The article discusses derivative-free algorithms with and without memory for solving numerically nonlinear systems. We proposed a family of fifth- and sixth-order schemes and extended them to algorithms with memory. We further discuss the convergence and computational efficiency of these algorithms. Numerical examples of mixed Hammerstein integral equation, discrete nonlinear ordinary differential equation, and Fisher's partial differential equation with Neumann's boundary conditions are discussed to demonstrate the convergence and efficiency of these schemes. Finally, some numerical results are included to examine the performance of the developed methods.

1 INTRODUCTION

Applications in science and engineering which lead to nonlinear systems justify the importance of finding solutions of

F (x) = 0

in the domain of computational mathematics, where

F : ℝ^{n} \to ℝ^{n}

is sufficiently many times differentiable. The Newton's method^1-6 is possibly one of the oldest schemes available in literature for determination of roots of nonlinear systems. The method is given by

\begin{align} x^{(j + 1)} = x^{(j)} - {[F^{'} (x^{(j)})]}^{- 1} F (x^{(j)}) . \end{align}

()

Here,

F^{'} (x)

is first-order Fr

é

chet derivative of the function

F (x)

and

{[F^{'} (x)]}^{- 1}

represents its inverse. This algorithm converges quadratically under the assumptions that function

F

should be continuously differentiable and the initial approximation considered to find this root is close enough to the solution. However, to handle the situations where

F^{'} (x)

either does not exist or cannot be evaluated easily, Traub⁶ proposed quadratically convergent derivative-free scheme given by

\begin{align} x^{(j + 1)} = x^{(j)} - {[t^{(j)}, x^{(j)}; F]}^{- 1} F (x^{(j)}) . \end{align}

()

In this case,

{[t^{(j)}, x^{(j)}; F]}^{- 1}

is the inverse of first-order divided difference

[t^{(j)}, x^{(j)}; F]

of the function F and

t^{(j)} = x^{(j)} + η F (x^{(j)})

, where

η \neq 0

represents an arbitrary constant. Here, we recall that the first-order divided difference operator of a multivariable function F is a map

[., .; F] : S \times S \subset ℝ^{n} \times ℝ^{n} \to L (ℝ^{n})

which is defined^{3, 7, 8} as given below:

\begin{align} [θ_{1}, θ_{2}; F] (θ_{1} - θ_{2}) = F (θ_{1}) - F (θ_{2}) \forall (θ_{1}, θ_{2}) \in ℝ^{n} \times ℝ^{n}, θ_{1} \neq θ_{2} . \end{align}

()

Furthermore, in order to obtain the Taylor series expansion of the divided difference operator, we use the Genocchi–Hermite formula^{7, 9-11} given by

\begin{align} [θ + h, θ; F] = \int_{0}^{1} F^{'} (θ + g h) d g, \forall (θ, h) \in ℝ^{n} \times ℝ^{n}, \end{align}

()

where

h = θ_{2} - θ_{1}

. The divided difference operator

[., .; F]

is a square matrix of order n and the definitions (3) and (4) are equivalent.⁷ In this article, for computational purposes we use the following definition proposed by Grau et al.¹² of first-order divided difference operator

[x, y; F]

, namely,

\begin{align} {[x, y; F]}_{i k} = \frac{f_{i} (x_{1}, \dots, x_{k - 1}, x_{k}, y_{k + 1}, \dots, y_{n}) - f_{i} (x_{1}, . . ., x_{k - 1}, y_{k}, y_{k + 1}, . . ., y_{n})}{(x_{k} - y_{k})}, 1 \leq i, k \leq n . \end{align}

()

This is a linear operator and satisfies (3). For $η = 1$ , algorithm in (2) is the famous Steffensen method.¹³

Different approaches have been used by researchers to construct computationally efficient root-finding algorithms for both scalar and systems of nonlinear equations. Several iterative methods have been constructed using compositional approach, quadrature approach, adomian decomposition approach, weight function approach to name a few.

Several higher order derivative-free schemes have been proposed in literature. Liu et al.¹⁴ and Zheng et al.¹⁵ suggested fourth-order derivative-free methods. Grau et al.¹⁶ suggested fourth- and sixth-order methods and Sharma and Gupta¹⁷ proposed a family of sixth-order derivative-free methods. A derivative-free family of seventh-order methods was proposed by Wang and Zhang.¹⁸ Recently, Qiu et al.¹⁹ proposed a hybrid genetic algorithm by combining the advantages of genetic algorithms and derivative-free iterative methods to solve a class of complex nonlinear functions with discontinuity, nondifferentiability and multiple roots. This suggests derivative-free algorithms can be extended to the area of soft computing.

Recently, a technique used to achieve higher efficiency, is to develop memory based methods. This concept is given by Traub,⁶ who introduced algorithms with and without memory for the first time. Though this area has been explored more for scalar nonlinear equations, but very little progress has been made for nonlinear systems. Some of the existing memory based derivative-free methods can be found in works of Ren and Argyros,²⁰ Chicharro et al.,²¹ and Ahmad et al.²² Here, we propose new multipoint schemes which are not only derivative-free but also with and without memory for solving numerically nonlinear systems.

2 DEVELOPMENT AND CONVERGENCE INVESTIGATION

In this section, we propose new derivative-free schemes using divided differences as follows:

\begin{align} \{\begin{array}{l} y^{(j)} = x^{(j)} - L_{j}^{- 1} F (x^{(j)}), \\ z^{(j)} = y^{(j)} - L_{j}^{- 1} F (y^{(j)}), \\ x^{(j + 1)} = z^{(j)} - (p I + L_{j}^{- 1} M_{j} (q I + r L_{j}^{- 1} M_{j})) L_{j}^{- 1} F (z^{(j)}) . \end{array} \end{align}

()

Here, $p, q$ , and r are arbitrary constants, I denotes an identity matrix of order n, $L_{j} = [u^{(j)}, v^{(j)}; F]$ , where $u^{(j)} = x^{(j)} - a F (x^{(j)})$ , $v^{(j)} = x^{(j)} + b F (x^{(j)})$ , and $M_{j} = [w^{(j)}, s^{(j)}; F]$ , where $w^{(j)} = z^{(j)} - c F (z^{(j)})$ and $s^{(j)} = z^{(j)} + d F (z^{(j)})$ . Here, $a, b, c$ and d are arbitrary nonzero parameters such that either $a \neq 0$ or $b \neq 0$ and either $c \neq 0$ or $d \neq 0$ and $L_{j}^{- 1}$ represents the inverse of $L_{j}$ .

To discuss the convergence of proposed schemes, we recall the following Taylor series expansion of vector functions.³

Lemma 1.Let F: $S \subseteq ℝ^{n} \to ℝ^{n}$ be t times Fr $é$ chet differentiable function in a set $S \subseteq ℝ^{n}$ , which is an open convex set. Then, for any $x, h \in ℝ^{n}$ , the following expression holds:

$F (x + h) = F (x) + F^{'} (x) h + \frac{1}{2!} F^{''} (x) h^{2} + \frac{1}{3!} F^{'''} (x) h^{3} + \dots + \frac{1}{(t - 1)!} F^{(t - 1)} (x) h^{t - 1} + R_{t},$ where $‖ R_{t} ‖ \leq \frac{1}{t!} \sup_{0 \leq δ \leq 1} ‖ F^{(t)} (x + δ h) ‖ ‖ h ‖^{t}$ and $h^{t} = (h, h, . .^{t} . ., h)$ .

For a sufficiently continuously differentiable function F on S, the Taylor series expansion of

F^{'} (x + t h)

about the point x is substituted in (4) and integration yields

\begin{align} [x + h, x; F] = \int_{0}^{1} F^{'} (x + t h) d t = F^{'} (x) + \frac{1}{2} F^{''} (x) h + \frac{1}{6} F^{'''} (x) h^{2} + O (h^{3}), \end{align}

()

where

h^{k} = (h, h, . .^{k} . ., h)

h \in ℝ^{n}

Let the error in the jth approximation of a solution “

α

” of a nonlinear system

F (x) = 0

e^{(j)} = x^{(j)} - α

. Under the assumption that

Γ = {[F^{'} (α)]}^{- 1}

exists, we develop

F (x^{(j)})

in a neighborhood of “

α

” to obtain

\begin{align} F (x^{(j)}) = F^{'} (α) [e^{(j)} + A_{2} {(e^{(j)})}^{2} + A_{3} {(e^{(j)})}^{3} + A_{4} {(e^{(j)})}^{4} + A_{5} {(e^{(j)})}^{5} + O ({(e^{(j)})}^{6})], \end{align}

()

where

A_{k} = \frac{1}{k!} Γ F^{(k)} (α) \in L_{k} (ℝ^{n}, ℝ^{n})

and

L_{k} (ℝ^{n}, ℝ^{n})

represents the set of bounded k-linear functions,

k = 2, 3, . . .,

and

{(e^{(j)})}^{k} = (e^{(j)}, e^{(j)}, . .^{k} ., e^{(j)}), e^{(j)} \in ℝ^{n}

and

Γ = {[F^{'} (α)]}^{- 1} \in L (ℝ^{n}, ℝ^{n})

. With help of (8), derivatives of function F in a neighborhood of “

α

” can be written as follows:²³

\begin{align} F^{'} (x^{(j)}) & = F^{'} (α) [I + 2 A_{2} e^{(j)} + 3 A_{3} {(e^{(j)})}^{2} + 4 A_{4} {(e^{(j)})}^{3} + 5 A_{5} {(e^{(j)})}^{4} + O ({(e^{(j)})}^{5})] . \end{align}

()

\begin{align} F^{''} (x^{(j)}) & = F^{'} (α) [2 A_{2} + 6 A_{3} e^{(j)} + 12 A_{4} {(e^{(j)})}^{2} + 20 A_{5} {(e^{(j)})}^{3} + O ({(e^{(j)})}^{4})] . \end{align}

()

\begin{align} F^{'''} (x^{(j)}) & = F^{'} (α) [6 A_{3} + 24 A_{4} e^{(j)} + 60 A_{5} {(e^{(j)})}^{2} + O ({(e^{(j)})}^{3})] . \end{align}

()

We now discuss the convergence of the algorithm (6) in the following theorem.

Theorem 1.Let $F$ : $S \subseteq ℝ^{n} \to ℝ^{n}$ be sufficiently Fr $é$ chet differentiable in set S, which is an open convex set and the solution “ $α$ ” of F lies in S. Assume that $F^{'} (x)$ is continuous and nonsingular at x= $α$ and $x^{(0)}$ is an initial approximation sufficiently close to “ $α$ .” Then, for all $a, b, c, d \in ℝ$ , such that either $a \neq 0$ or $b \neq 0$ and either $c \neq 0$ or $d \neq 0$ , the sequence of approximations ${x^{(j)}}_{j \geq 1}$ generated by scheme (6) has local order of convergence five for $q = 3 - 2 p, r = p - 2, p \neq 3$ and six for $p = 3, q = - 3, r = 1$ .

Proof.Let $e_{u}^{(j)} = u^{(j)} - α = e^{(j)} - a F (x^{(j)})$ and $e_{v}^{(j)} = v^{(j)} - α = e^{(j)} + b F (x^{(j)})$ .

Using $F (x^{(j)})$ from (8), one obtains

\begin{align} e_{u}^{(j)} = (I - a F^{'} (α)) e^{(j)} - a F^{'} (α) (A_{2} {(e^{(j)})}^{2} + A_{3} {(e^{(j)})}^{3} + A_{4} {(e^{(j)})}^{4} + O ({(e^{(j)})}^{5})), \end{align}

()

and

\begin{align} e_{v}^{(j)} = (I + b F^{'} (α)) e^{(j)} + b F^{'} (α) (A_{2} {(e^{(j)})}^{2} + A_{3} {(e^{(j)})}^{3} + A_{4} {(e^{(j)})}^{4} + O ({(e^{(j)})}^{5})) . \end{align}

()

Let $x + h = u^{(j)}$ , $x = v^{(j)}$ and $h = e_{u}^{(j)} - e_{v}^{(j)}$ in (7) and with help of (9)–(11), using the results in Reference 12, one gets

\begin{align} L_{j} & = [u^{(j)}, v^{(j)}; F] = F^{'} (α) [I + A_{2} (e_{u}^{(j)} + e_{v}^{(j)}) + A_{3} ({(e_{u}^{(j)})}^{2} + e_{u}^{(j)} e_{v}^{(j)} \\ + {(e_{v}^{(j)})}^{2}) + A_{4} ({(e_{u}^{(j)})}^{3} + {(e_{u}^{(j)})}^{2} e_{v}^{(j)} + e_{u}^{(j)} {(e_{v}^{(j)})}^{2} + {(e_{v}^{(j)})}^{3}) + O ({(e^{(j)})}^{5})], \end{align}

()

and

\begin{align} L_{j}^{- 1} & = {[u^{(j)}, v^{(j)}; F]}^{- 1} = [I - A_{2} (e_{u}^{(j)} + e_{v}^{(j)}) + (A_{2}^{2} - A_{3}) ({(e_{u}^{(j)})}^{2} \\ + {(e_{v}^{(j)})}^{2}) + (2 A_{2}^{2} - A_{3}) e_{u}^{(j)} e_{v}^{(j)} + O ({(e^{(j)})}^{5})] Γ . \end{align}

()

On premultiplying (8) by (15) and simplifying with help of (12) and (13), one obtains

\begin{align} L_{j}^{- 1} F (x^{(j)}) = e^{(j)} - A_{2} [I + (b - a) F^{'} (α)] {(e^{(j)})}^{2} + O ({(e^{(j)})}^{3}) . \end{align}

()

Using (16) in the first step of algorithm (6), one gets on simplification

\begin{align} e_{y}^{(j)} = y^{(j)} - α = A_{2} (I + (b - a) F^{'} (α)) {(e^{(j)})}^{2} + O ({(e^{(j)})}^{3}) . \end{align}

()

Taylor series expansion of $F (y^{(j)})$ about “ $α$ ” gives

\begin{align} F (y^{(j)}) = F^{'} (α) [e_{y}^{(j)} + A_{2} {(e_{y}^{(j)})}^{2} + A_{3} {(e_{y}^{(j)})}^{3} + O ({(e_{y}^{(j)})}^{4})] . \end{align}

()

Premultiplying (18) by (15) and substituting in second step of scheme (6), one obtains

\begin{align} e_{z}^{(j)} & = z^{(j)} - α = A_{2} (e_{u}^{(j)} + e_{v}^{(j)}) e_{y}^{(j)} - A_{2} {(e_{y}^{(j)})}^{2} - (A_{2}^{2} - A_{3}) \\ ({(e_{u}^{(j)})}^{2} + {(e_{v}^{(j)})}^{2}) e_{y}^{(j)} - (2 A_{2}^{2} - A_{3}) e_{u}^{(j)} e_{v}^{(j)} e_{y}^{(j)} + O ({(e^{(j)})}^{5}) . \end{align}

()

Now, the Taylor series of $F (z^{(j)})$ about “ $α$ ” gives

\begin{align} F (z^{(j)}) = F^{'} (α) (e_{z}^{(j)} + A_{2} {(e_{z}^{(j)})}^{2} + A_{3} {(e_{z}^{(j)})}^{3} + O ({(e_{z}^{(j)})}^{4})) . \end{align}

()

Let $e_{w}^{(j)} = w^{(j)} - α = e_{z}^{(j)} - c F (z^{(j)})$ , and $e_{s}^{(j)} = s^{(j)} - α = e_{z}^{(j)} + d F (z^{(j)})$ . Using (20), one gets

\begin{align} e_{w}^{(j)} = (I - c F^{'} (α)) e_{z}^{(j)} - c F^{'} (α) (A_{2} {(e_{z}^{(j)})}^{2} + A_{3} {(e_{z}^{(j)})}^{3} + O ({(e_{z}^{(j)})}^{4})), \end{align}

()

and

\begin{align} e_{s}^{(j)} = (I + d F^{'} (α)) e_{z}^{(j)} + d F^{'} (α) (A_{2} {(e_{z}^{(j)})}^{2} + A_{3} {(e_{z}^{(j)})}^{3} + O ({(e_{z}^{(j)})}^{4})) . \end{align}

()

Using $x + h = w^{(j)}$ , $x = s^{(j)}$ and $h = e_{w}^{(j)} - e_{s}^{(j)}$ in (7), and (9)–(11), one gets

\begin{align} M_{j} & = [w^{(j)}, s^{(j)}; F] \\ = F^{'} (α) [I + A_{2} (e_{w}^{(j)} + e_{s}^{(j)}) + A_{3} ({(e_{w}^{(j)})}^{2} + e_{w}^{(j)} e_{s}^{(j)} + {(e_{s}^{(j)})}^{2}) + O ({(e^{(j)})}^{7})] . \end{align}

()

Using (15), (20), and (23) in the third step of scheme (6), one can write

\begin{array}{lcr} e^{(j + 1)} & = & (1 - p - q - r) e_{z}^{(j)} + (p + 2 q + 3 r) A_{2} (e_{u}^{(j)} + e_{v}^{(j)}) e_{z}^{(j)} \\ - ((p + 3 q + 6 r) A_{2}^{2} - (p + 2 q + 3 r) A_{3}) ({(e_{u}^{(j)})}^{2} + {(e_{v}^{(j)})}^{2}) e_{z}^{(j)} \\ - (2 (p + 3 q + 6 r) A_{2}^{2} - (p + 2 q + 3 r) A_{3}) e_{u}^{(j)} e_{v}^{(j)} e_{z}^{(j)} \\ + ((p + 4 q + 10 r) A_{2}^{3} - (p + 3 q + 6 r) (A_{2} A_{3} + A_{3} A_{2}) + (p + 2 q + 3 r) A_{4}) \\ ({(e_{u}^{(j)})}^{3} + {(e_{v}^{(j)})}^{3}) e_{z}^{(j)} + (3 (p + 4 q + 10 r) A_{2}^{3} - 2 (p + 3 q + 6 r) (A_{2} A_{3} + A_{3} A_{2}) \\ + (p + 2 q + 3 r) A_{4}) e_{u}^{(j)} e_{v}^{(j)} (e_{u}^{(j)} + e_{v}^{(j)}) e_{z}^{(j)} - (p + q + r) A_{2} {(e_{z}^{(j)})}^{2} \\ - (q + 2 r) A_{2} (e_{w}^{(j)} + e_{s}^{(j)}) e_{z}^{(j)} + O ({(e^{(j)})}^{8}) . \end{array}

()

In order to attain higher order of convergence of the proposed scheme (6), the choice of parameters $p, q$ , and r should be such that the coefficients $1 - p - q - r$ , $p + 2 q + 3 r$ are zero. Solving these equations simultaneously, results in $q = 3 - 2 p, r = p - 2$ . The values of $q and r$ are substituted in the error equation (24), give

\begin{array}{lcr} e^{(j + 1)} & = - ((p - 3) A_{2}^{2}) ({(e_{u}^{(j)})}^{2} + {(e_{v}^{(j)})}^{2}) e_{z}^{(j)} - 2 (p - 3) A_{2}^{2} e_{u}^{(j)} e_{v}^{(j)} e_{z}^{(j)} + A_{2} (e_{w}^{(j)} + e_{s}^{(j)}) e_{z}^{(j)} \\ + [- (p - 3) (A_{2} A_{3} + A_{3} A_{2}) + (3 p - 8) A_{2}^{3})] ({(e_{u}^{(j)})}^{3} + {(e_{v}^{(j)})}^{3}) e_{z}^{(j)} \\ + [- 2 (p - 3) (A_{2} A_{3} + A_{3} A_{2}) + 2 (3 p - 8) A_{2}^{3})] ({(e_{u}^{(j)})}^{2} e_{v}^{(j)} + e_{u}^{(j)} {(e_{v}^{(j)})}^{2}) e_{z}^{(j)} \\ + O ({(e^{(j)})}^{7}) . \end{array}

()

It can be seen easily that the convergence order of the proposed algorithm is five for $p \neq 3$ and six for $p = 3$ . For $p \neq 3$ , the error equation (25) can be simplified with the help of (12), (13), (17), and (19) and one gets

\begin{align} e^{(j + 1)} & = - (p - 3) A_{2}^{2} {(e_{u}^{(j)} + e_{v}^{(j)})}^{2} e_{z}^{(j)} + O ({(e^{(j)})}^{6}), \\ = - (p - 3) A_{2}^{3} {(2 I + (b - a) F^{'} (α))}^{3} (I + (b - a) F^{'} (α)) {(e^{(j)})}^{5} + O ({(e^{(j)})}^{6}) . \end{align}

()

Using $p = 3$ , the error equation using (12), (13), (17), (19), (21), and (22) will become

\begin{align} e^{(j + 1)} & = A_{2} (e_{w}^{(j)} + e_{s}^{(j)}) e_{z}^{(j)} + A_{2}^{3} ({(e_{u}^{(j)})}^{3} + {(e_{v}^{(j)})}^{3}) e_{z}^{(j)} + 2 A_{2}^{3} ({(e_{u}^{(j)})}^{2} e_{v}^{(j)} \\ + e_{u}^{(j)} {(e_{v}^{(j)})}^{2}) e_{z}^{(j)} - A_{2} {(e_{z}^{(j)})}^{2} + O ({(e^{(j)})}^{7}) . \\ = - A_{2} {(e_{z}^{(j)})}^{2} + A_{2} (e_{w}^{(j)} + e_{s}^{(j)}) e_{z}^{(j)} + A_{2}^{3} {(e_{u}^{(j)} + e_{v}^{(j)})}^{3} e_{z}^{(j)} + O ({(e^{(j)})}^{7}) . \\ = A_{2}^{5} ((I + (d - c) F^{'} (α)) (I + (b - a) F^{'} (α)) + {(2 I + (b - a) F^{'} (α))}^{2}) \\ {(2 I + (b - a) F^{'} (α))}^{2} (I + (b - a) F^{'} (α)) {(e^{(j)})}^{6} + O ({(e^{(j)})}^{7}) . \end{align}

()

Remark 1.The proposed algorithms require evaluation of two divided differences, at most seven F and inversion of one matrix only per iteration.

3 METHODS WITH MEMORY

In literature, several techniques are used to increment the order of convergence. One such methodology is to convert a method without memory into a method with memory. As the name suggests, this is done with the assistance of current and preceding data stored in memory.

Now, if we are allowed to choose $b - a = - {[F^{'} (α)]}^{- 1}$ or $b - a = - 2 {[F^{'} (α)]}^{- 1}$ in the error equations (26) and (27), one can observe that the order of convergence of scheme becomes six and seven, respectively. This acceleration requires the knowledge of “ $α$ ” in advance, but in absence of this information, it is not viable to attain this acceleration. But, this acceleration can be achieved partially by considering an appropriate approximation of ${[F^{'} (α)]}^{- 1}$ based on the current and previous acquired information.

Hence, we consider the following value of varying parametric matrix

P^{(j)}

\begin{align} b - a = P^{(j)} = - {[u^{(j - 1)}, v^{(j - 1)}; F]}^{- 1} \approx - {[F^{'} (α)]}^{- 1}, \forall j = 1, 2, 3, \dots \end{align}

()

and define the iterative scheme as follows:

\begin{align} \{\begin{array}{l} y^{(j)} = x^{(j)} - L_{j}^{- 1} F (x^{(j)}), \\ z^{(j)} = y^{(j)} - L_{j}^{- 1} F (y^{(j)}), \\ x^{(j + 1)} = z^{(j)} - (p I + L_{j}^{- 1} M_{j} ((3 - 2 p) I + (p - 2) L_{j}^{- 1} M_{j})) L_{j}^{- 1} F (z^{(j)}) . \end{array} \end{align}

()

Here, $L_{j} = [u^{(j)}, v^{(j)}; F]$ , $v^{(j)} = x^{(j)} + δ P^{(j)} F (x^{(j)})$ , and $u^{(j)} = x^{(j)} - γ P^{(j)} F (x^{(j)})$ , where $δ$ and $γ$ are arbitrary constants which satisfy the condition $δ - γ = 1$ , and $P^{(j)}$ is given by (28). Note that the value of $P^{(j)}$ requires values that are based on preceding steps and hence (29) is a scheme with memory. In the undermentioned theorem, we establish the R-order of convergence of the scheme (29).

Theorem 2.Let F: $S \subseteq ℝ^{n} \to ℝ^{n}$ be sufficiently Fr $é$ chet differentiable function in a set $S \subseteq ℝ^{n}$ which is convex and contains the zero “ $α$ ” of $F (x)$ . Assume $F^{'} (x)$ is continuous and nonsingular at “ $α$ ” and the parameter matrix $P^{(j)}$ of the algorithm is recursively calculated using (28). Then, the R-order of convergence the scheme (29), for $p \neq 3$ and $b - a = P^{(j)}$ is 5.19, and for $p \neq 3$ and $b - a = 2 P^{(j)}$ is 5.54 where $c, d \in ℝ$ , such that either $c \neq 0$ or $d \neq 0$ . Further, for $p = 3$ and $b - a = P^{(j)}$ algorithm converges with R-order of convergence 6.16, and for $b - a = 2 P^{(j)}$ , the R-order of convergence is 6.31 where $c, d \in ℝ$ , such that either $c \neq 0$ or $d \neq 0$ . Furthermore, for $p = 3$ , $b - a = 2 P^{(j)}$ and $d - c = C^{(j)}$ , the R-order of convergence is 6.46, and for $b - a = 2 P^{(j)}$ and $d - c = C^{(j + 1)}$ , the R-order of convergence is 6.60.

Proof.Assume that the scheme (29) generates a sequence of approximations ${x^{(j)}}$ which approaches the zero “ $α$ ” of F and its R-order is $\geq σ$ , thus one can write

\begin{align} e^{(j + 1)} \sim Λ^{(j, σ)} {(e^{(j)})}^{σ} . \end{align}

()

Here, the sequence ${Λ^{(j, σ)}}$ approaches asymptotic error constant $Λ^{σ}$ of scheme when $j \to \infty$ and $ã \sim \tilde{b}$ symbolize that $ã$ and $\tilde{b}$ have magnitudes of same order. Thus, one can write

\begin{align} e^{(j + 1)} \sim Λ^{(j, σ)} {(Λ^{(j - 1, σ)} {(e^{(j - 1)})}^{σ})}^{σ} = Λ^{(j, σ)} {(Λ^{(j - 1, σ)})}^{σ} {(e^{(j - 1)})}^{σ^{2}} . \end{align}

()

Replacing j by $j - 1$ in (15), one obtains

\begin{align} b - a = P^{(j)} = - {[u^{(j - 1)}, v^{(j - 1)}; F]}^{- 1} = - (I - A_{2} (e_{u}^{(j - 1)} + e_{v}^{(j - 1)}) + O ({(e^{(j - 1)})}^{2})) Γ . \end{align}

()

Similarly, on replacing j by $j - 1$ in (12) and (13), one gets

\begin{align} I + P^{(j)} F^{'} (α) = A_{2} (e_{u}^{(j - 1)} + e_{v}^{(j - 1)}) + O ({(e^{(j - 1)})}^{2}) \sim e^{(j - 1)} . \end{align}

()

Using the value of $b - a = P^{(j)}$ , and rewriting the error equation (26), one can write

e^{(j + 1)} = - (p - 3) A_{2}^{3} {(2 I + P^{(j)} F^{'} (α))}^{3} (I + P^{(j)} F^{'} (α)) {(e^{(j)})}^{5} + O ({(e^{(j)})}^{6}) .

()

Using (33) in (34) and letting $j \to \infty$ , with the help of (31), one can write

\begin{align} e^{(j + 1)} \sim e^{(j - 1)} {(e^{(j)})}^{5} \sim e^{(j - 1)} {({(e^{(j - 1)})}^{σ})}^{5} \sim {(e^{(j - 1)})}^{5 σ + 1} . \end{align}

()

From (31) and (35), the exponents of $e^{(j - 1)}$ are compared and one gets the quadratic equation

\begin{align} σ^{2} = 5 σ + 1 . \end{align}

()

Solving (36), one gets the positive root as 5.19, which gives an estimate of the lower bound of R-order of convergence of the algorithm (29) for all $p, c, d \in ℝ$ such that either $c \neq 0$ or $d \neq 0$ and $p \neq 3$ .

Next, if the matrix $P^{(j)} = - 2 {[u^{(j - 1)}, v^{(j - 1)}; F]}^{- 1}$ then, replacing j by j-1 in (15), one obtains

\begin{align} b - a = P^{(j)} = - 2 [I - A_{2} (e_{u}^{(j - 1)} + e_{v}^{(j - 1)}) + O ({(e^{(j - 1)})}^{2})] Γ . \end{align}

()

Substituting $j - 1$ in (12) and (13), one obtains

\begin{align} 2 I + P^{(j)} F^{'} (α) = 2 A_{2} (e_{u}^{(j - 1)} + e_{v}^{(j - 1)}) + O ({(e^{(j - 1)})}^{2}) \sim e^{(j - 1)} . \end{align}

()

Using (34) in (38), as explained earlier, one can write

\begin{align} e^{(j + 1)} \sim {(e^{(j - 1)})}^{3} {(e^{(j)})}^{5} \sim {(e^{(j - 1)})}^{3} {({(e^{(j - 1)})}^{σ})}^{5} \sim {(e^{(j - 1)})}^{5 σ + 3} . \end{align}

()

On comparing exponents of $e^{(j - 1)}$ using (31) and (39), one gets

\begin{align} σ^{2} = 5 σ + 3 . \end{align}

()

The positive root of this quadratic equation (40) is 5.54. This gives the lower bound of the R-order of convergence of scheme (29) for all $p, c, d \in ℝ$ such that either $c \neq 0$ or $d \neq 0$ and $p \neq 3$ .

Next, to explore the convergence behavior of the algorithm (29) for $p = 3$ , we use (33) in the error equation (27) to obtain

\begin{array}{lcr} e^{(j + 1)} & = A_{2}^{5} ((I + (d - c) F^{'} (α)) (I + P^{(j)} F^{'} (α)) + {(2 I + P^{(j)} F^{'} (α))}^{2}) \\ {(2 I + P^{(j)} F^{'} (α))}^{2} (I + P^{(j)} F^{'} (α)) {(e^{(j)})}^{6} + O ({(e^{(j)})}^{7}) . \end{array}

()

From (33) and (41), one can write

\begin{align} e^{(j + 1)} \sim e^{(j - 1)} {(e^{(j)})}^{6} \sim e^{(j - 1)} {({(e^{(j - 1)})}^{σ})}^{6} \sim {(e^{(j - 1)})}^{6 σ + 1} . \end{align}

()

Comparison of exponents of $e^{(j - 1)}$ in (31) and (42), one can write the quadratic equation

\begin{align} σ^{2} = 6 σ + 1 . \end{align}

()

A positive solution of (43) is 6.16, which gives the lower bound of the R-order of convergence of the iterative scheme (29) for all $c, d \in ℝ$ such that either $c \neq 0$ or $d \neq 0$ .

Next, on substituting (38) in (41), we can write

\begin{align} e^{(j + 1)} \sim {(e^{(j - 1)})}^{2} {(e^{(j)})}^{6} \sim {(e^{(j - 1)})}^{2} {({(e^{(j - 1)})}^{σ})}^{6} \sim {(e^{(j - 1)})}^{6 σ + 2} . \end{align}

()

From (31) and (44), we compare the exponents of $e^{(j - 1)}$ . This yields the quadratic equation

\begin{align} σ^{2} = 6 σ + 2 . \end{align}

()

The positive root of the equation is 6.31, which is the lower bound of the R-order of convergence of the scheme (29) for all $c, d \in ℝ$ such that either $c \neq 0$ or $d \neq 0$ .

Now, for $d - c = C^{(j)} = - {[u^{(j - 1)}, v^{(j - 1)}; F]}^{- 1}$ and using (15) for $j - 1$ , one can write

\begin{align} d - c = C^{(j)} = - [I - A_{2} (e_{u}^{(j - 1)} + e_{v}^{(j - 1)}) + O ({(e^{(j - 1)})}^{2})] Γ . \end{align}

()

Using (12) and (13) for $j - 1$ , one gets

\begin{align} I + C^{(j)} F^{'} (α) = A_{2} (e_{u}^{(j - 1)} + e_{v}^{(j - 1)}) + O ({(e^{(j - 1)})}^{2}) \sim e^{(j - 1)} . \end{align}

()

Using (46), the error equation (41) can be written as

\begin{array}{lcr} e^{(j + 1)} & = A_{2}^{5} ((I + C^{(j)} F^{'} (α)) (I + P^{(j)} F^{'} (α)) + {(2 I + P^{(j)} F^{'} (α))}^{2}) \\ {(2 I + P^{(j)} F^{'} (α))}^{2} (I + P^{(j)} F^{'} (α)) {(e^{(j)})}^{6} + O ({(e^{(j)})}^{7}) . \end{array}

()

Using (38) and (47) in (48), one can write

\begin{align} e^{(j + 1)} & \sim (e^{(j - 1)} (I + P^{(j)} F^{'} (α)) + {(e^{(j - 1)})}^{2}) {(e^{(j - 1)})}^{2} {(e^{(j)})}^{6}, \\ \sim {(e^{(j - 1)})}^{3} {({(e^{(j - 1)})}^{σ})}^{6} \sim {(e^{(j - 1)})}^{6 σ + 3} . \end{align}

()

On comparing exponents of $e^{(j - 1)}$ in (31) and (49), one gets the quadratic equation

\begin{align} σ^{2} = 6 σ + 3 . \end{align}

()

Solving Equation (50), one gets the positive root as 6.46. This provides a lower bound of the R-order of convergence of the scheme (29).

Further, we consider $d - c = C^{(j + 1)} = - {[u^{(j)}, v^{(j)}; F]}^{- 1}$ and using (15), one can write

\begin{align} d - c = C^{(j + 1)} = - [I - A_{2} (e_{u}^{(j)} + e_{v}^{(j)}) + O ({(e^{(j)})}^{2})] Γ . \end{align}

()

Simplifying (51) and using (12) and (13), one gets

\begin{align} I + C^{(j + 1)} F^{'} (α) = A_{2} (e_{u}^{(j)} + e_{v}^{(j)}) + O ({(e^{(j)})}^{2}) \sim e^{(j)} . \end{align}

()

The error equation (41) with the help of (51) can be simplified as

\begin{array}{lcr} e^{(j + 1)} & = A_{2}^{5} ((I + C^{(j + 1)} F^{'} (α)) (I + P^{(j)} F^{'} (α)) + {(2 I + P^{(j)} F^{'} (α))}^{2}) \\ {(2 I + P^{(j)} F^{'} (α))}^{2} (I + P^{(j)} F^{'} (α)) {(e^{(j)})}^{6} + O ({(e^{(j)})}^{7}) . \end{array}

()

Using (38) and (52) in (53), one can write

\begin{align} e^{(j + 1)} & \sim (e^{(j)} (I + P^{(j)} F^{'} (α)) + {(e^{(j - 1)})}^{2}) {(e^{(j - 1)})}^{2} {(e^{(j)})}^{6}, \\ \sim ({(e^{(j - 1)})}^{σ} (I + P^{(j)} F^{'} (α)) + {(e^{(j - 1)})}^{2}) {(e^{(j - 1)})}^{2} {(e^{(j)})}^{6}, \\ \sim {(e^{(j - 1)})}^{4} {({(e^{(j - 1)})}^{σ})}^{6} (assuming σ \geq 2) \\ \sim {(e^{(j - 1)})}^{6 σ + 4} . \end{align}

()

Comparing the exponents of $e^{(j - 1)}$ from (31) and (54), yields the quadratic equation

\begin{align} σ^{2} = 6 σ + 4 . \end{align}

()

The positive root of (55) is 6.60, which estimates lower bound of R-order of convergence of the scheme (29).

3.1 Particular cases of family (29)

Here, let the ith scheme of order $ρ$ be represented by $Ω_{ρ, i}$ .

Case 1. For

p = 2

in (29), we get the scheme as given below:

\begin{align} \{\begin{array}{l} y^{(j)} = x^{(j)} - L_{j}^{- 1} F (x^{(j)}), z^{(j)} = y^{(j)} - L_{j}^{- 1} F (y^{(j)}), \\ x^{(j + 1)} = z^{(j)} - (2 I - L_{j}^{- 1} M_{j}) L_{j}^{- 1} F (z^{(j)}), \end{array} \end{align}

()

where

L_{j} = [u^{(j)}, v^{(j)}; F]

and

M_{j} = [w^{(j)}, s^{(j)}; F]

For $v^{(j)} = x^{(j)} + 2 δ_{1} P^{(j)} F (x^{(j)})$ and $u^{(j)} = x^{(j)} - 2 γ_{1} P^{(j)} F (x^{(j)})$ , where $γ_{1}$ and $δ_{1}$ are arbitrary constants such that $δ_{1} - γ_{1} = 1$ , and $P^{(j)}$ is given by (28), $w^{(j)} = z^{(j)} - c F (z^{(j)})$ , $s^{(j)} = z^{(j)} + d F (z^{(j)})$ , where c and d are arbitrary constants such that either $c \neq 0$ or $d \neq 0$ , the method is referred as $Ω_{5.54, i}$ . For $(γ_{1} = 0, δ_{1} = 1, c = 0, d \neq 0)$ , we denote the method by $Ω_{5.54, 1}$ .

Case 2. For

p = 3

in (29), we obtain the undermentioned algorithm:

\begin{align} \{\begin{array}{l} y^{(j)} = x^{(j)} - L_{j}^{- 1} F (x^{(j)}), z^{(j)} = y^{(j)} - L_{j}^{- 1} F (y^{(j)}), \\ x^{(j + 1)} = z^{(j)} - (3 I + L_{j}^{- 1} M_{j} (- 3 I + L_{j}^{- 1} M_{j})) L_{j}^{- 1} F (z^{(j)}), \end{array} \end{align}

()

where

L_{j} = [u^{(j)}, v^{(j)}; F]

and

M_{j} = [w^{(j)}, s^{(j)}; F]

Subcase (i) For $u^{(j)} = x^{(j)} - γ_{1} P^{(j)} F (x^{(j)})$ and $v^{(j)} = x^{(j)} + δ_{1} P^{(j)} F (x^{(j)})$ , where $γ_{1}$ and $δ_{1}$ are arbitrary constants that satisfy the condition $δ_{1} - γ_{1} = 1$ , and $P^{(j)}$ is given by (28), $w^{(j)} = z^{(j)} - c F (z^{(j)})$ , $s^{(j)} = z^{(j)} + d F (z^{(j)})$ , where c and d are arbitrary constants such that either $c \neq 0$ or $d \neq 0$ , the method is referred as $Ω_{6.16, i}$ . For $(γ_{1} = 0, δ_{1} = 1, c = 0, d \neq 0)$ , the method is addressed by $Ω_{6.16, 1}$ .

Subcase (ii) For $u^{(j)} = x^{(j)} - 2 γ_{1} P^{(j)} F (x^{(j)})$ , $v^{(j)} = x^{(j)} + 2 δ_{1} P^{(j)} F (x^{(j)})$ , $w^{(j)} = z^{(j)} - γ_{2} C^{(j)} F (z^{(j)})$ and $s^{(j)} = z^{(j)} + δ_{2} C^{(j)} F (z^{(j)})$ , where $γ_{1}$ , $δ_{1}$ , $γ_{2}$ and $δ_{2}$ are arbitrary constants satisfying the condition $δ_{1} - γ_{1} = 1$ and $δ_{2} - γ_{2} = 1$ , and $P^{(j)}$ and $C^{(j)}$ are given by (28), the method is denoted by $Ω_{6.46, i}$ . For $(γ_{1} = 0, δ_{1} = 1, γ_{2} = 0, δ_{2} = 1)$ , the method is referred as $Ω_{6.46, 1}$ .

Subcase (iii) For $u^{(j)} = x^{(j)} - 2 γ_{1} P^{(j)} F (x^{(j)})$ , $v^{(j)} = x^{(j)} + 2 δ_{1} P^{(j)} F (x^{(j)})$ , $w^{(j)} = z^{(j)} - γ_{2} C^{(j + 1)} F (z^{(j)})$ and $s^{(j)} = z^{(j)} + δ_{2} C^{(j + 1)} F (z^{(j)})$ where $γ_{1}$ , $δ_{1}$ , $γ_{2}$ , and $δ_{2}$ are arbitrary constants which satisfy the condition $δ_{1} - γ_{1} = 1$ and $δ_{2} - γ_{2} = 1$ , and $P^{(j)}$ and $C^{(j + 1)}$ are given by (28), the method is referred as $Ω_{6.60, i}$ . For $(γ_{1} = 0, δ_{1} = 1, γ_{2} = 0, δ_{2} = 1)$ , we address the method by $Ω_{6.60, 1}$ .

4 COMPUTATIONAL EFFICIENCY

Among various parameters that decide the quality of an algorithm, one parameter is its computational efficiency. It is discussed using efficiency index.⁴ The computational efficiency index is defined by $E = ρ^{\frac{1}{C}}$ , where $ρ$ is the convergence order and C denotes the computational cost per step. In case of a nonlinear system of n equations in n unknowns, the computational cost¹² per step has been modified and is calculated using

$C (μ, n, l) = A (n) μ + P (n, l) .$

Here, $A (n)$ is count of scalar functions while evaluating F and divided difference $[x, y; F]$ , and $P (n, l)$ counts number of products and quotients requisite in each step. To evaluate $C (μ, n, l)$ , we require the ratios $μ > 0$ and $l > 0$ . Here, $μ$ represents the ratio of number of products used for evaluation of the function F and number of scalar function in F, whereas $l \geq 1$ refers to the ratio of the time taken by one quotient and the time taken by one product.¹⁰

We recall that we calculate n scalar functions for F and $n (n - 1)$ for divided difference $[x, y; F]$ of F (refer (5)), and $F (x)$ and $F (y)$ are computed additionally. Also, we add $n^{2}$ quotients for each new divided difference, $n^{2}$ products are used in case of multiplication of a matrix and vector, whereas n products for multiplication of a vector and scalar.

Another aspect that contributes toward the computational cost is the calculation of number of products and quotients utilized for the evaluation of an inverse operator. Rather than calculating an inverse operator, we solve a linear system. The linear system uses $\frac{(n) (n - 1) (2 n - 1)}{6}$ products and $\frac{n (n - 1)}{2}$ quotients for LU decomposition, and further the resolution of two triangular linear systems requires $n (n - 1)$ products and n quotients.

The computational efficiency of the proposed methods is discussed by considering some existing derivative-free schemes with and without memory. The derivative-free algorithms without and with memory considered are SA $_{6, 2}$ , SA $_{6.16, 2}$ , SA $_{6.46, 2}$ , and SA $_{6.60, 2}$ proposed by Sharma and Arora,²⁴ which require five functions, two divided differences and one matrix inversion per step to be evaluated. Another derivative-free method GGN $_{6, 3}$ without memory is by Grau et al.¹² and this needs five functions, two divided differences and two matrix inverse per step to be evaluated. These schemes are given as follows:

Sharma and Arora method (SA $_{6, 2}$ ):

$y^{(j)} = x^{(j)} - {[u^{(j)}, x^{(j)}; F]}^{- 1} F (x^{(j)})$ ,

$z^{(j)} = y^{(j)} - (2 I - G^{(j)}) {[u^{(j)}, x^{(j)}; F]}^{- 1} F (y^{(j)})$ ,

$x^{(j + 1)} = z^{(j)} - (2 I - G^{(j)}) {[u^{(j)}, x^{(j)}; F]}^{- 1} F (z^{(j)}),$

where, $G^{(j)} = {[u^{(j)}, x^{(j)}; F]}^{- 1} [v^{(j)}, y^{(j)}; F]$ , $u^{(j)} = x^{(j)} + a F (x^{(j)}),$ $v^{(j)} = y^{(j)} + c F (y^{(j)})$ , and $a, c \neq 0$ are arbitrary constants.

Grau et al. method (GGN $_{6, 3}$ ):

$y^{(j)} = x^{(j)} - {[u_{1}^{(j)}, v_{1}^{(j)}; F]}^{- 1} F (x^{(j)})$ ,

$z^{(j)} = y^{(j)} - {(2 [y^{(j)}, x^{(j)}; F] - [u_{1}^{(j)}, v_{1}^{(j)}; F])}^{- 1} F (y^{(j)})$ ,

$x^{(j + 1)} = z^{(j)} - {(2 [y^{(j)}, x^{(j)}; F] - [u_{1}^{(j)}, v_{1}^{(j)}; F])}^{- 1} F (z^{(j)}),$

where $u_{1}^{(j)} = x^{(j)} + F (x^{(j)}),$ $v_{1}^{(j)} = x^{(j)} - F (x^{(j)})$ .

Sharma and Arora method (SA $_{6.16, 2}$ ):

$y^{(j)} = x^{(j)} - {[u^{(j)}, x^{(j)}; F]}^{- 1} F (x^{(j)})$ ,

$z^{(j)} = y^{(j)} - (2 I - G^{(j)}) {[u^{(j)}, x^{(j)}; F]}^{- 1} F (y^{(j)})$ ,

$x^{(j + 1)} = z^{(j)} - (2 I - G^{(j)}) {[u^{(j)}, x^{(j)}; F]}^{- 1} F (z^{(j)}) .$

Sharma and Arora method (SA $_{6.46, 2}$ ):

$y^{(j)} = x^{(j)} - {[u^{(j)}, x^{(j)}; F]}^{- 1} F (x^{(j)})$ ,

$z^{(j)} = y^{(j)} - (3 I - G^{(j)} (3 I - G^{(j)})) {[u^{(j)}, x^{(j)}; F]}^{- 1} F (y^{(j)}),$

$x^{(j + 1)} = z^{(j)} - (3 I - G^{(j)} (3 I - G^{(j)})) {[u^{(j)}, x^{(j)}; F]}^{- 1} F (z^{(j)}) .$

Sharma and Arora method (SA $_{6.60, 2}$ ):

$y^{(j)} = x^{(j)} - {[u^{(j)}, x^{(j)}; F]}^{- 1} F (x^{(j)})$ ,

$z^{(j)} = y^{(j)} - (3 I - G^{(j)} (3 I - G^{(j)})) {[u^{(j)}, x^{(j)}; F]}^{- 1} F (y^{(j)}),$

$x^{(j + 1)} = z^{(j)} - (3 I - G^{(j)} (3 I - G^{(j)})) {[u^{(j)}, x^{(j)}; F]}^{- 1} F (z^{(j)}) .$

In the methods, SA $_{6.16, 2}$ , SA $_{6.46, 2}$ , and SA $_{6.60, 2}$ mentioned above,

$G^{(j)} = {[u^{(j)}, x^{(j)}; F]}^{- 1} [v^{(j)}, y^{(j)}; F]$ , $u^{(j)} = x^{(j)} + B^{(j)} F (x^{(j)}),$ $v^{(j)} = y^{(j)} + c F (y^{(j)})$ , $B^{(j)} = - {[u^{(j - 1)}, x^{(j - 1)}; F]}^{- 1}$ and $c \neq 0$ is an arbitrary constant for SA $_{6.16, 2}$ and SA $_{6.46, 2}$ , and $v^{(j)} = y^{(j)} + C^{(j)} F (y^{(j)})$ , $B^{(j)} = C^{(j)} = - {[u^{(j - 1)}, x^{(j - 1)}; F]}^{- 1}$ for SA $_{6.60, 2}$ .

For $(p = 2, q = - 1, r = 0, a = 0, b \neq 0, c = 0, d \neq 0)$ and $(p = 3, q = - 3, r = 1, a = 0, b \neq 0, c = 0, d \neq 0)$ , the methods in (6) are referred as $Ω_{5, 1}, Ω_{6, 1}$ respectively.

From the definition, and taking into consideration all aspects of computational cost discussed above, the values of $C_{ρ, i}$ , $E_{ρ, i}$ (the computational cost and efficiency index of scheme $M_{ρ, i}$ , respectively) can be seen in Table 1.

TABLE 1. Values of

C_{ρ, i}

E_{ρ, i}

for various schemes

Methods	$C_{ρ, i}$	$E_{ρ, i}$
$Ω_{5, 1}$	$\frac{1}{6} n (- 17 + 2 n^{2} + 3 l (7 + 5 n) + 18 μ + 3 n (9 + 4 μ))$	${(5)}^{1 / C_{5, 1}}$
$Ω_{5.54, 1}$	$\frac{1}{6} n (- 17 + 2 n^{2} + 3 l (9 + 5 n) + 18 μ + 3 n (11 + 4 μ))$	${(5.54)}^{1 / C_{5.54, 1}}$
$Ω_{6, 1}$	$\frac{1}{6} n (- 23 + 2 n^{2} + 3 l (9 + 5 n) + 18 μ + 3 n (13 + 4 μ))$	${(6)}^{1 / C_{6, 1}}$
SA $_{6, 2}$	$\frac{1}{6} n (- 17 + 2 n^{2} + 3 l (9 + 5 n) + 18 μ + 3 n (13 + 4 μ))$	${(6)}^{1 / C_{6, 2}}$
GGN $_{6, 3}$	$\frac{1}{3} n (- 8 + 2 n^{2} + l (6 + 9 n) + 9 μ + n (9 + 6 μ))$	${(6)}^{1 / C_{6, 3}}$
$Ω_{6.16, 1}$	$\frac{1}{6} n (- 29 + 2 n^{2} + 3 l (11 + 5 n) + 18 μ + 3 n (15 + 4 μ))$	${(6.16)}^{1 / C_{6.16, 1}}$
SA $_{6.16, 2}$	$\frac{1}{6} n (- 23 + 2 n^{2} + 3 l (11 + 5 n) + 18 μ + 3 n (15 + 4 μ))$	${(6.16)}^{1 / C_{6.16, 2}}$
$Ω_{6.46, 1}$	$\frac{1}{6} n (- 29 + 2 n^{2} + 3 l (13 + 5 n) + 18 μ + 3 n (17 + 4 μ))$	${(6.46)}^{1 / C_{6.46, 1}}$
SA $_{6.46, 2}$	$\frac{1}{6} n (- 35 + 2 n^{2} + 15 l (3 + n) + 18 μ + 3 n (23 + 4 μ))$	${(6.46)}^{1 / C_{6.46, 2}}$
$Ω_{6.60, 1}$	$\frac{1}{6} n (- 23 + 2 n^{2} + 3 l (11 + 5 n) + 18 μ + 3 n (15 + 4 μ))$	${(6.60)}^{1 / C_{6.60, 1}}$
SA $_{6.60, 2}$	$\frac{1}{6} n (- 41 + 2 n^{2} + 3 l (17 + 5 n) + 18 μ + 3 n (25 + 4 μ))$	${(6.60)}^{1 / C_{6.60, 2}}$

4.1 Comparison between efficiencies

To carry out the comparison between computational efficiencies of any two schemes

M_{A}

and

M_{B}

, we consider the ratio

\begin{align} R_{M_{A}}^{M_{B}} = \frac{\log E_{M_{A}}}{\log E_{M_{B}}} = \frac{C_{M_{B}} \log (ρ_{A})}{C_{M_{A}} \log (ρ_{B})}, \end{align}

()

where

ρ_{A}

and

ρ_{B}

are the convergence order of schemes

M_{A}

and

M_{B}

, respectively. Clearly, if

R_{M_{A}}^{M_{B}} > 1

, the iterative method

M_{A}

is more efficient than

M_{B}

. The boundary between the computational efficiencies is given by the equation

R_{M_{A}}^{M_{B}} = 1

, which expresses

μ

as a function of

n

and l;

(μ > 0

, n is a positive integer

\geq 2

and

l \geq 1

The comparison of efficiencies of various new schemes with existing schemes has been tabulated in Table 2, whereas the comparison of new methods among themselves has been tabulated in Table 3. These results can be easily verified using the ratio given in (58).

TABLE 2. Comparison of efficiency index of new and existing schemes

	$S A_{6, 2}$	$G G N_{6, 3}$	SA $_{6.16, 2}$	$S A_{6.46, 2}$	$S A_{6.60, 2}$
$Ω_{5, 1}$	$E_{5, 1} < E_{6, 2} \forall n \geq 33$	$E_{5 ., 1} > E_{6, 3} \forall n \geq 7$	$E_{5, 1} < E_{6.16, 2} \forall n \geq 49$	$E_{5, 1} < E_{6.46, 2} \forall n \geq 112$	$E_{5, 1} < E_{6.60, 2} \forall n \geq 119$
$Ω_{5.54, 1}$	$E_{5.54, 1} < E_{6, 2} \forall n \geq 41$	$E_{5.54, 1} > E_{6, 3} \forall n \geq 9$	$E_{5.54, 1} < E_{6.16, 2} \forall n \geq 73$	$E_{5.54, 1} < E_{6.46, 2} \forall n \geq 177$	$E_{5.54, 1} < E_{6.60, 2} \forall n \geq 182$
$Ω_{6, 1}$	$E_{6, 1} > E_{6, 2} \forall n \geq 2$	$E_{6, 1} > E_{6, 3} \forall n \geq 2$	$E_{6, 1} < E_{6.16, 2} \forall n \geq 179$	$E_{6, 1} < E_{6.46, 2} \forall n \geq 338$	$E_{6, 1} < E_{6.60, 2} \forall n \geq 312$
$Ω_{6.16, 1}$	$E_{6.16, 1} > E_{6, 2} \forall n \geq 177$	$E_{6.16, 1} > E_{6, 3} \forall n \geq 12$	$E_{6.16, 1} > E_{6.16, 2} \forall n \geq 2$	$E_{6.16, 1} < E_{6.46, 2} \forall n \geq 430$	$E_{6.16, 1} < E_{6.60, 2} \forall n \geq 366$
$Ω_{6.46, 1}$	$E_{6.46, 1} > E_{6, 2} \forall n \geq 119$	$E_{6.46, 1} > E_{6, 3} \forall n \geq 14$	$E_{6.46, 1} > E_{6.16, 2} \forall n \geq 85$	$E_{6.46, 1} > E_{6.46, 2} \forall n \geq 2$	$E_{6.46, 1} < E_{6.60, 2} \forall n \geq 1012$
$Ω_{6.60, 1}$	$E_{6.60, 1} > E_{6, 2} \forall n \geq 30$	$E_{6.60, 1} > E_{6, 3} \forall n \geq 11$	$E_{6.60, 1} > E_{6.16, 2} \forall n \geq 2$	$E_{6.60, 1} > E_{6.46, 2} \forall n \geq 2$	$E_{6.60, 1} > E_{6.60, 2} \forall n \geq 2$

TABLE 3. Comparison of efficiency index of proposed methods among themselves

	$Ω_{5, 1}$	$Ω_{5.54, 1}$	$Ω_{6, 1}$	$Ω_{6.16, 1}$	$Ω_{6.46, 1}$	$Ω_{6.60, 1}$
$Ω_{5, 1}$	–	*	*	*	*	*
$Ω_{5.54, 1}$	$n \geq 28$	–	*	*	*	*
$Ω_{6, 1}$	$n \geq 32$	$n \geq 39$	–	*	*	*
$Ω_{6.16, 1}$	$n \geq 49$	$n \geq 73$	$n \geq 178$	–	*	*
$Ω_{6.46, 1}$	$n \geq 55$	$n \geq 77$	$n \geq 120$	$n \geq 87$	–	*
$Ω_{6.60, 1}$	$n \geq 32$	$n \geq 35$	$n \geq 32$	$n \geq 3$	$n \geq 2$	–

Note 1. In Table 3 “*” indicates that the comparison of method $M_{A}$ has been done with $M_{B}$ , so the comparison of $M_{B}$ with $M_{A}$ is not shown.

Note 2. The entries in Table 3 indicate the values of “n” for which one method is more efficient than the other method, for example, the method $Ω_{5.54, 1}$ is more efficient than the method $Ω_{5, 1}$ for all $n \geq 28$ .

Note 3. For the values of “n” other than the ones mentioned in table, the comparison depends on $μ$ and “n.”

5 NUMERICAL RESULTS

In this section, we discuss some numericals to examine convergence behavior of the proposed schemes. The new methods $Ω_{5, 1}$ , $Ω_{5.54, 1}$ , $Ω_{6, 1}$ , $Ω_{6.16, 1}$ , $Ω_{6.46, 1}$ , and $Ω_{6.60, 1}$ for particular values of the parameters $P^{(0)}$ have been compared with the existing schemes $S A_{6, 2}$ , $G G N_{6, 3}, S A_{6.16, 2}$ , $S A_{6.46, 2}$ , and $S A_{6.60, 2}$ . The values of parameters used for existing schemes $S A_{6.16, 2}$ , $S A_{6.46, 2}$ , and $S A_{6.60, 2}$ have been kept same as used by the authors in Reference 24.

The computational work is compiled in the programming software

M a t h e m a t i c a 7

,²⁵ using multiple-precision arithmetic with 4096 digits. For each problem and scheme, we tabulate the number of iterations (Itr) that the initial point uses to converge to the solution, by considering the stopping criterion

‖ x^{(j + 1)} - x^{(j)} ‖_{\infty} + ‖ F (x^{(j + 1)}) ‖_{\infty} < 1 0^{- 300}

, Error (

‖ x^{(j + 1)} - x^{(j)} ‖_{\infty} + ‖ F (x^{(j + 1)}) ‖_{\infty}

) at the end of fourth step, computational cost

C_{A}

and the efficiency index

E_{A}

using the values given in Table 1 and the values “n” and “

μ

” are given in each problem. To affirm the theoretical order of convergence, we measured the computational order of convergence (COC),²⁶ using the formula

COC = \frac{\log (‖ F (x^{(j + 1)}) ‖ / ‖ F (x^{(j)}) ‖)}{\log (‖ F (x^{(j)}) ‖ / ‖ F (x^{(j - 1)}) ‖)},

where the last three approximations are used to calculate COC. The e-Time reported in the last column is the mean CPU time. This is calculated by performing the program 25 times and taking its mean, by considering the above stopping criterion used in single execution of the program.

To compute the values $μ$ and l for numerical examples, we express the evaluation cost of elementary functions in terms of products. This depends on the arithmetic used, computer, and the software²⁷ (www.mpfr.org/mpfr-2.1.0/timings.html). These programs have been run in $M a t h e m a t i c a 7$ on a processor with specifications: Intel(R) Core (TM) i5-3210M CPU @2.50 GHz (64-bit machine) Microsoft Windows 7 (2009) and are given in Table 4. This shows the estimates of cost of the elementary functions in product units, where the running time of one product is measured in milliseconds (ms). From Table 4, one can observe that the time taken by one quotient is twice the time taken by product and hence $l = 2$ . The following numerical examples are considered to verify the results.

TABLE 4. CPU-time and estimate of computational cost of elementary functions for

x = \sqrt{3} - 1

and

y = \sqrt{5}

Function	$x * y$	$x / y$	$\sqrt{x}$	$\exp x$	$\ln x$	$\sin x$	$\cos x$	$\arccos x$	$\arcsin x$	$\arctan x$	$x^{y}$
CPU-time	0.04212	0.08424	0.03822	2.7924	2.4383	3.0093	3.0061	5.1387	4.9951	4.9702	5.6972
Cost	1	2	0.91	66.30	57.89	71.45	71.44	122	118.59	118.00	135.26

Problem 1.Here, we study the mixed Hammerstein's integral equation:³

x (s) = 1 + \frac{1}{5} \int_{0}^{1} G (s, t) x {(t)}^{3} d t, s, t \in [0, 1] where x \in C [0, 1]

and the kernel

G

G (s, t) = \{\begin{array}{lr} (1 - s) t, & t \leq s, \\ s (1 - t), & s \leq t . \end{array}

The above integral equation is discretized into a finite dimensional problem with the help of Gauss–Legendre quadrature formula for abscissas $t_{m}$ and weights ${\overline{ω}}_{m}$ determined for $n = 12$ . Let $x_{m}$ denote the approximation $x (t_{m}), (m = 1, 2, . . ., 12)$ . This results in a nonlinear system of equations given below:

5 x_{i} - 5 - \sum_{m = 1}^{12} a_{i m} x_{m}^{3} = 0, i = 1, 2, \dots, 12,

where

a_{i m} = \{\begin{array}{lcr} \overline{ω_{m}} t_{m} (1 - t_{i}) & i f & m \leq i, \\ \overline{ω_{m}} t_{i} (1 - t_{m}) & i f & i < m . \end{array}

The solution of the problem is

$α = {1.0009727166180117 \dots, 1.0048748186599682 \dots, 1.0109092367279116 \dots, 1.0176086786577538 \dots, 1.0233126345057937 \dots, 1.0265822324745664 \dots, 1.0265822324745664 \dots, 1.0233126345057937 \dots, {1.0176086786577538 \dots, 1.0109092367279116 \dots, 1.0048748186599682 \dots, 1.0009727166180117 \dots}}^{t}$ , which is correct up to 16 decimal places. The methods are tested using the parameters $(n, μ) = (12, 15)$ and the initial approximation of the solution as $x^{(0)} = {- 0.9, - 0.9, \dots, - 0.9}^{t}$ and the results are displayed in Table 5.

TABLE 5. Numerical results of different schemes for problem 1

Methods	$ρ$	Error	Itr	$C_{A}$	$E_{A}$	COC	e-Time
$Ω_{5, 1} (b = d = - 0.01)$	5	3.116 ( $-$ 181)	5	6854	1.000234844	5.000	7.190
$Ω_{5.54, 1} (P^{(0)} = - 0.01 I, d = - 0.01)$	5.54	6.859 ( $-$ 226)	5	7022	1.00024383	5.645	7.206
$Ω_{6, 1} (b = - 0.01, d = - 0.01)$	6	2.482 ( $-$ 283)	5	7154	1.00025049	6.000	6.806
SA $_{6, 2} (b = c = - 0.01)$	6	1.779 ( $-$ 245)	5	7166	1.00025007	6.000	7.610
GGN $_{6, 3}$	6	1.428 ( $-$ 7)	7	7324	1.00024467	6.000	10.364
$Ω_{6.16, 1} (P^{(0)} = - 0.01 I, d = - 0.01)$	6.16	5.490 ( $-$ 310)	4	7310	1.00024874	6.163	5.884
SA $_{6.16, 2} (B^{(0)} = - 0.01 I, c = - 0.01)$	6.16	3.190 ( $-$ 272)	5	7322	1.00024833	6.162	6.625
$Ω_{6.46, 1} (P^{(0)} = C^{(0)} = 0.01 I)$	6.46	3.881 ( $-$ 349)	4	7478	1.00024951	6.521	6.104
SA $_{6.46, 2} (B^{(0)} = - 0.01 I, c = - 0.01)$	6.46	1.800 ( $-$ 299)	5	7922	1.00023553	6.464	6.407
$Ω_{6.60, 1} (P^{(0)} = C^{(0)} = - 0.01 I)$	6.60	1.254 ( $-$ 363)	4	7322	1.00025776	6.701	5.203
SA $_{6.60, 2} (B^{(0)} = C^{(0)} = - 0.01 I)$	6.60	2.054 ( $-$ 318)	4	8078	1.00023363	6.615	6.121

Problem 2.Now, we discuss a system of fifty equations given by:

\arctan x_{i} + 1 - 2 \sum_{m = 1, m \neq i}^{50} x_{m}^{2} = 0, 1 \leq i \leq 50 .

The approximate solution is

α = {- 0.0960567972718922 \dots, - 0.0960567972718922 \dots, \dots, - 0.0960567972718922 {\dots}}^{t},

which is correct up to 16 digits. With initial guess as

x^{(0)} = {- \frac{1}{3}, - \frac{1}{3} \dots, - \frac{1}{3}}^{t}

and parameters as

(n, μ) = (50, 120)

, the methods are tested. Results for this problem are shown in Table 6.

TABLE 6. Numerical results of different schemes for problem 2

Methods	$ρ$	Error	Itr	$C_{A}$	$E_{A}$	COC	e-Time
$Ω_{5, 1} (b = d = - 0.01)$	5	2.828 ( $-$ 28)	6	683,625	1.0000023543	5.000	481.21
$Ω_{5.54, 1} (P^{(0)} = - 0.01 I, d = - 0.01)$	5.54	5.996 ( $-$ 55)	5	686,225	1.0000024948	5.646	509.44
$Ω_{6, 1} (b = d = - 0.01)$	6	1.975 ( $-$ 45)	6	688,675	1.0000026018	6.000	540.02
SA $_{6, 2} (b = c = - 0.01)$	6	1.887 ( $-$ 43)	6	688,725	1.0000026016	6.000	596.58
GGN $_{6, 3}$	6	2.866 ( $-$ 48)	6	723,900	1.0000024752	6.000	640.85
$Ω_{6.16, 1} (P^{(0)} = - 0.01 I, d = - 0.01)$	6.16	4.650 ( $-$ 49)	5	691,225	1.0000026302	6.162	346.76
SA $_{6.16, 2} (B^{(0)} = - 0.01 I, c = - 0.01)$	6.16	3.511 ( $-$ 47)	6	691,275	1.0000026300	6.162	481.42
$Ω_{6.46, 1} (P^{(0)} = C^{(0)} = - 0.01 I)$	6.46	6.862 ( $-$ 85)	5	693,825	1.0000026889	6.541	417.29
SA $_{6.46, 2} (B^{(0)} = - 0.01 I, c = - 0.01)$	6.46	2.310 ( $-$ 65)	5	701,375	1.0000026600	6.463	452.35
$Ω_{6.60, 1} (P^{(0)} = C^{(0)} = - 0.01 I)$	6.60	1.855 ( $-$ 86)	5	691,275	1.0000027298	6.707	443.78
SA $_{6.60, 2} (B^{(0)} = C^{(0)} = - 0.01 I)$	6.60	1.271 ( $-$ 67)	5	703,925	1.0000026808	6.619	489.79

Problem 3.Now, we examine the following boundary value problem:

y^{''} = \frac{1}{2} y^{3} + 3 y^{'} - \frac{3}{2 - x} + \frac{1}{2}, y (0) = 0, y (1) = 1 .

Assuming that the interval $[0, 1]$ is partitioned as, $x_{0} = 0 < x_{1} < x_{2} < \dots < x_{m} = 1$ , $x_{j + 1} = x_{j} + h$ , $h = \frac{1}{m},$ we define $y_{0} = y (x_{0}) = 0, y_{1} = y (x_{1}), \dots, y_{m - 1} = y (x_{m - 1}), y_{m} = y (x_{m}) = 1$ . The problem is discretized using the numerical formulas for first and second derivative given by:

y_{k}^{'} = \frac{y_{k + 1} - y_{k - 1}}{2 h}, y_{k}^{''} = \frac{y_{k - 1} - 2 y_{k} + y_{k + 1}}{h^{2}}, k = 1, 2, 3, . . ., m - 1 .

This yields a system of $m - 1$ nonlinear equations in $m - 1$ variables:

y_{k + 1} - 2 y_{k} + y_{k - 1} - \frac{h^{2}}{2} y_{k}^{3} - \frac{3}{2} h (y_{k + 1} - y_{k - 1}) + \frac{3}{2 - x_{k}} h^{2} - \frac{1}{2} h^{2} = 0, k = 1, 2, 3, \dots, m - 1 .

In this problem, we take

m = 200

. Hence, a system of

n = 199

nonlinear equations is solved. The initial guess is taken as

y^{(0)} = {\frac{43}{100}, \frac{43}{100}, . . ., \frac{43}{100}}^{t}

and we apply different methods to acquire approximate solution. The solution correct up to 16 decimal places is

α = {0.0025062505477845 \dots, 0.0050250953257465 \dots, \dots, {0.98019785632803 \dots, 0.99004966827654 \dots}}^{t} .

The values of the parameters are

(n, μ) = (199, 5) .

Numerical results for the problem can be seen in Table 7.

TABLE 7. Numerical results of different schemes for problem 3

Methods	$ρ$	Error	Itr	$C_{A}$	$E_{A}$	COC	e-Time
$Ω_{5, 1} (b = d = - 0.01)$	5	2.826 ( $-$ 250)	5	3,402,900	1.000000472961	5.000	76.48
$Ω_{5.54, 1} (P^{(0)} = - 0.01 I, d = - 0.01)$	5.54	7.837 ( $-$ 308)	4	3,442,899	1.000000497253	5.648	72.48
$Ω_{6, 1} (b = d = - 0.01)$	6	1.872 ( $-$ 412)	4	3,482,301	1.000000514533	6.001	76.15
SA $_{6, 2} (b = c = - 0.01)$	6	9.243 ( $-$ 385)	4	3,482,500	1.000000514504	6.000	84.03
GGN $_{6, 3}$	6	4.117 ( $-$ 325)	4	6,009,402	1.000000298159	6.003	86.31
$Ω_{6.16, 1} (P^{(0)} = - 0.01 I, d = - 0.01)$	6.16	1.411 ( $-$ 448)	4	3,522,101	1.000000516191	6.164	72.61
SA $_{6.16, 2} (B^{(0)} = - 0.01 I, c = - 0.01)$	6.16	5.067 ( $-$ 423)	4	3,522,300	1.000000516161	6.163	78.15
$Ω_{6.46, 1} (P^{(0)} = C^{(0)} = - 0.01 I)$	6.46	4.356 ( $-$ 504)	4	3,562,100	1.000000523744	6.539	73.74
SA $_{6.46, 2} (B^{(0)} = - 0.01 I, c = - 0.01)$	6.46	2.743 ( $-$ 488)	4	3,681,102	1.000000506812	6.466	85.24
$Ω_{6.60, 1} (P^{(0)} = C^{(0)} = - 0.01 I)$	6.60	1.453 ( $-$ 528)	4	3,522,300	1.000000535749	6.699	72.48
SA $_{6.60, 2} (B^{(0)} = C^{(0)} = - 0.01 I)$	6.60	4.591 ( $-$ 514)	4	3,720,902	1.000000507153	6.610	89.17

Problem 4.Now, we focus on the Van der Pol equation^{28, 29} which governs the flow of current in a vacuum tube is defined by:

y^{''} - \tilde{μ} (y^{2} - 1) y^{'} + y = 0, \tilde{μ} > 0 .

The boundary conditions are

y (0) = 0, y (2) = 1 .

The interval

[0, 2]

is partitioned as follows with nodes as

x_{0} = 0 < x_{1} < x_{2} < \dots < x_{m} = 2

x_{j + 1} = x_{0} + j h

h = \frac{2}{m},

and we define

y_{0} = y (x_{0}) = 0, y_{1} = y (x_{1}), \dots, y_{m - 1} = y (x_{m - 1}), y_{m} = y (x_{m}) = 1

. The problem is discretized using the numerical formulas for first and second derivative as given above:

This yields a system of nonlinear equations of size $(m - 1) \times (m - 1)$ :

\begin{align} (2 h^{2} - 4) y_{1} + (2 + \tilde{μ} h) y_{2} - \tilde{μ} h y_{1}^{2} y_{2} = 0, \\ (2 - \tilde{μ} h) y_{k - 1} + (2 h^{2} - 4) y_{k} + (2 + \tilde{μ} h) y_{k + 1} - \tilde{μ} h y_{k}^{2} (y_{k + 1} - y_{k - 1}) = 0, k = 2, 3, \dots, m - 2, \\ (2 - \tilde{μ} h) y_{m - 2} + 2 (2 h^{2} - 4) y_{m - 1} + (2 + \tilde{μ} h) + \tilde{μ} h y_{n - 1}^{2} (1 - y_{n - 2}) = 0 . \end{align}

In this problem, we take

\tilde{μ} = 1 / 2, m = 51

. Hence, a system of

n = 50

nonlinear equations is solved. The initial guess is taken as

y^{(0)} = Table [\frac{1}{i}, {i, 1, n}]

and we apply different methods to acquire approximate solution.

The values of the parameters are $(n, μ) = (50, 9) .$ Numerical results for the problem can be seen in Table 8.

TABLE 8. Numerical results of different schemes for Problem 4

Methods	$ρ$	Error	Itr	$C_{A}$	$E_{A}$	COC	e-Time
$Ω_{5, 1} (b = d = - 0.01)$	5	4.269 ( $-$ 104)	5	111,975	1.000014373	4.998	5.807
$Ω_{5.54, 1} (P^{(0)} = - 0.01 I, d = - 0.01)$	5.54	2.898 ( $-$ 114)	5	114,575	1.00001494224	5.012	6.306
$Ω_{6, 1} (b = d = - 0.01)$	6	4.419 ( $-$ 164)	5	117,025	1.0000153110	6.011	6.686
SA $_{6, 2} (b = c = - 0.01)$	6	2.712 ( $-$ 149)	5	117,075	1.0000153045	6.009	7.451
GGN $_{6, 3}$	6	1.692 ( $-$ 61)	6	152,250	1.00001176860	4.014	8.123
$Ω_{6.16, 1} (P^{(0)} = - 0.01 I, d = - 0.01)$	6.16	6.569 ( $-$ 185)	6	119,575	1.00001520460	6.165	6.258
SA $_{6.16, 2} (B^{(0)} = - 0.01 I, c = - 0.01)$	6.16	4.881 ( $-$ 174)	5	119,625	1.0000151983	6.164	7.125
$Ω_{6.46, 1} (P^{(0)} = C^{(0)} = - 0.01 I)$	6.46	7.672 ( $-$ 187)	5	122,175	1.00001527026	6.003	6.231
SA $_{6.46, 2} (B^{(0)} = - 0.01 I, c = - 0.01)$	6.46	1.495 ( $-$ 201)	5	129,725	1.0000143815	6.470	7.896
$Ω_{6.60, 1} (P^{(0)} = C^{(0)} = - 0.01 I)$	6.60	3.752 ( $-$ 194)	5	119,625	1.00001577500	5.999	6.208
SA $_{6.60, 2} (B^{(0)} = C^{(0)} = - 0.01 I)$	6.60	2.235 ( $-$ 213)	5	132,275	1.00001426636	6.614	7.458

Problem 5.A nonlinear partial differential equation can be solved with help of finite difference discretization. This yields a system of nonlinear algebraic equations. Here, we consider nonlinear PDE known as Fisher's equation³⁰ with homogeneous Neumann's boundary conditions and the diffusion coefficient D give below:

\begin{align} \{\begin{array}{l} u_{t} = D u_{x x} + u (1 - u), \\ u (x, 0) = 1.5 + 0.5 C o s (π x), 0 \leq x \leq 1, \\ u_{x} (0, t) = 0 \forall t \geq 0, u_{x} (1, t) = 0 \forall t \geq 0 . \end{array} \end{align}

()

Assume M and N to be number of partitions in x and t directions, and their respective step size as h and k. Let $u = u (x, t)$ denote the exact solution of the given system and its approximate solution be $w_{i, j} ≃ u (x_{i}, t_{j})$ at the grid points of the mesh created. Applying backward difference to $u_{t}$ and central difference to the other terms, that is, $u_{t} (x_{i}, t_{j}) ≃ \frac{(w_{i, j} - w_{i, j - 1})}{k}$ , $u_{x} (x_{i}, t_{j}) ≃ \frac{(w_{i + 1, j} - w_{i - 1, j})}{2 h}$ and $u_{x x} (x_{i}, t_{j}) ≃ \frac{(w_{i + 1, j} - 2 w_{i, j} + w_{i - 1, j})}{h^{2}}$ to solve the PDE with values $M = 21$ and $N = 21$ , and $D = 1$ gives rise to nonlinear system equations of size $m * n$ where $m = M - 1$ , $n = N - 1$ . Here, we take two different initial guesses $x^{(0)} = Table [\frac{5}{4} + \frac{i}{2 m n}, {i, 1, m * n}]$ and $x^{(0)} = Table [\frac{12}{5} + \frac{i}{2 m n}, {i, 1, m * n}]$ . This has resulted in a system of size 400, which is solved using schemes mentioned earlier.

In case of PDE's, since the precision of solution required is low, so the stopping criterion used is $‖ F (x) ‖_{2} < 1 0^{- 8}$ . We count iterations and mean CPU time (e-Time) used by schemes to arrive at the stopping criterion. We take average of 25 performances of the program for e-Time. The results are displayed in Table 9, and the approximate solution of the system can be seem in Figure 1.

Details are in the caption following the image — **FIGURE 1**
Open in figure viewer PowerPoint

Approximate solution of Fisher's equation

TABLE 9. Numerical results of different schemes for Problem 5

Methods	Itr	e-Time	Itr	e-Time
	$Table [\frac{5}{4} + \frac{i}{2 m n}, {i, 1, m * n}]$		$Table [\frac{12}{5} + \frac{i}{2 m n}, {i, 1, m * n}]$
$Ω_{5, 1} (b = d = - 0.01)$	3	9.81	7	89.25
$Ω_{5.54, 1} (P^{(0)} = - 0.1 I, d = - 0.1)$	3	8.52	4	80.24
$Ω_{6, 1} (b = d = - 0.01)$	3	8.06	Diverges	–
SA $_{6, 2} (b = c = - 0.01)$	6	11.65	Diverges	–
GGN $_{6, 3}$	3	10.64	4	100.25
$Ω_{6.16, 1} (P^{(0)} = - 0.1 I, d = - 0.01)$	3	7.1	Diverges	–
SA $_{6.16, 2} (B^{(0)} = - 0.01 I, c = - 0.01)$	Diverges	–	4	82.17
$Ω_{6.46, 1} (P^{(0)} = C^{(0)} = - 0.01 I)$	3	7.27	4	72.14
SA $_{6.46, 2} (B^{(0)} = - 0.01 I, c = - 0.01)$	Diverges	–	4	86.12
$Ω_{6.60, 1} (P^{(0)} = C^{(0)} = - 0.01 I)$	3	7.17	4	70.16
SA $_{6.60, 2} (B^{(0)} = C^{(0)} = - 0.01 I)$	Diverges	–	3	85.75

6 BASINS OF ATTRACTION

The dynamical behavior of the proposed algorithms is examined by generating their basins of attraction and comparing them with the existing methods mentioned earlier. We examine the following test problems which are systems of polynomials.

Example 1.

\begin{align} \{\begin{array}{l} {(x_{1} - 3)}^{2} - 16 x_{2}^{2} - 1 = 0, \\ x_{1}^{2} - x_{2}^{2} - 1 = 0, \end{array} \end{align}

with approximate solutions

{{- 1.48062, - 1.0919}, {- 1.48062, 1.0919}, {1.08062, - 0.409573}, {1.08062, 0.409573}}

Example 2.

\begin{align} \{\begin{array}{l} x_{1} x_{2} + x_{1} - x_{2} - 1 = 0, \\ x_{1} x_{2} - x_{1} + x_{2} - 1 = 0, \end{array} \end{align}

with solutions

{{1, 1}, {- 1, - 1}

Example 3.

\begin{align} \{\begin{array}{l} x_{1}^{2} - 1 = 0, \\ x_{2}^{2} - 1 = 0, \end{array} \end{align}

with its solutions as

{{1, 1}, {{1, - 1}, {{- 1, 1}, {- 1, - 1}

For generating basins of attraction, we take a square $[- 2, 2] \times [- 2, 2]$ of $401 \times 401$ points containing all solutions of system of equations. The iterative algorithm is then applied to all points in the square as initial points. We allocate color to each point depending on the root to which the corresponding orbit of the scheme, starting from the point, converges. In case the corresponding orbit does not approach any root of the polynomial system or tends to infinity, with tolerance $1 0^{- 3}$ and in a maximal of 50 iterations, black color is assigned to the point. The number of iterations required to attain the solution with the given stopping criterion is shown in brighter or darker colors. The brighter color means number of iterations required is lower.

The attraction basin of examples generated using respective schemes have been displayed in Figures 2, 3, and 4.

To compare the basins quantitatively, we calculate total number of initial points convergent in all test problem and then find the average number of iterations required per point. The average performance results of the schemes in comparison are displayed in Table 10. We can derive following conclusions on considering the average of three examples in Table 10:

TABLE 10. Numerical results of schemes

Ω_{5, 1}

Ω_{5.54, 1}

Ω_{6, 1}

, SA

_{6, 2}

, GGN

_{6, 3}

Ω_{6.16, 1}

, SA

_{6.16, 2}

Ω_{6.46, 1}

, SA

_{6.46, 2}

Ω_{6.60, 1}

, and SA

_{6.60, 2}

Methods	eg1	eg2	eg3	Average	%	eg1	eg2	eg3	Average
	Average convergent points					Average iterations
$Ω_{5, 1} (b = d = - 0.01)$	120,404	129,295	123,128	12,4275.67	77.29	15.46	12.80	14.64	14.3
$Ω_{5.54, 1} (P^{(0)} = - 0.01 I, d = - 0.01)$	123,222	132,716	127,998	127,978.67	79.59	14.74	11.85	13.38	13.32
$Ω_{6, 1} (P^{(0)} = - 0.01, d = - 0.01)$	115,285	122,779	118,233	118,765.67	73.86	16.68	14.48	15.79	15.65
SA $_{6, 2} (b = c = - 0.01)$	132,490	141,946	138,360	137,598.67	85.57	12.84	9.78	10.97	11.20
GGN $_{6, 3}$	159,998	160,668	160,000	160,222	99.63	3.38	3.48	3.16	3.34
$Ω_{6.16, 1} (P^{(0)} = - 0.01 I, d = - 0.01)$	112,973	124,303	116,878	118,051.33	73.41	17.23	9.25	16.09	14.19
SA $_{6.16, 2} (B^{(0)} = - 0.01 I, c = - 0.01)$	105,756	122,873	114,825	114,484.67	71.20	19.49	14.40	17.06	16.98
$Ω_{6.46, 1} (P^{(0)} = C^{(0)} = - 0.01 I)$	149,241	151,659	151,223	150,707.67	93.72	9.49	7.99	8.83	8.77
SA $_{6.46, 2} (B^{(0)} = 0.01 I, c = - 0.01)$	144,509	146,957	145,159	145,541.67	90.51	14.22	12.56	13.84	13.54
$Ω_{6.60, 1} (P^{(0)} = C^{(0)} = - 0.01 I)$	148,918	151,597	151,255	150,704	93.72	9.54	7.99	8.84	8.79
SA $_{6.60, 2} (B^{(0)} = C^{(0)} = - 0.01 I)$	144,509	146,957	145,160	145,542	90.51	14.22	12.56	13.84	13.54

(i) Both average number of convergent points and average number of iterations of schemes $Ω_{5, 1}$ and $Ω_{5.54, 1}$ indicate that the new algorithm with memory $Ω_{5.54, 1}$ is better than the method without memory $Ω_{5, 1}$ .

(ii) By considering convergent points and average number of iterations in case schemes of order 6 and above, methods with memory $Ω_{6.60, 1}$ and $Ω_{6.46, 1}$ are best and then followed by SA $_{6.60, 2}$ , SA $_{6.46, 2}$ . This is followed by $Ω_{6.16, 1}$ and further followed by SA $_{6.16, 2}$ .

Therefore, from Table 10 one can say that the new schemes $Ω_{6.60, 1}$ and $Ω_{6.46, 1}$ are best among all methods.

7 CONCLUSIONS

In this article, we proposed two families of fifth-order and sixth-order derivative-free schemes to solve nonlinear systems of equations, numerically. Further, from these families we derived several derivative-free schemes with memory. The proposed algorithms involve evaluation of only two divided differences and inverse of only one matrix per step. Consequently, the schemes are low in computational cost and result in high efficiency index. Some special members of the collection have been compared with some of the existing schemes using numerical examples and comparison of efficiencies has been shown. Various parameters like the error in approximation of the roots at end of fourth iterations, the number of iterations required, and the e-Time (CPU time) utilized by the scheme to accomplish the stopping criterion, verify that the new schemes give faster and closer approximation of roots than the already existing schemes. These results confirmed that the proposed schemes with memory $Ω_{6.60, 1}$ and $Ω_{6.46, 1}$ are best and more efficient than the existent schemes. Finally, these results have been revalidated with help of their basins of attraction. Thus, it can be ascertained that these novel schemes exhibit highly efficient nature particularly for large-scale systems.

ACKNOWLEDGMENT

The authors would like to thank all the five anonymous reviewers for their useful comments and suggestions, which have improved the final version of this manuscript.

CONFLICT OF INTEREST

The authors declare no potential conflict of interests.

Biographies

Mona Narang received her B.Sc.(Hons. School), M.Sc.(Hons. School), and Ph.D. in numerical analysis from Department of Mathematics, Panjab University, Chandigarh. Currently, she is working as an Associate Professor in the Department of Mathematics, D.A.V. College, Chandigarh. Her research interests include iterative schemes for numerical solution of system of nonlinear equations and its applications in solving real-life problems, dynamical study of iterative methods.
Saurabh Bhatia received his B.Sc.(Hons. School), M.Sc.(Hons. School), and Ph.D. in valuation theory from Department of Mathematics, Panjab University, Chandigarh. Currently, he is working in the Department of Mathematics, University Institute of Engineering and Technology, Panjab University, Chandigarh. His research interests include iterative schemes for numerical solution of nonlinear equations and system of nonlinear equations and its applications in solving real-life problems.
Vinay Kanwar did his Ph.D. in fluid dynamics in the year 1996 from Department of Mathematics, Himachal Pradesh University, Shimla, India. Currently, he is working as a Professor in Mathematics at University Institute of Engineering and Technology, Panjab University, Chandigarh. Presently, he is working in numerical analysis, fixed point theory, specifically in iterative techniques for solving nonlinear equations and nonlinear systems, numerical solution of differential equations, and their applications in solving real-life problems.

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An efficient family of Steffensen-type methods with memory for solving systems of nonlinear equations

Abstract

1 INTRODUCTION

2 DEVELOPMENT AND CONVERGENCE INVESTIGATION

3 METHODS WITH MEMORY

3.1 Particular cases of family (29)

4 COMPUTATIONAL EFFICIENCY

4.1 Comparison between efficiencies

5 NUMERICAL RESULTS

6 BASINS OF ATTRACTION

7 CONCLUSIONS

ACKNOWLEDGMENT

CONFLICT OF INTEREST

Biographies

REFERENCES

Citing Literature

Figures

References

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An efficient family of Steffensen-type methods with memory for solving systems of nonlinear equations

Abstract

1 INTRODUCTION

2 DEVELOPMENT AND CONVERGENCE INVESTIGATION

3 METHODS WITH MEMORY

3.1 Particular cases of family (29)

4 COMPUTATIONAL EFFICIENCY

4.1 Comparison between efficiencies

5 NUMERICAL RESULTS

6 BASINS OF ATTRACTION

7 CONCLUSIONS

ACKNOWLEDGMENT

CONFLICT OF INTEREST

Biographies

REFERENCES

Citing Literature

Figures

References

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