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"markdown": "# Symbolics.jl\n\n\n\n\nThere are a few options in `Julia` for symbolic math, for example, the `SymPy` package which wraps a Python library. This section describes a collection of native `Julia` packages providing many features of symbolic math.\n\n\n## About\n\n\nThe `Symbolics` package bills itself as a \"fast and modern Computer Algebra System (CAS) for a fast and modern programming language.\" This package relies on the `SymbolicUtils` package and is built upon by the `ModelingToolkit` package, which we don't describe here.\n\n\nWe begin by loading the `Symbolics` package which when loaded re-exports the `SymbolicUtils` package.\n\n``` {.julia .cell-code}\nusing Symbolics\n```\n\n\n## Symbolic variables\n\n\nSymbolic math at its core involves symbolic variables, which essentially defer evaluation until requested. The creation of symbolic variables differs between the two packages discussed here.\n\n\n`SymbolicUtils` creates variables which carry `Julia` type information (e.g. `Int`, `Float64`, ...). This type information carries through operations involving these variables. Symbolic variables can be created with the `@syms` macro. For example:\n\n::: {.cell execution_count=2}\n``` {.julia .cell-code}\n@syms x y::Int f(x::Real)::Real\n```\n\n::: {.cell-output .cell-output-display execution_count=3}\n```\n(x, y, f)\n```\n:::\n:::\n\n\nThis creates `x` a symbolic value with symbolic type `Number`, `y` a symbolic variable holding integer values, and `f` a symbolic function of a single real variable outputting a real variable.\n\n\nThe non-exported `symtype` function reveals the underlying type:\n\n::: {.cell execution_count=3}\n``` {.julia .cell-code}\nimport Symbolics.SymbolicUtils: symtype\n\nsymtype(x), symtype(y)\n```\n\n::: {.cell-output .cell-output-display execution_count=4}\n```\n(Number, Int64)\n```\n:::\n:::\n\n\nFor `y`, the symbolic type being real does not imply the type of `y` is a subtype of `Real`:\n\n::: {.cell execution_count=4}\n``` {.julia .cell-code}\nisa(y, Real)\n```\n\n::: {.cell-output .cell-output-display execution_count=5}\n```\nfalse\n```\n:::\n:::\n\n\nWe see that the function `f` when called with `y` would return a value of (symbolic) type `Real`:\n\n::: {.cell execution_count=5}\n``` {.julia .cell-code}\nf(y) |> symtype\n```\n\n::: {.cell-output .cell-output-display execution_count=6}\n```\nReal\n```\n:::\n:::\n\n\nAs the symbolic type of `x` is `Number` – which is not a subtype of `Real` – the following will error:\n\n::: {.cell execution_count=6}\n``` {.julia .cell-code}\nf(x)\n```\n\n::: {.cell-output .cell-output-error}\n```\nLoadError: Tuple{Number} is not a subtype of Tuple{Real}.\n```\n:::\n:::\n\n\n:::{.callout-note}\n## Note\nThe `SymPy` package also has an `@syms` macro to create variables. Though their names agree, they do different things. Using both packages together would require qualifying many shared method names. For `SymbolicUtils`, the `@syms` macro uses `Julia` types to parameterize the variables. In `SymPy` it is possible to specify *assumptions* on the variables, but that is different and not useful for dispatch without some extra effort.\n\n:::\n\nFor `Symbolics`, symbolic variables are created using a wrapper around an underlying `SymbolicUtils` object. This wrapper, `Num`, is a subtype of `Real` (the underlying `SymbolicUtils` object may have symbolic type `Real`, but it won't be a subtype of `Real`.)\n\n\nSymbolic values are created with the `@variables` macro. For example:\n\n::: {.cell execution_count=7}\n``` {.julia .cell-code}\n@variables x y::Int z[1:3]::Int f(..)::Int\n```\n\n::: {.cell-output .cell-output-display execution_count=8}\n```\n4-element Vector{Any}:\n x\n y\n z[1:3]\n f⋆\n```\n:::\n:::\n\n\nThis creates\n\n\n * a symbolic value `x` of `symtype` `Real`\n * a symbolic value `y` of `symtype` `Int`\n * a vector of symbolic values each of `symtype` `Int`\n * a symbolic function `f` returning an object of `symtype` `Int`\n\n\nThe symbolic type reflects that of the underlying object behind the `Num` wrapper:\n\n::: {.cell execution_count=8}\n``` {.julia .cell-code}\ntypeof(x), symtype(x), typeof(Symbolics.value(x))\n```\n\n::: {.cell-output .cell-output-display execution_count=9}\n```\n(Num, Real, SymbolicUtils.Sym{Real, Base.ImmutableDict{DataType, Any}})\n```\n:::\n:::\n\n\n(The `value` method unwraps the `Num` wrapper.)\n\n\n## Symbolic expressions\n\n\nSymbolic expressions are built up from symbolic variables through natural `Julia` idioms. `SymbolicUtils` privileges a few key operations: `Add`, `Mul`, `Pow`, and `Div`. For example:\n\n::: {.cell execution_count=9}\n``` {.julia .cell-code}\n@syms x y\ntypeof(x + y) # `Add`\n```\n\n::: {.cell-output .cell-output-display execution_count=10}\n```\nSymbolicUtils.Add{Number, Int64, Dict{Any, Number}, Nothing}\n```\n:::\n:::\n\n\n::: {.cell execution_count=10}\n``` {.julia .cell-code}\ntypeof(x * y) # `Mul`\n```\n\n::: {.cell-output .cell-output-display executioLine truncated
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229.141,96.8724 242.209,96.9164 255.277,97.0484 268.345,97.2683 281.412,97.5763 294.479,97.9721 307.544,98.456 320.608,99.0278 333.671,99.6875 346.732,100.435
|
||||
359.791,101.271 372.848,102.194 385.903,103.206 398.955,104.305 412.005,105.492 425.052,106.767 438.096,108.13 451.137,109.58 464.174,111.119 477.208,112.745
|
||||
490.238,114.459 503.263,116.26 516.285,118.15 529.301,120.127 542.314,122.191 555.321,124.343 568.323,126.583 581.32,128.91 594.312,131.325 607.298,133.827
|
||||
620.278,136.416 633.251,139.093 646.219,141.857 659.18,144.709 672.134,147.647 685.082,150.673 698.022,153.786 710.955,156.986 723.881,160.273 736.798,163.648
|
||||
749.708,167.109 762.61,170.657 775.504,174.291 788.389,178.013 801.265,181.821 814.132,185.716 826.991,189.698 839.839,193.766 852.679,197.92 865.509,202.161
|
||||
878.328,206.488 891.138,210.902 903.937,215.401 916.726,219.987 929.504,224.659 942.272,229.417 955.028,234.261 967.773,239.191 980.506,244.206 993.228,249.307
|
||||
1005.94,254.494 1018.63,259.766 1031.32,265.124 1043.99,270.567 1056.65,276.095 1069.3,281.708 1081.93,287.407 1094.56,293.191 1107.16,299.059 1119.76,305.012
|
||||
1132.34,311.05 1144.9,317.173 1157.45,323.38 1169.99,329.672 1182.51,336.048 1195.02,342.509 1207.51,349.053 1219.99,355.681 1232.45,362.394 1244.9,369.19
|
||||
1257.33,376.07 1269.74,383.034 1282.14,390.081 1294.52,397.212 1306.89,404.426 1319.24,411.723 1331.57,419.103 1343.88,426.566 1356.18,434.112 1368.46,441.741
|
||||
1380.72,449.452 1392.96,457.246 1405.19,465.122 1417.4,473.08 1429.58,481.121 1441.75,489.243 1453.91,497.447 1466.04,505.733 1478.15,514.101 1490.24,522.55
|
||||
1502.32,531.081 1514.37,539.693 1526.41,548.385 1538.42,557.159 1550.41,566.014 1562.38,574.949 1574.34,583.965 1586.27,593.061 1598.18,602.237 1610.07,611.494
|
||||
1621.93,620.83 1633.78,630.247 1645.6,639.743 1657.4,649.318 1669.18,658.973 1680.94,668.707 1692.67,678.52 1704.38,688.412 1716.07,698.383 1727.74,708.432
|
||||
1739.38,718.56 1751,728.766 1762.59,739.051 1774.16,749.413 1785.71,759.853 1797.23,770.371 1808.73,780.966 1820.2,791.639 1831.65,802.388 1843.07,813.215
|
||||
1854.47,824.119 1865.84,835.099 1877.19,846.156 1888.51,857.288 1899.81,868.498 1911.08,879.783 1922.32,891.143 1933.54,902.58 1944.73,914.092 1955.9,925.679
|
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1967.03,937.341 1978.14,949.078 1989.23,960.89 2000.28,972.776 2011.31,984.737 2022.31,996.772 2033.28,1008.88 2044.23,1021.06 2055.14,1033.32 2066.03,1045.65
|
||||
2076.89,1058.05 2087.72,1070.53 2098.52,1083.08 2109.29,1095.7 2120.03,1108.39 2130.75,1121.16 2141.43,1133.99 2152.08,1146.9 2162.71,1159.88 2173.3,1172.94
|
||||
2183.87,1186.06 2194.4,1199.26 2204.9,1212.52 2215.37,1225.86 2225.81,1239.26 2236.22,1252.74 2246.6,1266.29 2256.95,1279.9 2267.26,1293.59 2277.55,1307.35
|
||||
2287.8,1321.17 2298.02,1335.06 2308.21,1349.03 2318.36,1363.06 2328.48,1377.16 2338.57,1391.32 2348.63,1405.56 2358.65,1419.86 2368.64,1434.23 2378.6,1448.67
|
||||
2388.52,1463.17 2398.42,1477.74 2408.27,1492.38 2418.09,1507.08 2427.88,1521.85 2437.64,1536.69 2447.36,1551.59 2457.04,1566.55 2466.69,1581.58 2476.31,1596.68
|
||||
2485.89,1611.84 2495.43,1627.06 2504.94,1642.35 2514.42,1657.71 2523.86,1673.12 2533.26,1688.6 2542.63,1704.14 2551.96,1719.75 2561.26,1735.42 2570.51,1751.15
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Reference in new issue
Block a user