Step by Step solution about How to resolve the algorithm Roots of a function step by step in the Ruby programming language

The provided Ruby code defines two methods for finding roots of a function within a given range.

1. First Method: find_roots(f, range, step)

• f: A function represented as a lambda or proc.
• range: The range within which to search for roots.
• step: The step size for iterating through the range (optional, defaults to 0.001).

How it Works:

1. It calculates the sign of the function at the start of the range (range.begin), which is either -1, 0, or 1.
2. It iterates through the range with the specified step size.
3. For each value x, it calculates the value of the function at that point.
4. If the function value is 0, it prints "Root found at #{x}".
5. If the sign of the function value changes from the previous sign, it prints "Root found between #{x-step} and #{x}".
6. It updates the sign of the function value.

2. Second Method: find_roots(range, step)

• range: The range within which to search for roots.
• step: The step size for iterating through the range (optional, defaults to 1e-3).

How it Works:

1. It defines a Numeric extension that adds a sign method to numeric values.
2. It iterates through the range with the specified step size and applies the given block to each value.
3. The block should evaluate the function at that value and return the result.
4. For each value x, it calculates the sign of the function value.
5. If the function value is 0, it prints "Root found at #{x}".
6. If the sign of the function value changes from the previous sign, it prints "Root found between #{x-step} and #{x}".
7. It returns the sign of the function value, which is used to detect sign changes in the next iteration.

Usage Example:

The following code snippet shows an example of using the find_roots method:

f = lambda { |x| x**3 - 3*x**2 + 2*x }
find_roots(f, -1..3)

This code will find the roots of the function f(x) = x^3 - 3x^2 + 2x within the range [-1, 3] and print the results.

Note:

• Both methods assume that the function is continuous within the given range.
• The step size can be adjusted to improve accuracy and performance.
• The detection of roots between two points is approximate and may not be precise.

def sign(x)
  x <=> 0
end

def find_roots(f, range, step=0.001)
  sign = sign(f[range.begin])
  range.step(step) do |x|
    value = f[x]
    if value == 0
      puts "Root found at #{x}"
    elsif sign(value) == -sign
      puts "Root found between #{x-step} and #{x}"
    end
    sign = sign(value)
  end
end

f = lambda { |x| x**3 - 3*x**2 + 2*x }
find_roots(f, -1..3)


class Numeric
  def sign
    self <=> 0
  end
end

def find_roots(range, step = 1e-3)
  range.step( step ).inject( yield(range.begin).sign ) do |sign, x|
    value = yield(x)
    if value == 0
      puts "Root found at #{x}"
    elsif value.sign == -sign
      puts "Root found between #{x-step} and #{x}"
    end
    value.sign
  end
end

find_roots(-1..3) { |x| x**3 - 3*x**2 + 2*x }