學海無涯，思境無維；數乃六藝，理之始也。
或有一得足矣 愚千慮

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Newton's Method

$$\begin{gathered} f(z)=z^2⟹f^{\prime }(z)=2z \\ \mathrm{\Delta }y=z^2-x \\ {f}^{\prime }(z)=2z=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }z}⟹\mathrm{\Delta }z=\frac{\mathrm{\Delta }y}{2z} \\ {z}_{n+1}=z_n-\mathrm{\Delta }z \end{gathered}$$

package main

import (
	"fmt"
)

// x is the number we want to know it's sqaure root.
func Sqrt(x float64) float64 {

	z := 2.0 // z is the initial value we guess z^2 approches x

	for i := 0; i < 10; i++ {

		derivative := 2 * z  // since f(z) = z^2, then f'(z) = 2 * z
		delta_y := z * z - x
		delta_z := delta_y / derivative
		z = z - delta_z
		
		// Above codes are equal to the following one line code
		// z -= (z * z - x) / (2 * z)
		fmt.Println(z)
	}
	
	return z
}

func main() {
	
	fmt.Println(Sqrt(2))
}

tags: student homework python