Step by Step solution about How to resolve the algorithm Continued fraction step by step in the Go programming language

The provided code is written in Go and implements continued fraction (CF) expansions for approximating the values of sqrt(2), Napier's constant (Napier), and pi. Continued fractions represent real numbers as an infinite series of fractions, providing a way to approximate irrational numbers.

Here's a breakdown of the code:

1. Data Structures:

   • cfTerm: A struct representing a term in a continued fraction, consisting of two integers a and b.
   • cf: An array or slice of cfTerm structs, representing a continued fraction.

2. Continued Fraction Expansions:

   • cfSqrt2, cfNap, cfPi: These functions create and initialize continued fractions for approximating sqrt(2), Napier's constant, and pi, respectively. They set the initial terms based on specific formulas for each constant.

3. Real Value Calculation:

   • (f cf) real(): This method calculates the real (approximate) value of the continued fraction f. It uses an iterative process, starting with the last term and working backward to compute the fractional value.

4. Main Function:

   • In the main function, continued fractions are created for sqrt(2), Napier's constant, and pi, and their real values are printed to the console.

Here's how the code works:

• Continued Fraction Initialization: The cfSqrt2, cfNap, and cfPi functions initialize the continued fractions with appropriate terms based on formulas for each constant.

• Real Value Calculation: The real method of the cf type computes the real value of the continued fraction using an iterative algorithm. It starts with the last term and calculates each fractional value backward, resulting in an approximation of the real number.

• Main Function: In the main function, continued fractions for sqrt(2), Napier's constant, and pi are created with 20 terms each. Their real values are calculated and printed to the console.

This code provides a simplified implementation of continued fractions for approximating irrational numbers. By using a fixed number of terms, it provides a finite approximation of the constants sqrt(2), Napier's constant, and pi.

package main

import "fmt"

type cfTerm struct {
    a, b int
}

// follows subscript convention of mathworld and WP where there is no b(0).
// cf[0].b is unused in this representation.
type cf []cfTerm

func cfSqrt2(nTerms int) cf {
    f := make(cf, nTerms)
    for n := range f {
        f[n] = cfTerm{2, 1}
    }
    f[0].a = 1
    return f
}

func cfNap(nTerms int) cf {
    f := make(cf, nTerms)
    for n := range f {
        f[n] = cfTerm{n, n - 1}
    }
    f[0].a = 2
    f[1].b = 1
    return f
}

func cfPi(nTerms int) cf {
    f := make(cf, nTerms)
    for n := range f {
        g := 2*n - 1
        f[n] = cfTerm{6, g * g}
    }
    f[0].a = 3
    return f
}

func (f cf) real() (r float64) {
    for n := len(f) - 1; n > 0; n-- {
        r = float64(f[n].b) / (float64(f[n].a) + r)
    }
    return r + float64(f[0].a)
}

func main() {
    fmt.Println("sqrt2:", cfSqrt2(20).real())
    fmt.Println("nap:  ", cfNap(20).real())
    fmt.Println("pi:   ", cfPi(20).real())
}