
    "    # first we compute each column using Horner's rule\n",
    "    # then multiply by y^(j-1), we can represent matrix P by\n",
    "    # 1st col=y^0 * [f1_n(x)], 2nd col=y^1 * [f2_n(x)],... OR\n",
    "    # 1st row=x^0 * [g1_m(y)], 2nd row=x^1 * [g2_m(y)],... \n",
    "    # we use the first one to evaluate poly in 2D\n",
    "    n,m = size(p)\n",
    "    g = zero.(x)\n",
    "    for j = 1:m\n",
    "        p1 = zeros(n)\n",
    "        p1[:] = p[:,j]\n",
    "        f = one.(x) * p1[end]\n",
    "        for a in reverse(p1[1:end-1])\n",
    "            f = a .+ x .* f\n",
    "        end\n",
    "        g += f .* y.^(j-1)\n",
    "    end\n",
    "    return g\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$f(x,y) = 0.0 x^0 y^0 + 6.0 x^1 y^0 + 0.0 x^0 y^1 + 8.0 x^1 y^1$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = poly2d([0 2],[3 4])\n",
    "polydisp(\"f\",p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p1 = poly2d([1 2], [3 4])\n",
    "@test polyval2d(p1, [1], [1]) == [21]\n",
    "@test polyval2d(p1, [1;-1], [1;-1]) == [21;1]\n",
    "p2 = poly2d([1 2 -3], [4 5])\n",
    "@test polyval2d(p2, [1;-1], [1;-1]) == [0;4]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Addition and subtraction\n",
    "In the following we add/subtract $p(x,y)$ and $q(x,y)$ which they are created based on 1D basis $(p_x,p_y)$ and $(q_x,q_y)$ using poly2D function. lets assume the coefficients of 2D polynomial $p(x,y)$ and $q(x,y)$ are stored in matrices $\\boldsymbol{P}_{n_1\\times m_1}$ and $\\boldsymbol{Q}_{n_2\\times m_2}$, respectively. Then all we need is to add or subtract same entries of these two matrices and stored it in new matrix with size $(n,m)$ where $n=\\text{max}(n_1,n_2), m = \\text{max}(m_1,m_2)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "polyaddsub2d (generic function with 1 method)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function polyaddsub1d(p, q, mode)\n",
    "    \"\"\"\n",
    "    Addition/subtraction of two 1D polynomials p and q.\n",
    "    (maybe we don't need this function)\n",
    "    mode: \"add\" for addition \"sub\" for subtraction\n",
    "    \"\"\"\n",
    "    p = reshape(p, :)\n",
    "    q = reshape(q, :)\n",
    "    n = max(length(p), length(q))\n",
    "    r = zeros(n)\n",
    "    r[1:length(p)] = p\n",
    "    if mode == \"add\"\n",
    "        r[1:length(q)] += q\n",
    "    elseif mode == \"sub\"\n",
    "        r[1:length(q)] -= q\n",
    "    else\n",
    "        error(\"mode must be add or sub\")\n",
    "    end\n",
    "    reshape(r, 1, :)\n",
    "    \n",
    "    return r\n",
    "end\n",
    "\n",
    "function polyaddsub2d(p, q, mode)\n",
    "    \"\"\"\n",
    "    Add/subtract two polynomials p and q in 2D.\n",
    "    Input:\n",
    "    p: Coefficients of 2D polynomial created by poly2D(px,py)\n",
    "    q: Coefficients of 2D polynomial created by poly2D(qx,qy)\n",
    "    mode: \"add\" for addition \"sub\" for subtraction\n",
    "    Return:\n",
    "    P: a matrix which is the coefficients of p(x,y) +/- q(x,y)\n",
    "    \"\"\"\n",
    "    (n1,m1) = size(p)\n",
    "    (n2,m2) = size(q)\n",
    "    \n",
    "    # scatter pp in matrix P1 of size(n,m)\n",
    "    n = max(n1, n2)\n",
    "    m = max(m1, m2)\n",
    "    P1 = zeros(n,m)\n",
    "    P1[1:n1,1:m1] = p\n",
    "    # scatter qq in matrix P2 of size(n,m)\n",
    "    P2 = zeros(n,m)\n",
    "    P2[1:n2,1:m2] = q\n",
    "    \n",
    "    # add or subtract P1 and P2\n",
    "    if mode == \"add\"\n",
    "        P = P1 + P2\n",
    "    elseif mode == \"sub\"\n",
    "        P = P1 - P2\n",
    "    else\n",
    "        error(\"mode must be add or sub\")\n",
    "    end\n",
    "    \n",
    "    return P\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P1 = [4 5;14 18;-12 -15]\n",
    "p = poly2d([0 2],[3 4])\n",
    "q = poly2d([1 2 -3],[4 5])\n",
    "@test polyaddsub2d(p,q,\"add\") == P1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Multiplying Polynomials\n",
    "Consider $p(x,y)$ and $q(x,y)$ given by the coefficients $\\boldsymbol{P}_{n_1\\times m_1}$ and $\\boldsymbol{Q}_{n_2\\times m_2}$. We know column $j$ of $\\boldsymbol{P}$ represents $y^{j-1}f_{n_1-1}(x)$ and column $l$ of $\\boldsymbol{Q}$ is $y^{l-1}g_{n_2-1}(x)$, so we can multiply these two columns using 1D multiplying polynomial function, then add those columns which they represent same degree of $y$. For example, lets consider $\\boldsymbol{P}_{2\\times 2}$ and $\\boldsymbol{Q}_{3\\times 2}$,\n",
    "$$\n",
    "p(x,y) = y^0 \\,f_1(x) + y^1 \\, g_1(x), \\quad q(x,y) = y^0 \\,h_2(x) + y^1 \\, k_2(x)\n",
    "$$\n",
    "then,\n",
    "$$\n",
    "p(x,y) \\, q(x,y) = y^0 \\,f_1(x) \\, h_2(x) + y^1 \\, \\Big( f_1(x)\\,k_2(x) + g_1(x)\\,h_2(x) \\Big) + y^2 \\,g_1(x)\\,k_2(x)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "polymul2d (generic function with 1 method)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function polymul1d(p, q)\n",
    "    \"\"\"Multiply two polynomials in 1D, returinng a polynomial.\"\"\"\n",
    "    p = reshape(p, :)\n",
    "    q = reshape(q, :)\n",
    "    n = length(p) + length(q) - 1\n",
    "    r = zeros(n)\n",
    "    for (i, a) in enumerate(p)\n",
    "        for (j, b) in enumerate(q)\n",
    "            r[i+j-1] += a * b\n",
    "        end\n",
    "    end\n",
    "    # return a row vector (for easier viewing)\n",
    "    reshape(r, 1, :)\n",
    "    \n",
    "    return r\n",
    "end\n",
    "\n",
    "\n",
    "function polymul2d(p, q)\n",
    "    \"\"\"\n",
    "    Multiply two polynomials p and q in 2D.\n",
    "    Input:\n",
    "    p: Coefficients of 2D polynomial created by poly2D(px,py)\n",
    "    q: Coefficients of 2D polynomial created by poly2D(qx,qy)\n",
    "    Return:\n",
    "    P: a matrix which is the coefficients of p(x,y) * q(x,y)\n",
    "    \"\"\"\n",
    "    (n1,m1) = size(p)\n",
    "    (n2,m2) = size(q)\n",
    "\n",
    "    k1 = n1+n2-1\n",
    "    k2 = m1+m2-1\n",
    "    P1 = zeros(m1,k1,k2)\n",
    "    for i=1:m1\n",
    "        p1 = zeros(n1)\n",
    "        # get column i of p or y^(i-1)f_{n1-1}(x)\n",
    "        p1 = p[:,i]\n",
    "        for j=1:m2\n",
    "            q1 = zeros(n2)\n",
    "            q1 = q[:,j]\n",
    "            # multiply column i of p to all columns of q\n",
    "            r = polymul1d(p1, q1)\n",
    "            P1[i,:,j+(i-1)] = r\n",
    "        end\n",
    "    end\n",
    "    # add those columns with same degree of y\n",
    "    P = zeros(k1,k2)\n",
    "    for i=1:m1\n",
    "       P[:,:] += P1[i,:,:]\n",
    "    end\n",
    "\n",
    "    return P\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P1 = [12 31 20;48 124 80;12 31 20;-72 -186 -120]\n",
    "p = poly2d([1 2],[3 4])\n",
    "q = poly2d([1 2 -3],[4 5])\n",
    "@test polymul2d(p, q) == P1\n",
    "P2 = [0 0 0;24 62 40;48 124 80;-72 -186 -120]\n",
    "p = poly2d([0 2],[3 4])\n",
    "q = poly2d([1 2 -3],[4 5])\n",
    "@test polymul2d(p, q) == P2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inner products of polynomials\n",
    "\n",
    "We will use the $L^2$ inner product on general quadrilateral $E$\n",
    "$$ \\langle u, v \\rangle = \\int_{E} u(x,y) v(x,y) \\mathrm{d} x \\mathrm{d} y $$\n",
    "restricted to the case where $u(x,y)$ and $v(x,y)$ are polynomials. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "poly_norm2d (generic function with 1 method)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function BilinearMap(coord_E, xhat, yhat)\n",
    "    \"\"\" \n",
    "    This function maps [xhat,yhat] in Ehat=[-1,1]^2 \n",
    "    to (x,y) in quadrilateral E.\n",
    "    Input:\n",
    "    ------\n",
    "    coord_E: coordinates of quadrilateral E .\n",
    "    coord_E is 4x2 array\n",
    "    coord_E = [x1 y1;x2 y2;x3 y3;x4 y4] with vertices numbering\n",
    "    3----4\n",
    "    |    |\n",
    "    1----2\n",
    "    [xhat, yhat] in Ehat\n",
    "    Output:\n",
    "    ------\n",
    "    x, y: mapped vector in E.\n",
    "    DF_E: Jacobian matrix\n",
    "    J_E: det(DF_E)\n",
    "    \"\"\"\n",
    "    m = length(xhat)\n",
    "    P = 0.25 * [(1 .- xhat).*(1 .- yhat) (1 .+ xhat).*(1 .- yhat) (1 .- xhat).*(1 .+ yhat) (1 .+ xhat).*(1 .+ yhat)]\n",
    "    P_E = P * coord_E\n",
    "    # x=F_E(xhat)\n",
    "    X = P_E\n",
    "    x = X[:, 1]\n",
    "    y = X[:, 2]\n",
    "    # gradient of P, 1st row = dP/dxhat, 2nd row=dP/dyhat\n",
    "    # GradP_(2m x 4)\n",
    "    GradP = 0.25 * [-(1 .- yhat) (1 .- yhat) -(1 .+ yhat) (1 .+ yhat);\n",
    "                    -(1 .- xhat) -(1 .+ xhat) (1 .- xhat) (1 .+ xhat)]\n",
    "\n",
    "    # JT= [[dx/dxhat, dy/dxhat],\n",
    "    #      [dx/dyhat, dy/dyhat]] (2m x 2)\n",
    "    JT = GradP * coord_E\n",
    "    dxdxhat = JT[1:m,1]\n",
    "    dydxhat = JT[1:m,2]\n",
    "    dxdyhat = JT[m+1:2*m,1]\n",
    "    dydyhat = JT[m+1:2*m,2]\n",
    "    J_E = dxdxhat .* dydyhat .- dydxhat .* dxdyhat\n",
    "    \n",
    "    DF_E = [dxdxhat dxdyhat;dydxhat dydyhat]\n",
    "    \n",
    "    return x, y, DF_E, J_E\n",
    "end\n",
    "\n",
    "function GetQuadrature2d(Q)\n",
    "    \"\"\"\n",
    "    create Gauss points and weights on [-1, 1]^2\n",
    "    \"\"\"\n",
    "    # 1D quadrature on [-1, 1]\n",
    "    q = zgj(Q, 0.0, 0.0)\n",
    "    w = wgj(q, 0.0, 0.0)\n",
    "    \n",
    "    W2 = zeros(Q*Q)\n",
    "    Qx = zeros(Q*Q)\n",
    "    Qy = zeros(Q*Q)\n",
    "    for i=1:Q\n",
    "        for j=1:Q\n",
    "            k = (i-1)*Q +j\n",
    "            Qx[k] = q[j]\n",
    "            Qy[k] = q[i]\n",
    "            W2[k] = w[j]*w[i]\n",
    "        end\n",
    "    end\n",
    "    return W2, Qx, Qy\n",
    "end\n",
    "\n",
    "function poly_inner_product2d(u, v, coord_E)\n",
    "    \"\"\"the L2 inner product of u(x,y) and v(x,y) on quadrilateral E with coord_E\n",
    "    Both u and v are polynomials expressed as matrix in the usual way.\n",
    "    \"\"\"\n",
    "    P = polymul2d(u, v)\n",
    "    n,m = size(P)\n",
    "    # max degrees of poly in x or y\n",
    "    n = max(n,m)-1\n",
    "    # No. quadrature points in 1D\n",
    "    Q = Int(ceil((n+1)/2))\n",
    "    # get weights and quadrature points on [-1,1]^2\n",
    "    W2, Qx, Qy = GetQuadrature2d(Q)\n",
    "    # get J_E to updates weights and quadrature points on quadrilateral E\n",
    "    x, y, DF_E, J_E = BilinearMap(coord_E, Qx, Qy)\n",
    "    # evaluvate u*v at quadrature pts on E\n",
    "    R = polyval2d(P,x,y)\n",
    "    # loop over quadrature\n",
    "    inprod_uv = (J_E .* W2)' * R\n",
    "    \n",
    "    return inprod_uv \n",
    "end\n",
    "\n",
    "\n",
    "function poly_norm2d(u, coord_E)\n",
    "    \"\"\"Compute the norm of the polynomial u(x,y) induced by poly_inner_product2d().\n",
    "    \"\"\"\n",
    "    inprod_uu = poly_inner_product2d(u, u, coord_E)\n",
    "\n",
    "    return sqrt(inprod_uu)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\u001b[32m\u001b[1mTest Passed\u001b[22m\u001b[39m"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coord_E = [-1. -1.;1. -1.;-1. 1;1. 1.]\n",
    "err = 1e-14\n",
    "p = poly2d([0 2],[3 4])\n",
    "q = poly2d([1 2],[0 4])\n",
    "#exact value: int (p*q)dxdy = int(y[24x+48x^2] + y^2[32x+64x^2]) = 256/9\n",
    "@test isapprox(poly_inner_product2d(p, q, coord_E), 256/9, atol=err)\n",
    "p = poly2d([0 1],[0 0 1])\n",
    "q = poly2d([0 1 1],[0 0 1])\n",
    "#exact value: int (p*q)dxdy = int(y^4[x^2+x^3]) = 4/15\n",
    "@test isapprox(poly_inner_product2d(p, q, coord_E), 4/15, atol=err)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "monomials2d (generic function with 1 method)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function monomials(n)\n",
    "    \"\"\"Return a list of the first n monomials in order of increasing degree.\n",
    "    We will think of this as a \"tall\" matrix indexed by its n columns.\n",
    "    Each column contains a polynomial.\n",
    "    For example for n = 3, we have 1, x, x^2\n",
    "    \"\"\"\n",
    "    A = []\n",
    "    for j in 1:n\n",
    "        p = zeros(j)\n",
    "        p[end] = 1\n",
    "        push!(A, p)\n",
    "    end\n",
    "    return A\n",
    "end\n",
    "\n",
    "\n",
    "function monomials2d(n, m)\n",
    "    \"\"\"\n",
    "    Input:\n",
    "    n: number of basis in x direction in order of increasing degree\n",
    "    m: number of basis in y direction in order of increasing degree\n",
    "    Return:\n",
    "    A list which contains matrices for 2D polynomial\n",
    "    \"\"\"\n",
    "    A = monomials(n)\n",
    "    B = monomials(m)\n",
    "    P = []\n",
    "    for j = 1:m\n",
    "        q = copy(B[j])\n",
    "        for i = 1:n\n",
    "            p = copy(A[i])\n",
    "            pq = poly2d(p,q)\n",
    "            push!(P, pq)\n",
    "        end\n",
    "    end\n",
    "    return P\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "poly_QtQ (generic function with 1 method)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function poly_qr2d(n,m, coord_E)\n",
    "    \"\"\"\n",
    "    n: number of poly basis in x direction\n",
    "    m: number of poly basis in y direction\n",
    "    coord_E: coordinates of quadrilateral E\n",
    "    returns Q, R which Q is a list of size n*m of matrices\n",
    "    contains orthogonal poly on E\n",
    "    \"\"\"\n",
    "    \n",
    "    A = monomials2d(n,m)\n",
    "    Q = []\n",
    "    R = zeros(n*m, n*m)\n",
    "    for j = 1:m\n",
    "        for i = 1:n\n",
    "            k = i + n*(j-1) # kth column of A\n",
    "            v = copy(A[k])\n",
    "            for l = 1:k-1\n",
    "                r = poly_inner_product2d(Q[l],v, coord_E)\n",
    "                R[l, k] = r\n",
    "                v = polyaddsub2d(v,Q[l]*r,\"sub\")\n",
    "            end\n",
    "            r = poly_norm2d(v, coord_E)\n",
    "            R[k,k] = r\n",
    "            push!(Q,v/r)\n",
    "        end\n",
    "    end\n",
    "            \n",
    "    return Q, R\n",
    "end\n",
    "\n",
    "function poly_QtQ(Q, coord_E)\n",
    "    \"\"\"Compute Q.T Q where Q is a matrix of polynomials\n",
    "    expressed as a list of matrices. The result is an ordinary\n",
    "    square matrix.\n",
    "    \"\"\"\n",
    "    n = length(Q)\n",
    "    A = zeros(n, n)\n",
    "    for i in 1:n\n",
    "        for j in 1:n\n",
    "            A[i,j] = poly_inner_product2d(Q[i], Q[j], coord_E)\n",
    "        end\n",
    "    end\n",
    "    A\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Test Polynomial on $\\hat{E} = [-1,1]^2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "norm(QtQ - I) = 2.560618590774332e-15\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "2.560618590774332e-15"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coord_E = [-1. -1.;1. -1.;-1. 1;1. 1.]\n",
    "Q, R = poly_qr2d(3, 3, coord_E)\n",
    "QtQ = poly_QtQ(Q, coord_E)\n",
    "@show norm(QtQ - I)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$f(x,y) = -1.8911488521767566e-17 x^0 y^0 + 3.491063653003978e-17 x^1 y^0 + -3.7325306292962396e-18 x^2 y^0 + -0.9682458365518538 x^0 y^1 + 2.2377283801721832e-33 x^1 y^1 + 2.9047375096555625 x^2 y^1$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "polydisp(\"f\",Q[6])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "meshgrid(x, y) = (repeat(x, outer=length(y)), repeat(y, inner=length(x)))\n",
    "n = 10\n",
    "x, y = meshgrid(LinRange(-1,1,n), LinRange(-1,1,n))\n",
    "p = polyval2d(Q[6],x,y)\n",
    "plot(x, y, p, st = :surface, camera = (45,70))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Test on General Quad E"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "norm(QtQ - I) = 7.202928815612741e-13\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "7.202928815612741e-13"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coord_E = [0. 0.;2. 0.;0. 1;2. 1.]\n",
    "Q, R = poly_qr2d(3, 3, coord_E)\n",
    "QtQ = poly_QtQ(Q, coord_E)\n",
    "@show norm(QtQ - I)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$f(x,y) = -2.738612787525812 x^0 y^0 + 8.21583836257744 x^1 y^0 + -4.107919181288722 x^2 y^0 + 5.4772255750516665 x^0 y^1 + -16.431676725154997 x^1 y^1 + 8.215838362577498 x^2 y^1$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "polydisp(\"f\",Q[6])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "meshgrid(x, y) = (repeat(x, outer=length(y)), repeat(y, inner=length(x)))\n",
    "n = 10\n",
    "x, y = meshgrid(LinRange(0,2,n), LinRange(0,1,n))\n",
    "p = polyval2d(Q[6],x,y)\n",
    "plot(x, y, p, st = :surface, camera = (45,70))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
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