Solving Polynomial Equations using Julia

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.title[
# Solving Polynomial Equations using Julia
]
.subtitle[
## Julia Workshop
]

---

Julia offers several ways to solve polynomial equations, each with its strengths and weaknesses. Here's a breakdown of the common methods and when to use them:

**1. Using the `roots` Function (For Polynomials in Standard Form):**

The `roots` function from the `Polynomials` package is the most straightforward way to find all roots (including complex roots) of a polynomial expressed in standard form (e.g., `ax^2 + bx + c`).

---

```julia
using Polynomials

# Define the polynomial (e.g., x^2 - 5x + 6)
p = Polynomial([6, -5, 1])  # Coefficients in ascending order (constant term first)

# Find the roots
r = roots(p)

println(r)  # Output: [2.0, 3.0]

# Example with complex roots: x^2 + 1
p_complex = Polynomial([1, 0, 1])
r_complex = roots(p_complex)
println(r_complex) # Output: [-1.0im, 1.0im]
```

---

**Key Points about `roots`:**

*   **Standard Form:** The polynomial must be in standard form (coefficients arranged in ascending order of powers).
*   **All Roots:** It finds all roots, including real and complex roots.
*   **Numerical:** The roots are found numerically.
*   **Package:** You need to install the `Polynomials` package: `] add Polynomials`

**2. Using `solve` (For Symbolic Solutions - Limited):**

The `solve` function from the `Symbolics` package can sometimes find symbolic solutions to polynomial equations.  However, symbolic solutions are often only possible for relatively simple polynomials.  For higher-degree polynomials or more complex equations, `solve` might return a symbolic representation of the roots or fail to find a closed-form solution.

---

```julia
using Symbolics

@variables x

# Define the equation (e.g., x^2 - 5x + 6 = 0)
eq = x^2 - 5x + 6 ~ 0  # ~ is used for defining equations

# Solve the equation
sol = solve(eq, x)

println(sol)  # Output: [2, 3]

# Example where symbolic solutions may not be feasible
eq_complex = x^5 + x + 1 ~ 0
sol_complex = solve(eq_complex, x)
println(sol_complex) # Symbolic representation of roots, likely not a closed-form solution.
```

---

**Key Points about `solve`:**

*   **Symbolic:** It attempts to find symbolic solutions.
*   **Limited:** Symbolic solutions are not always possible.
*   **Package:** You need to install the `Symbolics` package: `] add Symbolics`

**3. Numerical Solvers (For General Equations):**

For equations that are not strictly polynomials or when symbolic solutions are not feasible, you can use numerical solvers from packages like `NLsolve` or `Roots`.  These solvers can find approximate solutions to a wide range of equations.

---

### using `NLsolve`

```julia
using NLsolve

# Example: Find a root of x^3 - 2x - 5 = 0 (Newton-Raphson method)
f(x) = x^3 - 2x - 5
df(x) = 3x^2 - 2 # Derivative

result = nlsolve((x) -> (f(x), df(x)), 3.0)  # Start with an initial guess (3.0)
root = result.zero

println(root)
```

---

### using `Roots`

```julia

# Using Roots.jl for polynomial equations.

using Roots
f(x) = x^3 - 2x - 5

root = find_zero(f, 3.0) # Find a zero near x = 3
println(root)

#Find all roots using polynomial interpolation.
f_poly(x) = Polynomial([5,2,0,1]) # Coefficients in ascending order.
roots_poly = find_zeros(f_poly,-3.0,3.0)
println(roots_poly)

```
---

**Key Points about Numerical Solvers:**

*   **Numerical:** They find approximate solutions.
*   **General:** They can handle a wider range of equations, not just polynomials.
*   **Initial Guess:** Many numerical solvers require an initial guess for the solution.
*   **Packages:** You might need to install packages like `NLsolve` (`] add NLsolve`) or `Roots` (`] add Roots`).

**Which Method to Choose:**

*   **`roots` (from `Polynomials`):**  Best for finding all roots of polynomials in standard form.
*   **`solve` (from `Symbolics`):** Use this if you need symbolic solutions and the polynomial is simple enough.
*   **Numerical Solvers (`NLsolve`, `Roots`):** Use these for general equations (including polynomials) when symbolic solutions are not available or practical.  `Roots` is particularly well-suited for finding zeros of functions, including polynomials.

---

### Summary

For most undergraduate probability problems involving polynomials, the `roots` function from the `Polynomials` package will be the most convenient and efficient.  If you have more complex equations or need symbolic solutions, then `solve` or numerical solvers would be more appropriate.