# L-5 MCS 471 Wed 31 Aug 2022 : newtonupdate.jl

"""
This script illustrates the definition of the update function

       dx = - f(x)/f'(x) 

for any function in x with SymPy,
to run one step with Newton's method x = x + dx.
"""

using SymPy
x = Sym("x")

"""
Returns the string representation of
the derivative of the expression in x,
given in the string s.
"""
function SymPyDerivative(s::String)
    evaluated = sympify(s)
    derivative = diff(evaluated, x)
    return string(derivative)
end

"""
Given a string representation of an expression in x,
returns the Julia function for the update -f(x)/f'(x),
where f(x) is the function defined by the input string.
"""
function NewtonUpdate(f::String, verbose::Bool=true)
    sdf = SymPyDerivative(f)
    sdx = string("-(", f, ")/(", sdf, ")")
    if verbose
        println("update : $sdx")
    end
    edx = sympify(sdx)
    return lambdify(edx)
end

# a simple test
dx = NewtonUpdate("x^3 + x - 1")
println("dx(1.0) : ", dx(1.0))