examples/TT Smoluchowski.ipynb

{
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  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
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   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import tt\n",
    "import tt.cross.rectcross as rect_cross\n",
    "#import rect_cross\n",
    "import time\n",
    "import scipy.special"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## Problem setting\n",
    "\n",
    "This code solves multicomponent Smoluchowski equation with source and sink terms\n",
    "\n",
    "\n",
    "$$\\frac{\\partial n(\\overline{v}, t)}{\\partial t} = \\frac{1}{2}\\int_0^{v_1} ... \\int_0^{v_d} K(\\overline{v}-\\overline{u}; \\overline{u}) n(\\overline{v} - \\overline{u}, t) n(\\overline{u}, t) du_1 ... du_d - \\\\\n",
    "\\\\ - n(\\overline{v},t) \\int_0^{V_{\\max}} ... \\int_0^{V_{\\max}} K(\\overline{u}; \\overline{v}) n(\\overline{u},t) du_1 ... du_d ~ + ~ q(v_1, \\ldots, v_d)$$\n",
    "\n",
    "details about the numerical scheme can be found in papers:$$\\\\$$\n",
    "1. Matveev SA, DA Zheltkov, EE Tyrtyshnikov, AP Smirnov, $$\\\\$$\n",
    "Tensor train versus Monte Carlo for the multicomponent Smoluchowski coagulation equation, Journal of Computational Physics (2016) 316, 164-179$$\\\\$$\n",
    "\n",
    "2. Smirnov AP, Matveev SA, Zheltkov DA, Tyrtyshnikov EE, $$\\\\$$\n",
    "Fast and accurate finite-difference method solving multicomponent Smoluchowski coagulation equation with source and sink terms, Procedia Computer Science  (2016) pp. 2141-2146"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setup the parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "d = 2\n",
    "N = 800\n",
    "r = 1\n",
    "h = 0.125\n",
    "tau = 0.025\n",
    "N_steps = 10\n",
    "tolerance = 1e-6\n",
    "print_res = 0\n",
    "T = N_steps * tau\n",
    "check_error = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
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   "outputs": [],
   "source": [
    "def Coag_K(x):#Coagulation kernel\n",
    "    if (x.size % 2) != 0:\n",
    "        print \"Kernel must depend on even number of indeces!\"\n",
    "        exit()\n",
    "    \n",
    "    u = (x[:, :d ] + 1e-2) * h\n",
    "    v = (x[:, d: ] + 1e-2) * h\n",
    "    # Ballistic kernel\n",
    "    # (u^(1/3) + v^(1/3))^2 * sqrt(1/u + 1/v)\n",
    "    return (np.power(u.sum(axis = 1), 1.0 / 3.0) + np.power(v.sum(axis = 1), 1.0 / 3.0))**2.0 *  np.power( 1.0 / u.sum(axis = 1) + 1.0 / v.sum(axis = 1), 0.5)\n",
    "    # Generalized multiplication a = 0.1\n",
    "    #return np.power(u.sum(axis = 1), 0.1) * np.power(v.sum(axis = 1), -0.1) + np.power(u.sum(axis = 1), -0.1) * np.power(v.sum(axis = 1), 0.1)\n",
    "\n",
    "def q(x):#Source\n",
    "    return np.exp(-x.sum(axis = 1) * h)"
   ]
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  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
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   "outputs": [],
   "source": [
    "def Check_n(x):\n",
    "    return np.exp(-x.sum(axis = 1) * h)\n",
    "\n",
    "def Check_Mass(x):\n",
    "    return (x.sum(axis = 1)) * h\n",
    "\n",
    "def Start_Cond(x):\n",
    "    return np.exp(-x.sum(axis = 1) * h)\n",
    "\n",
    "def Analytical(x):\n",
    "    return np.exp(-h * x.sum(axis=1)) / (1.0 + T / 2.0)**2 * scipy.special.i0(  2.0 * np.sqrt( h**d * np.prod(x, axis = 1)  * T / (2.0 + T)))"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "## TT-representation of operators\n",
    "\n",
    "the following cell contains implementation of TT-integration via trapezoids rule for the following integrals:\n",
    "    \n",
    "$$\\int_0^{v_1} ... \\int_0^{v_d} K(\\overline{v}-\\overline{u}; \\overline{u}) n(\\overline{v}) du_1 ... du_d$$\n",
    "\n",
    "How to perform its simplified version:\n",
    "    \n",
    "\\begin{align*}\\notag\n",
    "&q(v_1, \\ldots, v_d) =\\int_0^{v_1} \\ldots\\int_0^{v_d} f(v_1 - u_1, \\ldots, v_d - u_d) g(u_1, \\ldots, u_d) du_1 \\ldots du_d \\\\ \\notag\n",
    " &=\\sum_{\\alpha_0, \\ldots, \\alpha_d} \\sum_{\\beta_0, \\ldots, \\beta_d} \\int_0^{v_1} \\ldots\\int_0^{v_d} f_1(\\alpha_0, v_1 - u_1, \\alpha_1) \\ldots f_d(\\alpha_{d-1}, v_d - u_d, \\alpha_d) g_1(\\beta_1, u_1, \\beta_2) \\ldots g_d(\\beta_{d-1}, u_d, \\beta_d) du_1 \\ldots du_d\\\\ \\notag\n",
    "&=\\sum_{\\alpha_1, \\ldots, \\alpha_d} \\sum_{\\beta_0, \\ldots, \\beta_d} \\int_0^{v_1} f_1(\\alpha_0, v_1 - u_1, \\alpha_1)\\ g_1(\\beta_1, u_1, \\beta_2) du_1 \\ldots \\int_0^{v_d}  f_d(\\alpha_{d-1}, v_d - u_d, \\alpha_d)  \\ldots g_d(\\beta_{d-1}, u_d, \\beta_d) du_d.\n",
    "\\end{align*}\n",
    "\n",
    "One-dimensional convolutions will stand as:\n",
    "\n",
    "\\begin{equation}\\notag\n",
    " h \\begin{bmatrix}\n",
    "\\frac{1}{2}f_k(\\alpha_{k-1}, v_{k_0}, \\alpha_k) &  0  & 0 & 0 & 0 \\\\ \n",
    "f_k(\\alpha_{k-1}, v_{k_1}, \\alpha_k) &  \\frac{1}{2}f_k(\\alpha_{k-1}, v_{k_0}, \\alpha_k) & 0 & 0 & 0 \\\\ \n",
    "f_k(\\alpha_{k-1}, v_{k_2}, \\alpha_k) &  f(\\alpha_{k-1}, v_1, \\alpha_k) & \\frac{1}{2}f_k(\\alpha_{k-1}, v_{k_0}, \\alpha_k) & 0 & 0 \\\\ \n",
    "\\ldots & \\ldots & \\ldots& \\ldots & \\ldots\\\\ \n",
    "f_k(\\alpha_{k-1}, v_{k_N}, \\alpha_k) &  f_k(\\alpha_{k-1}, v_{k_{N-1}}, \\alpha_k)& \\ldots & \\ldots & \\frac{1}{2}f_k(\\alpha_{k-1}, v_{k_0}, \\alpha_k)\\\\\n",
    "\\end{bmatrix} \\times \n",
    "\\begin{bmatrix}\n",
    "\\frac{1}{2}g_k(\\beta_{k-1}, v_{k_0}, \\beta_k)  \\\\ \n",
    "g_k(\\beta_{k-1}, v_{k_1}, \\beta_k) \\\\ \n",
    "g_k(\\beta_{k-1}, v_{k_2}, \\beta_k)  \\\\ \n",
    "\\ldots \\\\ \n",
    "g_k(\\beta_{k-1}, v_{k_N}, \\beta_k) \\\\\n",
    "\\end{bmatrix}.\n",
    "\\end{equation}"
   ]
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   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
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   "source": [
    "def First_Integral(TT_solution, Coag_Kernel):\n",
    "    carriages = tt.tensor.to_list(Coag_Kernel)\n",
    "    \n",
    "    #list of last d - 1 carriages\n",
    "    list_u = carriages[d + 1  :]\n",
    "    #list of first d - 1 carriages\n",
    "    list_v = carriages[ : d - 1]\n",
    "    \n",
    "    for alpha in xrange(Coag_Kernel.r[d]):#Loop along the middle index of kernel\n",
    "        #print 'alpha', alpha\n",
    "        #insert one slice from d+1-th carriage and create TT\n",
    "        list_u.insert(0, carriages[d] [alpha : alpha + 1, :, :])\n",
    "        #append one slice from d-th carriage and create TT\n",
    "        list_v.append(carriages[d - 1][:, :, alpha : alpha + 1])\n",
    "        #get tensor trains from the lists\n",
    "        K_u_part = tt.tensor.from_list(list_u)\n",
    "        K_v_part = tt.tensor.from_list(list_v)\n",
    "        \n",
    "        #compute Integrands  \n",
    "        \n",
    "        K_u_part = tt.multifuncrs([K_u_part, TT_solution], lambda x: x.prod(axis = 1), eps = tolerance) # K_alpha (u) * n(u)\n",
    "        K_v_part = tt.multifuncrs([K_v_part, TT_solution], lambda x: x.prod(axis = 1), eps = tolerance) # K_alpha(v) * n(v)\n",
    "        K_u_part = tt.tensor.to_list(K_u_part)\n",
    "        K_v_part = tt.tensor.to_list(K_v_part)\n",
    "        \n",
    "        for i in xrange(d):\n",
    "            # add the required zeros, multiply by the integration mesh step\n",
    "            K_u_part[i][:, 0, :] = K_u_part[i][:, 0, :] / 2.0\n",
    "            K_v_part[i][:, 0, :] = K_v_part[i][:, 0, :] / 2.0\n",
    "            K_u_part[i] = np.concatenate( (K_u_part[i], np.zeros([K_u_part[i].shape[0], N - 1, K_u_part[i].shape[2]])), axis = 1)\n",
    "            # do the first FFT\n",
    "            K_u_part[i] = np.fft.fft(K_u_part[i], axis = 1, n = 2 * N - 1)\n",
    "            # add the required zeros and multiply by the integration mesh step\n",
    "            K_v_part[i] = np.concatenate( (K_v_part[i], np.zeros([K_v_part[i].shape[0], N - 1, K_v_part[i].shape[2]])), axis = 1)\n",
    "            # do the first FFT\n",
    "            K_v_part[i] = np.fft.fft(K_v_part[i], axis = 1, n = 2 * N - 1)\n",
    "\n",
    "        # Build tensor trains after the axis-wide FFTs\n",
    "        K_u_part = tt.tensor.from_list(K_u_part)\n",
    "        K_v_part = tt.tensor.from_list(K_v_part)\n",
    "        \n",
    "        \n",
    "        # Elementwise product of 2 TTs; Store the results in K_u_part(allows us not to make new TT but replase the old one)\n",
    "        K_u_part = tt.multifuncrs([K_u_part, K_v_part], lambda x: x.prod(axis = 1), eps = tolerance)\n",
    "        \n",
    "        # Back to list of TT-cores\n",
    "        K_u_part = tt.tensor.to_list(K_u_part)\n",
    "\n",
    "        for i in xrange(d):\n",
    "            # Inverse FFT along each phase-spase axis\n",
    "            K_u_part[i] = np.fft.ifft(K_u_part[i], axis = 1, n = 2 * N - 1)\n",
    "            # Truncation. Here we delete the additional (n-1)-zeros\n",
    "            K_u_part[i] = h *  np.abs(K_u_part[i][:, : N, :]) * np.sign(K_u_part[i][:, : N, :])\n",
    "            K_u_part[i][:,0,:] *= 0.0;\n",
    "        # Restore the TT\n",
    "        K_u_part = tt.tensor.from_list(K_u_part)\n",
    "        # Save the results in I1\n",
    "        if alpha == 0:\n",
    "            I1 = K_u_part\n",
    "        else:\n",
    "            I1 = tt.multifuncrs([I1, K_u_part], lambda x: x.sum(axis = 1), eps = tolerance)\n",
    "        #delete used slices before addition of the new ones\n",
    "        list_v.pop(d - 1)\n",
    "        list_u.pop(0)\n",
    "        #print 'I1'\n",
    "        #print I1\n",
    "    #============================================================================================\n",
    "    I1 = tt.multifuncrs([I1], lambda x: x.real, eps = tolerance)\n",
    "    return I1"
   ]
  },
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   "source": [
    "## More integrals\n",
    "The following cell contains implementation of TT-integration via trapezoids rule for the following integrals\n",
    "$$q(\\overline{v}) = \\int_0^{V_{\\max}} ... \\int_0^{V_{\\max}} K(\\overline{u}; \\overline{v}) f(\\overline{u}) du_1 ... du_d$$\n",
    "\n",
    "$$\\\\$$\n",
    "How to perform it:\n",
    "\\begin{align*}\n",
    "& q(\\overline{v}) = \\int_0^{V_{\\max}} ... \\int_0^{V_{\\max}} K(\\overline{u}; \\overline{v}) f(\\overline{u}) du_1 ... du_d = \\sum_{\\overline{\\alpha}, \\overline{\\beta}} K_1^v(\\alpha_0, v_1, \\alpha_1) \\ldots K_d^v(\\alpha_{d-1}, v_d, \\alpha_{d}) \n",
    "\\times \n",
    "\\\\ & \\times \\int_0^{V_{\\max}} K_1^u(\\alpha_{d}, u_1, \\alpha_{d+1}) f_1(\\beta_0, u_1, \\beta_1) du_1 \\ldots \\int_0^{V_{\\max}} K_d^u(\\alpha_{2d-1}, u_d, \\alpha_{2d}) f_d(\\beta_{d-1}, u_d, \\beta_d) du_d = \n",
    "\\\\ & \\sum_{\\overline{\\alpha}} K_1^v(\\alpha_0, v_1, \\alpha_1) \\ldots K_d^v(\\alpha_{d-1}, v_d, \\alpha_{d}) \\sum_{\\overline{\\beta}} \\int_0^{V_{\\max}} K_1^u(\\alpha_{d}, u_1, \\alpha_{d+1}) f_1(\\beta_0, u_1, \\beta_1) du_1 \\times \n",
    "\\\\ &\\ldots  \\times \\int_0^{V_{\\max}} K_d^u(\\alpha_{2d-1}, u_d, \\alpha_{2d}) f_d(\\beta_{d-1}, u_d, \\beta_d) du_d = \\sum_{\\alpha_d = 1}^{R_d} \\widetilde{K^v_{\\alpha_d}}(\\overline{v}) I_{\\alpha_d},\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
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   "outputs": [],
   "source": [
    "def Second_Integral(TT_solution, Coag_Kernel):\n",
    "    # create weights of rectangular quadrature like tt.ones * h**d;\n",
    "    # we multiply each carriage by h to avoid the machine zero\n",
    "    weights = tt.ones(N, d)\n",
    "    carriages = tt.tensor.to_list(weights)\n",
    "    for i in xrange(d):\n",
    "        carriages[i] = carriages[i] * h\n",
    "        carriages[i][:,0,:] /= 2\n",
    "        carriages[i][:,N-1,:] /= 2\n",
    "    weights = tt.tensor.from_list(carriages)\n",
    "    #create list of carriages of coagulation kernel\n",
    "\n",
    "    carriages = tt.tensor.to_list(Coag_Kernel)\n",
    "\n",
    "    #list of first d carriages\n",
    "    list_v = carriages[ : d]\n",
    "    #list if last d - 1 carriages\n",
    "    list_u = carriages[ d + 1 :]\n",
    "    Integral = np.zeros(Coag_Kernel.r[d])\n",
    "    #print 'Coagulation kernel in TT format'\n",
    "    #print Coag_Kernel \n",
    "    #print '============================================================'\n",
    "    for alpha in xrange(carriages[d - 1 ].shape[2]):\n",
    "        #print 'alpha', alpha\n",
    "        #insert one slice from d+1-th carriage to create TT\n",
    "        list_u.insert(0, carriages[d] [alpha : alpha + 1, :, :])\n",
    "        #get tensor trains from the lists\n",
    "        #K_v_part = tt.tensor.from_list(list_v)\n",
    "        K_u_part = tt.tensor.from_list(list_u)\n",
    "        #compute Integrand (u) = K_u(u) * n(u) \n",
    "        K_u_part = tt.multifuncrs([K_u_part, TT_solution], lambda x: np.prod(x, axis=1), eps = tolerance)\n",
    "        #compute I[alpha] = \\int_0^{v_{max}} Integrand[alpha](u) du will be a vector of size r_d\n",
    "        # sum of results over the middle d-th index\n",
    "        Integral[alpha] = tt.dot(K_u_part, weights)\n",
    "        list_u.pop(0)\n",
    "    #============================================================================================\n",
    "    list_v[d - 1] = list_v[d - 1].reshape(list_v[d - 1].shape[0] * list_v[d - 1].shape[1], list_v[d - 1].shape[2])\n",
    "    list_v[d - 1] = list_v[d - 1].dot(Integral)\n",
    "    list_v[d - 1] = list_v[d - 1].reshape(carriages[d - 1].shape[0], carriages[d - 1].shape[1], 1)\n",
    "    I2 = tt.tensor.from_list(list_v)\n",
    "    I2 = tt.multifuncrs([I2, TT_solution], lambda x: np.prod(x, axis = 1), eps = tolerance)\n",
    "\n",
    "    #print '============================================================'\n",
    "    #print 'Coagulation Kernel'\n",
    "    #print Coag_Kernel\n",
    "    return I2"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "## Finally\n",
    "\n",
    "The last cell contains implementation of predictor-corrector scheme solving the Cauchy problem for Smoluchowski coagulation equation\n",
    "$$\n",
    "\\begin{matrix}\n",
    "n^{k+\\frac{1}{2}}(\\overline{i})  = \n",
    "\\frac{\\tau}{2} \\left( L_{1}(\\overline{i}) (n^{k}) - n^{k}(\\overline{i}) \n",
    "L_{2}(\\overline{i}) (n^{k}) + q^k(\\overline{i})  \\right) + n^{k}(\\overline{i})  \\\\\n",
    "n^{k+1 }(\\overline{i}) = \\tau \\left( L_1(\\overline{i}) (n^{k+\\frac{1}{2}}) - \n",
    "                                    n^{k+\\frac{1}{2}}(\\overline{i}) L_{2}(\\overline{i}) (n^{k+\\frac{1}{2}}) + \n",
    "                                    q^{k+\\frac{1}{2}}(\\overline{i})\\right) + n^{k}(\\overline{i}).\n",
    "\\end{matrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Approximate starting condition\n",
      "swp: 0/5 er_rel = 1.7e+02 er_abs = 7.9e+02 erank = 3.0 fun_eval: 6400\n",
      "swp: 1/5 er_rel = 3.5e-16 er_abs = 1.6e-15 erank = 5.0 fun_eval: 19200\n",
      "Approximate TT for check of total mass\n",
      "swp: 0/5 er_rel = 1.0e+00 er_abs = 8.6e+04 erank = 3.0 fun_eval: 6400\n",
      "swp: 1/5 er_rel = 2.4e-12 er_abs = 2.1e-07 erank = 5.0 fun_eval: 19200\n",
      "Approximate source\n",
      "swp: 0/5 er_rel = 1.7e+02 er_abs = 7.9e+02 erank = 3.0 fun_eval: 6400\n",
      "swp: 1/5 er_rel = 3.5e-16 er_abs = 1.6e-15 erank = 5.0 fun_eval: 19200\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.136e+13, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 4.056e-14, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.056e-14, erank: 2.44949\n",
      "=multifuncrs= sweep 3{1}, max_dy: 0.000e+00, erank: 1.41421\n",
      "Starting mass =  1.99999796802\n",
      "Approximate kernel\n",
      "alpha =  1\n",
      "==============================================================\n",
      "Step 1\n",
      "==============================================================\n",
      "=multifuncrs= sweep 1{2}, max_dy: 4.419e+36, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 3.107e-16, erank: 1.73205\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.425e-16, erank: 2\n",
      "=multifuncrs= sweep 3{1}, max_dy: 0.000e+00, erank: 1\n",
      "=multifuncrs= sweep 1{2}, max_dy: 3.560e+43, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 5.370e-16, erank: 1.73205\n",
      "=multifuncrs= sweep 2{2}, max_dy: 5.370e-16, erank: 2\n",
      "=multifuncrs= sweep 3{1}, max_dy: 0.000e+00, erank: 1\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.010e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 4.819e-16, erank: 1.73205\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.819e-16, erank: 2\n",
      "=multifuncrs= sweep 3{1}, max_dy: 0.000e+00, erank: 1\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.487e+09, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.681e-16, erank: 1.73205\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.765e-16, erank: 2\n",
      "=multifuncrs= sweep 3{1}, max_dy: 0.000e+00, erank: 1\n",
      "=multifuncrs= sweep 1{2}, max_dy: 8.127e+25, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 2.601e-16, erank: 1.73205\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.394e-16, erank: 2\n",
      "=multifuncrs= sweep 3{1}, max_dy: 0.000e+00, erank: 1\n",
      "=multifuncrs= sweep 1{2}, max_dy: 6.233e+10, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 2.900e-16, erank: 1.73205\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.054e-16, erank: 2\n",
      "=multifuncrs= sweep 3{1}, max_dy: 0.000e+00, erank: 1\n",
      "First integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=1, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=1, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 9.334e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 8.114e-16, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 8.114e-16, erank: 2.44949\n",
      "=multifuncrs= sweep 3{1}, max_dy: 0.000e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 1{2}, max_dy: 5.004e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.056e-16, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.056e-16, erank: 2.44949\n",
      "=multifuncrs= sweep 3{1}, max_dy: 0.000e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 1{2}, max_dy: 5.818e+06, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 4.377e-16, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.377e-16, erank: 2.44949\n",
      "=multifuncrs= sweep 3{1}, max_dy: 0.000e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 1{2}, max_dy: 5.606e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 4.781e-03, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.781e-03, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 5.838e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 5.838e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 3.858e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 5.063e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 5.063e-02, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.279e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 2.296e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.907e+20, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 3.555e-10, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.555e-10, erank: 2.44949\n",
      "=multifuncrs= sweep 3{1}, max_dy: 0.000e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.925e+06, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 5.112e-16, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 5.112e-16, erank: 2.44949\n",
      "=multifuncrs= sweep 3{1}, max_dy: 0.000e+00, erank: 1.41421\n",
      "First integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=3, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=2, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 3.098e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 8.457e-03, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 8.457e-03, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 7.446e-09, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 7.446e-09, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.224e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 3.951e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.951e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.876e-15, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.876e-15, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "mass =  2.04980249602\n",
      "density = 1.0148514638\n",
      "Solution\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=3, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "==============================================================\n",
      "==============================================================\n",
      "Step 2\n",
      "==============================================================\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.217e+07, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 2.161e-03, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 2.161e-03, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.834e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.834e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.071e+19, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.221e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.221e-01, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 8.148e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 8.148e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.211e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.321e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.321e-02, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 5.022e-08, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.787e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.385e+02, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.348e-02, erank: 1.73205\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.348e-02, erank: 2.44949\n",
      "=multifuncrs= sweep 3{1}, max_dy: 6.006e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 6.006e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.122e+14, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.542e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.542e-02, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.225e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.225e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.948e+19, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 4.461e-03, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.461e-03, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.188e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.255e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "First integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=3, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=3, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.943e+19, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 3.624e-03, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.624e-03, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.514e-08, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.514e-08, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 6.855e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 5.748e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 5.748e-01, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 5.948e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 5.948e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.241e+19, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.208e+00, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.208e+00, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.961e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.961e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.078e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.978e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.978e-02, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.221e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.785e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 6.651e+05, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 2.080e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 2.080e-02, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 7.298e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 7.298e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 4.267e+16, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.079e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.079e-02, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.685e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.685e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 9.232e+03, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.564e-04, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.564e-04, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.485e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.786e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "First integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=3, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=3, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 3.518e+14, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 8.089e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 8.089e-02, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.881e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.881e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.258e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.817e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.817e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.254e-14, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.254e-14, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "mass =  2.09960415223\n",
      "density = 1.02679080751\n",
      "Solution\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=3, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "==============================================================\n",
      "==============================================================\n",
      "Step 3\n",
      "==============================================================\n",
      "=multifuncrs= sweep 1{2}, max_dy: 6.997e+08, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 4.289e-03, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.289e-03, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.026e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.026e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 4.665e+03, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 2.565e-03, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 2.565e-03, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.944e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.944e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 5.511e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 2.831e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 2.831e-02, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.629e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.811e-06, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 1.832e-06, erank: 3\n",
      "=multifuncrs= sweep 4{2}, max_dy: 2.189e-06, erank: 2.82843\n",
      "=multifuncrs= sweep 5{1}, max_dy: 2.224e-06, erank: 3\n",
      "=multifuncrs= sweep 5{2}, max_dy: 2.224e-06, erank: 2.82843\n",
      "=multifuncrs= sweep 6{1}, max_dy: 1.331e-06, erank: 3\n",
      "=multifuncrs= sweep 6{2}, max_dy: 1.331e-06, erank: 2.82843\n",
      "=multifuncrs= sweep 7{1}, max_dy: 3.802e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 7{2}, max_dy: 5.828e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 8{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 4.525e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 3.783e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.783e-02, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.974e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.636e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 9.725e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 5.338e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 5.338e-01, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.743e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.743e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.172e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 6.331e-04, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 6.331e-04, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.297e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.297e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "First integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=3, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=3, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 8.765e+16, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 3.246e-03, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.246e-03, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.810e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.810e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.956e+08, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 5.045e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 5.045e-02, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 6.203e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 6.203e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 4.666e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 4.474e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.474e-01, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.392e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.392e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.292e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 3.468e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.468e-02, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 6.652e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.522e-06, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 1.589e-06, erank: 3\n",
      "=multifuncrs= sweep 4{2}, max_dy: 2.303e-06, erank: 2.82843\n",
      "=multifuncrs= sweep 5{1}, max_dy: 2.377e-06, erank: 3\n",
      "=multifuncrs= sweep 5{2}, max_dy: 2.500e-06, erank: 2.82843\n",
      "=multifuncrs= sweep 6{1}, max_dy: 2.574e-06, erank: 3\n",
      "=multifuncrs= sweep 6{2}, max_dy: 2.574e-06, erank: 2.82843\n",
      "=multifuncrs= sweep 7{1}, max_dy: 1.393e-06, erank: 3\n",
      "=multifuncrs= sweep 7{2}, max_dy: 2.432e-06, erank: 2.82843\n",
      "=multifuncrs= sweep 8{1}, max_dy: 2.498e-06, erank: 3\n",
      "=multifuncrs= sweep 8{2}, max_dy: 2.498e-06, erank: 2.82843\n",
      "=multifuncrs= sweep 9{1}, max_dy: 1.213e-06, erank: 3\n",
      "=multifuncrs= sweep 9{2}, max_dy: 2.877e-06, erank: 2.82843\n",
      "=multifuncrs= sweep 10{1}, max_dy: 2.957e-06, erank: 3\n",
      "=multifuncrs= sweep 10{2}, max_dy: 2.957e-06, erank: 2.82843\n",
      "=multifuncrs= sweep 11{1}, max_dy: 2.946e-06, erank: 3\n",
      "=multifuncrs= sweep 1{2}, max_dy: 8.956e+16, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 9.375e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 9.375e-02, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 5.522e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 5.522e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 4.086e+04, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 9.884e-03, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 9.884e-03, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.327e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.327e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 4.594e+02, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 2.386e-03, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 2.386e-03, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 5.041e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 5.041e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "First integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=3, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=3, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 5.023e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.613e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.613e-02, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 5.514e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 5.514e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 7.882e+16, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 8.709e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 8.709e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 5.999e-14, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 5.999e-14, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "mass =  2.14939980131\n",
      "density = 1.03842725406\n",
      "Solution\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=3, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "==============================================================\n",
      "==============================================================\n",
      "Step 4\n",
      "==============================================================\n",
      "=multifuncrs= sweep 1{2}, max_dy: 6.984e+16, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.200e+00, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.200e+00, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.939e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.435e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 5.410e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 2.782e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 2.782e-02, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 9.225e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 9.225e-15, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.027e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 3.949e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.949e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 8.920e-09, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 8.920e-09, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.691e+02, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 3.397e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.397e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.742e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.742e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.127e+19, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 6.549e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 6.549e-01, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.390e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.815e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 5.870e+14, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 4.032e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.032e-01, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.920e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 2.920e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "First integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=3, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 5.062e+06, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 5.352e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 5.352e-02, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.085e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.085e-07, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.029e+11, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 2.233e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 2.233e-01, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.122e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 2.570e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.677e+19, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 6.865e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 6.865e-01, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.986e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.395e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.076e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 4.142e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.142e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.283e-08, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 2.283e-08, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 6.357e+15, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.734e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.734e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 8.862e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 8.862e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.189e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.015e+00, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.015e+00, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.633e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.470e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.237e+14, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 2.584e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 2.584e-01, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.726e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.726e-16, erank: 2.82843\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 1.73205\n",
      "First integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=3, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.370e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.511e+00, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.511e+00, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 7.764e-09, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 7.764e-09, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.991e+16, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 3.462e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.462e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.465e-07, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.465e-07, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "mass =  2.19920484084\n",
      "density = 1.04976637808\n",
      "Solution\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "==============================================================\n",
      "==============================================================\n",
      "Step 5\n",
      "==============================================================\n",
      "=multifuncrs= sweep 1{2}, max_dy: 9.601e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 6.666e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 6.666e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 5.860e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 5.860e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.221e+08, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.446e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.446e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.138e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.566e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.726e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 5.452e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 5.452e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 7.151e-08, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 7.438e-08, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 5.823e+04, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 2.278e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 2.278e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 7.302e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 7.302e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.578e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.227e+00, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.227e+00, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.239e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.239e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.099e+02, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 2.215e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 2.215e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.437e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.221e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "First integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 3.234e+15, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.160e+00, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.160e+00, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 9.498e-09, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 9.498e-09, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.361e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.064e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.064e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.318e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.318e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.511e+05, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 4.216e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.216e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.522e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 5.178e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.208e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 5.909e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 5.909e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.221e-07, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.221e-07, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 9.117e+07, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.032e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.032e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.322e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.322e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.169e+19, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.025e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.025e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.251e-15, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.251e-15, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.901e+16, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 6.346e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 6.346e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.294e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.294e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "First integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 8.434e+09, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 3.922e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.922e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.850e-08, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 2.850e-08, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 8.405e+16, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 2.721e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 2.721e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.003e-13, erank: 3.16228\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.003e-13, erank: 3.16228\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2.23607\n",
      "mass =  2.24901031613\n",
      "density = 1.0608119911\n",
      "Solution\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "==============================================================\n",
      "==============================================================\n",
      "Step 6\n",
      "==============================================================\n",
      "=multifuncrs= sweep 1{2}, max_dy: 9.187e+03, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.703e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.703e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.521e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.521e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.335e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.866e+00, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.866e+00, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.201e-15, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.201e-15, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.317e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 6.512e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 6.512e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.574e-07, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 2.574e-07, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 9.974e+16, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 4.374e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.374e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.112e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.112e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 4.439e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.564e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.564e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.824e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.824e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.460e+09, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 3.771e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.771e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.601e-15, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.601e-15, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "First integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 4.518e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 6.807e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 6.807e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.107e-08, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 2.107e-08, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 9.964e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.054e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.054e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 6.802e-15, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 6.802e-15, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 3.546e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 6.524e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 6.524e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.016e-15, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.016e-15, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.176e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.106e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.106e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.005e-07, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.085e-07, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 3.163e+16, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.802e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.802e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.842e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.842e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.104e+19, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 6.134e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 6.134e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.297e-15, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.297e-15, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 3.095e+10, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 5.815e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 5.815e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.858e-15, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.858e-15, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "First integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 7.903e+02, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 8.849e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 8.849e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 5.702e-08, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 5.702e-08, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 5.737e+02, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.787e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.787e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.748e-13, erank: 3.16228\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.748e-13, erank: 3.16228\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2.23607\n",
      "mass =  2.29881549398\n",
      "density = 1.07156877549\n",
      "Solution\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "==============================================================\n",
      "==============================================================\n",
      "Step 7\n",
      "==============================================================\n",
      "=multifuncrs= sweep 1{2}, max_dy: 5.076e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 3.498e-03, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.498e-03, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.291e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.291e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.267e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.736e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.736e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 5.580e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 5.580e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.623e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.469e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.469e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 5.066e-07, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 5.066e-07, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.746e+15, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 2.164e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 2.164e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.614e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 2.614e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.021e+08, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 4.702e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.702e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 7.353e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 7.353e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 9.606e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.263e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.263e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.007e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.007e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "First integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.155e+19, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 8.083e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 8.083e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.840e-08, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.840e-08, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.634e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.221e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.221e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.121e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 2.763e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 4.180e+05, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.048e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.048e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.482e-15, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.482e-15, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.354e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 8.450e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 8.450e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 5.413e-07, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 5.413e-07, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 3.612e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.844e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.844e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.880e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.880e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.199e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 5.305e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 5.305e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 7.130e-15, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 7.130e-15, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 3.091e+05, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.475e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.475e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 5.768e-15, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 5.768e-15, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "First integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 8.344e+15, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.324e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.324e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 9.822e-08, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 9.822e-08, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 4.397e+15, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 3.379e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.379e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.750e-13, erank: 3.16228\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.750e-13, erank: 3.16228\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2.23607\n",
      "mass =  2.34862036667\n",
      "density = 1.08204150277\n",
      "Solution\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "==============================================================\n",
      "==============================================================\n",
      "Step 8\n",
      "==============================================================\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.381e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.470e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.470e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.026e-15, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.026e-15, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.121e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 2.013e-03, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 2.013e-03, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.876e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.876e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.046e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 9.253e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 9.253e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 9.705e-07, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 9.705e-07, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 4.420e+02, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 6.465e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 6.465e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.366e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.366e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 3.098e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.514e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.514e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.711e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.900e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.610e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 6.518e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 6.518e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 5.976e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 5.976e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "First integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.715e+06, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 4.469e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.469e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 6.269e-08, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 6.269e-08, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.363e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.764e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.764e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.201e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.201e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 6.177e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.361e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.361e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.528e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.528e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.026e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 9.806e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 9.806e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.381e-06, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.381e-06, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 6.617e-07, erank: 3\n",
      "=multifuncrs= sweep 4{2}, max_dy: 1.058e-06, erank: 3\n",
      "=multifuncrs= sweep 5{1}, max_dy: 1.158e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 5{2}, max_dy: 1.158e-06, erank: 3\n",
      "=multifuncrs= sweep 6{1}, max_dy: 1.106e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 6{2}, max_dy: 1.106e-06, erank: 3\n",
      "=multifuncrs= sweep 7{1}, max_dy: 6.539e-07, erank: 3\n",
      "=multifuncrs= sweep 7{2}, max_dy: 8.886e-07, erank: 3\n",
      "=multifuncrs= sweep 8{1}, max_dy: 0.000e+00, erank: 2.23607\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.675e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.304e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.304e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 8.167e-12, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 8.168e-12, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.102e+19, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 3.899e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.899e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.096e-15, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.096e-15, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 3.371e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 6.462e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 6.462e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.069e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.069e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "First integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.430e+03, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.752e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.752e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.547e-07, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.547e-07, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 8.124e+16, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 4.291e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.291e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.145e-13, erank: 3.16228\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.145e-13, erank: 3.16228\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2.23607\n",
      "mass =  2.3984246333\n",
      "density = 1.09223503476\n",
      "Solution\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "==============================================================\n",
      "==============================================================\n",
      "Step 9\n",
      "==============================================================\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.238e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 6.929e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 6.929e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.418e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.744e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.849e+01, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.023e-03, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.023e-03, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.443e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.248e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.007e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.042e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.042e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.488e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.488e-06, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 1.214e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 4{2}, max_dy: 1.214e-06, erank: 3\n",
      "=multifuncrs= sweep 5{1}, max_dy: 1.349e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 5{2}, max_dy: 1.349e-06, erank: 3\n",
      "=multifuncrs= sweep 6{1}, max_dy: 1.168e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 6{2}, max_dy: 1.301e-06, erank: 3\n",
      "=multifuncrs= sweep 7{1}, max_dy: 1.439e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 7{2}, max_dy: 1.439e-06, erank: 3\n",
      "=multifuncrs= sweep 8{1}, max_dy: 1.236e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 8{2}, max_dy: 1.236e-06, erank: 3\n",
      "=multifuncrs= sweep 9{1}, max_dy: 1.220e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 9{2}, max_dy: 1.676e-06, erank: 3\n",
      "=multifuncrs= sweep 10{1}, max_dy: 1.791e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 10{2}, max_dy: 1.791e-06, erank: 3\n",
      "=multifuncrs= sweep 11{1}, max_dy: 1.486e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.941e+16, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 3.299e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 3.299e-01, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 6.356e-07, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 6.356e-07, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 5.941e+07, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.102e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.102e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.450e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.826e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 5.601e+10, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.458e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.458e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.220e-15, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.220e-15, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "First integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 6.378e+15, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.011e+00, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.011e+00, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 9.497e-08, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 9.497e-08, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 5.850e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.058e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.058e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.218e-15, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.218e-15, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.353e+16, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.505e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.505e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.762e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.391e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.135e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.112e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.112e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.741e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.741e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 4{1}, max_dy: 1.911e-08, erank: 3.16228\n",
      "=multifuncrs= sweep 4{2}, max_dy: 2.290e-08, erank: 3.16228\n",
      "=multifuncrs= sweep 5{1}, max_dy: 0.000e+00, erank: 2.23607\n",
      "=multifuncrs= sweep 1{2}, max_dy: 4.518e+02, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 5.202e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 5.202e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 5.754e-16, erank: 3.16228\n",
      "=multifuncrs= sweep 3{2}, max_dy: 5.754e-16, erank: 3.16228\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2.23607\n",
      "=multifuncrs= sweep 1{2}, max_dy: 3.322e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 6.739e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 6.739e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.727e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.817e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.037e+02, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.846e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.846e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 5.374e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 5.374e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "First integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=5, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 5.582e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.473e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.473e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.282e-07, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 2.282e-07, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.473e+15, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 4.188e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.188e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.945e-13, erank: 3.16228\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.945e-13, erank: 3.16228\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2.23607\n",
      "mass =  2.44822794007\n",
      "density = 1.10215431315\n",
      "Solution\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "==============================================================\n",
      "==============================================================\n",
      "Step 10\n",
      "==============================================================\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.109e+04, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.202e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.202e-01, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.958e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.065e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 3.450e+12, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.522e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.522e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.934e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 2.934e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.095e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.159e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.159e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.752e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 3{2}, max_dy: 2.752e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 4{1}, max_dy: 4.291e-08, erank: 3.16228\n",
      "=multifuncrs= sweep 4{2}, max_dy: 4.291e-08, erank: 3.16228\n",
      "=multifuncrs= sweep 5{1}, max_dy: 0.000e+00, erank: 2.23607\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.555e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 6.825e-02, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 6.825e-02, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.222e-15, erank: 3.16228\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.222e-15, erank: 3.16228\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2.23607\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.695e+12, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.184e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.184e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 4.244e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 4.244e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.586e+14, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.372e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.372e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 2.044e-15, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 2.044e-15, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "First integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=5, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral predictor\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 7.680e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 6.938e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 6.938e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.366e-07, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.366e-07, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.186e+16, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.901e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.901e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.669e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.669e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.222e+09, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 8.817e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 8.817e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.376e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.376e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.005e+00, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.226e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.226e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.800e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.800e-06, erank: 3.16228\n",
      "=multifuncrs= sweep 4{1}, max_dy: 3.588e-08, erank: 3.16228\n",
      "=multifuncrs= sweep 4{2}, max_dy: 4.245e-08, erank: 3.16228\n",
      "=multifuncrs= sweep 5{1}, max_dy: 0.000e+00, erank: 2.23607\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.067e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.264e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.264e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 1.707e-15, erank: 3.16228\n",
      "=multifuncrs= sweep 3{2}, max_dy: 1.707e-15, erank: 3.16228\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2.23607\n",
      "=multifuncrs= sweep 1{2}, max_dy: 8.721e+05, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.345e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.345e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.565e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.565e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 1.043e+18, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 7.183e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 7.183e-01, erank: 2.64575\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.023e-16, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.023e-16, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "First integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=5, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "Second integral corrector\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "=multifuncrs= sweep 1{2}, max_dy: 7.497e+17, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 1.224e+00, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 1.224e+00, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 3.217e-07, erank: 3\n",
      "=multifuncrs= sweep 3{2}, max_dy: 3.217e-07, erank: 3\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2\n",
      "=multifuncrs= sweep 1{2}, max_dy: 2.574e+12, erank: 1.41421\n",
      "=multifuncrs= sweep 2{1}, max_dy: 4.190e-01, erank: 2\n",
      "=multifuncrs= sweep 2{2}, max_dy: 4.190e-01, erank: 2.82843\n",
      "=multifuncrs= sweep 3{1}, max_dy: 6.352e-14, erank: 3.16228\n",
      "=multifuncrs= sweep 3{2}, max_dy: 6.352e-14, erank: 3.16228\n",
      "=multifuncrs= sweep 4{1}, max_dy: 0.000e+00, erank: 2.23607\n",
      "mass =  2.49802980483\n",
      "density = 1.11180434054\n",
      "Solution\n",
      "This is a 2-dimensional tensor \n",
      "r(0)=1, n(0)=800 \n",
      "r(1)=4, n(1)=800 \n",
      "r(2)=1 \n",
      "\n",
      "==============================================================\n",
      "time =  34.464517\n"
     ]
    }
   ],
   "source": [
    "#approximate starting condition\n",
    "print 'Approximate starting condition'\n",
    "x0 = tt.rand(N, d, r)\n",
    "TT_Solution = rect_cross.cross(Start_Cond, x0, nswp = 6, kickrank = 1, rf = 2)\n",
    "#TT_Solution = TT_Solution.round(tolerance)\n",
    "print 'Approximate TT for check of total mass'\n",
    "Check_mass  = rect_cross.cross(Check_Mass, x0, nswp = 6, kickrank = 1, rf = 2)\n",
    "#TT_Analyt   = rect_cross.cross(Analytical, x0, nswp = 6, kickrank = 1, rf = 2)\n",
    "#print TT_Analyt\n",
    "\n",
    "print 'Approximate source'\n",
    "source = rect_cross.cross(q, x0, nswp = 6, kickrank = 1, rf = 2)\n",
    "weights = tt.ones(N, d)\n",
    "carriages = tt.tensor.to_list(weights)\n",
    "for i in xrange(d):\n",
    "    carriages[i] = carriages[i] * h\n",
    "    carriages[i][:, 0, :] /= 2\n",
    "    carriages[i][:, N - 1, :] /= 2\n",
    "weights = tt.tensor.from_list(carriages)\n",
    "\n",
    "\n",
    "Mass_tt = tt.multifuncrs([TT_Solution, Check_mass], lambda x: np.prod(x, axis = 1 ), eps = tolerance)\n",
    "#Mass_tt = Mass_tt.round(tolerance/1e2)\n",
    "print 'Starting mass = ', tt.dot(Mass_tt, weights)\n",
    "print 'Approximate kernel'\n",
    "#test with ballistic kernel\n",
    "#\n",
    "#x0 = tt.rand(N, 2 * d, r)\n",
    "#Coag_Kernel = rect_cross.cross(Coag_K, x0, nswp = 6, kickrank = 1, rf = 2)\n",
    "#print Coag_Kernel\n",
    "#print '============================================================'\n",
    "#\n",
    "#test with K(u,v) = 1\n",
    "#\n",
    "Coag_Kernel = tt.ones(N, 2 * d)\n",
    "#Coag_Kernel = Coag_Kernel.round(tolerance)\n",
    "\n",
    "t1 = time.clock()\n",
    "print 'alpha = ', Coag_Kernel.r[d]\n",
    "density_file = open('density_time.log', 'w')\n",
    "for t in xrange(N_steps):\n",
    "    print '=============================================================='\n",
    "    print 'Step', t + 1\n",
    "    print '=============================================================='\n",
    "\n",
    "    First_integral = 0.25 * tau * First_Integral(TT_Solution, Coag_Kernel)\n",
    "\n",
    "    Second_integral = - tau * 0.5 * Second_Integral(TT_Solution, Coag_Kernel)\n",
    "    print 'First integral predictor'\n",
    "    print First_integral\n",
    "    print 'Second integral predictor'\n",
    "    print Second_integral\n",
    "\n",
    "    TT_Solution_predictor = tt.multifuncrs([TT_Solution, First_integral, Second_integral, 0.5 * tau * source], lambda x: np.sum(x, axis = 1) , eps = tolerance)\n",
    "\n",
    "    First_integral = 0.5 * tau * First_Integral(TT_Solution_predictor, Coag_Kernel)\n",
    "\n",
    "    Second_integral = - tau * Second_Integral(TT_Solution_predictor, Coag_Kernel)\n",
    "\n",
    "    print 'First integral corrector'\n",
    "    print First_integral\n",
    "    print 'Second integral corrector'\n",
    "    print Second_integral\n",
    "\n",
    "    TT_Solution = tt.multifuncrs([TT_Solution, First_integral, Second_integral, tau * source], lambda x:  np.sum(x, axis = 1), eps = tolerance)\n",
    "\n",
    "    Mass_tt = tt.multifuncrs([TT_Solution, Check_mass], lambda x: np.prod(x, axis = 1), eps = tolerance)\n",
    "    print 'mass = ', tt.dot(Mass_tt, weights)\n",
    "    print 'density =', tt.dot(TT_Solution, weights)\n",
    "    density_file.write(\"%f %f\\n\"%((t + 1) * tau, tt.dot(TT_Solution, weights)))\n",
    "    print 'Solution'\n",
    "    print TT_Solution\n",
    "    print '=============================================================='\n",
    "\n",
    "    if print_res and (N <= 20000):\n",
    "        print 'saving results into file %s.dat'%(t)\n",
    "        res = open(\"%s.dat\"%(t), 'w')\n",
    "        for x in xrange(N):\n",
    "            res.writelines(\"\\n\")\n",
    "            for y in xrange(N):\n",
    "                #print x*h, ' ', y*h, ' ', (x + y) * h * TT_Solution[x,y]\n",
    "                res.writelines(\"%s %s %s\\n\"%(y * h, x * h, np.abs((x + y) * h * TT_Solution[y,x].real)))\n",
    "        res.close()\n",
    "t2 = time.clock()\n",
    "print 'time = ', t2 - t1\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  }
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