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# Symbolics.jl
Incorporate:
Basics
https://github.com/SciML/ModelingToolkit.jl
https://github.com/JuliaSymbolics/Symbolics.jl
https://github.com/JuliaSymbolics/SymbolicUtils.jl
* Rewriting
https://github.com/JuliaSymbolics/SymbolicUtils.jl
* Plotting
Polynomials
Limits
XXX ... room here!
Derivatives
https://github.com/JuliaSymbolics/Symbolics.jl
Integration
https://github.com/SciML/SymbolicNumericIntegration.jl
The `Symbolics.jl` package is a Computer Algebra System (CAS) built entirely in `Julia`.
This package is under heavy development.
## Algebraic manipulations
### construction
@variables
SymbolicUtils.@syms assumptions
x is a `Num`, `Symbolics.value(x)` is of type `SymbolicUtils{Real, Nothing}
relation to SymbolicUtils
Num wraps things; Term
### Substitute
### Simplify
simplify
expand
rewrite rules
### Solving equations
solve_for
## Expressions to functions
build_function
## Derivatives
1->1: Symbolics.derivative(x^2 + cos(x), x)
1->3: Symbolics.derivative.([x^2, x, cos(x)], x)
3 -> 1: Symbolics.gradient(x*y^z, [x,y,z])
2 -> 2: Symbolics.jacobian([x,y^z], [x,y])
# higher order
1 -> 1: D(ex, x, n=1) = foldl((ex,_) -> Symbolics.derivative(ex, x), 1:n, init=ex)
2 -> 1: (2nd) Hessian
## Differential equations
## Integrals
WIP
## ----
# follow sympy tutorial
using Symbolics
import SymbolicUtils
@variables x y z
# substitution
ex = cos(x) + 1
substitute(ex, Dict(x=>y))
substitute(ex, Dict(x=>0)) # does eval
ex = x^y
substitute(ex, Dict(y=> x^y))
# expand trig
r1 = @rule sin(2 * ~x) => 2sin(~x)*cos(~x)
r2 = @rule cos(2 * ~x) => cos(~x)^2 - sin(~x)^2
expand_trig(ex) = simplify(ex, RuleSet([r1, r2]))
ex = sin(2x) + cos(2x)
expand_trig(ex)
## Multiple
@variables x y z
ex = x^3 + 4x*y -z
substitute(ex, Dict(x=>2, y=>4, z=>0))
# Converting Strings to Expressions
# what is sympify?
# evalf
# lambdify: symbolic expression -> function
ex = x^3 + 4x*y -z
λ = build_function(ex, x,y,z, expression=Val(false))
λ(2,4,0)
# pretty printing
using Latexify
latexify(ex)
# Simplify
@variables x y z t
simplify(sin(x)^2 + cos(x)^2)
simplify((x^3 + x^2 - x - 1) / (x^2 + 2x + 1)) # fails, no factor
simplify(((x+1)*(x^2-1))/((x+1)^2)) # works
import SpecialFunctions: gamma
simplify(gamma(x) / gamma(x-2)) # fails
# Polynomial
## expand
expand((x+1)^2)
expand((x+2)*(x-3))
expand((x+1)*(x-2) - (x-1)*x)
## factor
### not defined
## collect
COLLECT_RULES = [
@rule(~x*x^(~n::SymbolicUtils.isnonnegint) => (~x, ~n))
@rule(~x * x => (~x, 1))
]
function _collect(ex, x)
d = Dict()
exs = expand(ex)
if SymbolicUtils.operation(Symbolics.value(ex)) != +
d[0] => ex
else
for aᵢ ∈ SymbolicUtils.arguments(Symbolics.value(expand(ex)))
u = simplify(aᵢ, RuleSet(COLLECT_RULES))
if isa(u, Tuple)
a,n = u
else
a,n = u,0
end
d[n] = get(d, n, 0) + a
end
end
d
end
## cancel -- no factor
## apart -- no factor
## Trignometric simplification
INVERSE_TRIG_RUELS = [@rule(cos(acos(~x)) => ~x)
@rule(acos(cos(~x)) => abs(rem2pi(~x, RoundNearest)))
@rule(sin(asin(~x)) => ~x)
@rule(asin(sin(~x)) => abs(rem2pi(x + pi/2, RoundNearest)) - pi/2)
]
@variables θ
simplify(cos(acos(θ)), RuleSet(INVERSE_TRIG_RUELS))
# Copy from https://github.com/JuliaSymbolics/SymbolicUtils.jl/blob/master/src/simplify_rules.jl
# the TRIG_RULES are applied by simplify by default
HTRIG_RULES = [
@acrule(-sinh(~x)^2 + cosh(~x)^2 => one(~x))
@acrule(sinh(~x)^2 + 1 => cosh(~x)^2)
@acrule(cosh(~x)^2 + -1 => -sinh(~x)^2)
@acrule(tanh(~x)^2 + 1*sech(~x)^2 => one(~x))
@acrule(-tanh(~x)^2 + 1 => sech(~x)^2)
@acrule(sech(~x)^2 + -1 => -tanh(~x)^2)
@acrule(coth(~x)^2 + -1*csch(~x)^2 => one(~x))
@acrule(coth(~x)^2 + -1 => csch(~x)^2)
@acrule(csch(~x)^2 + 1 => coth(~x)^2)
@acrule(tanh(~x) => sinh(~x)/cosh(~x))
@acrule(sinh(-~x) => -sinh(~x))
@acrule(cosh(-~x) => -cosh(~x))
]
trigsimp(ex) = simplify(simplify(ex, RuleSet(HTRIG_RULES)))
trigsimp(sin(x)^2 + cos(x)^2)
trigsimp(sin(x)^4 -2cos(x)^2*sin(x)^2 + cos(x)^4) # no factor
trigsimp(cosh(x)^2 + sinh(x)^2)
trigsimp(sinh(x)/tanh(x))
EXPAND_TRIG_RULES = [
@acrule(sin(~x+~y) => sin(~x)*cos(~y) + cos(~x)*sin(~y))
@acrule(sinh(~x+~y) => sinh(~x)*cosh(~y) + cosh(~x)*sinh(~y))
@acrule(sin(2*~x) => 2sin(~x)*cos(~x))
@acrule(sinh(2*~x) => 2sinh(~x)*cosh(~x))
@acrule(cos(~x+~y) => cos(~x)*cos(~y) - sin(~x)*sin(~y))
@acrule(cosh(~x+~y) => cosh(~x)*cosh(~y) + sinh(~x)*sinh(~y))
@acrule(cos(2*~x) => cos(~x)^2 - sin(~x)^2)
@acrule(cosh(2*~x) => cosh(~x)^2 + sinh(~x)^2)
@acrule(tan(~x+~y) => (tan(~x) - tan(~y)) / (1 + tan(~x)*tan(~y)))
@acrule(tanh(~x+~y) => (tanh(~x) + tanh(~y)) / (1 + tanh(~x)*tanh(~y)))
@acrule(tan(2*~x) => 2*tan(~x)/(1 - tan(~x)^2))
@acrule(tanh(2*~x) => 2*tanh(~x)/(1 + tanh(~x)^2))
]
expandtrig(ex) = simplify(simplify(ex, RuleSet(EXPAND_TRIG_RULES)))
expandtrig(sin(x+y))
expandtrig(tan(2x))
# powers
# in genearl x^a*x^b = x^(a+b)
@variables x y a b
simplify(x^a*x^b - x^(a+b)) # 0
# x^a*y^a = (xy)^a When x,y >=0, a ∈ R
simplify(x^a*y^a - (x*y)^a)
## ??? How to specify such assumptions?
# (x^a)^b = x^(ab) only if b ∈ Int
@syms x a b
simplify((x^a)^b - x^(a*b))
@syms x a b::Int
simplify((x^a)^b - x^(a*b)) # nope
ispositive(x) = isa(x, Real) && x > 0
_isinteger(x) = isa(x, Integer)
_isinteger(x::SymbolicUtils.Sym{T,S}) where {T <: Integer, S} = true
POWSIMP_RULES = [
@acrule((~x::ispositive)^(~a::isreal) * (~y::ispositive)^(~a::isreal) => (~x*~y)^~a)
@rule(((~x)^(~a))^(~b::_isinteger) => ~x^(~a * ~b))
]
powsimp(ex) = simplify(simplify(ex, RuleSet(POWSIMP_RULES)))
@syms x a b::Int
simplify((x^a)^b - x^(a*b)) # nope
EXPAND_POWER_RULES = [
@rule((~x)^(~a + ~b) => (_~)^(~a) * (~x)^(~b))
@rule((~x*~y)^(~a) => (~x)^(~a) * (~y)^(~a))
## ... more on simplification...
## Calculus
@variables x y z
import Symbolics: derivative
derivative(cos(x), x)
derivative(exp(x^2), x)
# multiple derivative
Symbolics.derivative(ex, x, n::Int) = reduce((ex,_) -> derivative(ex, x), 1:n, init=ex) # helper
derivative(x^4, x, 3)
ex = exp(x*y*z)
using Chain
@chain ex begin
derivative(x, 3)
derivative(y, 3)
derivative(z, 3)
end
# using Differential operator
expr = exp(x*y*z)
expr |> Differential(x)^2 |> Differential(y)^3 |> expand_derivatives
# no integrate
# no limit
# Series
function series(ex, x, x0=0, n=5)
Σ = zero(ex)
for i ∈ 0:n
ex = expand_derivatives((Differential(x))(ex))
Σ += substitute(ex, Dict(x=>0)) * x^i / factorial(i)
end
Σ
end
# finite differences
# Solvers
@variables x y z a
eq = x ~ a
Symbolics.solve_for(eq, x)
eqs = [x + y + z ~ 1
x + y + 2z ~ 3
x + 2y + 3z ~ 3
]
vars = [x,y,z]
xs = Symbolics.solve_for(eqs, vars)
[reduce((ex, r)->substitute(ex, r), Pair.(vars, xs), init=ex.lhs) for ex ∈ eqs] == [eq.rhs for eq ∈ eqs]
A = [1 1; 1 2]
b = [1, 3]
xs = Symbolics.solve_for(A*[x,y] .~ b, [x,y])
A*xs - b
A = [1 1 1; 1 1 2]
b = [1,3]
A*[x,y,z] - b
Symbolics.solve_for(A*[x,y,z] .~ b, [x,y,z]) # fails, singular
# nonlinear solve
# use `λ = mk_function(ex, args, expression=Val(false))`
# polynomial roots
# differential equations

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# Symbolics.jl
There 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.
## About
The `Symbolics` package bill 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.
We begin by loading the `Symbolics` package which when loaded re-exports the `SymbolicUtils` package.
```julia
using Symbolics
```
## Symbolic variables
Symbolic math at its core involves symbolic variables, which essentially defer evaluation until requested. The creation of symbolic variables differs between the two package discussed here.
`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
```julia
@syms x y::Int f(x::Real)::Real
```
This creates `x` a symbolic value with type `Number`, `y` a symbolic variable holding integer values, and `f` a symbolic function of a single real variable outputing a real variable.
The `symtype` function reveals the underlying type:
```julia
import Symbolics.SymbolicUtils: symtype
symtype(x), symtype(y)
```
For `y`, the symbolic type being real does not imply the `y` has a subtype of `Real`:
```julia
isa(y, Real)
```
We see that the function `f`, when called with `y` would return a value of (symbolic) type `Real`:
```julia
f(y) |> symtype
```
As the symbolic type of `x` is `Number` -- which is not a subtype of `Real` -- the following will error:
```julia; error=true
f(x)
```
!!! note
The `SymPy` package also has an `@syms` macro to create variables. Though they 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.
For `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`.)
Symbolic values are created with the `@variables` macro. For example:
```julia
@variables x y::Int z[1:3]::Int f(..)::Int
```
This creates
* a symbolic value `x` of `symtype` `Real`
* a symbolic value `y` of `symtype` `Int`
* a vector of symbolic values each of `symtype` `Int`
* a symbolic function `f` returning an object of `symtype` `Int`
The symbolic type reflects that of the underlying object behind the `Num` wrapper:
```julia
typeof(x), symtype(x), typeof(Symbolics.value(x))
```
(The `value` method unwraps the `Num` wrapper.)
## Symbolic expressions
Symbolic expressions are built up from symbolic variables through natural `Julia` idioms. `SymbolicUtils` privileges a few key operations: `Add`, `Mul`, `Pow`, and `Div`. For examples:
```julia
@syms x y
typeof(x + y) # `Add`
```
```julia
typeof(x * y) # `Mul`
```
Whereas, applying a function leaves a different type:
```julia
typeof(sin(x))
```
The `Term` wrapper just represents the effect of calling a function (in this case `sin`) on its arguments (in this case `x`).
This happens in the background with symbolic variables in `Symbolics`:
```julia
@variables x
typeof(sin(x)), typeof(Symbolics.value(sin(x)))
```
### Tree structure to expressions
The `TermInterface` package is used by `SymbolicUtils` to explore the tree structdure of an expression. The main methods are (cf. [symbolicutils.jl](https://symbolicutils.juliasymbolics.org/#expression_interface)):
* `istree(ex)`: `true` if `ex` is not a *leaf* node (like a symbol or numeric literal)
* `operation(ex)`: the function being called (if `istree` returns `true`)
* `arguments(ex)`: the arguments to the function begin called
* `symtype(ex)`: the inferred type of the expression
In addition, the `issym` function, to determine if `x` is of type `Sym`, is useful to distinguish *leaf* nodes, as will be illustrated below.
These methods can be used to "walk" the tree:
```julia
@syms x y
ex = 1 + x^2 + y
operation(ex) # the outer function is `+`
```
```julia
arguments(ex) # `+` is n-ary, in this case with 3 arguments
```
```julia
ex1 = arguments(ex)[3] # terms have been reordered
operation(ex1) # operation for `x^2` is `^`
```
```julia
a, b = arguments(ex1)
```
```julia
istree(ex1), istree(a)
```
Here `a` is not a "tree", as it has no operation or arguments, it is just a variable (the `x` variable).
The value of `symtype` is the *inferred* type of an expression, which may not match the actual type. For example,
```julia
@variables x::Int
symtype(x), symtype(sin(x)), symtype(x/x), symtype(x / x^2)
```
The last one, is not likely to be an integer, but that is the inferred type in this case.
##### Example
As an example, we write a function to find the free symbols in a symbolic expression comprised of `SymbolicUtils` variables. (The `Symbolics.get_variables` also does this task.) To find the symbols involves walking the expression tree until a leaf node is found and then adding that to our collection if it matches `issym`.
```julia
import Symbolics.SymbolicUtils: issym
free_symbols(ex) = (s=Set(); free_symbols!(s, ex); s)
function free_symbols!(s, ex)
if istree(ex)
for a ∈ arguments(ex)
free_symbols!(s, a)
end
else
issym(ex) && push!(s, ex) # push new symbol onto set
end
end
```
```julia
@syms x y z
ex = sin(x + 1)*cos(z)
free_symbols(ex)
```
## Expression manipulation
### Substitute
The `substitute` command is used to replace values with other values. For examples:
```julia
@variables x y z
ex = 1 + x + x^2/2 + x^3/6
substitute(ex, x=>1)
```
This defines a symbolic expression, then substitutes the value `1` in for `x`. The `Pair` notation is useful for a *single* substitution. When there is more than one substitution, a dictionary is used:
```julia
w = x^3 + y^3 - 2z^3
substitute(w, Dict(x=>2, y=>3))
```
The `fold` argument can be passed `false` to inhibit evaluation of values. Compare:
```julia
ex = 1 + sqrt(x)
substitute(ex, x=>2), substitute(ex, x=>2, fold=false)
```
Or
```julia
ex = sin(x)
substitute(ex, x=>π), substitute(ex, x=>π, fold=false)
```
### Simplify
Algebraic operations with symbolic values can involve an exponentially increasing number of terms. As such, some simplification rules are applied after an operation to reduce the complexity of the computed value.
For example, `0+x` should simplify to `x`, `0+x`, `x^0`, or `x^1` should simplify, to some natural answer.
`SymbolicUtils` also [simplifies](https://symbolicutils.juliasymbolics.org/#simplification) several other expressions, including:
* `-x` becomes `(-1)*x`
* `x * x` becomes `x^2` (and `x^n` if more terms). Meaning this expression is represented as a power, not a product
* `x + x` becomes `2*x` (and `n*x` if more terms). Similarly, this represented as a product, not a sum.
* `p/q * x` becomes `(p*x)/q)`, similarly `p/q * x/y` becomes `(p*x)/(q*y)`
In `SymbolicUtils`, this *rewriting* is accomplished by means of *rewrite rules*. The package makes it easy to apply user-written rewrite rules.
### Rewriting
Many algebraic simplifications are done by the `simplify` command. For example, the basic trignometric identities are applied:
```julia
@variables x
ex = sin(x)^2 + cos(x)^2
ex, simplify(ex)
```
The `simplify` function applies a series of rewriting rule until the expression stabilizes. The rewrite rules can be user generated, if desired. For example, the Pythagorean identity of trigonometry, just used, can be implement with this rule:
```julia
r = @acrule(sin(~x)^2 + cos(~x)^2 => one(~x))
ex |> Symbolics.value |> r |> Num
```
The rewrite rule, `r`, is defined by the `@acrule` macro. The `a` is for associative, the `c` for commutative, assumptions made by the macro. (The `c` means `cos(x)^2 + sin(x)^2` will also simplify.) Rewrite rules are called on the underlying `SymbolicUtils` expression, so we first unwrap, then after re-wrap.
The above expression for `r` is fairly easy to appreciate. The value `~x` matches the same variable or expression. So the above rule will also simplify more complicated expressions:
```julia
@variables y z
ex1 = substitute(ex, x => sin(x + y + z))
ex1 |> Symbolics.value |> r |> Num
```
Rules involving two values are also easily created. This one, again, comes from the set of simplifications defined for trignometry and exponential simplifications:
```julia
r = @rule(exp(~x)^(~y) => exp(~x * ~y))
ex = exp(-x+z)^y
ex, ex |> Symbolics.value |> r |> Num
```
This rule is not commutative or associative, as `x^y` is not the same as `y^x` and `(x^y)^z` is not `x^(y^z)` in general.
The application of rules can be filtered through qualifying predicates. This artificial example uses `iseven` which returns `true` for even numbers. Here we subtract `1` when a number is not even, and otherwise leave the number alone. We do this with two rules:
```julia
reven = @rule ~x::iseven => ~x
rodd = @rule ~x::(!iseven) => ~x - 1
r = SymbolicUtils.Chain([rodd, reven])
r(2), r(3)
```
The `Chain` function conveniently allows the sequential application of rewrite rules.
The notation `~x` is called a "slot variable" in the [documentation](https://symbolicutils.juliasymbolics.org/rewrite/) for `SymbolicUtils`. It matches a single expression. To match more than one expression, a "segment variable", denoted with two `~`s is used.
### Creating functions
By utilizing the tree-like nature of a symbolic expression, a `Julia` expression can be built from an symbolic expression easily enough. The `Symbolics.toexpr` function does this:
```julia
ex = exp(-x + z)^y
Symbolics.toexpr(ex)
```
This output shows an internal representation of the steps for computing the value `ex` given different inputs.
Such `Julia` expressions are one step away from building `Julia` functions for evaluating symbolic expressions fast (though with some technical details about "world age" to be reckoned with). The `build_function` function with the argument `expression=Val(false)` will compile a `Julia` function:
```julia
h = build_function(ex, x, y, z; expression=Val(false))
h(1, 2, 3)
```
The above is *similar* to substitution:
```julia
substitute(ex, Dict(x=>1, y=>2, z=>3))
```
However, `build_function` will be **significantly** more performant, which when many function calls are used -- such as with plotting -- is a big advantage.
!!! note
The documentation colorfully says "`build_function` is kind of like if `lambdify` (from `SymPy`) ate its spinach."
The above, through passing ``3`` variables after the expression, creates a function of ``3`` variables. Functions of a vector of inputs can also be created, just by expressing the variables in that manner:
```juila
h1 = build_function(ex, [x, y, z]; expression=Val(false))
h1([1, 2, 3]) # not h1(1,2,3)
```
##### Example
As an example, here we use the `Roots` package to find a zero of a function defined symbolically:
```julia
import Roots
@variables x
ex = x^5 - x - 1
λ = build_function(ex, x; expression=Val(false))
Roots.find_zero(λ, (1, 2))
```
### Plotting
Using `Plots`, the plotting symbolic expressions is similar to the plotting of a function, as there is a plot recipe that converts the expression into a function via `build_function`.
For example,
```julia
using Plots
@variables x
plot(x^x^x, 0, 2)
```
A parametric plot is easily defined:
```julia
plot(sin(x), cos(x), 0, pi/4)
```
Expressions to be plotted can represent multivariate functions.
```julia
@variables x y
ex = 3*(1-x)^2*exp(-x^2 - (y+1)^2) - 10(x/5-x^3-y^5)*exp(-x^2-y^2) - 1/3*exp(-(x+1)^2-y^2)
xs = ys = range(-5, 5, length=100)
surface(xs, ys, ex)
```
The ordering of the variables is determined by `Symbolics.get_variables`:
```julia
Symbolics.get_variables(ex)
```
### Polynomial manipulations
There are some facilities for manipulating polynomial expressions in `Symbolics`. A polynomial, mathematically, is an expression involving one or more symbols with coefficients from a collection that has, at a minimum, addition and multiplication defined. The basic building blocks of polynomials are *monomials*, which are comprised of products of powers of the symbols. Mathematically, monomials are often allowed to have a multiplying coefficient and may be just a coefficient (if each symbol is taken to the power ``0``), but here we consider just expressions of the type ``x_1^{a_1} \cdot x_2^{a_2} \cdots x_k^{a_k}`` with the ``a_i > 0`` as monomials.
With this understanding, then an expression can be broken up into monomials with a possible leading coefficient (possibly ``1``) *and* terms which are not monomials (such as a constant or a more complicated function of the symbols). This is what is returned by the `polynomial_coeffs` function.
For example
```julia
@variables a b c x
d, r = polynomial_coeffs(a*x^2 + b*x + c, (x,))
```
The first term output is dictionary who's keys are the monomials and who's values are the coefficients. The second term, the residual, is all the remaining parts of the expression, in this case just the constant `c`.
The expression can then be reconstructed through
```julia
r + sum(v*k for (k,v) ∈ d)
```
The above has `a,b,c` as parameters and `x` as the symbol. This separation is designated by passing the desired polynomials symbols to `polynomial_coeff` as an iterable. (Above as a ``1``-element tuple.)
More complicated polynomials can be similarly decomposed:
```julia
@variables a b c x y z
ex = a*x^2*y*z + b*x*y^2*z + c*x*y*z^2
d, r = polynomial_coeffs(ex, (x, y, z))
```
The (sparse) decomposition of the polynomial is returned through `d`. The same pattern as above can be used to reconstruct the expression.
To extract the coefficient for a monomial term, indexing can be used. Of note, is an expression like `x^2*y*z` could *possibly* not equal the algebraically equal `x*y*z*x`, as they are only equal after some simplification, but the keys are in a canonical form, so this is not a concern:
```julia
d[x*y*z*x], d[z*y*x^2]
```
The residual term will capture any non-polynomial terms:
```julia
ex = sin(x) - x + x^3/6
d, r = polynomial_coeffs(ex, (x,))
r
```
To find the degree of a monomial expression, the `degree` function is available. Here it is applied to each monomial in `d`:
```julia
[degree(k) for (k,v) ∈ d]
```
The `degree` function will also identify the degree of more complicated terms:
```julia
degree(1 + x + x^2)
```
Mathematically the degree of the ``0`` polynomial may be ``-1`` or undefined, but here it is ``0``:
```julia
degree(0), degree(1), degree(x), degree(x^a)
```
The coefficients are returned as *values* of a dictionary, and dictionaries are unsorted. To have a natural map between polynomials of a single symbol in the standard basis and a vector, we could use a pattern like this:
```julia
@variables x a0 as[1:10]
p = a0 + sum(as[i]*x^i for i ∈ eachindex(collect(as)))
d, r = polynomial_coeffs(p, (x,))
d
```
To sort the values we can use a pattern like the following:
```julia
vcat(r, [d[k] for k ∈ sort(collect(keys(d)), by=degree)])
```
----
Rational expressions can be decomposed into a numerator and denominator using the following idiom, which ensures the outer operation is division (a binary operation):
```julia
@variables x
ex = (1 + x + x^2) / (1 + x + x^2 + x^3)
function nd(ex)
ex1 = Symbolics.value(ex)
(operation(ex1) == /) || return (ex, one(ex))
Num.(arguments(ex1))
end
nd(ex)
```
With this, the study of asymptotic behaviour of a univariate rational expression would involve an investigation like the following:
```julia
m,n = degree.(nd(ex))
m > n ? "limit is infinite" : m < n ? "limit is 0" : "limit is a constant"
```
### Algebraically solving equations
The `~` operator creates a symbolic equation. For example
```julia
@variables x y
x^5 - x ~ 1
```
or
```julia
ex = [5x + 2y, 6x + 3y] .~ [1, 2]
```
The `Symbolics.solve_for` function can solve *linear* equations. For example,
```julia
Symbolics.solve_for(ex, [x, y])
```
### Limits
As of writing, there is no extra functionality provided by `Symbolics` for computing limits.
### Derivatives
`Symbolics` provides the `derivative` function to compute the derivative of a function with respect to a variable:
```julia
@variables a b c x
ex = a*x^2 + b*x + c
Symbolics.derivative(ex, x)
```
The computation can also be broken up into an expression indicating the derivative and then a function to apply the derivative rules:
```julia
D = Differential(x)
D(ex)
```
and then
```julia
expand_derivatives(D(ex))
```
The differentials can be multiplied to create operators for taking higher-order derivatives:
```julia
@variables x y
ex = (x - y^2)/(x^2 + y^2)
Dx, Dy = Differential(x), Differential(y)
Dxx, Dxy, Dyy = Dx*Dx, Dx*Dy, Dy*Dy
[Dxx(ex) Dxy(ex); Dxy(ex) Dyy(ex)] .|> expand_derivatives
```
In addition to `Symbolics.derivative` there are also the helper functions, such as `hessian` which performs the above
```julia
Symbolics.hessian(ex, [x,y])
```
The `gradient` function is also available
```julia
@variables x y z
ex = x^2 - 2x*y + z*y
Symbolics.gradient(ex, [x, y, z])
```
The `jacobian` takes an array of expressions:
```julia
@variables x y
exs = [ x^2 - y^2, 2x*y]
Symbolics.jacobian(exs, [x,y])
```
### Integration
The `SymbolicNumericIntegration` package provides a means to integrate *univariate* expressions through its `integrate` function.
Symbolic integration can be approached in different ways. SymPy implements part of the Risch algorithm in addition to other algorithms. Rules-based algorithms could also be implemented.
For example, here is a simple rule that could be used to integrate a single integral
```julia
is_var(x) = (xs = Symbolics.get_variables(x); length(xs) == 1 && xs[1] === x)
@syms x ∫(x)
r = @rule ∫(~x::is_var) => x^2/2
r(∫(x))
```
The `SymbolicNumericIntegration` package includes many more predicates for doing rules-based integration, but it primarily approaches the task in a different manner.
If ``f(x)`` is to be integrated, a set of *candidate* answers is generated. The following is **proposed** as an answer: ``\sum q_i \Theta_i(x)``. Differentiating the proposed answer leads to a *linear system of equations* that can be solved.
The example in the [paper](https://arxiv.org/pdf/2201.12468v2.pdf) describing the method is with ``f(x) = x \sin(x)`` and the candidate thetas are ``{x, \sin(x), \cos(x), x\sin(x), x\cos(x)}`` so that we propose:
```math
\int f(x) dx = q_1 x + q_2 \sin(x) + q_3 \cos(x) + q_4 x \sin(x) + q_4 x \cos(x)
```
Differentiating both sides, yields a term ``x\sin(x)`` on the left, and equating coefficients gives:
```math
q_1 = q_4 = 0,\quad q_5 = -1, \quad q_4 - q_3 = q_2 - q_5 = 0
```
which can be solved with ``q_5=-1``, ``q_2=1``, and the other coefficients being ``0``. That is ``\int f(x) dx = 1 \sin(x) + (-1) x\cos(x)``.
The package provides an algorithm for the creation of candidates and the means to solve when possible. The `integrate` function is the main entry point. It returns three values: `solved`, `unsolved`, and `err`. The `unsolved` is the part of the integrand which can not be solved through this package. It is `0` for a given problem when `integrate` is successful in identifying an antiderivative, in which case `solved` is the answer. The value of `err` is a bound on the numerical error introduced by the algorithm.
To see, we have:
```julia
using SymbolicNumericIntegration
@variables x
integrate(x * sin(x))
```
The second term is `0`, as this has an identified antiderivative.
```julia
integrate(exp(x^2) + sin(x))
```
This returns `exp(x^2)` for the unsolved part, as this function has no simple antiderivative.
Powers of trig functions have antiderivatives, as can be deduced using integration by parts. When the fifth power is used, there is a numeric aspect to the algorithm that is seen:
```julia
u,v,w = integrate(sin(x)^5)
```
The derivative of `u` matches up to some numeric tolerance:
```julia
Symbolics.derivative(u, x) - sin(x)^5
```
The integration of rational functions (ratios of polynomials) can be done algorithmically, provided the underlying factorizations can be identified. The `SymbolicNumericIntegration` package has a function `factor_rational` that can identify factorizations.
```julia
import SymbolicNumericIntegration: factor_rational
@variables x
u = (1 + x + x^2)/ (x^2 -2x + 1)
v = factor_rational(u)
```
The summands in `v` are each integrable. We can see that `v` is a reexpression through
```julia
simplify(u - v)
```
The algorithm is numeric, not symbolic. This can be seen in these two factorizations:
```julia
u = 1 / expand((x^2-1)*(x-2)^2)
v = factor_rational(u)
```
or
```julia
u = 1 / expand((x^2+1)*(x-2)^2)
v = factor_rational(u)
```
As such, the integrals have numeric differences:
```julia
a,b,c = integrate(u)
```
We can see a bit of why through the following which needs a tolerance set to identify the rational numbers correctly:
```julia
cs = [first(arguments(term)) for term ∈ arguments(a)] # pick off coefficients
```
```julia
rationalize.(cs; tol=1e-8)
```

2
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@ -413,7 +413,7 @@ So the answer to the question is
This seems like a lot of work, and indeed it is more than is needed. The following would be more typical once the rules are learned:
```math
\int_0^\pi 100 \sin(x) dx = -100(-\cos(x)) \big|_0^{\pi} = 100 \cos(x) \big|_{\pi}^0 = 100(1) - 100(-1) = 200.
\int_0^\pi 100 \sin(x) dx = 100(-\cos(x)) \big|_0^{\pi} = 100 \cos(x) \big|_{\pi}^0 = 100(1) - 100(-1) = 200.
```
## The derivative of the integral

2
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@ -180,7 +180,7 @@ manner. This allows the `Julia` package `SymPy` to provide
functionality from SymPy within `Julia`.
!!! note
When `SymPy` is installed through the package manger, the underlying `Python`
When `SymPy` is installed through the package manager, the underlying `Python`
libraries will also be installed.
!!! note

6
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@ -33,7 +33,7 @@ To compile the pages through quarto
* This error
> fatal: 'gh-pages' is already checked out at '/Users/verzani/julia/CalculusWithJuliaNotes/quarto/f5611730'
was solved with
was solved with (https://waylonwalker.com/til/git-checkout-worktree/)
> git worktree remove f5611730
@ -45,9 +45,9 @@ Eventually, if this workflow seems to be settled:
* deprecate .jmd files
* deprecate need to make "pluto friendly"
* do something with JSXGraph
DONE? * do something with JSXGraph
* figure out why PlotlyLight doesn't work
* move to not use CalculusWithJulia.WeaveSupport
* use an include file not the "hack" in jmd2qmd
DONE * use an include file not the "hack" in jmd2qmd
* modify sympy's show method
* take advantage of mermaid, ojs, bibliography, ...

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@ -110,7 +110,7 @@ book:
- part: "Alternatives"
chapters:
# - alternatives/symbolics.qmd
- alternatives/symbolics.qmd
# - alternatives/sciML.qmd
# - alternatives/interval_arithmetic.qmd
- alternatives/plotly_plotting.qmd

778
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@ -0,0 +1,778 @@
# Symbolics.jl
There 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.
## About
The `Symbolics` package bill 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.
We begin by loading the `Symbolics` package which when loaded re-exports the `SymbolicUtils` package.
```{julia}
using Symbolics
```
## Symbolic variables
Symbolic math at its core involves symbolic variables, which essentially defer evaluation until requested. The creation of symbolic variables differs between the two package discussed here.
`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
```{julia}
@syms x y::Int f(x::Real)::Real
```
This creates `x` a symbolic value with type `Number`, `y` a symbolic variable holding integer values, and `f` a symbolic function of a single real variable outputing a real variable.
The `symtype` function reveals the underlying type:
```{julia}
import Symbolics.SymbolicUtils: symtype
symtype(x), symtype(y)
```
For `y`, the symbolic type being real does not imply the `y` has a subtype of `Real`:
```{julia}
isa(y, Real)
```
We see that the function `f`, when called with `y` would return a value of (symbolic) type `Real`:
```{julia}
f(y) |> symtype
```
As the symbolic type of `x` is `Number` – which is not a subtype of `Real` – the following will error:
```{julia}
#| error: true
f(x)
```
:::{.callout-note}
## Note
The `SymPy` package also has an `@syms` macro to create variables. Though they 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.
:::
For `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`.)
Symbolic values are created with the `@variables` macro. For example:
```{julia}
@variables x y::Int z[1:3]::Int f(..)::Int
```
This creates
* a symbolic value `x` of `symtype` `Real`
* a symbolic value `y` of `symtype` `Int`
* a vector of symbolic values each of `symtype` `Int`
* a symbolic function `f` returning an object of `symtype` `Int`
The symbolic type reflects that of the underlying object behind the `Num` wrapper:
```{julia}
typeof(x), symtype(x), typeof(Symbolics.value(x))
```
(The `value` method unwraps the `Num` wrapper.)
## Symbolic expressions
Symbolic expressions are built up from symbolic variables through natural `Julia` idioms. `SymbolicUtils` privileges a few key operations: `Add`, `Mul`, `Pow`, and `Div`. For examples:
```{julia}
@syms x y
typeof(x + y) # `Add`
```
```{julia}
typeof(x * y) # `Mul`
```
Whereas, applying a function leaves a different type:
```{julia}
typeof(sin(x))
```
The `Term` wrapper just represents the effect of calling a function (in this case `sin`) on its arguments (in this case `x`).
This happens in the background with symbolic variables in `Symbolics`:
```{julia}
@variables x
typeof(sin(x)), typeof(Symbolics.value(sin(x)))
```
### Tree structure to expressions
The `TermInterface` package is used by `SymbolicUtils` to explore the tree structdure of an expression. The main methods are (cf. [symbolicutils.jl](https://symbolicutils.juliasymbolics.org/#expression_interface)):
* `istree(ex)`: `true` if `ex` is not a *leaf* node (like a symbol or numeric literal)
* `operation(ex)`: the function being called (if `istree` returns `true`)
* `arguments(ex)`: the arguments to the function begin called
* `symtype(ex)`: the inferred type of the expression
In addition, the `issym` function, to determine if `x` is of type `Sym`, is useful to distinguish *leaf* nodes, as will be illustrated below.
These methods can be used to "walk" the tree:
```{julia}
@syms x y
ex = 1 + x^2 + y
operation(ex) # the outer function is `+`
```
```{julia}
arguments(ex) # `+` is n-ary, in this case with 3 arguments
```
```{julia}
ex1 = arguments(ex)[3] # terms have been reordered
operation(ex1) # operation for `x^2` is `^`
```
```{julia}
a, b = arguments(ex1)
```
```{julia}
istree(ex1), istree(a)
```
Here `a` is not a "tree", as it has no operation or arguments, it is just a variable (the `x` variable).
The value of `symtype` is the *inferred* type of an expression, which may not match the actual type. For example,
```{julia}
@variables x::Int
symtype(x), symtype(sin(x)), symtype(x/x), symtype(x / x^2)
```
The last one, is not likely to be an integer, but that is the inferred type in this case.
##### Example
As an example, we write a function to find the free symbols in a symbolic expression comprised of `SymbolicUtils` variables. (The `Symbolics.get_variables` also does this task.) To find the symbols involves walking the expression tree until a leaf node is found and then adding that to our collection if it matches `issym`.
```{julia}
import Symbolics.SymbolicUtils: issym
free_symbols(ex) = (s=Set(); free_symbols!(s, ex); s)
function free_symbols!(s, ex)
if istree(ex)
for a ∈ arguments(ex)
free_symbols!(s, a)
end
else
issym(ex) && push!(s, ex) # push new symbol onto set
end
end
```
```{julia}
@syms x y z
ex = sin(x + 1)*cos(z)
free_symbols(ex)
```
## Expression manipulation
### Substitute
The `substitute` command is used to replace values with other values. For examples:
```{julia}
@variables x y z
ex = 1 + x + x^2/2 + x^3/6
substitute(ex, x=>1)
```
This defines a symbolic expression, then substitutes the value `1` in for `x`. The `Pair` notation is useful for a *single* substitution. When there is more than one substitution, a dictionary is used:
```{julia}
w = x^3 + y^3 - 2z^3
substitute(w, Dict(x=>2, y=>3))
```
The `fold` argument can be passed `false` to inhibit evaluation of values. Compare:
```{julia}
ex = 1 + sqrt(x)
substitute(ex, x=>2), substitute(ex, x=>2, fold=false)
```
Or
```{julia}
ex = sin(x)
substitute(ex, x=>π), substitute(ex, x=>π, fold=false)
```
### Simplify
Algebraic operations with symbolic values can involve an exponentially increasing number of terms. As such, some simplification rules are applied after an operation to reduce the complexity of the computed value.
For example, `0+x` should simplify to `x`, `0+x`, `x^0`, or `x^1` should simplify, to some natural answer.
`SymbolicUtils` also [simplifies](https://symbolicutils.juliasymbolics.org/#simplification) several other expressions, including:
* `-x` becomes `(-1)*x`
* `x * x` becomes `x^2` (and `x^n` if more terms). Meaning this expression is represented as a power, not a product
* `x + x` becomes `2*x` (and `n*x` if more terms). Similarly, this represented as a product, not a sum.
* `p/q * x` becomes `(p*x)/q)`, similarly `p/q * x/y` becomes `(p*x)/(q*y)`
In `SymbolicUtils`, this *rewriting* is accomplished by means of *rewrite rules*. The package makes it easy to apply user-written rewrite rules.
### Rewriting
Many algebraic simplifications are done by the `simplify` command. For example, the basic trignometric identities are applied:
```{julia}
@variables x
ex = sin(x)^2 + cos(x)^2
ex, simplify(ex)
```
The `simplify` function applies a series of rewriting rule until the expression stabilizes. The rewrite rules can be user generated, if desired. For example, the Pythagorean identity of trigonometry, just used, can be implement with this rule:
```{julia}
r = @acrule(sin(~x)^2 + cos(~x)^2 => one(~x))
ex |> Symbolics.value |> r |> Num
```
The rewrite rule, `r`, is defined by the `@acrule` macro. The `a` is for associative, the `c` for commutative, assumptions made by the macro. (The `c` means `cos(x)^2 + sin(x)^2` will also simplify.) Rewrite rules are called on the underlying `SymbolicUtils` expression, so we first unwrap, then after re-wrap.
The above expression for `r` is fairly easy to appreciate. The value `~x` matches the same variable or expression. So the above rule will also simplify more complicated expressions:
```{julia}
@variables y z
ex1 = substitute(ex, x => sin(x + y + z))
ex1 |> Symbolics.value |> r |> Num
```
Rules involving two values are also easily created. This one, again, comes from the set of simplifications defined for trignometry and exponential simplifications:
```{julia}
r = @rule(exp(~x)^(~y) => exp(~x * ~y))
ex = exp(-x+z)^y
ex, ex |> Symbolics.value |> r |> Num
```
This rule is not commutative or associative, as `x^y` is not the same as `y^x` and `(x^y)^z` is not `x^(y^z)` in general.
The application of rules can be filtered through qualifying predicates. This artificial example uses `iseven` which returns `true` for even numbers. Here we subtract `1` when a number is not even, and otherwise leave the number alone. We do this with two rules:
```{julia}
reven = @rule ~x::iseven => ~x
rodd = @rule ~x::(!iseven) => ~x - 1
r = SymbolicUtils.Chain([rodd, reven])
r(2), r(3)
```
The `Chain` function conveniently allows the sequential application of rewrite rules.
The notation `~x` is called a "slot variable" in the [documentation](https://symbolicutils.juliasymbolics.org/rewrite/) for `SymbolicUtils`. It matches a single expression. To match more than one expression, a "segment variable", denoted with two `~`s is used.
### Creating functions
By utilizing the tree-like nature of a symbolic expression, a `Julia` expression can be built from an symbolic expression easily enough. The `Symbolics.toexpr` function does this:
```{julia}
ex = exp(-x + z)^y
Symbolics.toexpr(ex)
```
This output shows an internal representation of the steps for computing the value `ex` given different inputs.
Such `Julia` expressions are one step away from building `Julia` functions for evaluating symbolic expressions fast (though with some technical details about "world age" to be reckoned with). The `build_function` function with the argument `expression=Val(false)` will compile a `Julia` function:
```{julia}
h = build_function(ex, x, y, z; expression=Val(false))
h(1, 2, 3)
```
The above is *similar* to substitution:
```{julia}
substitute(ex, Dict(x=>1, y=>2, z=>3))
```
However, `build_function` will be **significantly** more performant, which when many function calls are used – such as with plotting – is a big advantage.
:::{.callout-note}
## Note
The documentation colorfully says "`build_function` is kind of like if `lambdify` (from `SymPy`) ate its spinach."
:::
The above, through passing $3$ variables after the expression, creates a function of $3$ variables. Functions of a vector of inputs can also be created, just by expressing the variables in that manner:
```{juila}
h1 = build_function(ex, [x, y, z]; expression=Val(false))
h1([1, 2, 3]) # not h1(1,2,3)
```
##### Example
As an example, here we use the `Roots` package to find a zero of a function defined symbolically:
```{julia}
import Roots
@variables x
ex = x^5 - x - 1
λ = build_function(ex, x; expression=Val(false))
Roots.find_zero(λ, (1, 2))
```
### Plotting
Using `Plots`, the plotting symbolic expressions is similar to the plotting of a function, as there is a plot recipe that converts the expression into a function via `build_function`.
For example,
```{julia}
using Plots
@variables x
plot(x^x^x, 0, 2)
```
A parametric plot is easily defined:
```{julia}
plot(sin(x), cos(x), 0, pi/4)
```
Expressions to be plotted can represent multivariate functions.
```{julia}
@variables x y
ex = 3*(1-x)^2*exp(-x^2 - (y+1)^2) - 10(x/5-x^3-y^5)*exp(-x^2-y^2) - 1/3*exp(-(x+1)^2-y^2)
xs = ys = range(-5, 5, length=100)
surface(xs, ys, ex)
```
The ordering of the variables is determined by `Symbolics.get_variables`:
```{julia}
Symbolics.get_variables(ex)
```
### Polynomial manipulations
There are some facilities for manipulating polynomial expressions in `Symbolics`. A polynomial, mathematically, is an expression involving one or more symbols with coefficients from a collection that has, at a minimum, addition and multiplication defined. The basic building blocks of polynomials are *monomials*, which are comprised of products of powers of the symbols. Mathematically, monomials are often allowed to have a multiplying coefficient and may be just a coefficient (if each symbol is taken to the power $0$), but here we consider just expressions of the type $x_1^{a_1} \cdot x_2^{a_2} \cdots x_k^{a_k}$ with the $a_i > 0$ as monomials.
With this understanding, then an expression can be broken up into monomials with a possible leading coefficient (possibly $1$) *and* terms which are not monomials (such as a constant or a more complicated function of the symbols). This is what is returned by the `polynomial_coeffs` function.
For example
```{julia}
@variables a b c x
d, r = polynomial_coeffs(a*x^2 + b*x + c, (x,))
```
The first term output is dictionary who's keys are the monomials and who's values are the coefficients. The second term, the residual, is all the remaining parts of the expression, in this case just the constant `c`.
The expression can then be reconstructed through
```{julia}
r + sum(v*k for (k,v) ∈ d)
```
The above has `a,b,c` as parameters and `x` as the symbol. This separation is designated by passing the desired polynomials symbols to `polynomial_coeff` as an iterable. (Above as a $1$-element tuple.)
More complicated polynomials can be similarly decomposed:
```{julia}
@variables a b c x y z
ex = a*x^2*y*z + b*x*y^2*z + c*x*y*z^2
d, r = polynomial_coeffs(ex, (x, y, z))
```
The (sparse) decomposition of the polynomial is returned through `d`. The same pattern as above can be used to reconstruct the expression. To extract the coefficient for a monomial term, indexing can be used. Of note, is an expression like `x^2*y*z` could *possibly* not equal the algebraically equal `x*y*z*x`, as they are only equal after some simplification, but the keys are in a canonical form, so this is not a concern:
```{julia}
d[x*y*z*x], d[z*y*x^2]
```
The residual term will capture any non-polynomial terms:
```{julia}
ex = sin(x) - x + x^3/6
d, r = polynomial_coeffs(ex, (x,))
r
```
To find the degree of a monomial expression, the `degree` function is available. Here it is applied to each monomial in `d`:
```{julia}
[degree(k) for (k,v) ∈ d]
```
The `degree` function will also identify the degree of more complicated terms:
```{julia}
degree(1 + x + x^2)
```
Mathematically the degree of the $0$ polynomial may be $-1$ or undefined, but here it is $0$:
```{julia}
degree(0), degree(1), degree(x), degree(x^a)
```
The coefficients are returned as *values* of a dictionary, and dictionaries are unsorted. To have a natural map between polynomials of a single symbol in the standard basis and a vector, we could use a pattern like this:
```{julia}
@variables x a0 as[1:10]
p = a0 + sum(as[i]*x^i for i ∈ eachindex(collect(as)))
d, r = polynomial_coeffs(p, (x,))
d
```
To sort the values we can use a pattern like the following:
```{julia}
vcat(r, [d[k] for k ∈ sort(collect(keys(d)), by=degree)])
```
---
Rational expressions can be decomposed into a numerator and denominator using the following idiom, which ensures the outer operation is division (a binary operation):
```{julia}
@variables x
ex = (1 + x + x^2) / (1 + x + x^2 + x^3)
function nd(ex)
ex1 = Symbolics.value(ex)
(operation(ex1) == /) || return (ex, one(ex))
Num.(arguments(ex1))
end
nd(ex)
```
With this, the study of asymptotic behaviour of a univariate rational expression would involve an investigation like the following:
```{julia}
m,n = degree.(nd(ex))
m > n ? "limit is infinite" : m < n ? "limit is 0" : "limit is a constant"
```
### Algebraically solving equations
The `~` operator creates a symbolic equation. For example
```{julia}
@variables x y
x^5 - x ~ 1
```
or
```{julia}
ex = [5x + 2y, 6x + 3y] .~ [1, 2]
```
The `Symbolics.solve_for` function can solve *linear* equations. For example,
```{julia}
Symbolics.solve_for(ex, [x, y])
```
### Limits
As of writing, there is no extra functionality provided by `Symbolics` for computing limits.
### Derivatives
`Symbolics` provides the `derivative` function to compute the derivative of a function with respect to a variable:
```{julia}
@variables a b c x
ex = a*x^2 + b*x + c
Symbolics.derivative(ex, x)
```
The computation can also be broken up into an expression indicating the derivative and then a function to apply the derivative rules:
```{julia}
D = Differential(x)
D(ex)
```
and then
```{julia}
expand_derivatives(D(ex))
```
The differentials can be multiplied to create operators for taking higher-order derivatives:
```{julia}
@variables x y
ex = (x - y^2)/(x^2 + y^2)
Dx, Dy = Differential(x), Differential(y)
Dxx, Dxy, Dyy = Dx*Dx, Dx*Dy, Dy*Dy
[Dxx(ex) Dxy(ex); Dxy(ex) Dyy(ex)] .|> expand_derivatives
```
In addition to `Symbolics.derivative` there are also the helper functions, such as `hessian` which performs the above
```{julia}
Symbolics.hessian(ex, [x,y])
```
The `gradient` function is also available
```{julia}
@variables x y z
ex = x^2 - 2x*y + z*y
Symbolics.gradient(ex, [x, y, z])
```
The `jacobian` takes an array of expressions:
```{julia}
@variables x y
exs = [ x^2 - y^2, 2x*y]
Symbolics.jacobian(exs, [x,y])
```
### Integration
The `SymbolicNumericIntegration` package provides a means to integrate *univariate* expressions through its `integrate` function.
Symbolic integration can be approached in different ways. SymPy implements part of the Risch algorithm in addition to other algorithms. Rules-based algorithms could also be implemented.
For example, here is a simple rule that could be used to integrate a single integral
```{julia}
is_var(x) = (xs = Symbolics.get_variables(x); length(xs) == 1 && xs[1] === x)
@syms x ∫(x)
r = @rule ∫(~x::is_var) => x^2/2
r(∫(x))
```
The `SymbolicNumericIntegration` package includes many more predicates for doing rules-based integration, but it primarily approaches the task in a different manner.
If $f(x)$ is to be integrated, a set of *candidate* answers is generated. The following is **proposed** as an answer: $\sum q_i \Theta_i(x)$. Differentiating the proposed answer leads to a *linear system of equations* that can be solved.
The example in the [paper](https://arxiv.org/pdf/2201.12468v2.pdf) describing the method is with $f(x) = x \sin(x)$ and the candidate thetas are ${x, \sin(x), \cos(x), x\sin(x), x\cos(x)}$ so that we propose:
$$
\int f(x) dx = q_1 x + q_2 \sin(x) + q_3 \cos(x) + q_4 x \sin(x) + q_4 x \cos(x)
$$
Differentiating both sides, yields a term $x\sin(x)$ on the left, and equating coefficients gives:
$$
q_1 = q_4 = 0,\quad q_5 = -1, \quad q_4 - q_3 = q_2 - q_5 = 0
$$
which can be solved with $q_5=-1$, $q_2=1$, and the other coefficients being $0$. That is $\int f(x) dx = 1 \sin(x) + (-1) x\cos(x)$.
The package provides an algorithm for the creation of candidates and the means to solve when possible. The `integrate` function is the main entry point. It returns three values: `solved`, `unsolved`, and `err`. The `unsolved` is the part of the integrand which can not be solved through this package. It is `0` for a given problem when `integrate` is successful in identifying an antiderivative, in which case `solved` is the answer. The value of `err` is a bound on the numerical error introduced by the algorithm.
To see, we have:
```{julia}
using SymbolicNumericIntegration
@variables x
integrate(x * sin(x))
```
The second term is `0`, as this has an identified antiderivative.
```{julia}
integrate(exp(x^2) + sin(x))
```
This returns `exp(x^2)` for the unsolved part, as this function has no simple antiderivative.
Powers of trig functions have antiderivatives, as can be deduced using integration by parts. When the fifth power is used, there is a numeric aspect to the algorithm that is seen:
```{julia}
u,v,w = integrate(sin(x)^5)
```
The derivative of `u` matches up to some numeric tolerance:
```{julia}
Symbolics.derivative(u, x) - sin(x)^5
```
The integration of rational functions (ratios of polynomials) can be done algorithmically, provided the underlying factorizations can be identified. The `SymbolicNumericIntegration` package has a function `factor_rational` that can identify factorizations.
```{julia}
import SymbolicNumericIntegration: factor_rational
@variables x
u = (1 + x + x^2)/ (x^2 -2x + 1)
v = factor_rational(u)
```
The summands in `v` are each integrable. We can see that `v` is a reexpression through
```{julia}
simplify(u - v)
```
The algorithm is numeric, not symbolic. This can be seen in these two factorizations:
```{julia}
u = 1 / expand((x^2-1)*(x-2)^2)
v = factor_rational(u)
```
or
```{julia}
u = 1 / expand((x^2+1)*(x-2)^2)
v = factor_rational(u)
```
As such, the integrals have numeric differences:
```{julia}
a,b,c = integrate(u)
```
We can see a bit of why through the following which needs a tolerance set to identify the rational numbers correctly:
```{julia}
cs = [first(arguments(term)) for term ∈ arguments(a)] # pick off coefficients
```
```{julia}
rationalize.(cs; tol=1e-8)
```

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# This is *if* we have CI publish; currently we use `quarto render gh-pages`
on:
push:
branches: main