{
  "cells": [
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      "cell_type": "markdown",
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      "source": [
        "# Einstein summation in numpy"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "It turns out that `numpy` actually has several more mini-languages embedded in it. This next example is borrowed and slightly generalized from mathematics, where it is called Einstein summation."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Recall that matrix-matrix multiplication can be expressed by:\n",
        "$$\n",
        "(AB)_{ij} = \\sum_k A_{ik} B_{kj}$$\n",
        "\n",
        "Einstein summation is a relatively natural way of generalizing this to arrays with multiple dimensions. The above matrix-matrix multiplication expression, for example, becomes:\n",
        "\n",
        "$$ A_{ij} = B_{ik} C_{kj}$$\n",
        "\n",
        "Where the implied rule is that repeated indices that are not part of the output will be summed over.\n",
        "\n",
        "numpy simply takes this convention and turns it into a way of expressing array operations:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import numpy as np"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "A = np.random.randn(15, 20)\n",
        "B = np.random.randn(20, 25)\n",
        "\n",
        "AB1 = A.dot(B)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "1.21357255039e-14\n"
          ]
        }
      ],
      "source": [
        "AB2 = np.einsum(\"ik,kj->ij\", A, B)\n",
        "\n",
        "print(np.linalg.norm(AB1 - AB2))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": true
      },
      "outputs": [],
      "source": []
    }
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    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
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      "file_extension": ".py",
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