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# [Mixed Symbolic-Numeric Perturbation Theory](@id perturb_alg)

## Background
[**Symbolics.jl**](https://github.com/JuliaSymbolics/Symbolics.jl) is a fast and modern Computer Algebra System (CAS) written in the Julia Programming Language. It is an integral part of the [SciML](https://sciml.ai/) ecosystem of differential equation solvers and scientific machine learning packages. While **Symbolics.jl** is primarily designed for modern scientific computing (e.g. automatic differentiation and machine learning), it is also a powerful CAS that can be used for *classic* scientific computing. One such application is using *perturbation theory* to solve algebraic and differential equations.

[**Symbolics.jl**](https://github.com/JuliaSymbolics/Symbolics.jl) is a fast and modern Computer Algebra System (CAS) written in the Julia Programming Language. It is an integral part of the [SciML](https://sciml.ai/) ecosystem of differential equation solvers and scientific machine learning packages. While **Symbolics.jl** is primarily designed for modern scientific computing (e.g., auto-differentiation, machine learning), it is a powerful CAS that can also be useful for *classic* scientific computing. One such application is using the *perturbation* theory to solve algebraic and differential equations.
Perturbation methods are a collection of techniques to solve hard problems that generally don't have a closed solution, but depend on a tunable parameter and have closed or easy solutions for some values of the parameter. The main idea is to assume a solution that is a power series in the tunable parameter (say $ϵ$), such that $ϵ = 0$ corresponds to an easy solution, and then solve iteratively for higher-order corrections.

Perturbation methods are a collection of techniques to solve intractable problems that generally don't have a closed solution but depend on a tunable parameter and have closed or easy solutions for some values of the parameter. The main idea is to assume a solution as a power series in the tunable parameter (say $ϵ$), such that $ϵ = 0$ corresponds to an easy solution.
The hallmark of perturbation methods is the generation of long and convoluted intermediate equations, which are subjected to algorithmic and mechanical manipulations. Therefore, these problems are well suited for CAS. In fact, CAS software packages have been used to help with the perturbation calculations since the early 1970s.

We will discuss the general steps of the perturbation methods to solve algebraic (this tutorial) and [differential equations](https://docs.sciml.ai/ModelingToolkit/stable/examples/perturbation/)
This tutorial shows how to mix symbolic manipulations and numerical methods to solve algebraic equations with perturbation theory. [Another tutorial applies it to differential equations](https://docs.sciml.ai/ModelingToolkit/stable/examples/perturbation/).

The hallmark of the perturbation method is the generation of long and convoluted intermediate equations, which are subjected to algorithmic and mechanical manipulations. Therefore, these problems are well suited for CAS. In fact, CAS software packages have been used to help with the perturbation calculations since the early 1970s.

In this tutorial, our goal is to show how to use a mix of symbolic manipulations (**Symbolics.jl**) and numerical methods to solve simple perturbation problems.

## Solving the Quintic

We start with the “hello world!” analog of the perturbation problems, solving the quintic (fifth-order) equations. We want to find a real valued $x$ such that $x^5 + x = 1$. According to the Abel's theorem, a general quintic equation does not have a closed form solution. Of course, we can easily solve this equation numerically; for example, by using the Newton's method. We use the following implementation of the Newton's method:
## Solving the quintic equation

We start with the “hello world!” analog of perturbation problems: finding a real solution $x$ to the quintic (fifth-order) equation
```@example perturb
using Symbolics, SymbolicUtils
function solve_newton(f, x, x₀; abstol=1e-8, maxiter=50)
xₙ = Float64(x₀)
fₙ₊₁ = x - f / Symbolics.derivative(f, x)
using Symbolics # load Symbolics.jl
@variables x # create a symbolic variable x
quintic = x^5 + x ~ 1 # create a symbolic representation of the quintic equation
```
According to Abel's theorem, a general quintic equation does not have a closed form solution. But we can easily solve it numerically using Newton's method (here implemented for simplicity, and not performance):
```@example perturb
function solve_newton(eq, x, x₀; abstol=1e-8, maxiters=50)
# symbolic expressions for f(x) and f′(x)
f = eq.lhs - eq.rhs # want to find root of f(x)
f′ = Symbolics.derivative(f, x)
for i = 1:maxiter
xₙ₊₁ = substitute(fₙ₊₁, Dict(x => xₙ))
xₙ = x₀ # numerical value of the initial guess
for i = 1:maxiters
# calculate new guess by numerically evaluating symbolic expression at previous guess
xₙ₊₁ = substitute(x - f / f′, x => xₙ)
if abs(xₙ₊₁ - xₙ) < abstol
return xₙ₊₁
return xₙ₊₁ # converged
else
xₙ = xₙ₊₁
end
end
return xₙ₊₁
error("Newton's method failed to converge")
end
```

In this code, `Symbolics.derivative(eq, x)` does exactly what it names implies: it calculates the symbolic derivative of `eq` (a **Symbolics.jl** expression) with respect to `x` (a **Symbolics.jl** variable). We use `Symbolics.substitute(eq, D)` to evaluate the update formula by substituting variables or sub-expressions (defined in a dictionary `D`) in `eq`. It should be noted that `substitute` is the workhorse of our code and will be used multiple times in the rest of these tutorials. `solve_newton` is written with simplicity and clarity in mind, and not performance.

Let's go back to our quintic. We can define a Symbolics variable as `@variables x` and then solve the equation `solve_newton(x^5 + x - 1, x, 1.0)` (here, `x₀ = 1.0` is our first guess). The answer is 0.7549. Now, let's see how we can solve the same problem using the perturbation methods.

We introduce a tuning parameter $\epsilon$ into our equation: $x^5 + \epsilon x = 1$. If $\epsilon = 1$, we get our original problem. For $\epsilon = 0$, the problem transforms to an easy one: $x^5 = 1$ which has an exact real solution $x = 1$ (and four complex solutions which we ignore here). We expand $x$ as a power series on $\epsilon$:

```math
x(\epsilon) = a_0 + a_1 \epsilon + a_2 \epsilon^2 + O(\epsilon^3)
```

$a_0$ is the solution of the easy equation, therefore $a_0 = 1$. Substituting into the original problem,

```math
(a_0 + a_1 \epsilon + a_2 \epsilon^2)^5 + \epsilon (a_0 + a_1 \epsilon + a_2 \epsilon^2) - 1 = 0
```

Expanding the equations, we get

```math
\epsilon (1 + 5 a_1) + \epsilon^2 (a_1 + 5 a_2 + 10 a_1^2) + 𝑂(\epsilon^3) = 0
```

This equation should hold for each power of $\epsilon$. Therefore,

```math
1 + 5 a_1 = 0
```

and
```math
a_1 + 5 a_2 + 10 a_1^2 = 0
x₀ = 1.0 # initial guess
solve_newton(quintic, x, x₀)
```

This system of equations does not initially seem to be linear because of the presence of terms like $10 a_1^2$, but upon closer inspection is found to be linear (this is a feature of the perturbation methods). In addition, the system is in a triangular form, meaning the first equation depends only on $a_1$, the second one on $a_1$ and $a_2$, such that we can replace the result of $a_1$ from the first one into the second equation and remove the non-linear term. We solve the first equation to get $a_1 = -\frac{1}{5}$. Substituting in the second one and solve for $a_2$:

```math
a_2 = \frac{(-\frac{1}{5} + 10(-(\frac{1}{5})²)}{5} = -\frac{1}{25}
```

Finally,

```math
x(\epsilon) = 1 - \frac{\epsilon}{5} - \frac{\epsilon^2}{25} + O(\epsilon^3)
```

Solving the original problem, $x(1) = 0.76$, compared to 0.7548 calculated numerically. We can improve the accuracy by including more terms in the expansion of $x$. However, the calculations, while straightforward, become messy and intractable to do manually very quickly. This is why a CAS is very helpful to solve perturbation problems.

Now, let's see how we can do these calculations in Julia. Let $n$ be the order of the expansion. We start by defining the symbolic variables:

Let us now solve the same problem with perturbation theory. First, we introduce a tuning parameter $\epsilon$ into our equation:
```@example perturb
n = 2
@variables ϵ a[1:n]
@variables ϵ # perturbation expansion parameter
quintic = x^5 + ϵ*x ~ 1
```

Then, we define

If $\epsilon = 1$, we get our original problem. With $\epsilon = 0$, the problem transforms to the easy quintic equation $x^5 = 1$ with the trivial real solution $x = 1$ (and four complex solutions which we ignore). We expand $x$ as a power series up to eighth order in $\epsilon$:
```@example perturb
x = 1 + a[1]*ϵ + a[2]*ϵ^2
x_taylor = series(x, ϵ, 0:8)
```

The next step is to substitute `x` in the problem equation

Then we insert it into the quintic equation and expand it, too, to the same order:
```@example perturb
eq = x^5 + ϵ*x - 1
quintic_taylor = substitute(quintic, x => x_taylor)
quintic_taylor = taylor(quintic_taylor, ϵ, 0:8)
```

The expanded form of `eq` is

This equation must hold for each power of $\epsilon$, so we can separate it into one equation per order:
```@example perturb
expand(eq)
taylor_coeff(quintic_taylor, ϵ, 0:8)
```

We need a way to get the coefficients of different powers of `ϵ`. Function `collect_powers(eq, x, ns)` returns the powers of variable `x` in expression `eq`. Argument `ns` is the range of the powers.

Note closely three important features of perturbation methods:
1. The $0$-th order equation is trivial: here $x_0^5 = 1$ has the trivial real solution $x_0 = 1$.
2. Except the trivial $0$-th order equation, the $n$-th order equation is *linear* in $x_i$.
3. The system of equations is *triangular* in the unknowns $x_n$, in the sense that the $n$-th order equation can be solved for for $x_n$ given only the lower-order solutions $x_{m<n}$.
This is what makes the perturbation theory so attractive: we can simply start with the trivial solution $x_0 = 1$, then solve for the $x_n$ order-by-order by simply substituting the already obtained solutions $x_{m<n}$ at each step. Let us write a function that solves a general equation `eq` for the variable `x` with this *cascading* process:
```@example perturb
function collect_powers(eq, x, ns)
eq = expand(eq)
[Symbolics.coeff(eq, x^i) for i in ns]
end
```

To return the coefficients of $ϵ$ and $ϵ^2$ in `eq`, we can write
function solve_perturbed(eq, x, x₀, ϵ, order)
x_taylor = series(x, ϵ, 0:order) # expand unknown in a taylor series
x_coeffs = taylor_coeff(x_taylor, ϵ, 0:order) # array of coefficients
eq_taylor = substitute(eq, x => x_taylor) # expand equation in taylor series
eqs = taylor_coeff(eq_taylor, ϵ, 0:order) # separate into order-by-order equations
```@example perturb
eqs = collect_powers(eq, ϵ, 1:2)
```

Having the coefficients of the powers of `ϵ`, we can set each equation in `eqs` to 0 (remember, we rearrange the problem such that `eq` is 0) and solve the system of linear equations to find the numerical values of the coefficients. **Symbolics.jl** has a function `symbolic_linear_solve` that can solve systems of linear equations. However, the presence of higher-order terms in `eqs` prevents `symbolic_linear_solve(eqs, a)` from workings properly. Instead, we can exploit the fact that our system is in a triangular form and start by solving `eqs[1]` for `a₁` and then substitute this in `eqs[2]` and solve for `a₂`, and so on. This *cascading* process is done by function `solve_coef(eqs, ps)`:

```@example perturb
function solve_coef(eqs, ps)
vals = Dict()
for i = 1:length(ps)
eq = substitute(eqs[i], vals)
vals[ps[i]] = symbolic_linear_solve(eq, ps[i])
sol = [x_coeffs[1] => x₀] # store solutions in a symbolic-numeric map
# solve equations order-by-order
for (eq, x_coeff) in zip(eqs[2:end], x_coeffs[2:end])
x_coeff_expr = Symbolics.symbolic_linear_solve(eq, x_coeff) # solve linear n-th order equation for x_n (in terms of lower n)
x_coeff_val = substitute(x_coeff_expr, sol) # substitute lower-order solutions to get numerical value
sol = [sol; x_coeff => x_coeff_val] # store solution
end
vals
end
```

Here, `eqs` is an array of expressions (assumed to be equal to 0) and `ps` is an array of variables. The result is a dictionary of *variable* => *value* pairs. We apply `solve_coef` to `eqs` to get the numerical values of the parameters:

```@example perturb
vals = solve_coef(eqs, a)
```

Finally, we substitute back the values of `a` in the definition of `x` as a function of `𝜀`. Note that `𝜀` is a number (usually Float64), whereas `ϵ` is a symbolic variable.
```@example perturb
X = 𝜀 -> 1 + vals[a[1]]*𝜀 + vals[a[2]]*𝜀^2
```

Therefore, the solution to our original problem becomes `X(1)`, which is equal to 0.76. We can use larger values of `n` to improve the accuracy of estimations.

| n | x |
|---|----------------|
|1 |0.8 |
|2 |0.76|
|3 |0.752|
|4 |0.752|
|5 |0.7533|
|6 |0.7543|
|7 |0.7548|
|8 |0.7550|

Remember, the numerical value is 0.7549. The two functions `collect_powers` and `solve_coef(eqs, a)` are used in all the examples in this and the next tutorial.

## Solving the Kepler's Equation

Historically, the perturbation methods were first invented to solve orbital calculations of the Moon and the planets. In homage to this history, our second example has a celestial theme. Our goal is solving the Kepler's equation:
return substitute(x_taylor, sol) # evalaute series with solved coefficients
end
```math
E - e\sin(E) = M
x_sol = solve_perturbed(quintic, x, 1, ϵ, 8)
```

where $e$ is the *eccentricity* of the elliptical orbit, $M$ is the *mean anomaly*, and $E$ (unknown) is the *eccentric anomaly* (the angle between the position of a planet in an elliptical orbit and the point of periapsis). This equation is central to solving two-body Keplerian orbits.

Similar to the first example, it is easy to solve this problem using the Newton's method. For example, let $e = 0.01671$ (the eccentricity of the Earth) and $M = \pi/2$. We have `solve_newton(x - e*sin(x) - M, x, M)` equals to 1.5875 (compared to π/2 = 1.5708). Now, we try to solve the same problem using the perturbation techniques (see function `test_kepler`).

For $e = 0$, we get $E = M$. Therefore, we can use $e$ as our perturbation parameter. For consistency with other problems, we also rename $e$ to $\epsilon$ and $E$ to $x$.

From here on, we use the helper function `def_taylor` to define Taylor's series by calling it as `x = def_taylor(ϵ, a, 1)`, where the arguments are, respectively, the perturbation variable, which is an array of coefficients (starting from the coefficient of $\epsilon^1$), and an optional constant term.

The $n$-th order solution of the quintic equation is the sum of this series up to the $\epsilon^n$-th order term, evaluated at $\epsilon=1$ (to recover the original quintic):
```@example perturb
def_taylor(x, ps) = sum([a*x^i for (i,a) in enumerate(ps)])
def_taylor(x, ps, p₀) = p₀ + def_taylor(x, ps)
for n in 0:8
println("$n-th order solution: x = ", substitute(taylor(x_sol, ϵ, 0:n), ϵ => 1.0))
end
```
This is close to the solution from Newton's method!

We start by defining the variables (assuming `n = 3`):
## Solving Kepler's Equation

Historically, perturbation methods were first invented to calculate orbits of the Moon and the planets. In homage to this history, our second example is to solve Kepler's equation
```@example perturb
n = 3
@variables ϵ M a[1:n]
x = def_taylor(ϵ, a, M)
@variables e E M
kepler = E - e * sin(E) ~ M
```
Here $e$ is the *eccentricity* of the elliptical orbit, $M$ is the *mean anomaly*, and $E$ (unknown) is the *eccentric anomaly* (the angle between the position of a planet in an elliptical orbit and the point of periapsis). This equation is central to solving two-body Keplerian orbits.

We further simplify by substituting `sin` with its power series using the `expand_sin` helper function:

Similar to the first example, it is easy to solve this problem with Newton's method. For example, with Earth's eccentricity $e = 0.01671$ and $M = \pi/2$:
```@example perturb
expand_sin(x, n) = sum([(isodd(k) ? -1 : 1)*(-x)^(2k-1)/factorial(2k-1) for k=1:n])
solve_newton(substitute(kepler, Dict(e => 0.01671, M => π/2)), E, π/2)
```

To test,

Next, let us solve the same problem with our perturbative solver. For Kepler's equation, it is most common to expand in $M$ (when $M=0$, the trivial solution is $E=0$):
```@example perturb
expand_sin(0.1, 10) ≈ sin(0.1)
E_sol = solve_perturbed(kepler, E, 0, M, 5)
```

The problem equation is

Numerically, we get almost the same answer as with Newton's method:
```@example perturb
eq = x - ϵ * expand_sin(x, n) - M
substitute(E_sol, Dict(e => 0.01671, M => π/2))
```

We follow the same process as the first example. We collect the coefficients of the powers of `ϵ`

As we see, however, the power of perturbation theory is that it also gives us the full symbolic series solution for $E$ (*before* numbers for $e$ and $M$ are inserted). Upon inspection, we see that our series matches [the result from Wikipedia](https://en.wikipedia.org/wiki/Kepler%27s_equation#Inverse_Kepler_equation):
```@example perturb
eqs = collect_powers(eq, ϵ, 1:n)
E_wiki = 1/(1-e)*M - e/(1-e)^4 * M^3/factorial(3) + (9e^2+e)/(1-e)^7 * M^5/factorial(5)
```

and then solve for `a`:

Alternatively, it is possible to solve Kepler's equation by expanding in $e$ instead of $M$ (when $e = 0$, the trivial solution is $E=M$):
```@example perturb
vals = solve_coef(eqs, a)
E_sol′ = solve_perturbed(kepler, E, M, e, 5)
```

Finally, we substitute `vals` back in `x`:

We can expand the trigonometric functions in $M$, and see that the combined $e$-$M$ series matches the result from Wikipedia at least up to order $e^5 M^5$:
```@example perturb
x′ = substitute(x, vals)
X = (𝜀, 𝑀) -> substitute(x′, Dict(ϵ => 𝜀, M => 𝑀))
X(0.01671, π/2)
```

The result is 1.5876, compared to the numerical value of 1.5875. It is customary to order `X` based on the powers of `𝑀` instead of `𝜀`. We can calculate this series as `collect_powers(x′, M, 0:5)`. The result (after manual cleanup) is

E_sol′ = taylor(E_sol′, M, 0:5)
E_wiki′ = taylor(taylor(E_wiki, e, 0:5), M, 0:5)
isequal(E_wiki′, E_sol′)
```
(1 + 𝜀 + 𝜀^2 + 𝜀^3)*𝑀
- (𝜀 + 4*𝜀^2 + 10*𝜀^3)*𝑀^3/6
+ (𝜀 + 16*𝜀^2 + 91*𝜀^3)*𝑀^5/120
```

Comparing the formula to the one for 𝐸 in the [Wikipedia article on the Kepler's equation](https://en.wikipedia.org/wiki/Kepler%27s_equation):

```math
E = \frac{1}{1-\epsilon}M
-\frac{\epsilon}{(1-\epsilon)^4} \frac{M^3}{3!} + \frac{(9\epsilon^2
+ \epsilon)}{(1-\epsilon)^7}\frac{M^5}{5!}\cdots
```

The first deviation is in the coefficient of $\epsilon^3 M^5$.

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