# An Iterative Hybrid Algorithm for Roots of Non-Linear Equations

## Abstract

**:**

## 1. Introduction

## 2. Background, Definitions

_{k}$-$ x

_{k−1}| + |f(x

_{k})|) < $\epsilon $, and upper bound on iterations as 100: for function f:[a, b] → R, such that

**Definition**

**(Convergence)**

**Definition**(

**Order of Convergence**).

_{n,}and $\alpha $ ∈ R, n ≥ 0, sequence {x

_{n}} converge to $\alpha $.

_{n}} is said to converge to $\alpha $ with convergence order p, and C is the asymptotic error constant [10,12,18,19].

_{n}= $\alpha $ + e

_{n}, the error equation becomes

_{n+1}= C e

_{n}

^{p}+ O(e

_{n}

^{p+1})

**Definition**

**(Computational**

**Order**

**of**

**Convergence).**

**Definition**(

**Efficiency Index**).

#### 2.1. Bisection Method

#### 2.2. False Position (Regula Falsi) Method

#### 2.3. Dekker’s Method

_{k}} is the sequence of best estimates, and the root enclosing interval is always [a

_{k}, b

_{k}] or [b

_{k}, a

_{k}]. Dekker’s algorithm maintain three values: b is the current best approximation, c is the previous approximation b

_{k−1}, and a is the contrapoint so that the root lies in the interval [a

_{k}, b

_{k}]$\text{}{\displaystyle \cup}\text{}$[b

_{k}, a

_{k}]. Both the secant point, s, and midpoint, m, are computed; b

_{k}is s or m, whichever is closest to b

_{k−1}. Though it is better than the False position method, nonetheless, it has some road blocks, handled by Brent.

#### 2.4. Brent’s Algorithm

#### Detour to Reverse Quadratic Interpolation

_{0}. The simplest such technique is the Newton–Raphson method, which is slower than its variations. These algorithms outperform the conventional Newton–Raphson method. The variations include decomposition based [16], quadrature based, and undetermined coefficients based [11,13,14]. These methods are quite complex and detailed. The reader may wish to refer to the full papers for details. For completeness, we describe the iteration formulas to show how these methods iterate to get to the root.

#### 2.5. Newton-Raphson (1760)

^{2}), where $\u03f5$= |${x}_{n+1}-{x}_{n}$|.

#### 2.6. Oghovese-John Method (2014)

_{n+1}, are defined in terms of ${\mathrm{v}}_{\mathrm{n}}$ instead of ${\mathrm{x}}_{\mathrm{n}}$

_{n+1}=${\text{}\mathrm{x}}_{\mathrm{n}}\u2013\text{}\frac{\mathrm{f}({\mathrm{x}}_{\mathrm{n}})}{\text{}\mathrm{f}{}^{\prime}({\mathrm{x}}_{\mathrm{n}})}$

#### 2.7. Grau-Diaz-Barero (2006)

#### 2.8. Sharma-Guha (2007)

#### 2.9. Khattri-Abbasbandy (2011)

#### 2.10. Fang-Chen-Tian-Sun-Chen (2011)

_{n}, b

_{n}, c

_{n}are real numbers chosen in such a way that 0 ≤ |a

_{n}|, |b

_{n}|, |c

_{n}| ≤ 1, and sign(a

_{n}f(x

_{n})) = sign(f′(x

_{n})), sign(b

_{n}f(y

_{n})) = sign(f′(y

_{n})),

_{n}f(Z

_{n})) = sign(f′(z

_{n})),

#### 2.11. Jayakumar (2013)

#### 2.12. Nora-Imran-Syamsudhuha (2018)

^{2}− 3ab + a

^{2}), a = z

_{n}− x

_{n}, b = y

_{n}− x

_{n}.

#### 2.13. Weerakon et al. (2000)

#### 2.14. Edmond Halley (1995)

^{3}, three times continuously differentiable. The root iterations have cubic convergence. The function f(x) is expanded to approximate the quadratic in two ways and cancelling the second degree term to arrive at the linear formula

_{n+1}) = f(x

_{n}) + (x

_{n+1}− x

_{n})f′(x

_{n}) +O((x

_{n+1}− x

_{n})

^{2})

_{n+1}− x

_{n})f″(x

_{n}), (29) by 2f′(x

_{n}),

Method | Year | Name in Table 2, Table 3, Table 4 and Table 5 | EFF |
---|---|---|---|

Newton–Rapson | 1760 | MN_R | 1.4142 |

Edmond Halley | 1995 | NHAL | 1.4422 |

Weerkoon–Fernando | 2000 | NWFhm3 | 1.4310 |

Grau–Diaz–Barero | 2006 | MGDB | 1.5651 |

Sharma–Guha | 2007 | MSG | 1.5651 |

Fang–Chen–Tian–Sun–Chen | 2011 | FCTSC | 1.3480 |

Khattri–Abbasbandy | 2011 | MKA | 1.4860 |

Jayakumar | 2013 | MJKs | 1.3161 |

Jayakumar | 2013 | MJKhs | 1.3161 |

Oghovese–John | 2014 | MOJmis | 1.2599 |

Nora–Imran–Syamsudhuha | 2018 | MNIS | 1.5651 |

Hybrid Method | 2021 | Hybridn | 1.5874 |

## 3. Three Way Hybrid Algorithm

#### Hybrid Algorithm

_{k+1}, b

_{k+1}]

_{1}= a, b

_{1}= b

_{k+1}= ba

_{k+1}= a

_{1}; fb

_{k+1}= bb

_{k+1}= b

_{1}

_{1}= a

_{1}− f(a

_{1})/$\text{}\mathrm{f}{}^{\prime}$(a

_{1});

_{k+1}= ba

_{k+1}= a

_{k}; fb

_{k+1}= bb

_{k+1}= b

_{k}

_{k+1}= n

_{k}− f(n

_{k})/f′(n

_{k})

_{=}$\frac{{a}_{k}+{b}_{k}}{2}$, and ∈

_{m =}|f(m)|

_{k}− $\frac{f({a}_{k})({b}_{k}-{a}_{k})}{f({b}_{k})-f({a}_{k})}$− and ∈

_{f}= |f(s)|

_{a}= ∈

_{f}

_{k+1}, bb

_{k+1}]

_{m}= f(m)

_{a}= ∈

_{m}

_{k})·f(r) > 0,

_{k+1}= r; bb

_{k+1}= b

_{k};

_{k+1}= a

_{k}; bb

_{k+1}= r;

_{k+1}, fb

_{k+1}]

_{f}= f(s)

_{a}= ∈

_{f}

_{k})·f(r) > 0,

_{k+1}= r; fb

_{k+1}= b

_{k};

_{k+1}= a

_{k}; fb

_{k+1}= r;

_{k+1}, bb

_{k+1}] and [fa

_{k+1}, fb

_{k+1}], define

_{k+1}, b

_{k+1}] = [ba

_{k+1}, bb

_{k+1}] ∩ [fa

_{k+1}, fb

_{k+1}]

_{k+1}= max(ba

_{k+1}, fa

_{k+1});

_{k+1}= min(bb

_{k+1}, fb

_{k+1});

_{k+1}if f(n

_{k+1}) < min(f(a

_{k+1}), f(b

_{k+1}))

_{k+1.}or b

_{k+1}by n

_{k+1}resulting in further smaller interval, with new root r = n

_{k+1}

_{k+1}= r

_{a}= |f(r

_{k})|+|b

_{k}− a

_{k}|

_{a}< ∈ or k > maxIterations

## 4. Convergence of Hybrid Algorithm

_{s}, is used as the stopping criteria. For the Bisection method, on interval [a, b], the upper bound n

_{b}(∈) on the number of iterations can be found from $\frac{b-a}{{2}^{n}}$ < ${\in}_{s}$ and is lg ((b − a)/∈

_{s}). Since e

_{n+1}= 1/2 e

_{n}, it has linear convergence. For the False Position method, it depends on the location of the root near the endpoint of the bracketing interval and the convexity of the function. Thus, the bound n

_{f}(∈

_{s}) for the number of iterations for the False Position method cannot be predetermined, it can be less, n

_{f}(∈

_{s}) < n

_{b}(∈) = lg ((b − a)/∈

_{s}), or can be greater, n

_{f}(∈

_{s}) > n

_{b}(∈

_{s}) = lg ((b − a)/∈

_{s}). The number of iterations, n(∈

_{s}), in the hybrid algorithm is less than min(n

_{f}(∈

_{s}), n

_{b}(∈

_{s})). The introduction of the Newton–Raphson in the Hybrid further reduces the complexity of computations, resulting in fewer iterations. The Newton–Raphson algorithm of the quadratic order of convergence is given in Section 2.5: The convergence analysis of Newton is trivial; however, for completeness it is as follows.

**Proposition.**

_{0}is sufficiently close to$\alpha $, then the Newton-Raphson as defined in Section 2.5 has second order convergence.

**Proof.**

_{n}is the approximation, then f($\alpha $) = 0, x

_{n}= $\alpha $ + e

_{n}with error e

_{n}. Let ${c}_{k}$ = $\frac{1}{k!}\frac{{f}^{(k)}(\alpha )}{{f}^{\prime}(\alpha )}$, then by Taylor series

## 5. Empirical Results of Simulations

^{3}, the initial interval value is [0.5, 1]; in Table 4, the function is 0.7x

^{5}− 8x

^{4}+ 44x

^{3}− 90x

^{2}+ 82x − 25, the initial interval value is [0, 1], and in Table 5, the function is x

^{3}+ log(x); the initial interval value is [0.1, 2].

**Table 2.**

**One Parameter.**Comparison of overall number of function evaluations (NOFE) in hybrid algorithm and all other algorithms on all functions for the number of function evaluations required for the solution.

Function | x^{3} − x^{2} − x − 1 | exp(x) + x − 6 | x^{3} + log(x) | sin(x) − x/2 | (x − 1)^{3} − 1 | x^{3} + 4x^{2} − 10 | (x − 2)^{23} −1 | sin(x)^{2} − x^{2} + 1 | 8 − x + log2(x) | 1.0/(x − 3) − 6 | x^{9} − 8 | x^{0.5} − 1 | 0.7x^{5} − 8x^{4} +4 4x^{3} − 90x^{2} + 82x − 25 | 5x^{3} − 5x^{2} + 6x − 2 | 0.5x^{3} −4x^{2} +5.5 x − 1 | 5x^{3} − 5x^{2} + 6x − 2 | −0.6x^{2} + 2.4x + 5.5 | x^{8} − 1 | sin(x) − x^{3} | 8exp(−x)sin(x) − 1 | sin(x) + x | (0.8 − 0.3*x)/x |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Method | ||||||||||||||||||||||

MN_R | 34 | 14 | 10 | 10 | 6 | 8 | 22 | 16 | 14 | 18 | 42 | 14 | 12 | 8 | 12 | 8 | 6 | 52 | 14 | 6 | 6 | 10 |

MNIS | 40 | 44 | 60 | 76 | 28 | 28 | 84 | 76 | 64 | 60 | 180 | 220 | 44 | 60 | 76 | 76 | 68 | 84 | 44 | 56 | 28 | 60 |

MKA | 27 | 27 | 39 | 51 | 15 | 18 | 57 | 51 | 42 | 42 | 153 | 114 | 42 | 39 | 63 | 63 | 45 | 51 | 21 | 39 | 21 | 33 |

MSG | 28 | 12 | 39 | 8 | 8 | 8 | 64 | 28 | 12 | 12 | 36 | 12 | 12 | 8 | 32 | 8 | 8 | 248 | 16 | 8 | 12 | 12 |

FCTSC | 600 | 12 | 12 | 12 | 6 | 6 | 18 | 12 | 36 | 18 | 60 | 18 | 12 | 12 | 18 | 12 | 12 | 18 | 12 | 6 | 18 | 18 |

MGDB | 20 | 8 | 32 | 8 | 8 | 8 | 16 | 12 | 12 | 12 | 36 | 16 | 12 | 8 | 32 | 8 | 8 | 36 | 12 | 4 | 12 | 12 |

MOJmis | 81 | 15 | 9 | 9 | 6 | 6 | 39 | 21 | 21 | 18 | 42 | 12 | 12 | 6 | 9 | 6 | 6 | 3 | 15 | 6 | 12 | 12 |

MWFhm3 | 50 | 10 | 10 | 10 | 5 | 10 | 15 | 15 | 20 | 15 | 40 | 15 | 10 | 10 | 10 | 10 | 5 | 40 | 10 | 5 | 10 | 10 |

MJKs | 36 | 20 | 12 | 12 | 8 | 8 | 72 | 28 | 28 | 24 | 56 | 20 | 16 | 12 | 20 | 12 | 8 | 4 | 24 | 8 | 8 | 16 |

MJKhs | 36 | 16 | 12 | 12 | 8 | 8 | 28 | 24 | 28 | 20 | 52 | 16 | 16 | 8 | 12 | 8 | 8 | 4 | 12 | 8 | 12 | 12 |

NHAL | 27 | 12 | 27 | 12 | 9 | 9 | 15 | 18 | 27 | 9 | 198 | 45 | 12 | 12 | 18 | 12 | 15 | 18 | 18 | 9 | 30 | 24 |

Hybridn | 15 | 6 | 6 | 6 | 3 | 6 | 12 | 9 | 9 | 9 | 24 | 9 | 6 | 6 | 6 | 6 | 3 | 3 | 9 | 3 | 3 | 9 |

**Table 3.**

**All Parameters highlighting NOFE.**Comparison of hybrid and all algorithms on all parameters used for the solution for a function.

Summary for Comparison of Methods for | |||||||||
---|---|---|---|---|---|---|---|---|---|

Function sin(x) − x^{3}, Intial value 0.500 | |||||||||

Max Iterations = 100 | Error tolerance = 0.0000001000 | ||||||||

Method | Order | nofe | NIters | NOFE | COC | EFF | Root | |x_{n} − x_{n−1}| | Function Value |

MN_R | 2 | 2 | 7 | 14 | 2.01 | 1.4142136 | −0.928626 | 6.547 × 10^{−5} | 0.000000014 |

MNIS | 6 | 4 | 11 | 44 | 1 | 1.5650846 | −0.928626 | 1.04 × 10^{−7} | −0.000000085 |

MKA | 4 | 3 | 7 | 21 | 1 | 1.5874011 | 0.9286263 | 9.7 × 10^{−8} | 0.000000077 |

MSG | 6 | 4 | 4 | 16 | 4.29 | 1.5650846 | −0.928626 | 1.13 × 10^{−6} | 0 |

FCTSC | 6 | 6 | 2 | 12 | 4.29 | 1.3480062 | 0 | 0 | 0 |

MGDB | 6 | 4 | 3 | 12 | 4.29 | 1.5650846 | −0.928626 | 2.56 × 10^{−7} | 0 |

MOJmis | 2 | 3 | 5 | 15 | 3.93 | 1.2599211 | 0.9286263 | 0.000183 | 0 |

MWFhm3 | 6 | 5 | 2 | 10 | 3.93 | 1.4309691 | −0.928626 | 0 | 0 |

MJKs | 3 | 4 | 6 | 24 | 3.42 | 1.316074 | 0.9286263 | 1.302 × 10^{−5} | 0 |

MJKhs | 3 | 4 | 3 | 12 | 4.79 | 1.316074 | 0.9286263 | 0.0022525 | 0.000000039 |

NHAL | 3 | 3 | 6 | 18 | 3 | 1.4422496 | 0 | 0 | 0 |

Hybridn | 4 | 3 | 3 | 9 | 4.83 | 1.5874011 | −0.928626 | 6.547 × 10^{−5} | 0 |

**Table 4.**

**All Parameters highlighting**computational order of convergence

**(COC).**Comparison of hybrid and all algorithms on all parameters used for the solution for another function.

Summary for Comparison of Methods for | |||||||||
---|---|---|---|---|---|---|---|---|---|

Function 0.7*x^{5} − 8*x^{4} + 44*x^{3} − 90*x^{2} + 82*x − 25, Initial Value 00 | |||||||||

Max Iterations = 100 | Error tolerance = 0.0000001000 | ||||||||

Method | Order | nofe | NIters | NOFE | COC | EFF | Root | |x_{n} − x_{n−1}| | Function Value |

MN_R | 2 | 2 | 6 | 12 | 2.01 | 1.414214 | 0.579409 | 1 × 10^{−7} | 0 |

MNIS | 6 | 4 | 11 | 44 | 1 | 1.565085 | 0.579409 | 1.5 × 10^{−8} | −9.9 × 10^{−8} |

MKA | 4 | 3 | 13 | 39 | 1 | 1.587401 | 0.579409 | 1.2 × 10^{−8} | −7.9 × 10^{−8} |

MSG | 6 | 4 | 3 | 12 | 5.79 | 1.565085 | 0.579409 | 9.5 × 10^{−8} | 0 |

FCTSC | 6 | 6 | 2 | 12 | 5.79 | 1.348006 | 0.579409 | 7.9 × 10^{−8} | −7.9 × 10^{−8} |

MGDB | 6 | 4 | 3 | 12 | 5.79 | 1.565085 | 0.579409 | 1.1 × 10^{−8} | 0 |

MOJmis | 2 | 3 | 4 | 12 | 2.94 | 1.259921 | 0.579409 | 7.83 × 10^{−7} | 0 |

MWFhm3 | 6 | 5 | 2 | 10 | 2.94 | 1.430969 | 0.579409 | 0 | 0 |

MJKs | 3 | 4 | 4 | 16 | 2.93 | 1.316074 | 0.579409 | 1.16 × 10^{−6} | 0 |

MJKhs | 3 | 4 | 4 | 16 | 2.96 | 1.316074 | 0.579409 | 1.02 × 10^{−7} | 0 |

NHAL | 3 | 3 | 4 | 12 | 3.01 | 1.44225 | 0.579409 | 1.6 × 10^{−8} | 0 |

Hybridn | 4 | 3 | 2 | 6 | 6.76 | 1.587401 | 0.579409 | 0 | 0 |

**Table 5.**

**All Parameters highlighting**efficiency index

**(EFF).**Comparison of hybrid and all algorithms on all parameters used for the solution for another different function.

Summary for Comparison of Methods for | |||||||||
---|---|---|---|---|---|---|---|---|---|

Function x^{3} + log(x), Initial value 0.100 | |||||||||

Max Iterations = 100 | Error tolerance = 0.0000001000 | ||||||||

Method | Order | nofe | NIters | NOFE | COC | EFF | Root | |x_{n} − x_{n−1}| | FunctionValue |

MN_R | 2 | 2 | 5 | 10 | 1.94 | 1.4142 | 0.704709 | 3.57 × 10^{−7} | 0 |

MNIS | 6 | 4 | 15 | 60 | 1 | 1.5651 | 0.036264 | 4.7 × 10^{−8} | 2.9 × 10^{−8} |

MKA | 4 | 3 | 13 | 39 | 1 | 1.5874 | 0.704709 | 6 × 10^{−8} | −0.00000007 |

MSG | 6 | 4 | 7 | 28 | 6.62 | 1.5651 | 0.036264 | 3.53 × 10^{−7} | 0 |

FCTSC | 6 | 6 | 2 | 12 | 6.62 | 1.3480 | 0.704709 | 0 | 0 |

MGDB | 6 | 4 | 8 | 32 | 3.22 | 1.5651 | 0.704709 | 0.059996 | 5 × 10^{−9} |

MOJmis | 2 | 3 | 3 | 9 | 4.59 | 1.2599 | 0.704709 | 0.00025 | 0 |

MWFhm3 | 6 | 5 | 2 | 10 | 5.79 | 1.4310 | 0.704709 | 0 | 0 |

MJKs | 3 | 4 | 3 | 12 | 6.34 | 1.3161 | 0.704709 | 0.000155 | 0 |

MJKhs | 3 | 4 | 3 | 12 | 5.71 | 1.3161 | 0.704709 | 8.56 × 10^{−5} | 0 |

NHAL | 3 | 3 | 9 | 27 | 3.04 | 1.4422 | 0.704709 | 0 | 0 |

Hybridn | 4 | 3 | 2 | 6 | 6.87 | 1.5874 | 0.704709 | 0 | 0 |

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**In Summary, we have**

**Algorithm**(Oghovese–John)

_{0}be given, Newton–Raphson approximation iterates become

_{0}= v

_{0}= x

_{0}.

_{n,}u

_{n}, v

_{n}are known from

_{0}= x

_{0}− $\frac{\mathrm{f}({\mathrm{x}}_{0})}{{f}^{\prime}({\mathrm{x}}_{0})}$

_{0}= u

_{0}− $\frac{\mathrm{f}({\mathrm{u}}_{0})}{{f}^{\prime}({\mathrm{u}}_{0})}$

_{n+1}instead of x

_{n+1}$\cong $ x

_{n}− $\frac{\mathrm{f}({\mathrm{x}}_{\mathrm{n}})}{{\mathrm{f}}^{\prime}({\mathrm{x}}_{\mathrm{n}})}$

_{n+1}is acceptable using

_{n+1}$-$ x

_{n}|+|f(x

_{n+1})|) < $\epsilon $ or maxIteration reached

## Appendix B

f | the function |

a, xl | lower value bracket |

b, xu | upper value bracket |

es | error stopping critia |

imax | upper bound on the number of iterations |

iter | the number of iterations |

root | approxmate final root |

roots | approxmate iterated rootss |

ea | error at each iteration |

bl | lower value brack at each iteration |

br | upper value brack at each iteration |

**for**bisection,

**false**position and

**for**newtons methods for evaluation with additional documention is standard call to keep the hybrid code simple

**if**f(a)*f(b) > 0 %

**if**guesses

**do**not bracket

**for**Bisection,False Postion methods

**return**

**for**i = 1:imax

**false**position predicted point

**if**(abs(f(rootb)) < abs(f(rootse)))

**else**

**if**(abs(f(rootn)) < min(abs(f(xl)), abs(f(xu))))

**if**(f(rootn)*f(xu) < 0))

**if**(xl < rootn) && (xu > rootn)

**else**

**if**(xl < rootn) && (xu > rootn)

**for**documentation

**if**ea(i) < es

**break**;

**for**next iteration

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**Figure 1.**(

**a**) Convex function concave up, left endpoint fixed. (

**b**) Convex function concave up, right endpoint fixed. (

**c**) Convex function concave down, left endpoint fixed. (

**d**) Convex function concave down, right endpoint fixed.

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**MDPI and ACS Style**

Sabharwal, C.L.
An Iterative Hybrid Algorithm for Roots of Non-Linear Equations. *Eng* **2021**, *2*, 80-98.
https://doi.org/10.3390/eng2010007

**AMA Style**

Sabharwal CL.
An Iterative Hybrid Algorithm for Roots of Non-Linear Equations. *Eng*. 2021; 2(1):80-98.
https://doi.org/10.3390/eng2010007

**Chicago/Turabian Style**

Sabharwal, Chaman Lal.
2021. "An Iterative Hybrid Algorithm for Roots of Non-Linear Equations" *Eng* 2, no. 1: 80-98.
https://doi.org/10.3390/eng2010007