{
"cells": [
{
"cell_type": "raw",
"id": "2990f581",
"metadata": {},
"source": [
"---\n",
"title: \"Operators and axis\"\n",
"author: \"Jeremy Howard\"\n",
"date: \"2022-07-05\"\n",
"categories: [Dyalog, APL, Glyphs]\n",
"---"
]
},
{
"cell_type": "markdown",
"id": "5d6af942",
"metadata": {},
"source": [
"This post will explore some introductory operators (aka \"higher order functions\") in APL."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "84e8fa35",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────────────────────────────────────┐\n",
"│Was ON -style=max -trains=tree -fns=on│\n",
"└──────────────────────────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"]box on -style=max -trains=tree -fns=on"
]
},
{
"cell_type": "markdown",
"id": "a627597a",
"metadata": {},
"source": [
"## Axis"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "987a8fd9",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"↓1 2│\n",
"│3 4│\n",
"│5 6│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"3 2⍴⍳6"
]
},
{
"cell_type": "markdown",
"id": "40f2e3e4-4c36-4d16-b121-ae78b8c925ae",
"metadata": {},
"source": [
"Axis changes how the equal operator consumes the right argument. We specify a dimesion and we broadcast the operator across that dimension. In this case we are comparing the columns of the array to the left argument."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b883b3f9",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"↓1 0│\n",
"│0 1│\n",
"│1 0│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 4 5 =[1] 3 2⍴⍳6"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "5161cb08",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───────┐\n",
"↓10 20 30│\n",
"│40 50 60│\n",
"└~───────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←mat ← 2 3 ⍴ 10 20 30 40 50 60 "
]
},
{
"cell_type": "markdown",
"id": "33585019-cee3-48aa-9e51-736e6b3b72aa",
"metadata": {},
"source": [
"Similarly we can modify the plus operator to broadcast along the columns."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8845a1fe",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───────┐\n",
"↓11 21 31│\n",
"│42 52 62│\n",
"└~───────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mat+[1]1 2 ⍝ add along first axis"
]
},
{
"cell_type": "markdown",
"id": "c5e58a1e",
"metadata": {},
"source": [
"## Comma functions"
]
},
{
"cell_type": "markdown",
"id": "a445ebe1",
"metadata": {},
"source": [
"### `,` (Comma)"
]
},
{
"cell_type": "markdown",
"id": "738a9d5e",
"metadata": {},
"source": [
"#### monadic `,` (Ravel)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "459d3868",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌┌→──┐\n",
"↓↓1 2│\n",
"││3 4│\n",
"││ │\n",
"││5 6│\n",
"││7 8│\n",
"└└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕ ← cube ← 2 2 2 ⍴ ⍳8"
]
},
{
"cell_type": "markdown",
"id": "11048033-fdb2-4b95-955a-5a4bccecc05b",
"metadata": {},
"source": [
"Returns the elemenst as a vector"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "04f7bdd0",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────────────┐\n",
"│1 2 3 4 5 6 7 8│\n",
"└~──────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
", cube"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1cea85bb",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────────┐\n",
"│ ┌→──┐ ┌→──┐ │\n",
"│ │1 2│ │1 2│ │\n",
"│ └~──┘ └~──┘ │\n",
"└∊────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
", (1 2)(1 2)"
]
},
{
"cell_type": "markdown",
"id": "f89344a6",
"metadata": {},
"source": [
"#### monadic `,` (Ravel) with axis"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "4abc4506",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│ABC│\n",
"└───┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"'ABC'"
]
},
{
"cell_type": "markdown",
"id": "b7a18e88-4c0f-4600-98e7-d99a9a9a7f14",
"metadata": {},
"source": [
"We can use the axis modify insert a new dimension with ravel. We started with a single dimension, and we inserted a new one after it"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "dbb8811f",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"↓ABC│\n",
"└───┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
",[0.5]'ABC'"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c5e72477-dc3b-47d6-a414-43201356c711",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→┐\n",
"│3│\n",
"└~┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍴'ABC'"
]
},
{
"cell_type": "markdown",
"id": "bfa13d4b-aee0-49ad-a385-c630203e620d",
"metadata": {},
"source": [
"The fraction indicates where the new dimension is inserted"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "6e73a139-fbab-450c-9917-bfeb5402bfe6",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│1 3│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍴,[0.5]'ABC'"
]
},
{
"cell_type": "markdown",
"id": "6f85c20a-fb6b-4ced-adbe-cfbf780fab33",
"metadata": {},
"source": [
"If we specify a null dimension the dimension is added at the end"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "f14396b7-187a-4b4a-8cae-126215e1c336",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│3 1│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍴,[⍬]'ABC'"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8c7c1bcd-2584-41b3-ae81-a2aa4a8eafd2",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌┌→──┐\n",
"↓↓1 2│\n",
"││3 4│\n",
"││ │\n",
"││5 6│\n",
"││7 8│\n",
"└└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 2 2⍴⍳8"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d347d249",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌┌→──────────┐\n",
"↓↓ 1 2 3 4│\n",
"││ 5 6 7 8│\n",
"││ 9 10 11 12│\n",
"││ │\n",
"││13 14 15 16│\n",
"││17 18 19 20│\n",
"││21 22 23 24│\n",
"└└~──────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←M ← 2 3 4 ⍴ ⍳24"
]
},
{
"cell_type": "markdown",
"id": "fd03f4b8-12d4-4e4a-8198-3f7dde0b7228",
"metadata": {},
"source": [
"Specifying more than one dimension will merge the two given dimensions"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "dbe8d81d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────────┐\n",
"↓ 1 2 3 4│\n",
"│ 5 6 7 8│\n",
"│ 9 10 11 12│\n",
"│13 14 15 16│\n",
"│17 18 19 20│\n",
"│21 22 23 24│\n",
"└~──────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
",[1 2]M"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "17fb1de1",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│6 4│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍴,[1 2]M"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1ef4b061",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────────────────────────────────┐\n",
"↓ 1 2 3 4 5 6 7 8 9 10 11 12│\n",
"│13 14 15 16 17 18 19 20 21 22 23 24│\n",
"└~──────────────────────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
",[2 3]M"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "0d19a8dd",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───┐\n",
"│2 12│\n",
"└~───┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍴,[2 3]M"
]
},
{
"cell_type": "markdown",
"id": "69c3dff5",
"metadata": {},
"source": [
"#### dyadic `,` (Catenate/Laminate (Join))"
]
},
{
"cell_type": "markdown",
"id": "265f3d8a-6a6a-41f1-a549-407e06fc3fde",
"metadata": {},
"source": [
"Joins arrays together"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "da1548a8",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────────┐\n",
"│1 2 3 4 5 6│\n",
"└~──────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 2 3 , 4 5 6"
]
},
{
"cell_type": "markdown",
"id": "7e9abd02-3f61-4e0f-a379-5d584864c8d7",
"metadata": {},
"source": [
"Will be broadcast along the first dimension."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1ef39154",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌┌→─────┐\n",
"↓↓1 2 99│\n",
"││3 4 99│\n",
"││ │\n",
"││5 6 99│\n",
"││7 8 99│\n",
"└└~─────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"cube ← 2 2 2 ⍴ ⍳8\n",
"cube , 99"
]
},
{
"cell_type": "markdown",
"id": "799f29eb-53dc-4bbb-9412-dc67a90847d3",
"metadata": {},
"source": [
"#### dyadic `,` (Ravel) with axis"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "56f49ed4-ad6c-4ff0-8800-8f05dd0d263d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"↓1 2│\n",
"│3 4│\n",
"│5 6│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rect←3 2⍴⍳6\n",
"rect"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "f0ed7222-4619-4332-8073-22623934751f",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│3 2│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍴rect"
]
},
{
"cell_type": "markdown",
"id": "c6f86ad7-3622-46bb-bb75-e07e81fff150",
"metadata": {},
"source": [
"Add the element 10 to the end of each row"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "262a3227-b740-4400-9705-05cdd18d354d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────┐\n",
"↓1 2 10│\n",
"│3 4 10│\n",
"│5 6 10│\n",
"└~─────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rect,10"
]
},
{
"cell_type": "markdown",
"id": "ad4634e1-1983-419d-857d-eec56d444904",
"metadata": {},
"source": [
"We can modify laminate to add the given element to the columns (axis 1) instead of the rows (axis 2)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "064f6362-ea44-4813-905c-6810f3cca280",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"↓ 1 2│\n",
"│ 3 4│\n",
"│ 5 6│\n",
"│10 10│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rect,[1]10"
]
},
{
"cell_type": "markdown",
"id": "c69ac56a-5fb4-4a0f-9723-f4d5b05966b7",
"metadata": {},
"source": [
"A fractional axis creates a new axis and laminates that onto the existing rect"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "7f789a3b-2223-4cda-826d-112a3800682f",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌┌→────┐\n",
"↓↓ 1 2│\n",
"││ 3 4│\n",
"││ 5 6│\n",
"││ │\n",
"││10 10│\n",
"││10 10│\n",
"││10 10│\n",
"└└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rect,[0.5]10"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "7ffc9c0b",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────┐\n",
"↓HEADING│\n",
"│-------│\n",
"└───────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"'HEADING',[0.5]'-'"
]
},
{
"cell_type": "markdown",
"id": "393ff183",
"metadata": {},
"source": [
"### `⍪` (Comma bar)"
]
},
{
"cell_type": "markdown",
"id": "e39ced39",
"metadata": {},
"source": [
"#### monadic `⍪` (Table / Ravel items)\n",
"\n",
"Very similar to ravel. Operates on the first axis by default instead of the last axis."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "a7fff912",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→┐\n",
"↓1│\n",
"└~┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍪1"
]
},
{
"cell_type": "markdown",
"id": "31a833c1-d54c-4cc1-8d2b-ea0bee670e93",
"metadata": {},
"source": [
"This can be used to create columns instead of rows"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "98de0b8e",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"│ABCDE│\n",
"└─────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
"┌→┐\n",
"↓A│\n",
"│B│\n",
"│C│\n",
"│D│\n",
"│E│\n",
"└─┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←,5⍴⎕A\n",
"⎕←⍪5⍴⎕A"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "4b35b1a5",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌┌→───┐\n",
"↓↓ABCD│\n",
"││EFGH│\n",
"││IJKL│\n",
"││ │\n",
"││MNOP│\n",
"││QRST│\n",
"││UVWX│\n",
"└└────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 3 4⍴⎕A"
]
},
{
"cell_type": "markdown",
"id": "8beb410c-365a-4dd5-bb99-6c78603f8777",
"metadata": {},
"source": [
"Ravels the dimensions down to 2 leaving the size of the first dimension unchanged and merging the others."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8a18842f-7a69-40c1-a3ba-0680e9c97f61",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───┐\n",
"│2 12│\n",
"└~───┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍴⍪2 3 4⍴⎕A"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "037190ca",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───────────┐\n",
"↓ABCDEFGHIJKL│\n",
"│MNOPQRSTUVWX│\n",
"└────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⍪2 3 4⍴⎕A"
]
},
{
"cell_type": "markdown",
"id": "6d2fc7fb",
"metadata": {},
"source": [
"#### dyadic `⍪` (Catenate first)"
]
},
{
"cell_type": "markdown",
"id": "98fd1916-3562-49c6-9b67-9810d2d788df",
"metadata": {},
"source": [
"Operates by catenating the first dimensions"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "9a7ff5ba",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────────┐\n",
"│1 2 3 4 5 6│\n",
"└~──────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 2 3 ⍪ 4 5 6"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "ece4e465",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌┌→────┐\n",
"↓↓ 1 2│\n",
"││ 3 4│\n",
"││ │\n",
"││ 5 6│\n",
"││ 7 8│\n",
"││ │\n",
"││99 99│\n",
"││99 99│\n",
"└└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"cube ← 2 2 2 ⍴ ⍳8\n",
"cube ⍪ 99"
]
},
{
"cell_type": "markdown",
"id": "aff32832",
"metadata": {},
"source": [
"## Operator glyphs"
]
},
{
"cell_type": "markdown",
"id": "4a5637bb",
"metadata": {},
"source": [
"### `/` (Slash)"
]
},
{
"cell_type": "markdown",
"id": "2280a745",
"metadata": {},
"source": [
"#### monadic `/` (Reduce / N-wise Reduce)"
]
},
{
"cell_type": "markdown",
"id": "81deb293",
"metadata": {},
"source": [
"##### monadic function (Reduce)"
]
},
{
"cell_type": "markdown",
"id": "4d50e619-25d8-42d0-83f5-886b9403c7e6",
"metadata": {},
"source": [
"inserts the function on the right between the elements of the given array\n",
"\n",
"1 2 3 4 5 ---> 1+2+3+4+5 "
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "fe10c50b",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────┐\n",
"│1 2 3 4 5│\n",
"└~────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
" \n",
"15\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕ ← a ← ⍳5\n",
"+/a"
]
},
{
"cell_type": "markdown",
"id": "35faa2a5-b69f-4a3e-92da-b7df15055454",
"metadata": {},
"source": [
"Can be used to multiply all elements together"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "bdefcfd6",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"120\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a ← ⍳5\n",
"×/a"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "cd0a05f3",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1.5\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a ← ⍳3\n",
"÷/a"
]
},
{
"cell_type": "markdown",
"id": "8d194d1d-d95b-4363-b1ce-412ee09b412e",
"metadata": {},
"source": [
"Find the maximum element in an array"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "55830e33",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"6\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a ← 4 6 2\n",
"⌈/ a"
]
},
{
"cell_type": "markdown",
"id": "4e9da966-92c9-4a62-8293-822e6e773a6f",
"metadata": {},
"source": [
"Find the minimum element in the array"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "9bc76d4b",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"2\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a ← 4 6 2\n",
"⌊/ a"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1722938f",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"120\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"×/⍳5"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "6adcb7d7",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"120\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"!5"
]
},
{
"cell_type": "markdown",
"id": "d75ce5c5",
"metadata": {},
"source": [
"##### dyadic function (N-wise Reduce)"
]
},
{
"cell_type": "markdown",
"id": "8d680e4f",
"metadata": {},
"source": [
"Windowed sum with window of `3`:"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "a4886242",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│6 9│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"3+/⍳4 ⍝ (1+2+3) (2+3+4)"
]
},
{
"cell_type": "markdown",
"id": "d9925a15",
"metadata": {},
"source": [
"Windowed sum with window of `2`:"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "ccb02a6d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"│3 5 7│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2+/⍳4 ⍝ (1+2) (2+3) (3+4)"
]
},
{
"cell_type": "markdown",
"id": "4a08d2a9",
"metadata": {},
"source": [
"Moving average"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "f38bbd1e",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│2 3│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"3÷⍨3+/⍳4"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "daf6d8d7",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────┐\n",
"│0 0 0 0 0│\n",
"└~────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"0+/⍳4 ⍝ Identity element for +"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "f7f84184",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────┐\n",
"│1 1 1 1 1│\n",
"└~────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"0×/⍳4 ⍝ Identity element for ×"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "a38d586d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────────────────┐\n",
"│ ┌→──┐ ┌→──┐ ┌→──┐ │\n",
"│ │2 1│ │3 2│ │4 3│ │\n",
"│ └~──┘ └~──┘ └~──┘ │\n",
"└∊──────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"¯2,/⍳4⍝ (2,1) (3,2) (4,3)"
]
},
{
"cell_type": "markdown",
"id": "9e75d568",
"metadata": {},
"source": [
"#### Axis (with Monadic Operand)"
]
},
{
"cell_type": "markdown",
"id": "31dd5a83-b6bd-4631-8e91-9d1713f00ae5",
"metadata": {},
"source": [
"Combining reduce with axis allows us to specify the dimension to reduce. \n",
"\n",
"Sum the columns"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "3f53c64f",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"│5 7 9│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mat←2 3⍴⍳6\n",
"+/[1]mat"
]
},
{
"cell_type": "markdown",
"id": "57a1e5d3-978e-4ba4-a8f2-d6f125e5cfe8",
"metadata": {},
"source": [
"Sum the rows"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8afa5145",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───┐\n",
"│6 15│\n",
"└~───┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mat←2 3⍴⍳6\n",
"+/[2]mat"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "9fa2ebcb",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───┐\n",
"│6 15│\n",
"└~───┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mat←2 3⍴⍳6\n",
"+/mat"
]
},
{
"cell_type": "markdown",
"id": "773b883b",
"metadata": {},
"source": [
"### `\\` (Slope)"
]
},
{
"cell_type": "markdown",
"id": "89401d1e",
"metadata": {},
"source": [
"#### monadic `\\` (Scan)"
]
},
{
"cell_type": "markdown",
"id": "228481ac-b655-4c6a-88b8-790ee11fb81f",
"metadata": {},
"source": [
"Show all the intermediate steps of reduce\n",
"\n",
"1 (1+2) (1+2+3) (1+2+3+4) (1+2+3+4+5)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "cad6c863",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────────┐\n",
"│1 3 6 10 15│\n",
"└~──────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a ← ⍳5\n",
"+\\a"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "3e093072",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→───────────┐\n",
"│1 2 6 24 120│\n",
"└~───────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a ← ⍳5\n",
"×\\a"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "5ea8bd1b",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"│1 2 3│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
"┌→────────┐\n",
"│1 0.5 1.5│\n",
"└~────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕ ← a ← ⍳3\n",
"÷\\a"
]
},
{
"cell_type": "markdown",
"id": "fc3c7818",
"metadata": {},
"source": [
"### `⌿` (Slash Bar)"
]
},
{
"cell_type": "markdown",
"id": "aeb4f2da",
"metadata": {},
"source": [
"#### monadic `⌿` (Reduce First)\n",
"\n",
"Works similar to reduce. Operates on the first dimension rather than the last."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "152dad7d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"↓1 2 3│\n",
"│4 5 6│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
"┌→───┐\n",
"│6 15│\n",
"└~───┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←mat ← 2 3 ⍴ ⍳6\n",
"+/mat"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d65f9e59",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"↓1 2 3│\n",
"│4 5 6│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
"┌→────┐\n",
"│5 7 9│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←mat ← 2 3 ⍴ ⍳6\n",
"+/[1]mat"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "5129c061",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"│5 7 9│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"+⌿mat"
]
},
{
"cell_type": "markdown",
"id": "8437a76a",
"metadata": {},
"source": [
"### `⍀` (Slope Bar)"
]
},
{
"cell_type": "markdown",
"id": "dbfdb8bc",
"metadata": {},
"source": [
"#### monadic `⍀` (Scan first)"
]
},
{
"cell_type": "markdown",
"id": "48b9a87b-f141-489d-bf6f-312df0ddb702",
"metadata": {},
"source": [
"Similar to scan but operates on the first dimension rather than the last"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "0d0d0e4b",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"↓1 2 3│\n",
"│4 5 6│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
"┌→────┐\n",
"↓1 2 3│\n",
"│5 7 9│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕ ← mat ← 2 3 ⍴ ⍳6\n",
"+⍀mat"
]
},
{
"cell_type": "markdown",
"id": "c683f047",
"metadata": {},
"source": [
"### `⍤` (Jot Diaresis)"
]
},
{
"cell_type": "markdown",
"id": "5bc89ad4",
"metadata": {},
"source": [
"#### dyadic `⍤` (Rank)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e1615c86-fccd-4d7c-b131-e8dac79b3dcd",
"metadata": {},
"outputs": [],
"source": [
"trace←{⍺←⊢⋄⎕←'⍺: '⍺ '⍵: '⍵⋄ ⍺ ⍺⍺ ⍵} ⍝ explainer function"
]
},
{
"cell_type": "markdown",
"id": "46b6787b-09f9-4e5e-8bba-3d8b4ca762ac",
"metadata": {},
"source": [
"⍤ lets you specify the dimensions of the matrix to pass into the next function. In this case we only want plus reduce first to operate across rows."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8101e7f2",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌┌→──────────┐\n",
"↓↓ 1 2 3 4│\n",
"││ 5 6 7 8│\n",
"││ 9 10 11 12│\n",
"││ │\n",
"││13 14 15 16│\n",
"││17 18 19 20│\n",
"││21 22 23 24│\n",
"└└~──────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
"┌→───────┐\n",
"↓10 26 42│\n",
"│58 74 90│\n",
"└~───────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←cube ← 2 3 4 ⍴ ⍳24\n",
"(+⌿⍤1)cube"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "f25f52cc-403d-46ff-8228-f6edfc1615ca",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────────────┐\n",
"│ ┌→──┐ ┌→──────┐ │\n",
"│ │⍵: │ │1 2 3 4│ │\n",
"│ └───┘ └~──────┘ │\n",
"└∊────────────────┘\n",
"┌→────────────────┐\n",
"│ ┌→──┐ ┌→──────┐ │\n",
"│ │⍵: │ │5 6 7 8│ │\n",
"│ └───┘ └~──────┘ │\n",
"└∊────────────────┘\n",
"┌→───────────────────┐\n",
"│ ┌→──┐ ┌→─────────┐ │\n",
"│ │⍵: │ │9 10 11 12│ │\n",
"│ └───┘ └~─────────┘ │\n",
"└∊───────────────────┘\n",
"┌→────────────────────┐\n",
"│ ┌→──┐ ┌→──────────┐ │\n",
"│ │⍵: │ │13 14 15 16│ │\n",
"│ └───┘ └~──────────┘ │\n",
"└∊────────────────────┘\n",
"┌→────────────────────┐\n",
"│ ┌→──┐ ┌→──────────┐ │\n",
"│ │⍵: │ │17 18 19 20│ │\n",
"│ └───┘ └~──────────┘ │\n",
"└∊────────────────────┘\n",
"┌→────────────────────┐\n",
"│ ┌→──┐ ┌→──────────┐ │\n",
"│ │⍵: │ │21 22 23 24│ │\n",
"│ └───┘ └~──────────┘ │\n",
"└∊────────────────────┘\n",
"┌→───────┐\n",
"↓10 26 42│\n",
"│58 74 90│\n",
"└~───────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(+⌿trace⍤1)cube ⍝ show the input to pluse reduce first"
]
},
{
"cell_type": "markdown",
"id": "b910a81b-0aef-45de-a534-03bb6b071e32",
"metadata": {},
"source": [
"Sum the columns"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "abfe1dad",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────────┐\n",
"↓1 2 3 4│\n",
"│5 6 7 8│\n",
"│9 10 11 12│\n",
"└~─────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
"┌→──────────┐\n",
"│15 18 21 24│\n",
"└~──────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←mat ← 3 4 ⍴ ⍳12\n",
"+⌿mat"
]
},
{
"cell_type": "markdown",
"id": "a19736ef-768a-465d-9667-f7d68e539add",
"metadata": {},
"source": [
"For each of the 3 4 (the first 2 dimensions) matrices in the larger cube apply plus reduce first"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "3caf9370",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌┌→──────────┐\n",
"↓↓ 1 2 3 4│\n",
"││ 5 6 7 8│\n",
"││ 9 10 11 12│\n",
"││ │\n",
"││13 14 15 16│\n",
"││17 18 19 20│\n",
"││21 22 23 24│\n",
"└└~──────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
"┌→──────────┐\n",
"↓15 18 21 24│\n",
"│51 54 57 60│\n",
"└~──────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←cube\n",
"(+⌿⍤2)cube"
]
},
{
"cell_type": "markdown",
"id": "bc636c48-b859-443a-be50-bee227a444d4",
"metadata": {},
"source": [
"Given ⍺ and ⍵ arguments we can specify dimensions for each. In the following we take the ⍺ argument by element and the ⍵ argument by row"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "63cefbe6",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────────┐\n",
"↓1 2 3 4│\n",
"│5 6 7 8│\n",
"│9 10 11 12│\n",
"└~─────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
"┌→────────────────────────┐\n",
"│ ┌→──┐ ┌→──┐ ┌→──────┐ │\n",
"│ │⍺: │ 1 │⍵: │ │1 2 3 4│ │\n",
"│ └───┘ └───┘ └~──────┘ │\n",
"└∊────────────────────────┘\n",
"┌→────────────────────────┐\n",
"│ ┌→──┐ ┌→──┐ ┌→──────┐ │\n",
"│ │⍺: │ 2 │⍵: │ │5 6 7 8│ │\n",
"│ └───┘ └───┘ └~──────┘ │\n",
"└∊────────────────────────┘\n",
"┌→───────────────────────────┐\n",
"│ ┌→──┐ ┌→──┐ ┌→─────────┐ │\n",
"│ │⍺: │ 3 │⍵: │ │9 10 11 12│ │\n",
"│ └───┘ └───┘ └~─────────┘ │\n",
"└∊───────────────────────────┘\n",
"┌→──────────┐\n",
"↓ 2 3 4 5│\n",
"│ 7 8 9 10│\n",
"│12 13 14 15│\n",
"└~──────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←mat\n",
"1 2 3 (+trace⍤0 1) mat"
]
},
{
"cell_type": "markdown",
"id": "10d5153b",
"metadata": {},
"source": [
"#### dyadic `⍤` (Atop)"
]
},
{
"cell_type": "markdown",
"id": "1d61412e-1612-4414-8b9b-9d7832614717",
"metadata": {},
"source": [
"⍤ used dyadically is a function application rule\n",
"\n",
" f⍤g X → f(gX) → fgX\n",
"X f⍤g Y → f X g Y"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "96480209",
"metadata": {},
"outputs": [],
"source": [
"f ← *⍤÷"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "07dfbebb",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1.395612425\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
" \n",
"1.395612425\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←*(÷3)\n",
"⎕←f 3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "35d3df26",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1.947734041\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
" \n",
"1.947734041\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←*2÷3\n",
"⎕←2 f 3"
]
},
{
"cell_type": "markdown",
"id": "618ab13f",
"metadata": {},
"source": [
"### `∘` (Jot)"
]
},
{
"cell_type": "markdown",
"id": "75b436ce",
"metadata": {},
"source": [
"#### dyadic `∘` (Bind)"
]
},
{
"cell_type": "markdown",
"id": "d6cc9efc-5074-440a-a60b-a50fdd7bae16",
"metadata": {},
"source": [
"Can be used for partial function application. Binds the right argument 2 to the power function returning a new function that takes only a single argument"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c231621a",
"metadata": {},
"outputs": [],
"source": [
"sqr ← *∘2"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b7cbacfb",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"9\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sqr 3"
]
},
{
"cell_type": "markdown",
"id": "976e5348-92e4-4aca-8f2a-4f9be263eaf6",
"metadata": {},
"source": [
"Can bind the left argument as well"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d0b9f0b1",
"metadata": {},
"outputs": [],
"source": [
"pow2 ← 2∘*"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "2d432fe5",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"8\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pow2 3"
]
},
{
"cell_type": "markdown",
"id": "e6c3fd06",
"metadata": {},
"source": [
"#### dyadic `∘` (Beside)"
]
},
{
"cell_type": "markdown",
"id": "3e1d27a7-41a1-4877-addf-18e5815da6f7",
"metadata": {},
"source": [
"Another function composition rule\n",
"\n",
" f∘g X → f(gX) → fgX\n",
"X f∘g Y → X f g Y"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "514cfaf5",
"metadata": {},
"outputs": [],
"source": [
"f ← *∘÷"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "547aa92a",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1.395612425\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"*(÷3)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "6c9ab7db",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1.395612425\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f 3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1c92fcab",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1.25992105\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 f 3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "88b61671",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1.25992105\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 * (÷3)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "be94480f",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1.25992105\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2*÷3"
]
},
{
"cell_type": "markdown",
"id": "cd58ec37",
"metadata": {},
"source": [
"### `⍥` (Circle diaresis)"
]
},
{
"cell_type": "markdown",
"id": "3397ffbc",
"metadata": {},
"source": [
"#### dyadic `⍥` (Over)"
]
},
{
"cell_type": "markdown",
"id": "4829d654-8512-4b55-b581-73af812150dd",
"metadata": {},
"source": [
"Another function composition rule. \n",
"\n",
"\n",
" f⍥g Y → f(gX) → fgX\n",
"X f⍥g Y → (g X)f(g Y)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "15503ed8",
"metadata": {},
"outputs": [],
"source": [
"f ← *⍥÷"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "39400248",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1.395612425\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"*(÷3)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "4b9b1a63",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1.395612425\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f 3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "ebb9611d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"0.793700526\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 f 3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e9005d9d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"0.793700526\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(÷2)*÷3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "f983a246",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"5040\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"10 (÷⍥!) 6 ⍝ P(10,4)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "85da6f8e",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"5040\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(!10)÷!(10-4) ⍝ P(10,4)"
]
},
{
"cell_type": "markdown",
"id": "13f6bdc4",
"metadata": {},
"source": [
"### `⍣` (Star Diaeresis)"
]
},
{
"cell_type": "markdown",
"id": "06e52bcd",
"metadata": {},
"source": [
"#### dyadic `⍣` (Power operator)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e5aeb826",
"metadata": {},
"outputs": [],
"source": [
"S ← +∘1"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "94a44e68",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"S 0"
]
},
{
"cell_type": "markdown",
"id": "da840ec8-0488-4e54-864a-920ad26f1293",
"metadata": {},
"source": [
"⍣ Calls a function the given number of times.\n",
"\n",
"S(S(S 0))"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "0f8dd750",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"3\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(S⍣3) 0"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "9f9daf8a",
"metadata": {},
"outputs": [],
"source": [
"add ← {(S⍣⍺) ⍵}"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b9e9a45c",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"5\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 add 3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "96858617",
"metadata": {},
"outputs": [],
"source": [
"mult ← {⍺ (add⍣⍵) 0}"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "fcefb1db",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"12\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"3 mult 4"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "804d9c16",
"metadata": {},
"outputs": [],
"source": [
"P ← S⍣¯1"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "faa5c23e",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"2\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P 3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "bb4c25cf",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"2\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(S⍣¯3) 5"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d66c5206",
"metadata": {},
"outputs": [],
"source": [
"sqr ← *∘2"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "f2e08283",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"3\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(sqr⍣¯1)9"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8fc3dc3e",
"metadata": {},
"outputs": [],
"source": [
"pow ← {⍺ (mult⍣⍵) 1}"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "64d80892",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"8\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 pow 3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "b929e29d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1.618033989\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 +∘÷⍣= 1"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c008024b",
"metadata": {},
"outputs": [],
"source": [
"f ← +∘÷"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "0fef609d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"2\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 f 1"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "26420a6e",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1.5\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 f 2"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "abbd9af4",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1.666666667\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 f 1.5"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "61e500d3",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1.618034448\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 (f⍣15) 1"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "7749b6a9",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1.618033989\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 (f⍣=) 1"
]
},
{
"cell_type": "markdown",
"id": "6d32ee42",
"metadata": {},
"source": [
"## Linear Algebra"
]
},
{
"cell_type": "markdown",
"id": "85f3061a",
"metadata": {},
"source": [
"### `.` (Dot)"
]
},
{
"cell_type": "markdown",
"id": "471ba9ed",
"metadata": {},
"source": [
"#### Dyadic `.` (Inner Product)"
]
},
{
"cell_type": "markdown",
"id": "f0c8e79e-1aaf-47b2-980d-1d5c590b513c",
"metadata": {},
"source": [
"The ⍵⍵ operator specifies how elements are combined. The ⍺⍺ argument specifies how the results of the ⍵⍵ operation are combined."
]
},
{
"cell_type": "markdown",
"id": "67ed7a8f-5da7-42d0-a97e-447f8e6afe7c",
"metadata": {},
"source": [
"(1×4) + (2×5) + (3×6)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "68dbc16f",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"32\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 2 3 +.× 4 5 6 ⍝ Dot product"
]
},
{
"cell_type": "markdown",
"id": "fa57e7fc-1496-4a14-a9fa-97393edf1bb2",
"metadata": {},
"source": [
"(3=3)∧(3=3)∧(3=3)∧(3=3)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "afb1c2cf",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"3 ∧.= 3 3 3 3 ⍝ All-equal"
]
},
{
"cell_type": "markdown",
"id": "7df63fab-6cf8-4983-a0b5-9bf8d3c95213",
"metadata": {},
"source": [
"(1×1)+(2×3) (1×2)+(2×4)\n",
"\n",
"(3×1)+(4×3) (3×2)+(4×4)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "a74f3e9c",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"↓1 2│\n",
"│3 4│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"text/html": [
"┌→────┐\n",
"↓ 7 10│\n",
"│15 22│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←mat←2 2⍴⍳4\n",
"mat +.× mat ⍝ matrix product"
]
},
{
"cell_type": "markdown",
"id": "0f466a96-4746-4484-84da-8ac7d9aa4c98",
"metadata": {},
"source": [
"In this case we are going to duplicate the ⍵ argument. Once for each element of the ⍺ argument. The first one is multiplied by the first element of the ⍺ argument. The second by the second element, etc.\n",
"\n",
"1× 4 5 6 7\n",
"\n",
"2× 4 5 6 7\n",
"\n",
"3× 4 5 6 7"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c79a3d12",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────────┐\n",
"↓ 4 5 6 7│\n",
"│ 8 10 12 14│\n",
"│12 15 18 21│\n",
"└~──────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1 2 3 ∘.× 4 5 6 7 ⍝ Special case: outer prodct"
]
},
{
"cell_type": "markdown",
"id": "d60de628",
"metadata": {},
"source": [
"### `⌹` (Domino;Quad Divide)"
]
},
{
"cell_type": "markdown",
"id": "499ddf80",
"metadata": {},
"source": [
"#### Monadic `⌹` (Matrix Inverse Of)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "3ae7e56b",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────┐\n",
"↓¯2 1 │\n",
"│ 1.5 ¯0.5│\n",
"└~────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mat←2 2⍴⍳4\n",
"⎕←inv←⌹ mat"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "465cc80b",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"↓1 0│\n",
"│0 1│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"inv +.× mat ⍝ Identity"
]
},
{
"cell_type": "markdown",
"id": "da77be8f",
"metadata": {},
"source": [
"#### Dyadic `⌹` (Matrix Division By)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "04795293",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────┐\n",
"│¯4 4.5│\n",
"└~─────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕←div←5 6 ⌹ mat"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8f78bb2c",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──┐\n",
"│5 6│\n",
"└~──┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mat +.× div"
]
},
{
"cell_type": "markdown",
"id": "88dc2b77",
"metadata": {},
"source": [
"## Custom operators"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "034e07cf",
"metadata": {},
"outputs": [],
"source": [
"f ← *∘2"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "12b4ae31",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"6.01\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"d ← 0.01\n",
"x ← 3\n",
"((f (x+d)) - f x) ÷ d"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "a03eb5d9",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"6.0001\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"d ← 0.0001\n",
"((f (x+d)) - f x) ÷ d"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "701f4ec2",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"6.01\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"grad ← {((⍺⍺ ⍺+⍵) - ⍺⍺ ⍺) ÷ ⍵}\n",
"3 f grad 0.01"
]
},
{
"cell_type": "markdown",
"id": "e871a76b",
"metadata": {},
"source": [
"## Diaeresis and Tilde Diaeresis"
]
},
{
"cell_type": "markdown",
"id": "f209adf1",
"metadata": {},
"source": [
"### `⍨` (Tilde Diaeresis)"
]
},
{
"cell_type": "markdown",
"id": "3e097c27",
"metadata": {},
"source": [
"#### dyadic `⍨` (Commute)"
]
},
{
"cell_type": "markdown",
"id": "ea9112b0-0901-4e8d-a081-3e51f399bc98",
"metadata": {},
"source": [
"Modifies the given function so its arguments are swapped."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "cabed390",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"1\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"3-2"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c5b94832",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"¯1\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2-3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "da470b37",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"¯1\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"3-⍨2"
]
},
{
"cell_type": "markdown",
"id": "7783d49c-2503-4639-a6bf-79c1146d812e",
"metadata": {},
"source": [
"Normally to use a mask to select elements the mask is the ⍺ argument to /. If we wanted to right this out we would need to calculate the mask first.\n",
"\n",
"(≠v)/v"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "17cfef7d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→─────────┐\n",
"│22 10 21 5│\n",
"└~─────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v←22 10 22 22 21 10 5 10\n",
"v/⍨≠v"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "f0bc56b3",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"6.01\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"grad ← {⍵ ÷⍨ (⍺⍺ ⍺+⍵) - ⍺⍺ ⍺}\n",
"3 f grad 0.01"
]
},
{
"cell_type": "markdown",
"id": "c522a81a-f038-4403-8b24-2153ee0a24ad",
"metadata": {},
"source": [
"Can also be used to reflect the ⍵ argument to become the ⍺ argument as well.\n",
"\n",
"3 × 3"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "7569f72e",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"9\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pow ← ×⍨\n",
"pow 3"
]
},
{
"cell_type": "markdown",
"id": "de3a07f5",
"metadata": {},
"source": [
"#### dyadic `⍨` (Constant)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "396b95e8",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
" \n",
"0\n",
" \n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"zero ← 0⍨\n",
"2 zero 5"
]
},
{
"cell_type": "markdown",
"id": "8eea99a0",
"metadata": {},
"source": [
"### `¨` (Diaresis)"
]
},
{
"cell_type": "markdown",
"id": "35482065",
"metadata": {},
"source": [
"#### monadic `¨` (Each)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "65e7b966",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→──────────────────┐\n",
"│ ┌→──────┐ ┌→────┐ │\n",
"│ │1 2 3 4│ │5 6 7│ │\n",
"│ └~──────┘ └~────┘ │\n",
"└∊──────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕ ← a ← (1 2 3 4)(5 6 7)"
]
},
{
"cell_type": "markdown",
"id": "990a53df-822a-4cd4-870e-6912d8a3f81a",
"metadata": {},
"source": [
"Apply +/ to each of the elements of a. Each element is itself an array. So we sum those up."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "fee3843a",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────┐\n",
"│10 18│\n",
"└~────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"+/¨a"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c2f8b0be",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────────────┐\n",
"│ ┌→────┐ ┌→────┐ │\n",
"│ │1 2 3│ │4 5 6│ │\n",
"│ └~────┘ └~────┘ │\n",
"└∊────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"⎕ ← b ← (1 2 3)(4 5 6)"
]
},
{
"cell_type": "markdown",
"id": "aa8b86d3-bf4b-41f0-924b-376596067525",
"metadata": {},
"source": [
"Distribute the plus and an element to each of the elements of b. \n",
"\n",
"2 + 1 2 3 \n",
"\n",
"3 + 4 5 6"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "1f256e34",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"┌→────────────────┐\n",
"│ ┌→────┐ ┌→────┐ │\n",
"│ │3 4 5│ │7 8 9│ │\n",
"│ └~────┘ └~────┘ │\n",
"└∊────────────────┘\n",
""
]
},
"execution_count": null,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2 3 +¨ b"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Dyalog APL",
"language": "apl",
"name": "dyalog-kernel"
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"nbformat": 4,
"nbformat_minor": 5
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