NonLinearEigen.ipynb

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    "colab": {
      "name": "NonLinearEigen.ipynb",
      "provenance": [],
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
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  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/dnguyend/lagrange_rayleigh/blob/master/NonLinearEigen.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "kbJFQ2jJt-_u",
        "colab_type": "text"
      },
      "source": [
        "$\\newcommand{\\bP}{\\boldsymbol{P}}$\n",
        "$\\newcommand{\\bL}{\\boldsymbol{L}}$\n",
        "$\\newcommand{\\bx}{\\boldsymbol{x}}$\n",
        "$\\newcommand{\\bLx}{\\boldsymbol{L}_{\\bx}}$\n",
        "$\\newcommand{\\blbd}{\\boldsymbol{\\lambda}}$\n",
        "$\\newcommand{\\bLlbd}{\\boldsymbol{L}_{\\blbd}}$\n",
        "$\\newcommand{\\Plbd}{\\boldsymbol{P}_{\\blbd}}$\n",
        "$\\newcommand{\\bJC}{\\boldsymbol{J}_{\\boldsymbol{C}}}$\n",
        "# The non linear eigenvalue problem#\n",
        "\n",
        "We show how the frameworks in the paper applies to the non linear eigen value problem. \n",
        "\n",
        "Since classical eigenvalue problem is a special case of nonlinear eigenvalue problem, the analysis here works for classical eigenvalue problem.\n",
        "\n",
        "Much of this result is known, but we want to cast it under the framework of the paper. We do a comparison of Rayleigh Chebyshev versus regular RQI, and two-sided RQI. As expected, because of the extra calculations, Rayleigh Chebyshev underperform two-sided RQI. It outperforms regular RQI. \n",
        "\n",
        "A special feature is since $\\bL(\\bx, \\blbd) = \\bP(\\blbd)\\bx$ is linear in $\\bx$, it turns out\n",
        "$$\\nu = \\bLx^{-1}\\bL(\\bx, \\blbd) = \\bx$$.\n",
        "$$\\zeta = \\bP^{-1}(\\blbd)\\Plbd(\\blbd)\\bx$$\n",
        "Therefore\n",
        "$$\\bx + \\eta = \\bx -\\bx + \\zeta\\bJC(\\zeta)\\bJC(\\bx) = \\zeta\\bJC(\\zeta)\\bJC(\\bx) $$\n",
        "is proportional to $\\zeta$.\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "cHtVO5tdt1Ig",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 136
        },
        "outputId": "fa6b9e5d-9e72-43a4-a31b-f497fd948df4"
      },
      "source": [
        "!git clone https://github.com/dnguyend/lagrange_rayleigh"
      ],
      "execution_count": 1,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Cloning into 'lagrange_rayleigh'...\n",
            "remote: Enumerating objects: 138, done.\u001b[K\n",
            "remote: Counting objects: 100% (138/138), done.\u001b[K\n",
            "remote: Compressing objects: 100% (97/97), done.\u001b[K\n",
            "remote: Total 138 (delta 65), reused 99 (delta 37), pack-reused 0\u001b[K\n",
            "Receiving objects: 100% (138/138), 1.25 MiB | 22.81 MiB/s, done.\n",
            "Resolving deltas: 100% (65/65), done.\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-qSRNIZuu-Gc",
        "colab_type": "text"
      },
      "source": [
        "First we introduce a number of algorithms and support functions. the matrix and its eigenvectors are could all be complex numbers:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "d3cxWnZ70ADG",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "from __future__ import print_function\n",
        "\n",
        "import numpy as np\n",
        "import time\n",
        "from numpy import conjugate\n",
        "from numpy.linalg import solve, norm\n",
        "\n",
        "\n",
        "def gen_nonlinear_symmetric(n, m):\n",
        "    # a polynomial matrix of degree m-1\n",
        "    # [:, :, 0] correspond to highest degree\n",
        "    # [:, :, m-1] is the constant term\n",
        "    # symmetric\n",
        "    mat = np.random.randn(n, n, m)\n",
        "    for i in range(m):\n",
        "        mat[:, :, i] = 0.5 * (\n",
        "            mat[:, :, i] + mat[:, :, i].T)\n",
        "    return mat\n",
        "\n",
        "\n",
        "def gen_nonlinear(n, m, hermitian=False):\n",
        "    # a polynomial matrix of degree m-1\n",
        "    # [:, :, 0] correspond to highest degree\n",
        "    # [:, :, m-1] is the constant term\n",
        "    # symmetric\n",
        "    mat = np.random.randn(n, n, m) + np.random.randn(n, n, m)*1.j\n",
        "    if hermitian:\n",
        "        for i in range(m):\n",
        "            mat[:, :, i] = 0.5 * (\n",
        "                mat[:, :, i] + conjugate(mat[:, :, i]).T)\n",
        "    return mat\n",
        "\n",
        "\n",
        "def deriv(mat):\n",
        "    if mat.shape[2] == 1:\n",
        "        return np.zeros_like(mat)\n",
        "    ret = np.zeros((\n",
        "        mat.shape[0], mat.shape[1], mat.shape[2]-1), dtype=mat.dtype)\n",
        "    for i in range(mat.shape[2]-1):\n",
        "        ret[:, :, i] = mat[:, :, i] * (mat.shape[2]-1-i)\n",
        "    return ret\n",
        "\n",
        "\n",
        "def eval_mat(mat, lbd):\n",
        "    ret = np.zeros_like(mat[:, :, 0])\n",
        "    for i in range(mat.shape[2]):\n",
        "        ret = ret * lbd + mat[:, :, i]\n",
        "    return ret\n",
        "\n",
        "\n",
        "def eval_pol(mat, v, w):\n",
        "    return np.poly1d(\n",
        "        [conjugate(w) @ (mat[:, :, i] @ v) for i in range(mat.shape[2])])\n",
        "\n",
        "\n",
        "def rayleigh_chebyshev_linear(\n",
        "        mat, v0, lbd0, max_itr, tol, u, debug_print=False, do_chebyshev=True):\n",
        "    \"\"\" Works for both real and complex field\n",
        "    \"\"\"\n",
        "    der_mat = deriv(mat)\n",
        "    der_mat2 = deriv(der_mat)\n",
        "    lbd = lbd0\n",
        "    v_k = v0 / (u @ v0)\n",
        "    for k in range(max_itr):\n",
        "        err = norm((eval_mat(mat, lbd) @ v_k))\n",
        "        if err < tol:\n",
        "            break\n",
        "\n",
        "        F_lbd = eval_mat(mat, lbd)\n",
        "        Fp_lbd = eval_mat(der_mat, lbd)\n",
        "        Fp2_lbd = eval_mat(der_mat2, lbd)\n",
        "        zeta = solve(F_lbd, (Fp_lbd @ v_k))\n",
        "        # we do not want the second degree term\n",
        "        # to dominate so we add it only if norm\n",
        "        zeta /= (u @ zeta)\n",
        "        eta = zeta - v_k\n",
        "        # if False and (norm(eta) < .001):\n",
        "        if do_chebyshev and (norm(eta < 1e-2)):\n",
        "            # pdb.set_trace()\n",
        "            xH_P_lbd_x = np.sum(u * (Fp_lbd @ v_k))\n",
        "            xPb = u @ (F_lbd @ eta)\n",
        "            J_R_eta = -xPb / xH_P_lbd_x\n",
        "            L_x_lbd = (Fp_lbd @ eta) * J_R_eta\n",
        "            L_lbd_lbd = (Fp2_lbd @ v_k) * (J_R_eta * J_R_eta)\n",
        "            T_adj = solve(F_lbd, -L_x_lbd - 0.5 * L_lbd_lbd)\n",
        "            \n",
        "            v_k_n = zeta + T_adj\n",
        "            v_k_n /= (u @ v_k_n)\n",
        "            v_k = v_k_n\n",
        "            if False:\n",
        "                v_k_n = zeta\n",
        "                v_k = v_k_n\n",
        "        else:\n",
        "            v_k_n = zeta\n",
        "            v_k = v_k_n\n",
        "        rr = np.poly1d(\n",
        "            [u @ (mat[:, :, i] @ v_k) for i in range(mat.shape[2])]).roots\n",
        "        lbd = rr[np.argsort(np.abs(rr - lbd))[0]]\n",
        "        if debug_print:\n",
        "            print(\"rr=%s lbd=%s\" % (str(rr), str(lbd)))\n",
        "\n",
        "    return v_k, lbd, k, err < tol\n",
        "\n",
        "\n",
        "def _rayleigh_chebyshev_trans_not_used(\n",
        "        mat, v0, lbd0, max_itr, tol, debug_print=False, do_chebyshev=True):\n",
        "    \"\"\"mat, v, lbd are all real or complex we use zTz=1 so non compact\n",
        "    for complex. SHOULD NOT USE\n",
        "    \"\"\"\n",
        "    der_mat = deriv(mat)\n",
        "    der_mat2 = deriv(der_mat)\n",
        "    lbd = lbd0\n",
        "    v_k = v0 / np.sqrt(v0.T @ v0)\n",
        "    for k in range(max_itr):\n",
        "        err = norm(eval_mat(mat, lbd) @ v_k)\n",
        "        if err < tol:\n",
        "            break\n",
        "\n",
        "        F_lbd = eval_mat(mat, lbd)\n",
        "        Fp_lbd = eval_mat(der_mat, lbd)\n",
        "        Fp2_lbd = eval_mat(der_mat2, lbd)\n",
        "        zeta = solve(F_lbd, (Fp_lbd @ v_k))\n",
        "        # we do not want the second degree term\n",
        "        # to dominate so we add it only if norm\n",
        "        zeta /= np.sqrt(zeta.T @ zeta)\n",
        "        eta = zeta - v_k\n",
        "        if do_chebyshev and (norm(eta) < .01):\n",
        "            # pdb.set_trace()\n",
        "            xH_P_lbd_x = np.sum(v_k * (Fp_lbd @ v_k))\n",
        "            xPb = v_k @ ((F_lbd + F_lbd.T) @ eta)\n",
        "            J_R_eta = -xPb / xH_P_lbd_x\n",
        "            L_x_lbd = (Fp_lbd @ eta) * J_R_eta\n",
        "            L_lbd_lbd = (Fp2_lbd @ v_k) * (J_R_eta * J_R_eta)\n",
        "            T_adj = solve(F_lbd, -L_x_lbd - 0.5 * L_lbd_lbd)\n",
        "            if True or (norm(T_adj) < 0.1 * norm(eta)):\n",
        "                v_k_n = zeta + T_adj\n",
        "                v_k_n /= np.sqrt(v_k_n.T @ v_k_n)\n",
        "                v_k = v_k_n\n",
        "            else:\n",
        "                v_k_n = zeta\n",
        "                v_k = v_k_n\n",
        "        else:\n",
        "            v_k_n = zeta\n",
        "            v_k = v_k_n\n",
        "        rr = np.poly1d(\n",
        "            [v_k @ (mat[:, :, i] @ v_k) for i in range(mat.shape[2])]).roots\n",
        "        # rr = eval_pol(mat, v_k, v_k).roots\n",
        "        lbd = rr[np.argsort(np.abs(rr - lbd))[0]]\n",
        "        if debug_print:\n",
        "            print(\"rr=%s lbd=%s\" % (str(rr), str(lbd)))\n",
        "\n",
        "    return v_k, lbd, k, err < tol\n",
        "            \n",
        "\n",
        "def two_sided_schreiber(\n",
        "        mat, v0, lbd0, max_itr, tol, w0=None, debug_print=False):\n",
        "    der_mat = deriv(mat)\n",
        "    lbd = lbd0\n",
        "    v_k = v0 / norm(v0)\n",
        "    if w0 is None:\n",
        "        w0 = v0.T / norm(v0)\n",
        "    w_k = w0\n",
        "    for k in range(max_itr):\n",
        "        err = norm(eval_mat(mat, lbd) @ v_k)\n",
        "        if err < tol:\n",
        "            break\n",
        "\n",
        "        F_lbd = eval_mat(mat, lbd)\n",
        "        Fp_lbd = eval_mat(der_mat, lbd)\n",
        "        v_k_n = solve(F_lbd, (Fp_lbd @ v_k))\n",
        "        v_k_n /= norm(v_k_n)\n",
        "\n",
        "        w_k_n = solve(\n",
        "            conjugate(F_lbd).T,\n",
        "            conjugate(Fp_lbd).T @ w_k)\n",
        "        w_k_n /= norm(w_k_n)\n",
        "        v_k = v_k_n\n",
        "        w_k = w_k_n\n",
        "\n",
        "        rr = eval_pol(mat, v_k, w_k).roots\n",
        "        lbd = rr[np.argsort(np.abs(rr - lbd))[0]]\n",
        "        if debug_print:\n",
        "            print(\"rr=%s lbd=%s v_k=%s w_k=%s\" % (str(rr), str(lbd), str(v_k), str(w_k)))\n",
        "    return v_k_n, lbd, k, err < tol\n",
        "\n",
        "\n",
        "def inverse_iteration(mat, v0, lbd0, max_itr, tol, u):\n",
        "    \"\"\" inverse_iteration\n",
        "    the constraint set is u @ vt = 1\n",
        "    return v, lbd, nbr_iter, convergence\n",
        "    \"\"\"\n",
        "    v = v0.copy()\n",
        "    der_mat = deriv(mat)\n",
        "    lbd = lbd0\n",
        "    for k in range(max_itr):\n",
        "        err = norm(eval_mat(mat, lbd) @ v)\n",
        "        if err < tol:\n",
        "            break\n",
        "        vt = solve(\n",
        "            eval_mat(mat, lbd), eval_mat(der_mat, lbd)) @ v\n",
        "        lbd -= (u.T @ v) / (np.sum(u * vt))\n",
        "        v = vt/norm(vt)\n",
        "    return v, lbd, k, err < tol\n"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "AYvW8zYS0jbl",
        "colab_type": "text"
      },
      "source": [
        "We compare all runs in the following:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "kufM_x800sSX",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 1000
        },
        "outputId": "9b2307d5-e697-454c-9f27-be3addd962d0"
      },
      "source": [
        "\n",
        "def compare_all():\n",
        "    tol = 1e-8\n",
        "    max_itr = 200\n",
        "    lbd0 = 1+0j\n",
        "    n = 100\n",
        "    m = 4\n",
        "    mat = gen_nonlinear(n, m, hermitian=False)\n",
        "    u = np.array([1.+0j] + (n-1)*[0.j])\n",
        "    n_test = 500\n",
        "    sr_lbd_arr = np.zeros((n_test), dtype=np.complex128)\n",
        "    sc_lbd_arr = np.zeros((n_test), dtype=np.complex128)\n",
        "    scb_lbd_arr = np.zeros((n_test), dtype=np.complex128)\n",
        "    sr_k_arr = np.zeros((n_test), dtype=float)\n",
        "    sc_k_arr = np.zeros((n_test), dtype=float)\n",
        "    scb_k_arr = np.zeros((n_test), dtype=float)\n",
        "    \n",
        "    sr_conv_arr = np.zeros((n_test), dtype=float)\n",
        "    sc_conv_arr = np.zeros((n_test), dtype=float)\n",
        "    scb_conv_arr = np.zeros((n_test), dtype=float)\n",
        "\n",
        "    sr_time_arr = np.zeros((n_test), dtype=float)\n",
        "    sc_time_arr = np.zeros((n_test), dtype=float)\n",
        "    scb_time_arr = np.zeros((n_test), dtype=float)\n",
        "\n",
        "    sr_err_arr = np.zeros((n_test), dtype=float)\n",
        "    sc_err_arr = np.zeros((n_test), dtype=float)\n",
        "    scb_err_arr = np.zeros((n_test), dtype=float)\n",
        "\n",
        "    for ii in range(n_test):\n",
        "        v0 = np.random.randn(n) + np.random.randn(n)*1.j\n",
        "        v0 /= np.sum(u*v0)\n",
        "        # i_v, i_lbd, i_k, i_conv = inverse_iteration(\n",
        "        # mat, v0, lbd0, max_itr, tol, u)\n",
        "        # print(\"DOING SR\")\n",
        "        t_start = time.time()\n",
        "        sr_v, sr_lbd, sr_k, sr_conv = two_sided_schreiber(\n",
        "            mat, v0.copy(), lbd0, max_itr, tol, w0=None, debug_print=False)\n",
        "        t_end = time.time()\n",
        "        sr_time_arr[ii] = t_end - t_start\n",
        "        # print(\"DOING SC\")\n",
        "        t_start = time.time()\n",
        "        if True:\n",
        "            sc_v, sc_lbd, sc_k, sc_conv = rayleigh_chebyshev_linear(\n",
        "                mat, v0.copy(), lbd0, max_itr, tol,\n",
        "                u, debug_print=False, do_chebyshev=False)\n",
        "            \"\"\"\n",
        "        else:\n",
        "            sc_v, sc_lbd, sc_k, sc_conv = rayleigh_chebyshev_trans(\n",
        "                mat, v0, lbd0, max_itr, tol,\n",
        "                debug_print=False, do_chebyshev=True)\n",
        "            \"\"\"\n",
        "\n",
        "        t_end = time.time()\n",
        "        sc_time_arr[ii] = t_end - t_start\n",
        "        \n",
        "        t_start = time.time()\n",
        "        scb_v, scb_lbd, scb_k, scb_conv = rayleigh_chebyshev_linear(\n",
        "            mat, v0.copy(), lbd0, max_itr, tol,\n",
        "            u, debug_print=False, do_chebyshev=True)\n",
        "        t_end = time.time()\n",
        "        scb_time_arr[ii] = t_end - t_start\n",
        "        \n",
        "        sr_lbd_arr[ii] = sr_lbd\n",
        "        sc_lbd_arr[ii] = sc_lbd\n",
        "        scb_lbd_arr[ii] = scb_lbd\n",
        "        \n",
        "        sr_k_arr[ii] = sr_k\n",
        "        sc_k_arr[ii] = sc_k\n",
        "        scb_k_arr[ii] = scb_k\n",
        "        \n",
        "        sr_conv_arr[ii] = sr_conv\n",
        "        sc_conv_arr[ii] = sc_conv\n",
        "        scb_conv_arr[ii] = scb_conv\n",
        "\n",
        "        sr_err_arr[ii] = np.linalg.norm(eval_mat(mat, sr_lbd) @ sr_v)\n",
        "        sc_err_arr[ii] = np.linalg.norm(eval_mat(mat, sc_lbd) @ sc_v)\n",
        "        scb_err_arr[ii] = np.linalg.norm(eval_mat(mat, scb_lbd) @ scb_v)\n",
        "        \n",
        "        print(\"sr ii=%d lbd=%s k=%d conv=%d\" % (\n",
        "            ii, str(sr_lbd), sr_k, sr_conv))\n",
        "        print(\"sc ii=%d lbd=%s k=%d conv=%d\" % (\n",
        "            ii, str(sc_lbd), sc_k, sc_conv))\n",
        "        print(\"scb ii=%d lbd=%s k=%d conv=%d\" % (\n",
        "            ii, str(scb_lbd), scb_k, scb_conv))\n",
        "\n",
        "    import pandas as pd\n",
        "    pdF = pd.DataFrame({'sr_lbd_arr': sr_lbd_arr,\n",
        "                        'sc_lbd_arr': sc_lbd_arr,\n",
        "                        'scb_lbd_arr': scb_lbd_arr,\n",
        "                        'sr_k_arr': sr_k_arr,\n",
        "                        'sc_k_arr': sc_k_arr,\n",
        "                        'scb_k_arr': scb_k_arr,\n",
        "                        'sr_conv_arr': sr_conv_arr,\n",
        "                        'sc_conv_arr': sc_conv_arr,\n",
        "                        'scb_conv_arr': scb_conv_arr,\n",
        "                        \n",
        "                        'sr_time_arr': sr_time_arr,\n",
        "                        'sc_time_arr': sc_time_arr,\n",
        "                        'scb_time_arr': scb_time_arr,\n",
        "\n",
        "                        'sr_err_arr': sr_err_arr,\n",
        "                        'sc_err_arr': sc_err_arr,\n",
        "                        'scb_err_arr': scb_err_arr},\n",
        "                       columns=['sr_lbd_arr',\n",
        "                                'sc_lbd_arr',\n",
        "                                'scb_lbd_arr',\n",
        "                                'sr_k_arr',\n",
        "                                'sc_k_arr',\n",
        "                                'scb_k_arr',\n",
        "                                'sr_conv_arr',\n",
        "                                'sc_conv_arr',\n",
        "                                'scb_conv_arr',\n",
        "                                'sr_time_arr',\n",
        "                                'sc_time_arr',\n",
        "                                'scb_time_arr',\n",
        "                                'sr_err_arr',\n",
        "                                'sc_err_arr',\n",
        "                                'scb_err_arr'])\n",
        "    print(pdF[[a for a in pdF.columns if '_k_' in a]].describe())\n",
        "    print(pdF[[a for a in pdF.columns if '_time_' in a]].describe())\n",
        "    print(pdF[[a for a in pdF.columns if '_lbd_' in a]].apply(\n",
        "        np.real, axis=0).describe())\n",
        "    print(pdF[[a for a in pdF.columns if '_lbd_' in a]].apply(\n",
        "        np.imag, axis=0).describe())\n",
        "    print(pdF[[a for a in pdF.columns if '_err_' in a]].describe())\n",
        "\n",
        "    pdF[[a for a in pdF.columns if '_lbd_' in a]].apply(np.real, axis=0).hist()\n",
        "    pdF[[a for a in pdF.columns if '_lbd_' in a]].apply(np.imag, axis=0).hist()\n",
        "    return pdF\n",
        "    \n",
        "def check_eigen(mat, lbd, v):\n",
        "    return eval_mat(mat, lbd) @ v, norm(eval_mat(mat, lbd) @ v)\n",
        "\n",
        "\n",
        "def check_deriv(mat, lbd):\n",
        "    dd = deriv(mat)\n",
        "    h = 1e-5\n",
        "    print((eval_mat(mat, lbd+h) - eval_mat(mat, lbd))/h-eval_mat(dd, lbd))\n",
        "\n",
        "\n",
        "\n",
        "pdF = compare_all()"
      ],
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "sr ii=0 lbd=(0.9433429035751112-0.061986666921747834j) k=6 conv=1\n",
            "sc ii=0 lbd=(0.8167044437631178-0.6732783815818864j) k=9 conv=1\n",
            "scb ii=0 lbd=(1.4286704832958583-0.4950528849821565j) k=7 conv=1\n",
            "sr ii=1 lbd=(0.9433429035751114-0.06198666692174804j) k=5 conv=1\n",
            "sc ii=1 lbd=(0.9433429035751063-0.0619866669217468j) k=8 conv=1\n",
            "scb ii=1 lbd=(0.9433429035751215-0.06198666692173842j) k=4 conv=1\n",
            "sr ii=2 lbd=(0.9433429035751112-0.0619866669217479j) k=5 conv=1\n",
            "sc ii=2 lbd=(0.9433429035751328-0.06198666692172852j) k=7 conv=1\n",
            "scb ii=2 lbd=(0.9154334548814698-0.14915189154423203j) k=5 conv=1\n",
            "sr ii=3 lbd=(0.9433429035751095-0.06198666692174825j) k=5 conv=1\n",
            "sc ii=3 lbd=(0.5553704522022601-0.5212007153273682j) k=11 conv=1\n",
            "scb ii=3 lbd=(0.8299850994895788-0.4032607607777108j) k=6 conv=1\n",
            "sr ii=4 lbd=(0.9433429035751101-0.06198666692174822j) k=5 conv=1\n",
            "sc ii=4 lbd=(0.7434301332761796+0.2772223100922925j) k=7 conv=1\n",
            "scb ii=4 lbd=(1.0364920838369902-0.20949445042647374j) k=8 conv=1\n",
            "sr ii=5 lbd=(1.0582032208004417+0.1455038668630644j) k=6 conv=1\n",
            "sc ii=5 lbd=(1.0930214450828297-0.21692968297439763j) k=10 conv=1\n",
            "scb ii=5 lbd=(1.0930214450828448-0.21692968297440432j) k=6 conv=1\n",
            "sr ii=6 lbd=(0.9433429035751114-0.061986666921748236j) k=5 conv=1\n",
            "sc ii=6 lbd=(0.943342903575109-0.061986666921751095j) k=7 conv=1\n",
            "scb ii=6 lbd=(0.9433429035751169-0.06198666692174547j) k=5 conv=1\n",
            "sr ii=7 lbd=(1.0930214450828428-0.2169296829744046j) k=8 conv=1\n",
            "sc ii=7 lbd=(0.9433429035752064-0.06198666692298975j) k=7 conv=1\n",
            "scb ii=7 lbd=(0.9164388992154012+0.18145550905886146j) k=4 conv=1\n",
            "sr ii=8 lbd=(0.9433429035751111-0.06198666692174793j) k=7 conv=1\n",
            "sc ii=8 lbd=(0.3790924080095755-0.24604234374001915j) k=17 conv=1\n",
            "scb ii=8 lbd=(0.9164388992148981+0.18145550905946894j) k=4 conv=1\n",
            "sr ii=9 lbd=(0.9433429035751106-0.0619866669217485j) k=5 conv=1\n",
            "sc ii=9 lbd=(0.9433429035751064-0.06198666692174713j) k=9 conv=1\n",
            "scb ii=9 lbd=(1.0364920838369915-0.20949445042647385j) k=5 conv=1\n",
            "sr ii=10 lbd=(0.9433429035751111-0.06198666692174811j) k=5 conv=1\n",
            "sc ii=10 lbd=(0.9154334548803978-0.14915189154267375j) k=9 conv=1\n",
            "scb ii=10 lbd=(0.569152653989525+0.12363164654205487j) k=8 conv=1\n",
            "sr ii=11 lbd=(0.8347432209035668+0.15463087235736664j) k=6 conv=1\n",
            "sc ii=11 lbd=(0.9433429035752525-0.06198666692206673j) k=9 conv=1\n",
            "scb ii=11 lbd=(0.9433429035751102-0.0619866669217464j) k=5 conv=1\n",
            "sr ii=12 lbd=(0.9433429035751099-0.0619866669217481j) k=4 conv=1\n",
            "sc ii=12 lbd=(0.916438899233828+0.18145550909117555j) k=8 conv=1\n",
            "scb ii=12 lbd=(0.7975067415890389-0.2848420423077904j) k=7 conv=1\n",
            "sr ii=13 lbd=(0.9433429035751109-0.06198666692174838j) k=5 conv=1\n",
            "sc ii=13 lbd=(0.9433429035914106-0.06198666691984853j) k=8 conv=1\n",
            "scb ii=13 lbd=(0.9433429035751103-0.061986666921744205j) k=5 conv=1\n",
            "sr ii=14 lbd=(0.9433429035751101-0.06198666692174806j) k=5 conv=1\n",
            "sc ii=14 lbd=(0.6626488936933432+0.15003463231260164j) k=9 conv=1\n",
            "scb ii=14 lbd=(0.943342903572157-0.06198666692179747j) k=4 conv=1\n",
            "sr ii=15 lbd=(1.0930214450828433-0.2169296829744048j) k=9 conv=1\n",
            "sc ii=15 lbd=(0.4970302295031423+0.35543287154417175j) k=8 conv=1\n",
            "scb ii=15 lbd=(1.0582032208004393+0.14550386686306993j) k=8 conv=1\n",
            "sr ii=16 lbd=(0.9433429035751102-0.06198666692174832j) k=5 conv=1\n",
            "sc ii=16 lbd=(0.9433429035749623-0.06198666692168166j) k=5 conv=1\n",
            "scb ii=16 lbd=(0.9433429035751122-0.061986666921749894j) k=4 conv=1\n",
            "sr ii=17 lbd=(1.0364920838369907-0.2094944504264747j) k=7 conv=1\n",
            "sc ii=17 lbd=(0.7434301332761805+0.27722231009229115j) k=9 conv=1\n",
            "scb ii=17 lbd=(1.0996521817685168-0.03267157308285691j) k=6 conv=1\n",
            "sr ii=18 lbd=(0.9433429035751112-0.061986666921748104j) k=6 conv=1\n",
            "sc ii=18 lbd=(0.8347432209035668+0.1546308723573739j) k=12 conv=1\n",
            "scb ii=18 lbd=(0.9164388992149001+0.18145550905947047j) k=7 conv=1\n",
            "sr ii=19 lbd=(0.9433429035751091-0.06198666692174799j) k=5 conv=1\n",
            "sc ii=19 lbd=(0.7975067415890362-0.28484204230779203j) k=12 conv=1\n",
            "scb ii=19 lbd=(1.0582032208004415+0.14550386686306682j) k=5 conv=1\n",
            "sr ii=20 lbd=(0.9433429035751113-0.06198666692174809j) k=5 conv=1\n",
            "sc ii=20 lbd=(0.9154334548804013-0.14915189154266786j) k=10 conv=1\n",
            "scb ii=20 lbd=(0.9154334548803997-0.14915189154266417j) k=5 conv=1\n",
            "sr ii=21 lbd=(0.9433429035751114-0.06198666692174783j) k=5 conv=1\n",
            "sc ii=21 lbd=(1.0364920838369909-0.20949445042647688j) k=8 conv=1\n",
            "scb ii=21 lbd=(0.9433429035751111-0.06198666692174988j) k=4 conv=1\n",
            "sr ii=22 lbd=(0.9433429035751124-0.061986666921748895j) k=6 conv=1\n",
            "sc ii=22 lbd=(0.6158779463724693+0.027490685432328264j) k=10 conv=1\n",
            "scb ii=22 lbd=(0.9433429035752444-0.06198666692179959j) k=9 conv=1\n",
            "sr ii=23 lbd=(0.943342903575111-0.06198666692174844j) k=5 conv=1\n",
            "sc ii=23 lbd=(0.9433429035751069-0.06198666692174694j) k=7 conv=1\n",
            "scb ii=23 lbd=(0.943342903575113-0.061986666921747514j) k=5 conv=1\n",
            "sr ii=24 lbd=(1.0582032208004417+0.145503866863064j) k=10 conv=1\n",
            "sc ii=24 lbd=(1.0996521817685283-0.032671573082858486j) k=9 conv=1\n",
            "scb ii=24 lbd=(1.2992802280393785-0.060799123100279934j) k=4 conv=1\n",
            "sr ii=25 lbd=(0.9433429035751121-0.06198666692174852j) k=6 conv=1\n",
            "sc ii=25 lbd=(0.5337798856090255-0.20178608032880888j) k=9 conv=1\n",
            "scb ii=25 lbd=(0.9433429035750781-0.061986666920551894j) k=4 conv=1\n",
            "sr ii=26 lbd=(0.9433429035751105-0.06198666692174789j) k=4 conv=1\n",
            "sc ii=26 lbd=(0.9433429035751107-0.06198666692174809j) k=7 conv=1\n",
            "scb ii=26 lbd=(0.9433429035750596-0.06198666692016412j) k=7 conv=1\n",
            "sr ii=27 lbd=(0.943342903575111-0.06198666692174844j) k=4 conv=1\n",
            "sc ii=27 lbd=(0.943342903576993-0.061986666920715465j) k=8 conv=1\n",
            "scb ii=27 lbd=(0.9433429035751254-0.06198666692175801j) k=4 conv=1\n",
            "sr ii=28 lbd=(0.9433429035751103-0.06198666692174783j) k=6 conv=1\n",
            "sc ii=28 lbd=(1.036492083836988-0.2094944504264749j) k=8 conv=1\n",
            "scb ii=28 lbd=(1.093021445082841-0.21692968297440512j) k=6 conv=1\n",
            "sr ii=29 lbd=(0.9433429035751103-0.061986666921747965j) k=5 conv=1\n",
            "sc ii=29 lbd=(0.9433429035750819-0.061986666921720154j) k=8 conv=1\n",
            "scb ii=29 lbd=(1.0930214450828208-0.21692968297441426j) k=8 conv=1\n",
            "sr ii=30 lbd=(0.9164388992148997+0.18145550905946967j) k=6 conv=1\n",
            "sc ii=30 lbd=(0.943342903575109-0.0619866669217506j) k=8 conv=1\n",
            "scb ii=30 lbd=(0.9164388992149012+0.18145550905946892j) k=5 conv=1\n",
            "sr ii=31 lbd=(0.9433429035751121-0.061986666921747986j) k=10 conv=1\n",
            "sc ii=31 lbd=(0.9433429035750864-0.0619866669216956j) k=7 conv=1\n",
            "scb ii=31 lbd=(0.9433429035921003-0.061986666908458984j) k=5 conv=1\n",
            "sr ii=32 lbd=(0.943342903575112-0.06198666692174816j) k=4 conv=1\n",
            "sc ii=32 lbd=(0.943342903574337-0.061986666912101265j) k=6 conv=1\n",
            "scb ii=32 lbd=(0.9433429035751156-0.061986666921753975j) k=4 conv=1\n",
            "sr ii=33 lbd=(0.9433429035751111-0.06198666692174807j) k=4 conv=1\n",
            "sc ii=33 lbd=(0.5691526539895296+0.12363164654205484j) k=18 conv=1\n",
            "scb ii=33 lbd=(0.9433429035748614-0.061986666920981495j) k=4 conv=1\n",
            "sr ii=34 lbd=(0.9433429035751121-0.06198666692174782j) k=6 conv=1\n",
            "sc ii=34 lbd=(1.0996521817685203-0.03267157308285641j) k=9 conv=1\n",
            "scb ii=34 lbd=(1.058203220800441+0.14550386686306874j) k=6 conv=1\n",
            "sr ii=35 lbd=(0.9433429035751112-0.06198666692174719j) k=6 conv=1\n",
            "sc ii=35 lbd=(0.9433429035749344-0.061986666921658634j) k=8 conv=1\n",
            "scb ii=35 lbd=(0.797506741589038-0.28484204230779125j) k=9 conv=1\n",
            "sr ii=36 lbd=(0.94334290357511-0.061986666921747424j) k=5 conv=1\n",
            "sc ii=36 lbd=(0.9433429035747984-0.061986666923460526j) k=7 conv=1\n",
            "scb ii=36 lbd=(0.9433429035750984-0.06198666692175801j) k=5 conv=1\n",
            "sr ii=37 lbd=(1.0996521817685245-0.03267157308285651j) k=8 conv=1\n",
            "sc ii=37 lbd=(1.0930214450828424-0.2169296829744054j) k=8 conv=1\n",
            "scb ii=37 lbd=(1.0364920838266665-0.20949445041168122j) k=5 conv=1\n",
            "sr ii=38 lbd=(0.9433429035751104-0.061986666921747834j) k=5 conv=1\n",
            "sc ii=38 lbd=(0.9164388992149489+0.18145550905948762j) k=9 conv=1\n",
            "scb ii=38 lbd=(0.9433429035751133-0.061986666921748104j) k=5 conv=1\n",
            "sr ii=39 lbd=(1.0996521817685225-0.03267157308285651j) k=7 conv=1\n",
            "sc ii=39 lbd=(0.9164388992149006+0.18145550905947028j) k=9 conv=1\n",
            "scb ii=39 lbd=(0.9164388992148992+0.18145550905946892j) k=6 conv=1\n",
            "sr ii=40 lbd=(0.9433429035751116-0.06198666692174799j) k=5 conv=1\n",
            "sc ii=40 lbd=(1.0364920838369924-0.20949445042647585j) k=10 conv=1\n",
            "scb ii=40 lbd=(0.9433429035751111-0.06198666692174608j) k=6 conv=1\n",
            "sr ii=41 lbd=(1.0996521817685252-0.032671573082855954j) k=10 conv=1\n",
            "sc ii=41 lbd=(0.9433429035772416-0.061986666924100306j) k=11 conv=1\n",
            "scb ii=41 lbd=(1.0996521817685223-0.032671573082856745j) k=5 conv=1\n",
            "sr ii=42 lbd=(0.9433429035751103-0.06198666692174764j) k=5 conv=1\n",
            "sc ii=42 lbd=(0.3871876197416802-0.09154704917405657j) k=11 conv=1\n",
            "scb ii=42 lbd=(0.9433429035751142-0.06198666692175079j) k=6 conv=1\n",
            "sr ii=43 lbd=(1.0364920838369889-0.20949445042647413j) k=7 conv=1\n",
            "sc ii=43 lbd=(0.7084100986148998+0.40180271484232294j) k=13 conv=1\n",
            "scb ii=43 lbd=(1.0582032208130157+0.14550386688643388j) k=5 conv=1\n",
            "sr ii=44 lbd=(0.9433429035751113-0.06198666692174807j) k=5 conv=1\n",
            "sc ii=44 lbd=(0.9433429035751134-0.06198666692174987j) k=7 conv=1\n",
            "scb ii=44 lbd=(1.0364920838369889-0.20949445042647633j) k=8 conv=1\n",
            "sr ii=45 lbd=(0.9433429035751116-0.06198666692174831j) k=4 conv=1\n",
            "sc ii=45 lbd=(0.9433429035751089-0.0619866669217466j) k=8 conv=1\n",
            "scb ii=45 lbd=(1.0364920838369875-0.20949445042647746j) k=6 conv=1\n",
            "sr ii=46 lbd=(0.9433429035751113-0.06198666692174791j) k=4 conv=1\n",
            "sc ii=46 lbd=(0.6280609732556826-0.2424485063933972j) k=12 conv=1\n",
            "scb ii=46 lbd=(0.8347432209037292+0.1546308723572453j) k=5 conv=1\n",
            "sr ii=47 lbd=(1.0582032208004422+0.1455038668630633j) k=7 conv=1\n",
            "sc ii=47 lbd=(1.099652181768522-0.03267157308285604j) k=7 conv=1\n",
            "scb ii=47 lbd=(1.0996521817744747-0.032671573076941386j) k=4 conv=1\n",
            "sr ii=48 lbd=(0.9433429035751121-0.061986666921748354j) k=5 conv=1\n",
            "sc ii=48 lbd=(0.9433429035885667-0.061986666941402174j) k=9 conv=1\n",
            "scb ii=48 lbd=(0.943342903575221-0.06198666692854479j) k=3 conv=1\n",
            "sr ii=49 lbd=(0.9433429035751093-0.06198666692174829j) k=5 conv=1\n",
            "sc ii=49 lbd=(0.9433429036095524-0.06198666689974869j) k=7 conv=1\n",
            "scb ii=49 lbd=(0.9433429035751095-0.061986666921748035j) k=4 conv=1\n",
            "sr ii=50 lbd=(0.6158779463724692+0.027490685432329826j) k=6 conv=1\n",
            "sc ii=50 lbd=(0.9433429035751109-0.06198666692175304j) k=8 conv=1\n",
            "scb ii=50 lbd=(1.036492083836989-0.2094944504264705j) k=7 conv=1\n",
            "sr ii=51 lbd=(0.9433429035751109-0.06198666692174841j) k=6 conv=1\n",
            "sc ii=51 lbd=(0.9433429035751111-0.06198666692174758j) k=6 conv=1\n",
            "scb ii=51 lbd=(0.9433429035750921-0.061986666921774895j) k=7 conv=1\n",
            "sr ii=52 lbd=(0.915433454880397-0.1491518915426669j) k=6 conv=1\n",
            "sc ii=52 lbd=(0.9433429035751095-0.06198666692175768j) k=7 conv=1\n",
            "scb ii=52 lbd=(0.9433429035751094-0.06198666692174755j) k=4 conv=1\n",
            "sr ii=53 lbd=(0.9433429035751107-0.061986666921747986j) k=4 conv=1\n",
            "sc ii=53 lbd=(1.0364920838370448-0.2094944504264866j) k=12 conv=1\n",
            "scb ii=53 lbd=(1.0364920838347826-0.20949445042548068j) k=5 conv=1\n",
            "sr ii=54 lbd=(0.9433429035751103-0.061986666921747806j) k=5 conv=1\n",
            "sc ii=54 lbd=(0.9433429035751092-0.061986666921750214j) k=7 conv=1\n",
            "scb ii=54 lbd=(1.0996521817685216-0.032671573082854906j) k=7 conv=1\n",
            "sr ii=55 lbd=(0.9433429035751104-0.06198666692174771j) k=5 conv=1\n",
            "sc ii=55 lbd=(0.9433429035751126-0.061986666921752115j) k=8 conv=1\n",
            "scb ii=55 lbd=(1.093021445084171-0.2169296829737891j) k=8 conv=1\n",
            "sr ii=56 lbd=(0.9433429035751116-0.06198666692174827j) k=5 conv=1\n",
            "sc ii=56 lbd=(0.9433429035746476-0.06198666692056603j) k=8 conv=1\n",
            "scb ii=56 lbd=(0.7553962598470535-0.07859215343527542j) k=7 conv=1\n",
            "sr ii=57 lbd=(0.943342903575112-0.0619866669217483j) k=4 conv=1\n",
            "sc ii=57 lbd=(0.9433429036061958-0.06198666692173813j) k=6 conv=1\n",
            "scb ii=57 lbd=(0.9433429035927323-0.061986666923763666j) k=4 conv=1\n",
            "sr ii=58 lbd=(0.9433429035751103-0.061986666921748375j) k=4 conv=1\n",
            "sc ii=58 lbd=(0.9154334548664692-0.14915189154149722j) k=7 conv=1\n",
            "scb ii=58 lbd=(0.9433429035751105-0.061986666921753045j) k=7 conv=1\n",
            "sr ii=59 lbd=(0.94334290357511-0.06198666692174797j) k=5 conv=1\n",
            "sc ii=59 lbd=(1.2992802280386826-0.060799123097368235j) k=8 conv=1\n",
            "scb ii=59 lbd=(1.0582032208004333+0.14550386686306876j) k=5 conv=1\n",
            "sr ii=60 lbd=(0.9433429035751115-0.061986666921748354j) k=5 conv=1\n",
            "sc ii=60 lbd=(0.8347432209035665+0.15463087235736603j) k=14 conv=1\n",
            "scb ii=60 lbd=(0.9433429035751071-0.061986666921747854j) k=5 conv=1\n",
            "sr ii=61 lbd=(1.0582032208004402+0.145503866863063j) k=6 conv=1\n",
            "sc ii=61 lbd=(0.834743220903519+0.15463087235698028j) k=12 conv=1\n",
            "scb ii=61 lbd=(0.9433429035751127-0.06198666692175285j) k=4 conv=1\n",
            "sr ii=62 lbd=(0.9433429035751102-0.06198666692174818j) k=5 conv=1\n",
            "sc ii=62 lbd=(0.9154334548798805-0.14915189154296096j) k=9 conv=1\n",
            "scb ii=62 lbd=(1.099652181768522-0.03267157308286024j) k=4 conv=1\n",
            "sr ii=63 lbd=(0.9154334548803976-0.1491518915426672j) k=5 conv=1\n",
            "sc ii=63 lbd=(0.94334290357511-0.06198666692174815j) k=8 conv=1\n",
            "scb ii=63 lbd=(0.9433429035753569-0.06198666692151669j) k=6 conv=1\n",
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            "sc ii=64 lbd=(0.755396259847062-0.07859215343524986j) k=7 conv=1\n",
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            "sr ii=459 lbd=(0.9433429035751114-0.061986666921748146j) k=5 conv=1\n",
            "sc ii=459 lbd=(1.0582032208004675+0.14550386686311959j) k=7 conv=1\n",
            "scb ii=459 lbd=(1.0930214450827378-0.21692968297454057j) k=6 conv=1\n",
            "sr ii=460 lbd=(0.9433429035751115-0.06198666692174852j) k=4 conv=1\n",
            "sc ii=460 lbd=(0.9433429035751109-0.06198666692175016j) k=7 conv=1\n",
            "scb ii=460 lbd=(0.94334290357698-0.0619866669217633j) k=4 conv=1\n",
            "sr ii=461 lbd=(0.94334290357511-0.061986666921747584j) k=5 conv=1\n",
            "sc ii=461 lbd=(0.8347432209035665+0.1546308723573675j) k=8 conv=1\n",
            "scb ii=461 lbd=(0.943342903575115-0.06198666692175017j) k=6 conv=1\n",
            "sr ii=462 lbd=(0.9433429035751103-0.06198666692174794j) k=5 conv=1\n",
            "sc ii=462 lbd=(0.37909240800955474-0.24604234373996253j) k=9 conv=1\n",
            "scb ii=462 lbd=(1.05820322080044+0.1455038668630592j) k=5 conv=1\n",
            "sr ii=463 lbd=(0.8347432209035667+0.15463087235736672j) k=7 conv=1\n",
            "sc ii=463 lbd=(1.0364920838369904-0.20949445042647363j) k=9 conv=1\n",
            "scb ii=463 lbd=(1.09965218176852-0.03267157308285239j) k=5 conv=1\n",
            "sr ii=464 lbd=(0.9433429035751103-0.06198666692174843j) k=6 conv=1\n",
            "sc ii=464 lbd=(0.540146865309448+0.740963837225382j) k=10 conv=1\n",
            "scb ii=464 lbd=(0.9154334548804011-0.14915189154266567j) k=7 conv=1\n",
            "sr ii=465 lbd=(0.9433429035751106-0.06198666692174758j) k=5 conv=1\n",
            "sc ii=465 lbd=(0.9433429035786013-0.06198666691846731j) k=7 conv=1\n",
            "scb ii=465 lbd=(0.9154334548801647-0.149151891542758j) k=6 conv=1\n",
            "sr ii=466 lbd=(0.9433429035751091-0.061986666921747896j) k=6 conv=1\n",
            "sc ii=466 lbd=(0.9433429035749752-0.06198666692153545j) k=6 conv=1\n",
            "scb ii=466 lbd=(0.9433429035751137-0.061986666921751615j) k=4 conv=1\n",
            "sr ii=467 lbd=(0.9433429035751113-0.061986666921748194j) k=5 conv=1\n",
            "sc ii=467 lbd=(0.9433429035737937-0.06198666692150079j) k=8 conv=1\n",
            "scb ii=467 lbd=(0.9433429035743193-0.061986666921567034j) k=5 conv=1\n",
            "sr ii=468 lbd=(0.9154334548803971-0.14915189154266717j) k=6 conv=1\n",
            "sc ii=468 lbd=(0.3763640795337512+0.04682551129329654j) k=10 conv=1\n",
            "scb ii=468 lbd=(0.7022421625738495+0.46665968078458075j) k=8 conv=1\n",
            "sr ii=469 lbd=(0.9433429035751099-0.06198666692174795j) k=5 conv=1\n",
            "sc ii=469 lbd=(0.9433429035751097-0.0619866669217469j) k=7 conv=1\n",
            "scb ii=469 lbd=(0.9154334548803958-0.1491518915426658j) k=7 conv=1\n",
            "sr ii=470 lbd=(0.9433429035751107-0.06198666692174796j) k=4 conv=1\n",
            "sc ii=470 lbd=(0.7975067415890633-0.2848420423078112j) k=9 conv=1\n",
            "scb ii=470 lbd=(0.6158779463724687+0.0274906854323277j) k=6 conv=1\n",
            "sr ii=471 lbd=(0.9433429035751102-0.06198666692174802j) k=5 conv=1\n",
            "sc ii=471 lbd=(1.0996521817685223-0.032671573082853254j) k=8 conv=1\n",
            "scb ii=471 lbd=(0.4479984228903567+0.5704502832041487j) k=6 conv=1\n",
            "sr ii=472 lbd=(0.8926646100897666-0.41985443611704676j) k=5 conv=1\n",
            "sc ii=472 lbd=(0.755396259847062-0.0785921534352505j) k=13 conv=1\n",
            "scb ii=472 lbd=(0.9433429035474743-0.0619866669742914j) k=4 conv=1\n",
            "sr ii=473 lbd=(0.8347432209035668+0.15463087235736656j) k=9 conv=1\n",
            "sc ii=473 lbd=(0.8347432209035663+0.15463087235736755j) k=12 conv=1\n",
            "scb ii=473 lbd=(0.8347432209020441+0.15463087235575546j) k=6 conv=1\n",
            "sr ii=474 lbd=(0.9154334548803972-0.14915189154266706j) k=6 conv=1\n",
            "sc ii=474 lbd=(0.6158779463533157+0.02749068542963735j) k=7 conv=1\n",
            "scb ii=474 lbd=(0.9433429035751146-0.061986666921746564j) k=6 conv=1\n",
            "sr ii=475 lbd=(0.9433429035751116-0.061986666921748j) k=5 conv=1\n",
            "sc ii=475 lbd=(0.943342903581732-0.061986666939919874j) k=8 conv=1\n",
            "scb ii=475 lbd=(0.9433429035735805-0.06198666692320387j) k=5 conv=1\n",
            "sr ii=476 lbd=(0.9433429035751116-0.061986666921748375j) k=7 conv=1\n",
            "sc ii=476 lbd=(0.743430133276178+0.2772223100922923j) k=15 conv=1\n",
            "scb ii=476 lbd=(0.9433429035758926-0.061986666921920036j) k=6 conv=1\n",
            "sr ii=477 lbd=(0.9433429035751112-0.06198666692174776j) k=6 conv=1\n",
            "sc ii=477 lbd=(0.7434301332761777+0.2772223100922899j) k=10 conv=1\n",
            "scb ii=477 lbd=(1.0582032208004544+0.14550386686305553j) k=5 conv=1\n",
            "sr ii=478 lbd=(0.9433429035751106-0.06198666692174783j) k=4 conv=1\n",
            "sc ii=478 lbd=(1.09965218176852-0.032671573082857806j) k=11 conv=1\n",
            "scb ii=478 lbd=(1.0930214450826914-0.2169296829743739j) k=6 conv=1\n",
            "sr ii=479 lbd=(0.9433429035751127-0.06198666692174807j) k=5 conv=1\n",
            "sc ii=479 lbd=(0.6158779463844687+0.027490685434311785j) k=7 conv=1\n",
            "scb ii=479 lbd=(0.6262746717703549+0.5785423295598706j) k=6 conv=1\n",
            "sr ii=480 lbd=(0.9433429035751111-0.061986666921747834j) k=5 conv=1\n",
            "sc ii=480 lbd=(0.9433429035756544-0.06198666692094956j) k=6 conv=1\n",
            "scb ii=480 lbd=(0.9433429035751064-0.06198666692174763j) k=6 conv=1\n",
            "sr ii=481 lbd=(1.0996521817685239-0.03267157308285662j) k=6 conv=1\n",
            "sc ii=481 lbd=(1.0996521817684874-0.032671573082876194j) k=15 conv=1\n",
            "scb ii=481 lbd=(0.9164388992157269+0.1814555090591274j) k=5 conv=1\n",
            "sr ii=482 lbd=(1.0996521817685216-0.032671573082856065j) k=5 conv=1\n",
            "sc ii=482 lbd=(0.7434301332761827+0.27722231009231324j) k=9 conv=1\n",
            "scb ii=482 lbd=(1.0996521817685236-0.032671573082859874j) k=4 conv=1\n",
            "sr ii=483 lbd=(0.9433429035751101-0.06198666692174836j) k=5 conv=1\n",
            "sc ii=483 lbd=(1.0996521817685068-0.03267157308292155j) k=7 conv=1\n",
            "scb ii=483 lbd=(0.9433429035751135-0.0619866669217484j) k=6 conv=1\n",
            "sr ii=484 lbd=(0.9164388992149+0.18145550905946967j) k=7 conv=1\n",
            "sc ii=484 lbd=(0.9433429035751142-0.061986666921742664j) k=13 conv=1\n",
            "scb ii=484 lbd=(0.9433429035751062-0.06198666692174685j) k=5 conv=1\n",
            "sr ii=485 lbd=(0.9433429035751114-0.06198666692174857j) k=5 conv=1\n",
            "sc ii=485 lbd=(0.9433429035751099-0.06198666692175009j) k=10 conv=1\n",
            "scb ii=485 lbd=(1.0930214450828404-0.2169296829744074j) k=7 conv=1\n",
            "sr ii=486 lbd=(0.9433429035751113-0.0619866669217483j) k=5 conv=1\n",
            "sc ii=486 lbd=(1.0582032208004404+0.14550386686305983j) k=10 conv=1\n",
            "scb ii=486 lbd=(1.0582032207994652+0.14550386686262554j) k=5 conv=1\n",
            "sr ii=487 lbd=(0.9433429035751101-0.061986666921747674j) k=5 conv=1\n",
            "sc ii=487 lbd=(0.7975067415896722-0.28484204230717486j) k=10 conv=1\n",
            "scb ii=487 lbd=(1.487559015552828+0.6965331590927615j) k=5 conv=1\n",
            "sr ii=488 lbd=(0.9433429035751113-0.06198666692174875j) k=5 conv=1\n",
            "sc ii=488 lbd=(0.943342903575112-0.06198666692174741j) k=8 conv=1\n",
            "scb ii=488 lbd=(0.9433429035771643-0.061986666921943j) k=6 conv=1\n",
            "sr ii=489 lbd=(0.9164388992149+0.1814555090594698j) k=7 conv=1\n",
            "sc ii=489 lbd=(1.0582032208004437+0.14550386686306022j) k=8 conv=1\n",
            "scb ii=489 lbd=(0.5553704522018184-0.5212007153282396j) k=5 conv=1\n",
            "sr ii=490 lbd=(0.9433429035751123-0.061986666921747605j) k=5 conv=1\n",
            "sc ii=490 lbd=(0.9433429035670057-0.06198666686390859j) k=7 conv=1\n",
            "scb ii=490 lbd=(0.9433429035854142-0.06198666692560346j) k=4 conv=1\n",
            "sr ii=491 lbd=(0.9433429035751097-0.061986666921747806j) k=4 conv=1\n",
            "sc ii=491 lbd=(0.9154334548803904-0.14915189154260391j) k=8 conv=1\n",
            "scb ii=491 lbd=(0.9433429035751121-0.06198666692174995j) k=5 conv=1\n",
            "sr ii=492 lbd=(0.9433429035751102-0.06198666692174833j) k=4 conv=1\n",
            "sc ii=492 lbd=(0.9433429035751282-0.06198666692173767j) k=7 conv=1\n",
            "scb ii=492 lbd=(0.94334290358673-0.06198666689880977j) k=4 conv=1\n",
            "sr ii=493 lbd=(0.9433429035751103-0.06198666692174807j) k=5 conv=1\n",
            "sc ii=493 lbd=(0.9433429035750951-0.061986666921751685j) k=7 conv=1\n",
            "scb ii=493 lbd=(0.9433429035722088-0.06198666692121721j) k=4 conv=1\n",
            "sr ii=494 lbd=(1.0996521817685223-0.032671573082856176j) k=9 conv=1\n",
            "sc ii=494 lbd=(0.9433429035751094-0.06198666692174805j) k=7 conv=1\n",
            "scb ii=494 lbd=(0.943342903575106-0.06198666692174824j) k=5 conv=1\n",
            "sr ii=495 lbd=(0.9433429035751112-0.061986666921747535j) k=5 conv=1\n",
            "sc ii=495 lbd=(0.9433429035742252-0.06198666692232001j) k=10 conv=1\n",
            "scb ii=495 lbd=(1.093021445082841-0.21692968297440257j) k=6 conv=1\n",
            "sr ii=496 lbd=(1.0364920838369893-0.20949445042647447j) k=8 conv=1\n",
            "sc ii=496 lbd=(0.9154334548809261-0.14915189154270586j) k=8 conv=1\n",
            "scb ii=496 lbd=(1.0930214450828393-0.2169296829744036j) k=6 conv=1\n",
            "sr ii=497 lbd=(1.0996521817685248-0.03267157308285629j) k=6 conv=1\n",
            "sc ii=497 lbd=(1.0930214450856421-0.21692968296872958j) k=11 conv=1\n",
            "scb ii=497 lbd=(0.9433429035751099-0.061986666921750394j) k=8 conv=1\n",
            "sr ii=498 lbd=(0.9433429035751106-0.06198666692174783j) k=6 conv=1\n",
            "sc ii=498 lbd=(0.9433429035751099-0.06198666692174883j) k=9 conv=1\n",
            "scb ii=498 lbd=(1.09965218176852-0.032671573082861005j) k=5 conv=1\n",
            "sr ii=499 lbd=(0.9433429035751113-0.061986666921747896j) k=6 conv=1\n",
            "sc ii=499 lbd=(0.8347432209035665+0.15463087235736622j) k=10 conv=1\n",
            "scb ii=499 lbd=(1.0996521817685223-0.03267157308285796j) k=8 conv=1\n",
            "         sr_k_arr    sc_k_arr   scb_k_arr\n",
            "count  500.000000  500.000000  500.000000\n",
            "mean     5.440000    9.062000    5.720000\n",
            "std      1.176933    2.210867    1.386161\n",
            "min      4.000000    5.000000    3.000000\n",
            "25%      5.000000    7.000000    5.000000\n",
            "50%      5.000000    9.000000    6.000000\n",
            "75%      6.000000   10.000000    6.000000\n",
            "max     11.000000   18.000000   11.000000\n",
            "       sr_time_arr  sc_time_arr  scb_time_arr\n",
            "count   500.000000   500.000000    500.000000\n",
            "mean      0.017428     0.024496      0.020807\n",
            "std       0.005386     0.008487      0.007522\n",
            "min       0.010098     0.011714      0.009220\n",
            "25%       0.013801     0.018427      0.015299\n",
            "50%       0.016364     0.022387      0.018885\n",
            "75%       0.019071     0.027989      0.023747\n",
            "max       0.056507     0.082285      0.066353\n",
            "       sr_lbd_arr  sc_lbd_arr  scb_lbd_arr\n",
            "count  500.000000  500.000000   500.000000\n",
            "mean     0.945408    0.891698     0.953490\n",
            "std      0.067745    0.181077     0.161928\n",
            "min      0.569153    0.068714    -0.137387\n",
            "25%      0.943343    0.834743     0.916439\n",
            "50%      0.943343    0.943343     0.943343\n",
            "75%      0.943343    0.943343     1.058203\n",
            "max      1.196366    1.428670     1.778552\n",
            "       sr_lbd_arr  sc_lbd_arr  scb_lbd_arr\n",
            "count  500.000000  500.000000   500.000000\n",
            "mean    -0.045514   -0.028731    -0.053475\n",
            "std      0.071609    0.207210     0.165554\n",
            "min     -0.419854   -0.715617    -0.715617\n",
            "25%     -0.061987   -0.149152    -0.149152\n",
            "50%     -0.061987   -0.061987    -0.061987\n",
            "75%     -0.061987    0.027491    -0.032672\n",
            "max      0.277222    0.850344     0.696533\n",
            "         sr_err_arr    sc_err_arr   scb_err_arr\n",
            "count  5.000000e+02  5.000000e+02  5.000000e+02\n",
            "mean   4.508447e-10  6.237234e-10  4.157590e-10\n",
            "std    1.506566e-09  1.609424e-09  1.325885e-09\n",
            "min    3.351361e-14  2.175588e-13  1.714604e-13\n",
            "25%    5.888858e-14  8.398791e-13  7.228035e-13\n",
            "50%    7.553821e-14  2.686553e-12  1.409562e-12\n",
            "75%    1.052182e-11  1.932154e-10  1.641834e-11\n",
            "max    9.831600e-09  9.735094e-09  9.438456e-09\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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            "text/plain": [
              "<Figure size 432x288 with 4 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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            "text/plain": [
              "<Figure size 432x288 with 4 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "WOEMXIKr1w-0",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 317
        },
        "outputId": "a20478c1-0d08-49bc-d83b-5dba7b55f874"
      },
      "source": [
        "pdF.describe()"
      ],
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/html": [
              "<div>\n",
              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
              "    }\n",
              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>sr_lbd_arr</th>\n",
              "      <th>sc_lbd_arr</th>\n",
              "      <th>scb_lbd_arr</th>\n",
              "      <th>sr_k_arr</th>\n",
              "      <th>sc_k_arr</th>\n",
              "      <th>scb_k_arr</th>\n",
              "      <th>sr_conv_arr</th>\n",
              "      <th>sc_conv_arr</th>\n",
              "      <th>scb_conv_arr</th>\n",
              "      <th>sr_time_arr</th>\n",
              "      <th>sc_time_arr</th>\n",
              "      <th>scb_time_arr</th>\n",
              "      <th>sr_err_arr</th>\n",
              "      <th>sc_err_arr</th>\n",
              "      <th>scb_err_arr</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>count</th>\n",
              "      <td>(500+0j)</td>\n",
              "      <td>(500+0j)</td>\n",
              "      <td>(500+0j)</td>\n",
              "      <td>500.000000</td>\n",
              "      <td>500.000000</td>\n",
              "      <td>500.000000</td>\n",
              "      <td>500.0</td>\n",
              "      <td>500.0</td>\n",
              "      <td>500.0</td>\n",
              "      <td>500.000000</td>\n",
              "      <td>500.000000</td>\n",
              "      <td>500.000000</td>\n",
              "      <td>5.000000e+02</td>\n",
              "      <td>5.000000e+02</td>\n",
              "      <td>5.000000e+02</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>mean</th>\n",
              "      <td>(0.9454078677563117-0.04551380221201718j)</td>\n",
              "      <td>(0.8916982350415611-0.02873139249182711j)</td>\n",
              "      <td>(0.9534902215304613-0.053474511400243j)</td>\n",
              "      <td>5.440000</td>\n",
              "      <td>9.062000</td>\n",
              "      <td>5.720000</td>\n",
              "      <td>1.0</td>\n",
              "      <td>1.0</td>\n",
              "      <td>1.0</td>\n",
              "      <td>0.017428</td>\n",
              "      <td>0.024496</td>\n",
              "      <td>0.020807</td>\n",
              "      <td>4.508447e-10</td>\n",
              "      <td>6.237234e-10</td>\n",
              "      <td>4.157590e-10</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>std</th>\n",
              "      <td>(0.0985765274573385+0j)</td>\n",
              "      <td>(0.27518146872830235+0j)</td>\n",
              "      <td>(0.23157877118383025+0j)</td>\n",
              "      <td>1.176933</td>\n",
              "      <td>2.210867</td>\n",
              "      <td>1.386161</td>\n",
              "      <td>0.0</td>\n",
              "      <td>0.0</td>\n",
              "      <td>0.0</td>\n",
              "      <td>0.005386</td>\n",
              "      <td>0.008487</td>\n",
              "      <td>0.007522</td>\n",
              "      <td>1.506566e-09</td>\n",
              "      <td>1.609424e-09</td>\n",
              "      <td>1.325885e-09</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>min</th>\n",
              "      <td>(0.569152653989528+0.12363164654205451j)</td>\n",
              "      <td>(0.06871389386778284+0.01810412090690429j)</td>\n",
              "      <td>(-0.13738684895496872+0.3174870572108329j)</td>\n",
              "      <td>4.000000</td>\n",
              "      <td>5.000000</td>\n",
              "      <td>3.000000</td>\n",
              "      <td>1.0</td>\n",
              "      <td>1.0</td>\n",
              "      <td>1.0</td>\n",
              "      <td>0.010098</td>\n",
              "      <td>0.011714</td>\n",
              "      <td>0.009220</td>\n",
              "      <td>3.351361e-14</td>\n",
              "      <td>2.175588e-13</td>\n",
              "      <td>1.714604e-13</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>25%</th>\n",
              "      <td>(0.9433429035751102-0.06198666692174817j)</td>\n",
              "      <td>(0.8347432209035539+0.15463087235726886j)</td>\n",
              "      <td>(0.9164388992148994+0.1814555090594699j)</td>\n",
              "      <td>5.000000</td>\n",
              "      <td>7.000000</td>\n",
              "      <td>5.000000</td>\n",
              "      <td>1.0</td>\n",
              "      <td>1.0</td>\n",
              "      <td>1.0</td>\n",
              "      <td>0.013801</td>\n",
              "      <td>0.018427</td>\n",
              "      <td>0.015299</td>\n",
              "      <td>5.888858e-14</td>\n",
              "      <td>8.398791e-13</td>\n",
              "      <td>7.228035e-13</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>50%</th>\n",
              "      <td>(0.943342903575111-0.06198666692174812j)</td>\n",
              "      <td>(0.9433429035751064-0.061986666921746966j)</td>\n",
              "      <td>(0.943342903575112-0.06198666692174945j)</td>\n",
              "      <td>5.000000</td>\n",
              "      <td>9.000000</td>\n",
              "      <td>6.000000</td>\n",
              "      <td>1.0</td>\n",
              "      <td>1.0</td>\n",
              "      <td>1.0</td>\n",
              "      <td>0.016364</td>\n",
              "      <td>0.022387</td>\n",
              "      <td>0.018885</td>\n",
              "      <td>7.553821e-14</td>\n",
              "      <td>2.686553e-12</td>\n",
              "      <td>1.409562e-12</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>75%</th>\n",
              "      <td>(0.9433429035751116-0.06198666692174884j)</td>\n",
              "      <td>(0.9433429035822748-0.06198666693425757j)</td>\n",
              "      <td>(1.0582032207996783+0.14550386686274944j)</td>\n",
              "      <td>6.000000</td>\n",
              "      <td>10.000000</td>\n",
              "      <td>6.000000</td>\n",
              "      <td>1.0</td>\n",
              "      <td>1.0</td>\n",
              "      <td>1.0</td>\n",
              "      <td>0.019071</td>\n",
              "      <td>0.027989</td>\n",
              "      <td>0.023747</td>\n",
              "      <td>1.052182e-11</td>\n",
              "      <td>1.932154e-10</td>\n",
              "      <td>1.641834e-11</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>max</th>\n",
              "      <td>(1.1963663271672798+0.24790590631302645j)</td>\n",
              "      <td>(1.4286704832959418-0.49505288498211664j)</td>\n",
              "      <td>(1.7785516855555492-0.3745587505597521j)</td>\n",
              "      <td>11.000000</td>\n",
              "      <td>18.000000</td>\n",
              "      <td>11.000000</td>\n",
              "      <td>1.0</td>\n",
              "      <td>1.0</td>\n",
              "      <td>1.0</td>\n",
              "      <td>0.056507</td>\n",
              "      <td>0.082285</td>\n",
              "      <td>0.066353</td>\n",
              "      <td>9.831600e-09</td>\n",
              "      <td>9.735094e-09</td>\n",
              "      <td>9.438456e-09</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "                                      sr_lbd_arr  ...   scb_err_arr\n",
              "count                                   (500+0j)  ...  5.000000e+02\n",
              "mean   (0.9454078677563117-0.04551380221201718j)  ...  4.157590e-10\n",
              "std                      (0.0985765274573385+0j)  ...  1.325885e-09\n",
              "min     (0.569152653989528+0.12363164654205451j)  ...  1.714604e-13\n",
              "25%    (0.9433429035751102-0.06198666692174817j)  ...  7.228035e-13\n",
              "50%     (0.943342903575111-0.06198666692174812j)  ...  1.409562e-12\n",
              "75%    (0.9433429035751116-0.06198666692174884j)  ...  1.641834e-11\n",
              "max    (1.1963663271672798+0.24790590631302645j)  ...  9.438456e-09\n",
              "\n",
              "[8 rows x 15 columns]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 5
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "7jAxtjSn17ZU",
        "colab_type": "text"
      },
      "source": [
        "The result shows Rayleigh Chebyshev underperforms two-sided RQI, but outperform regular RQI."
      ]
    }
  ]
}