Mixed Symbolic-Numeric Perturbation Theory

Background

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 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 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.

We will discuss the general steps of the perturbation methods to solve algebraic (this tutorial) and differential equations

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:

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)

    for i = 1:maxiter
        xₙ₊₁ = substitute(fₙ₊₁, Dict(x => xₙ))
        if abs(xₙ₊₁ - xₙ) < abstol
            return xₙ₊₁
        else
            xₙ = xₙ₊₁
        end
    end
    return xₙ₊₁
end

solve_newton (generic function with 1 method)

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 $\epsilon$:

\[ x(\epsilon) = a_0 + a_1 \epsilon + a_2 \epsilon^2 + O(\epsilon^3)\]

\[a_0\]

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

\[ (a_0 + a_1 \epsilon + a_2 \epsilon^2)^5 + \epsilon (a_0 + a_1 \epsilon + a_2 \epsilon^2) - 1 = 0\]

Expanding the equations, we get

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

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

\[ 1 + 5 a_1 = 0\]

and

\[ a_1 + 5 a_2 + 10 a_1^2 = 0\]

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 $a_2$:

\[ a_2 = \frac{(-\frac{1}{5} + 10(-(\frac{1}{5})²)}{5} = -\frac{1}{25}\]

Finally,

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

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:

n = 2
@variables ϵ a[1:n]

2-element Vector{Any}:
 ϵ
  a[1:2]

Then, we define

x = 1 + a[1]*ϵ + a[2]*ϵ^2

\[ \begin{equation} 1 + a_{1} \epsilon + \epsilon^{2} a_{2} \end{equation} \]

The next step is to substitute x in the problem equation

  eq = x^5 + ϵ*x - 1

\[ \begin{equation} -1 + \left( 1 + a_{1} \epsilon + \epsilon^{2} a_{2} \right) \epsilon + \left( 1 + a_{1} \epsilon + \epsilon^{2} a_{2} \right)^{5} \end{equation} \]

The expanded form of eq is

expand(eq)

\[ \begin{equation} \epsilon + 5 a_{1} \epsilon + \epsilon^{2} a_{1} + 5 \epsilon^{2} a_{2} + 10 \epsilon^{2} a_{1}^{2} + \epsilon^{3} a_{2} + 20 \epsilon^{3} a_{1} a_{2} + 10 \epsilon^{3} a_{1}^{3} + 10 \epsilon^{4} a_{2}^{2} + 30 \epsilon^{4} a_{1}^{2} a_{2} + 5 \epsilon^{4} a_{1}^{4} + 30 \epsilon^{5} a_{2}^{2} a_{1} + 20 \epsilon^{5} a_{1}^{3} a_{2} + 10 \epsilon^{6} a_{2}^{3} + \epsilon^{5} a_{1}^{5} + 30 \epsilon^{6} a_{2}^{2} a_{1}^{2} + 5 \epsilon^{6} a_{1}^{4} a_{2} + 20 \epsilon^{7} a_{2}^{3} a_{1} + 10 \epsilon^{7} a_{2}^{2} a_{1}^{3} + 5 \epsilon^{8} a_{2}^{4} + 10 \epsilon^{8} a_{2}^{3} a_{1}^{2} + 5 \epsilon^{9} a_{2}^{4} a_{1} + \epsilon^{10} a_{2}^{5} \end{equation} \]

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.

function collect_powers(eq, x, ns; max_power=100)
    eq = substitute(expand(eq), Dict(x^j => 0 for j=last(ns)+1:max_power))

    eqs = []
    for i in ns
        powers = Dict(x^j => (i==j ? 1 : 0) for j=1:last(ns))
        push!(eqs, substitute(eq, powers))
    end
    eqs
end

collect_powers (generic function with 1 method)

To return the coefficients of $ϵ$ and $ϵ^2$ in eq, we can write

eqs = collect_powers(eq, ϵ, 1:2)

2-element Vector{Any}:
                 1 + 5a[1]
 a[1] + 5a[2] + 10(a[1]^2)

A few words on how collect_powers works, It uses substitute to find the coefficient of a given power of x by passing a Dict with all powers of x set to 0, except the target power which is set to 1. For example, the following expression returns the coefficient of ϵ^2 in eq,

substitute(expand(eq), Dict(
  ϵ => 0,
  ϵ^2 => 1,
  ϵ^3 => 0,
  ϵ^4 => 0,
  ϵ^5 => 0,
  ϵ^6 => 0,
  ϵ^7 => 0,
  ϵ^8 => 0)
)

\[ \begin{equation} a_{1} + 5 a_{2} + 10 a_{1}^{2} \end{equation} \]

Back to our problem. 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 Symbolics.solve_for that can solve systems of linear equations. However, the presence of higher-order terms in eqs prevents Symbolics.solve_for(eqs .~ 0, 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):

function solve_coef(eqs, ps)
    vals = Dict()

    for i = 1:length(ps)
        eq = substitute(eqs[i], vals)
        vals[ps[i]] = Symbolics.solve_for(eq ~ 0, ps[i])
    end
    vals
end

solve_coef (generic function with 1 method)

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:

vals = solve_coef(eqs, a)

Dict{Any, Any} with 2 entries:
  a[2] => -0.04
  a[1] => -0.2

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.

X = 𝜀 -> 1 + vals[a[1]]*𝜀 + vals[a[2]]*𝜀^2

#7 (generic function with 1 method)

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.

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:

\[ E - e\sin(E) = M\]

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.

def_taylor(x, ps) = sum([a*x^i for (i,a) in enumerate(ps)])
def_taylor(x, ps, p₀) = p₀ + def_taylor(x, ps)

def_taylor (generic function with 2 methods)

We start by defining the variables (assuming n = 3):

n = 3
@variables ϵ M a[1:n]
x = def_taylor(ϵ, a, M)

\[ \begin{equation} M + a_{1} \epsilon + \epsilon^{2} a_{2} + \epsilon^{3} a_{3} \end{equation} \]

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

expand_sin(x, n) = sum([(isodd(k) ? -1 : 1)*(-x)^(2k-1)/factorial(2k-1) for k=1:n])

expand_sin (generic function with 1 method)

To test,

expand_sin(0.1, 10) ≈ sin(0.1)

true

The problem equation is

eq = x - ϵ * expand_sin(x, n) - M

\[ \begin{equation} a_{1} \epsilon + \epsilon^{2} a_{2} + \epsilon^{3} a_{3} - \left( M + a_{1} \epsilon + \epsilon^{2} a_{2} + \epsilon^{3} a_{3} + \frac{1}{6} \left( - M - a_{1} \epsilon - \epsilon^{2} a_{2} - \epsilon^{3} a_{3} \right)^{3} - \frac{1}{120} \left( - M - a_{1} \epsilon - \epsilon^{2} a_{2} - \epsilon^{3} a_{3} \right)^{5} \right) \epsilon \end{equation} \]

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

eqs = collect_powers(eq, ϵ, 1:n)

3-element Vector{Any}:
                                                          -M + a[1] + (1//6)*(M^3) - (1//120)*(M^5)
                                              -a[1] + a[2] + (1//2)*(M^2)*a[1] - (1//24)*(M^4)*a[1]
 -a[2] + a[3] + (1//2)*(M^2)*a[2] + (1//2)*M*(a[1]^2) - (1//24)*(M^4)*a[2] - (1//12)*(M^3)*(a[1]^2)

and then solve for a:

vals = solve_coef(eqs, a)

Dict{Any, Any} with 3 entries:
  a[2] => M - (1//6)*(M^3) + (1//120)*(M^5) - (1//2)*(M^2)*(M - (1//6)*(M^3) + …
  a[3] => M - (1//6)*(M^3) + (1//120)*(M^5) - (1//2)*(M^2)*(M - (1//6)*(M^3) + …
  a[1] => M - (1//6)*(M^3) + (1//120)*(M^5)

Finally, we substitute vals back in x:

x′ = substitute(x, vals)
X = (𝜀, 𝑀) -> substitute(x′, Dict(ϵ => 𝜀, M => 𝑀))
X(0.01671, π/2)

\[ \begin{equation} 1.5876 \end{equation} \]

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(sol, M, 0:3). The result (after manual cleanup) is

(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:

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

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