sympy のサンプルコード : x * exp(-(x**2+y**2+z**2))の偏微分の評価

test_20201030_01.ipynb

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    {
      "cell_type": "code",
      "metadata": {
        "id": "swa8yuvNSLJi"
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      "source": [
        "# modules to import\n",
        "import sympy"
      ],
      "execution_count": 1,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "1gha9VzKT7NJ"
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      "source": [
        "# declaration of sympy symbols\n",
        "x      = sympy.symbols('x')\n",
        "y      = sympy.symbols('y')\n",
        "z      = sympy.symbols('z')\n",
        "r_pow2 = x ** 2 + y ** 2 + z ** 2"
      ],
      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "eivSCHwRUTQ_"
      },
      "source": [
        "# declaration of the objective function\n",
        "objective_function = x * sympy.exp(-r_pow2)"
      ],
      "execution_count": 3,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "bC7LOphrT-8H"
      },
      "source": [
        "# evaluation of the objective function of the differential\n",
        "pdv1_dx = sympy.diff( objective_function, x, 1 )\n",
        "pdv1_dy = sympy.diff( objective_function, y, 1 )\n",
        "pdv1_dz = sympy.diff( objective_function, z, 1 )"
      ],
      "execution_count": 4,
      "outputs": []
    },
    {
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        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "source": [
        "print( \"d/dx:\", pdv1_dx )\n",
        "print( \"d/dy:\", pdv1_dy )\n",
        "print( \"d/dz:\", pdv1_dz )"
      ],
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "d/dx: -2*x**2*exp(-x**2 - y**2 - z**2) + exp(-x**2 - y**2 - z**2)\n",
            "d/dy: -2*x*y*exp(-x**2 - y**2 - z**2)\n",
            "d/dz: -2*x*z*exp(-x**2 - y**2 - z**2)\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "aanjR4hlSxJl"
      },
      "source": [
        "# evaluation of the objective function of the differential\n",
        "\n",
        "pdv2_dxdx = sympy.diff(pdv1_dx, x)\n",
        "pdv2_dydx = sympy.diff(pdv1_dx, y)\n",
        "pdv2_dzdx = sympy.diff(pdv1_dx, z)\n",
        "\n",
        "pdv2_dxdy = sympy.diff(pdv1_dy, x)\n",
        "pdv2_dydy = sympy.diff(pdv1_dy, y)\n",
        "pdv2_dzdy = sympy.diff(pdv1_dy, z)\n",
        "\n",
        "pdv2_dxdz = sympy.diff(pdv1_dz, x)\n",
        "pdv2_dydz = sympy.diff(pdv1_dz, y)\n",
        "pdv2_dzdz = sympy.diff(pdv1_dz, z)"
      ],
      "execution_count": 6,
      "outputs": []
    },
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        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "source": [
        " print( \"d2/dxdx:\", pdv2_dxdx )\n",
        " print( \"d2/dydx:\", pdv2_dydx )\n",
        " print( \"d2/dzdx:\", pdv2_dzdx )\n",
        " \n",
        " print(\"\")\n",
        " \n",
        " print( \"d2/dxdy:\", pdv2_dxdy )\n",
        " print( \"d2/dydy:\", pdv2_dydy )\n",
        " print( \"d2/dzdy:\", pdv2_dzdy )\n",
        " \n",
        " print(\"\")\n",
        " \n",
        " print( \"d2/dxdz:\", pdv2_dxdz )\n",
        " print( \"d2/dydz:\", pdv2_dydz )\n",
        " print( \"d2/dzdz:\", pdv2_dzdz )"
      ],
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "d2/dxdx: 4*x**3*exp(-x**2 - y**2 - z**2) - 6*x*exp(-x**2 - y**2 - z**2)\n",
            "d2/dydx: 4*x**2*y*exp(-x**2 - y**2 - z**2) - 2*y*exp(-x**2 - y**2 - z**2)\n",
            "d2/dzdx: 4*x**2*z*exp(-x**2 - y**2 - z**2) - 2*z*exp(-x**2 - y**2 - z**2)\n",
            "\n",
            "d2/dxdy: 4*x**2*y*exp(-x**2 - y**2 - z**2) - 2*y*exp(-x**2 - y**2 - z**2)\n",
            "d2/dydy: 4*x*y**2*exp(-x**2 - y**2 - z**2) - 2*x*exp(-x**2 - y**2 - z**2)\n",
            "d2/dzdy: 4*x*y*z*exp(-x**2 - y**2 - z**2)\n",
            "\n",
            "d2/dxdz: 4*x**2*z*exp(-x**2 - y**2 - z**2) - 2*z*exp(-x**2 - y**2 - z**2)\n",
            "d2/dydz: 4*x*y*z*exp(-x**2 - y**2 - z**2)\n",
            "d2/dzdz: 4*x*z**2*exp(-x**2 - y**2 - z**2) - 2*x*exp(-x**2 - y**2 - z**2)\n"
          ],
          "name": "stdout"
        }
      ]
    },
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          "base_uri": "https://localhost:8080/"
        }
      },
      "source": [
        "print( \"d2/dxdy == d2/dydx:\", (pdv2_dydx - pdv2_dxdy) == 0 )\n",
        "print( \"d2/dzdy == d2/dydz:\", (pdv2_dzdy - pdv2_dydz) == 0 )\n",
        "print( \"d2/dxdz == d2/dzdx:\", (pdv2_dxdz - pdv2_dzdx) == 0 )"
      ],
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "d2/dxdy == d2/dydx: True\n",
            "d2/dzdy == d2/dydz: True\n",
            "d2/dxdz == d2/dzdx: True\n"
          ],
          "name": "stdout"
        }
      ]
    }
  ]
}