Higher order multi-step interval iterative methods for solving nonlinear equations in $R^n$

Abstract

In this paper, higher order multi-step interval iterative methods are proposed for solving nonlinear equations in $R^n$. Each method leads to an an interval vector enclosing the approximate solution along with the rigorous error bounds automatically. These methods require solving linear interval systems of equations. Interval extension of Gaussian elimination algorithm is described and used for solving them. The convergence analysis of both the methods is established to show their third and fourth order of convergence. A number of numerical examples are worked out and the performance in terms of iterations count and diameters of resulting interval vectors are measured.

1 Introduction

One of the most difficult and challenging problems of numerical analysis is to have iterative methods of high orders along with their convergence analysis and error estimations for solutions of nonlinear equations [4, 20, 21]. These methods under suitable convergence conditions lead to the approximate solutions. However, the validity of the solutions obtained by these methods can be verified by a separate error analysis carried out either statistically or in some other random ways. One such way is the use of interval analysis [1, 2, 11, 12, 15] which automatically and simultaneously approximates the solutions and validates the results by giving rigorous error bounds on accumulated rounding errors, approximation errors, truncation errors and errors due to uncertainties in the initial data. Thus, interval versions of these methods are developed which not only give an approximation to the solution but also rigorous error bounds automatically. An interval iterative method begins with a starting interval vector including a solution and generates a sequence of improved monotonic interval vectors each enclosing the solution and converging either to an interval vector including the solution or to the solution itself. Generally, a starting interval vector is not known and one tries to obtain it either by conditions on appropriate operators which form the basis of these methods or by some other bisection techniques discussed in [17]. Next, convergence analysis is carried out to provide conditions which ensures the sequence of interval vectors converging monotonically to the solution. The aim of this paper is to propose higher order multi-step interval iterative methods for enclosing solutions of nonlinear equations in $R^n$ given by

$$\begin{aligned} F(x) = 0 \end{aligned}$$

(1.1)

where $F:D \subset R^n \rightarrow R^n$ is a continuously differentiable operator in the sense of Fréchet and D be a convex domain. Due to the development of advanced computer H/W and S/W, this problem has gained an added advantage. These problems have been extensively studied due to their wide range of appearances in a number of fields. The Kinetic theory of gases and elasticity involve boundary value problems [6, 18] whose solutions are obtained by solving (1.1). The solutions of difference or differential equations arising in mathematical modeling of the dynamical systems give rise to a number of such equations whose solutions represent the stability of these systems. Optimization problems such as the locations of extremal points of functions also require solutions of (1.1). Other applied areas such as chemical engineering, transportation, operations research also involve solving these equations either individually or collectively. Many one point and multi points iterative methods of various orders to solve (1.1) along with their convergence analysis are discussed in [18, 19, 23, 26]. Most of these methods require the convexity conditions on F. The quadratically convergent Newton’s method for (1.1) is given by

$$\begin{aligned} x^{(k+1)} = x^{(k)}-F'(x^{(k)})^{-1} F(x^{(k)}), k \ge 0 \end{aligned}$$

(1.2)

where $x^{(0)}$ be a suitably chosen initial vector and $F'(x)^{-1}$ is the inverse of the derivative in the sense of Fréchet of F(x). The most famous theoretical results on its convergence is the Kantorovich theorem [18] which gives sufficient conditions to guarantee the existence and uniqueness as well as convergence to a solution $x^*$. These sufficient conditions are very difficult to check in practice. The quadratically convergent interval extension of (1.2) is proposed in [16]. Starting with an interval vector $\hat{X}^{(0)}$ containing $x^{*}$, this is given by

$$\begin{aligned} N(\hat{X}^{(k)})= & {} m(\hat{X}^{(k)})- \hat{F'}(\hat{X}^{(k)})^{-1} F(m(\hat{X}^{(k)})),\nonumber \\ \hat{X}^{(k+1)}= & {} N(\hat{X}^{(k)}) \cap \hat{X}^{(k)}, k \ge 0 \end{aligned}$$

(1.3)

where $N(\hat{X}) = m(\hat{X}) - \hat{F'}(\hat{X})^{-1} F(m(\hat{X}))$ and $m(\hat{X})$ are the interval Newton operator and mid point of $\hat{X}$. Another version of interval Newton method using Newton operator $NN(\hat{X}^{(k)})$ is described in [16]. Assuming that the interval extension of Gaussian elimination algorithm (IGEA) is feasible, it is given for $k = 0,1,2, \ldots$ by

$$\begin{aligned} NN(\hat{X}^{(k)})= & {} m(\hat{X}^{(k)})-IGEA(\hat{F'}(\hat{X}^{(k)}), F(m(\hat{X}^{(k)}))), \nonumber \\ \hat{X}^{(k+1)}= & {} NN(\hat{X}^{(k)}) \cap \hat{X}^{(k)}, \end{aligned}$$

(1.4)

where $IGEA(\hat{A}, b)$ is applied on $\hat{A}$ and fixed b. If the starting interval vector contains no solution then the above interval iterative method breaks down after a finite number of steps. A linearly convergent interval iterative method proposed in [13] for solving (1.1) can be given for $k = 0,1,2, \ldots$ by

$$\begin{aligned} K(\hat{X}^{(k)})= & {} m(\hat{X}^{(k)})- R^{(k)} m(\hat{X}^{(k)})+(I-R^{(k)}\hat{F'}(\hat{X}^{(k)}))(\hat{X}^{(k)}-m(\hat{X}^{(k)})), \nonumber \\ \hat{X}^{(k+1)}= & {} K(\hat{X}^{(k)}) \cap \hat{X}^{(k)}, \end{aligned}$$

(1.5)

where $R^{(k)} = (m(\hat{F'}(\hat{X}^{(k)})))^{-1}$ and $\hat{X}^{(0)}$ be a suitable initial interval vector containing the root. Krawczyk’s operator is given by $K(\hat{X}) = m(\hat{X})- R~ m(\hat{X}) + (I-R~ \hat{F'}(\hat{X}))(\hat{X}-m(\hat{X}))$, where, R be an arbitrary non singular real matrix. Later, Alefeld et al. [3] extended the method (1.5) to second order under suitable conditions. For other modifications of (1.5), one can refer to the works of [10, 14]. Recently, Petković [22] developed a third and fourth order interval iterative methods in I(R) for enclosing the simple roots of nonlinear equations. Motivated from his work, we propose higher order multi-step interval iterative methods for enclosing the roots of (1.1).

The aim of this paper is to propose higher order multi-step interval iterative methods for solving nonlinear equations in $R^n$. Each method leads to an an interval vector enclosing the approximate solution along with the rigorous error bounds automatically. These methods require solving linear interval systems of equations. Interval extension of Gaussian elimination algorithm is described and used for solving them. The convergence analysis of both the methods is established to show their third and fourth order of convergence. A number of numerical examples are worked out and the performance in terms of iterations count and diameters of resulting interval vectors are measured.

The organization of the paper is arranged as follows. The basic concepts used in the paper are given in Sect. 2. The proposed interval iterative methods of order three and four in $R^n$ and their convergence analysis are established in Sects. 3 and 4, respectively. In Sect. 5, a number of examples are worked out to demonstrate the efficiency of our methods. Finally, conclusions are included in Sect. 6.

2 Basic concepts

In this section, the concepts of interval analysis is described in brief. For detail explanations, one can refer to [1, 16]. Let I(R) is the set of real compact intervals over the set of reals R defined as $I(R) = \{[x,y] | x \le y, x , y \in R\}$. Operations in I(R) are defined by $Z = X * Y = \{ z = x * y | x \in X, y \in Y \}$, where, $X, Y \in I(R)$, and the symbol $*$ stands for $+,-,\times ,$ and  /  in R. We shall use the same operator in I(R) and differentiate it from the context in which it is used. It is to be noted that the algebraic properties of I(R) are different from those of real arithmetic operations, for instance, the subtraction and division are not the inverse operations of addition and multiplication, respectively. Another main difference given by the fact that the distribution law of real arithmetic is not valid in general. Only the following so called subdistributive law given for $X,Y,Z \in I(R)$ by $X ( Y + Z ) \subseteq X Y + X Z$ holds in I(R). The distance and diameter between two intervals $X = [\underline{x}, \overline{x}]$ and $Y = [\underline{y}, \overline{y}]$ are defined as $q(X, Y) = \max \{|\underline{x}-\underline{y}|, |\overline{x}-\overline{y}|\}$ and $d(X) = (\overline{x}-\underline{x}).$ The absolute value of an interval X can be also be defined in terms of distance and is given by $| X | = q(X, 0).$ In order to consider multidimensional problems, the above concepts can easily be extended to interval vectors and interval matrices applied component wise. Let $I(R^n), M(R^{n \times n})$, and $L(R^{n \times n})$ denote the set of $n \times 1$ interval vectors, the set of $n \times n$ interval matrices and the set of $n \times n$ real matrices respectively. If $F:R^n \rightarrow R^n$ is a differentiable mapping then $\hat{F} : I(R^n) \rightarrow I(R^n)$ and $\hat{F'}:I(R^n) \rightarrow M(R^{n \times n})$ denote an interval extension of F and an interval extension of $F'$ respectively. In our work, any interval vector $X \in I(R^n)$ and any interval matrix $A \in M(R^{n \times n})$ is represented by $\hat{X}$ and $\hat{A}$, respectively. An interval vector, its diameter and its absolute value are denoted $\hat{X} \in I(R^n) = (X_1,X_2,\ldots ,X_n)$, where $X_i \in I(R)$, $d(\hat{X}) = (d(X_1),d(X_2), \ldots d(X_n))$ and $|\hat{X}| = (|X_1|,|X_2|, \ldots |X_n|)$ , for $i = 1, 2, \ldots ,n$. In a similar manner, if $\hat{A} \in M(R^{n \times n})$, then $\hat{A} =(A_{ij})$, where $A_{ij} \in I(R)$, $d(\hat{A}) = (d(A_{ij}))$ and $|\hat{A}| = (|A_{ij}|)$ for $1\le i,j \le n$. The following properties can easily be proved

• 1. $(\hat{A} + \hat{B})C = \hat{A}C+\hat{B}C.$

• 2. $d(\hat{A} \pm \hat{B}) = d(\hat{A}) + d(\hat{B}).$

• 3. $d(\hat{A} b) = d(\hat{A})|b|.$

• 4. $\hat{A}b = \{ Ab | A \in \hat{A}\}.$

where, $\hat{A}$, $\hat{B}$ be interval matrices, A,C be real matrices and b be a real vector. In general, $\hat{A}(\hat{B}b) \subseteq (\hat{A} \hat{B})b$ with equality for unit vector b.

2.1 Interval Gaussian elimination algorithm

In this subsection, we shall review the salient features of interval Gaussian elimination algorithm (IGEA) as described in [2]. Let $\hat{A} = (A_{ij})$ be an interval matrix of dimension $n \times n$ and $\hat{b} = (b_{i})$ be an interval vector with n components. The problem is to obtain an interval vector $\hat{X} = (X_{i}) , 1\le i\le n$ with n components by using the interval extension of Gaussian elimination algorithm on $\hat{A} = (A_{ij}), 1\le i,j \le n$ and $\hat{b} = (b_{i}), 1\le i\le n$ with n components satisfying $\{ x = A^{-1}b | A \in \hat{A}, b \in \hat{b} \} \subseteq \hat{X}.$ If we set $A^{(1)}_{ij} = A_{ij}$ and ${b}^{(1)}_{i} = b_{i}$, then the algorithm is given as follows.

The above algorithm is applicable only if there is no division by an interval which contains zero. If it is assumed that this restriction holds in our case then the feasibility of the interval Gaussian elimination algorithm is guaranteed without depending on $\hat{b}.$ If we define the following interval matrices

$$\begin{aligned}&\hat{L}^{(k)} := \left[ \begin{array}{cccccc} 1 &{} &{} &{} &{} &{} \\ &{} \ddots &{} &{} &{} O &{} \\ &{} &{} 1 &{} &{} &{} \\ &{} &{} -\frac{A_{k+1,k}^{(k)}}{A_{kk}^{(k)}} &{} 1 &{} &{} \\ &{} O &{} \vdots &{} O &{} \ddots &{} \\ &{} &{} -\frac{A_{nk}^{(k)}}{A_{kk}^{(k)}}&{} &{} &{} 1 \\ \end{array} \right] , \quad 1 \le k \le n-1, \quad \\&\hat{D}^{(k)} := \left[ \begin{array}{ccccccc} 1 &{} &{} &{} &{} &{} &{}\\ &{} \ddots &{} &{} &{} &{} O &{}\\ &{} &{} 1 &{} &{} &{} &{}\\ &{} &{} &{} \frac{1}{A_{kk}^{(n)}} &{} &{} &{} \\ &{} &{} &{} &{} 1 &{} &{}\\ &{} O &{} &{} &{} &{} \ddots &{}\\ &{} &{} &{} &{} &{} &{} 1 \\ \end{array} \right] , \quad 1 \le k \le n, \\&\hat{U}^{(k)} := \left[ \begin{array}{cccccc} 1 &{} &{} &{} O &{} &{} \\ &{} \ddots &{} &{} &{} &{} \\ &{} &{} 1 &{}-A_{k,k+1}^{(n)} &{} \cdots &{} -A_{kn}^{(n)} \\ &{} &{} &{} 1 &{} O &{} \\ &{} O &{} &{} &{} \ddots &{} \\ &{} &{} &{} &{} &{} 1 \\ \end{array} \right] , \quad 1 \le k \le n-1, \end{aligned}$$

then, the interval vector $\hat{X}$ obtained by Gaussian elimination algorithm is given by

$$\begin{aligned} \hat{X} = \hat{D}^{(1)}(\hat{U}^{(1)}(\hat{D}^{(2)}(\hat{U}^{(2)}( \ldots ( \hat{D}^{(n-1)}(\hat{U}^{(n-1)}(\hat{D}^{(n)}\hat{c}) \ldots ) \end{aligned}$$

where $\hat{c} = \hat{L}^{(n-1)}(\hat{L}^{(n-2)}( \ldots (\hat{L}^{(2)}(\hat{L}^{(1)}\hat{b}) \ldots ).$ The interval vector $\hat{X}$ is usually denoted by $\hat{X} = IGEA(\hat{A}, \hat{b})$ ( ‘Interval Gaussian elimination algorithm of $\hat{A}$ and $\hat{b}$’). The interval matrix $IGEA(\hat{A})$ is the product of the interval matrices arising in the definition of $IGEA(\hat{A}, \hat{b})$ and is given by

$$\begin{aligned} IGEA(\hat{A}) = \hat{D}^{(1)}(\hat{U}^{(1)}( \ldots (\hat{D}^{(n-1)}(\hat{U}^{(n-1)}( \hat{D}^{(n)} ( \hat{L}^{(n-1)}(\ldots (\hat{L}^{(1)})\ldots ). \end{aligned}$$

Note that it is not possible to omit the parenthesis in the expression for $IGEA(\hat{A}, \hat{b})$ and $IGEA(\hat{A})$, respectively.

Lemma 1

For an interval matrix $\hat{A}$ and a point vector b, it always holds that $IGEA(\hat{A}, b) \subseteq IGEA(\hat{A}) b.$

Proof

The proof is obvious and can be omitted. $\square$

Corollary 1

If $b = e_{i}$, where $e_{i}$ be the unit vector then $IGEA(\hat{A}, e_{i}) = IGEA(\hat{A}) e_{i}$.

3 Proposed method-1

In this section, using the concepts of interval analysis, we shall propose an interval iterative method denoted by PM1 for solving (1.1) in $R^n$ and establish its third order of convergence. The interval extensions of iterative methods used for solving (1.1) in $R^n$ based on the concepts of interval analysis are discussed by several researchers [3, 5, 8, 15]. A third order two-step modification of second order Krwaczyk’s algorithm for solving (1.1) is given in [27]. It is given for $k =0,1,2, \ldots$ by

$$\begin{aligned} K(\hat{X}^{(k)})= & {} m(\hat{X}^{(k)})- R^{(k)} F(m(\hat{X}^{(k)}))+(I-R^{(k)}\hat{F'}(\hat{X}^{(k)}))(\hat{X}^{(k)}-m(\hat{X}^{(k)})),\nonumber \\ \hat{T}^{(k)}= & {} K(\hat{X}^{(k)}) \cap \hat{X}^{(k)}, \nonumber \\ U(\hat{X}^{(k)})= & {} m(\hat{T}^{(k)})- R^{(k)} F(m(\hat{T}^{(k)}))+(I-R^{(k)}\hat{F'}(\hat{X}^{(k)}))(\hat{T}^{(k)}-m(\hat{T}^{(k)})), \nonumber \\ \hat{X}^{(k+1)}= & {} U(\hat{X}^{(k)}) \cap \hat{X}^{(k)}, \end{aligned}$$

(3.1)

where $\hat{X}^{(0)}$ is the starting interval vector containing the solution, $m(\hat{X}^{(k)})$ and $m(\hat{T}^{(k)})$ are the midpoints of interval vectors $\hat{X}^{(k)}$ and $\hat{T}^{(k)}$ respectively. Using the concepts of sub inverses and super inverses of bounded linear operators [18], a third order two sided iterative method for solving (1.1) is described in [9]. A two-step modification of Newton’s method [18] for solving (1.1) is given for $k = 0,1,2, \ldots$ by

$$\begin{aligned} y^{(k)} = x^{(k)}-F'(x^{(k)})^{-1} F(x^{(k)}), \nonumber \\ x^{(k+1)} = y^{(k)}-F'(x^{(k)})^{-1} F(y^{(k)}), \end{aligned}$$

(3.2)

Per iteration, it requires two functions and one inverse of the Fréchet derivative evaluations. Our proposed interval iterative method PM1 is the interval extension of (3.2). Let $\hat{F}^{'}(\hat{X})$ be an inclusion monotonic interval extension of $F'(x), ~ x \in \hat{X} \subseteq D$. Let $x^{*} \in {R}^{n}$ be the root of F(x) such that $F(x^{*}) = 0, ~~ x^{*} \in \hat{X}^{(0)} \subseteq D$, where $\hat{X}^{(0)}$ be a starting interval vector. PM1 is given for $k =0,1,2, \ldots$ by

$$\begin{aligned} NN(\hat{X}^{(k)})= & {} m(\hat{X}^{(k)})- IGEA (\hat{F}^{'}(\hat{X}^{(k)}),F(m(\hat{X}^{(k)}))), \nonumber \\ \hat{Y}^{(k)}= & {} NN(\hat{X}^{(k)}) \cap \hat{X}^{(k)},\nonumber \\ P(\hat{X}^{(k)})= & {} m(\hat{Y}^{(k)})-IGEA (\hat{F}^{'}(\hat{X}^{(k)}),F(m(Y^{(k)}))), \nonumber \\ \hat{X}^{(k+1)}= & {} P(\hat{X}^{(k)})\cap \hat{X}^{(k)}, \end{aligned}$$

(3.3)

where $m(\hat{X}^{(k)})$ and $m(\hat{Y}^{(k)})$ are the midpoints of interval vectors $\hat{X}^{(k)}$ and $\hat{Y}^{(k)}$ respectively. The following theorem establishes the third order convergence of the sequence of inclusion monotone interval vectors $\{\hat{X}^{(k)}\}^{\infty }_{k = 0 }$ generated by (3.3) and each enclosing the solution.

Theorem 1

Let $x^{*} \in \hat{X}^{(0)}$. The sequence of interval vectors $\{\hat{X}^{(k)}\}^{\infty }_{k = 0 }$ generated by (3.3) satisfies the following.

• 1. $x^{*} \in \hat{X}^{(k)}$    $\forall ~~~ k\ge 0$.

• 2. For $A = \,\mid I- IGEA (F^{'}(\hat{X}^{(0)})). F^{'}(\hat{X}^{(0)}) \mid$, if the spectral radius $\rho (A)< 1$ then $q(\hat{X}^{(k+1)},x^{*}) \rightarrow 0$ as $k \rightarrow \infty .$

• 3. For all $1\le i, j \le n, L\ge 0, \hat{X} \subseteq \hat{X}^{(0)}$, if $d(\hat{F}^{'}(\hat{X})_{ij})\le L \Vert d(\hat{X})\Vert$ and $P(\hat{X}^{(0)}) \subseteq NN(\hat{X}^{(0)}) \subseteq \hat{X}^{(0)}$ then $\{\hat{X}^{(k)}\}^{\infty }_{k = 0 }$ converges to $x^*$ with cubic rate of convergence given by

  $$\begin{aligned} \Vert d(\hat{X}^{(k+1)})\Vert \le \beta \Vert d(\hat{X}^{(k)})\Vert ^3, ~~\beta \ge 0. \end{aligned}$$

  (3.4)

Proof

(1) Using (1.4), if $x^{*} \in \hat{X}^{(k)}$ then it implies $x^{*} \in \hat{Y}^{(k)}$. Since, $x^{*}, ~m(\hat{Y}^{(k)}) \in \hat{Y}^{(k)}$, by using mean value theorem, we get

$$\begin{aligned} F(m(\hat{Y}^{(k)})) = F(m(\hat{Y}^{(k)}))- F(x^{*}) = J(m(\hat{Y}^{(k)}),x^{*})(m(\hat{Y}^{(k)})-x^{*}), \end{aligned}$$

(3.5)

where $J(m(\hat{Y}^{(k)}),x^{*}) = \int _{0}^{1} F^{'}(m(\hat{Y}^{(k)})+ t(x^{*}-m(\hat{Y}^{(k)}))) dt, ~~~~t \in [0,1].$ This implies $J(m(\hat{Y}^{(k)}),x^{*}) \in \hat{F}^{'}(\hat{Y}^{(k)}) \subseteq \hat{F}^{'}(\hat{X}^{(k)})$. Simplifying (3.5), we get

$$\begin{aligned} x^{*}= & {} m(\hat{Y}^{((k))})-J^{-1}(m(\hat{Y}^{(k)}),x^{*}) F(m(\hat{Y}^{(k)}))\\\in & {} m(\hat{Y}^{(k)})- IGEA(\hat{{F}}^{'}({Y}^{(k)}), F(m(\hat{Y}^{(k)}))) \end{aligned}$$

since, $J^{-1}(m(\hat{Y}^{(k)}),x^{*}) F(m(\hat{Y}^{(k)})) \in IGEA(\hat{{F}}^{'}({Y}^{(k)}), F(m(\hat{Y}^{(k)})))$. This gives $x^{*} \in m(\hat{Y}^{(k)})- IGEA(\hat{F}^{'}(\hat{Y}^{(k)}), F(m(\hat{Y}^{(k)}))) \subseteq m(\hat{Y}^{(k)})- IGEA(\hat{F}^{'}(\hat{X}^{(k)}), F(m(\hat{Y}^{(k)}))) = P(\hat{X}^{(k)})$. Therefore, $x^{*} \in P(\hat{X}^{(k)}) \cap \hat{X}^{(k)} =\hat{X}^{(k+1)}.$ This gives $x^{*} \in \hat{X}^{(k)}~ \forall ~k ~\ge ~ 0.$

(2) Since $\rho (A)<1$, we get

$$\begin{aligned} P(\hat{X}^{(k)})-x^{*}= & {} m(\hat{Y}^{(k)})-x^{*}-IGEA(\hat{F}^{'}(\hat{X}^{(k)}), F(m(\hat{Y}^{(k)}))) \\\subseteq & {} m(\hat{Y}^{(k)})-x^{*}- IGEA(\hat{F}^{'}(\hat{X}^{(k)})) F(m(\hat{Y}^{(k)})) \\= & {} m(\hat{Y}^{(k)})-x^{*}- IGEA(\hat{F}^{'}(\hat{X}^{(k)})) J(m(\hat{Y}^{(k)}),x^{*}) (m(\hat{Y}^{(k)})-x^{*}) \\\subseteq & {} m(\hat{Y}^{(k)})-x^{*}- IGEA(\hat{F}^{'}(\hat{X}^{(k)}))\hat{F}'(\hat{Y}^{(k)}) (m(\hat{Y}^{(k)})-x^{*}) \\\subseteq & {} m(\hat{Y}^{(k)})-x^{*}- IGEA(\hat{F}^{'}(\hat{X}^{(k)})) \hat{F}^{'}(\hat{X}^{(k)})(m(\hat{Y}^{(k)})-x^{*}) \\= & {} (I- IGEA (\hat{F}'(\hat{X}^{(k)})) \hat{F}^{'}(\hat{X}^{(k)})) (m(\hat{Y}^{(k)})-x^{*}) \end{aligned}$$

Now, $\mid P(\hat{X}^{(k)})-x^{*} \mid \,\,\le \,\mid I- IGEA (F^{'}(\hat{X}^{(0)})) F^{'}(\hat{X}^{(0)}) \mid \mid m(\hat{Y}^{(k)})-x^{*} \mid \,\,\, \le \,A \mid m(\hat{X}^{(k)})-x^{*} \mid$. This gives $q(P(\hat{X}^{(k)}), x^{*}) \le A q(\hat{X}^{(k)},x^{*})$. Since, $x^* \in P(\hat{X}^{(k)}) \cap \hat{X}^{(k)} = \hat{X}^{(k+1)} \subseteq P(\hat{X}^{(k)})$. Thus, $q(\hat{X}^{(k+1)}, x^{*}) \le q(P(\hat{X}^{(k)}), x^{*}) \le A q(\hat{X}^{(k)},x^{*}).$ Therefore, $q(\hat{X}^{(k+1)}, x^{*}) \le A^{(k+1)} q(\hat{X}^{(0)}, x^{*}).$ Since, $\rho (A) < 1$, this leads to $q(\hat{X}^{(k+1)},x^{*}) \rightarrow 0~~ \hbox {as}~~ k \rightarrow \infty .$

(3) Taking diameter on both sides on (3.3), we get

$$\begin{aligned} d(\hat{X}^{(k+1)})\le & {} d(P(\hat{X}^{(k)})) \\= & {} d \big ( IGEA (F^{'}(\hat{X}^{(k)}),F(m(\hat{Y}^{(k)}))) \big ) \\\le & {} d\big (IGEA(F^{'}(\hat{X}^{(k)}))\big ) \mid F(m(\hat{Y}^{(k)}))\mid \\\le & {} d\big (IGEA(F^{'}(\hat{X}^{(k)}))\big ) \mid J(m(\hat{Y}^{(k)}),x^{*}) \mid \mid m(\hat{Y}^{(k)}) - x^{*} \mid \\\le & {} d\big (IGEA(F^{'}(\hat{X}^{(k)}))\big ) \mid F^{'}(\hat{X}^{(k)}) \mid d(\hat{Y}^{(k)}\\\le & {} d\big (IGEA(F^{'}(\hat{X}^{(k)}))\big ) \mid F^{'}(\hat{X}^{(0)}) \mid d(\hat{Y}^{(k)}). \end{aligned}$$

Using a monotone vector norm, we get

$$\begin{aligned} \Vert d(\hat{X}^{(k+1)})\Vert \le \Vert d\big (IGEA(F'(\hat{X}^{(k)}))\big )\Vert ~~ \Vert \mid F^{'}(\hat{X}^{(0)}) \mid \Vert ~~ \Vert d(\hat{Y}^{(k)})\Vert . \end{aligned}$$

(3.6)

 Since, the sequence of interval vectors $\{\hat{Y}^{(k)}\}_{k=0}^{\infty }$ generated by (1.5) converges quadratically, so for a scaler $\alpha \ge 0,$ we get

$$\begin{aligned} \Vert d(\hat{Y}^{(k)})\Vert \le \alpha \Vert d(\hat{X}^{(k)})\Vert ^2. \end{aligned}$$

(3.7)

Using (3.6), (3.7) and from the result $\Vert d\big (IGEA(F^{'}(\hat{X}^{(k)}))\big )\Vert \le \beta \Vert d(\hat{X}^{(k)})\Vert ~~ \beta \ge 0,$ we get $\Vert d(\hat{X}^{(k+1)}) \Vert \le \gamma \Vert d(\hat{X}^{(k)})\Vert ^3$, where, $\gamma = \alpha \beta \omega \Vert \mid F^{'}(\hat{X}^{(0)}) \mid \Vert$ and $\omega$ is a consequences of the equivalence of all the norms. Hence the cubic rate of convergence of the PM1 is established. $\square$

4 Proposed method-2

In this section, using the concepts of interval analysis, we shall propose an interval iterative method denoted by PM2 for solving (1.1) in $R^n$ and establish its fourth order of convergence. A three-step modification of Newton’s method [7] is given for $k = 0,1,2, \ldots$ by

$$\begin{aligned} y^{(k)}= & {} x^{(k)}-F^{'}(x^{(k)})^{-1} F(x^{(k)}), \nonumber \\ z^{(k)}= & {} y^{(k)}-F^{'}(x^{(k)})^{-1} F(y^{(k)}), \nonumber \\ x^{(k+1)}= & {} z^{(k)}-F^{'}(x^{(k)})^{-1} F(z^{(k)}). \end{aligned}$$

(4.8)

Per iteration this method takes three functions and one inverse of first Fréchet derivative. Now, we shall review an interval extension of fourth order iterative method given in [27]. It is a three-step modification of second order Krwaczyk’s algorithm for solving (1.1). It is given for $k =0,1,2, \ldots$ by

$$\begin{aligned} K(\hat{X}^{(k)})= & {} m(\hat{X}^{(k)})- R^{(k)} F(m(\hat{X}^{(k)}))+(I-R^{(k)}\hat{F'}(\hat{X}^{(k)}))(\hat{X}^{(k)}-m(\hat{X}^{(k)})),\nonumber \\ \hat{T}^{(k)}= & {} K(\hat{X}^{(k)}) \cap \hat{X}^{(k)}, \nonumber \\ U(\hat{X}^{(k)})= & {} m(\hat{T}^{(k)})- R^{(k)} F(m(\hat{T}^{(k)}))+(I-R^{(k)}\hat{F'}(\hat{X}^{(k)}))(\hat{T}^{(k)}-m(\hat{T}^{(k)})), \nonumber \\ \hat{V}^{(k)}= & {} U(\hat{X}^{(k)}) \cap \hat{X}^{(k)},\nonumber \\ W(\hat{X}^{(k)})= & {} m(\hat{V}^{(k)})- R^{(k)} F(m(\hat{V}^{(k)}))+(I-R^{(k)}\hat{F'}(\hat{X}^{(k)}))(\hat{V}^{(k)}-m(\hat{V}^{(k)})), \nonumber \\ \hat{X}^{(k+1)}= & {} W(\hat{X}^{(k)}) \cap \hat{X}^{(k)}, \end{aligned}$$

(4.9)

where $\hat{X}^{(0)}$ is the starting interval vector containing the solution, $m(\hat{X}^{(k)})$, $m(\hat{T}^{(k)})$ and $m(\hat{V}^{(k)})$ are the midpoints of interval vectors $\hat{X}^{(k)}$, $\hat{T}^{(k)}$ and $\hat{V}^{(k)}$, respectively. Now, we shall discuss our proposed method PM2 which is the interval extension of (4.8) using the concepts of interval analysis. Starting with a interval vector $\hat{X}^{(0)}$ containing the root $x^{*}$, the three-step interval iterative method is given for $k =0,1, 2, \ldots$ by

$$\begin{aligned} NN(\hat{X}^{(k)})= & {} m(\hat{X}^{(k)})- IGEA (F^{'}(\hat{X}^{(k)}),F(m(\hat{X}^{(k)}))), \nonumber \\ \hat{Y}^{(k)}= & {} NN(\hat{X}^{(k)}) \cap \hat{X}^{(k)},\nonumber \\ P(\hat{X}^{(k)})= & {} m(Y^{(k)})-IGEA (F^{'}(\hat{X}^{(k)}),F(m(\hat{Y}^{(k)}))),\nonumber \\ \hat{Z}^{(k)}= & {} P(\hat{X}^{(k)})\cap \hat{X}^{(k)}, \nonumber \\ S(\hat{X}^{(k)})= & {} m(\hat{Z}^{(k)})-IGEA (F^{'}(\hat{X}^{(k)}),F(m(\hat{Z}^{(k)}))), \nonumber \\ \hat{X}^{(k+1)}= & {} S(\hat{X}^{(k)})\cap \hat{X}^{(k)}, \end{aligned}$$

(4.10)

where $m(\hat{X}^{(k)})$, $m(\hat{Y}^{(k)})$ and $m(\hat{Z}^{(k)})$ are the midpoints of interval vectors $\hat{X}^{(k)}$, $\hat{Y}^{(k)}$ and $\hat{Z}^{(k)}$ respectively. In a similar manner, we can prove the first two parts of the Theorem 1 for the method (4.10). The following theorem establish the fourth order of convergence of the sequence of interval vectors $\{\hat{X}^{(k)}\}^{\infty }_{k = 0 }$ generated by (4.10).

Theorem 2

Let $x^{*} \in \hat{X}^{(0)}$. The sequence of interval vectors $\{\hat{X}^{(k)}\}^{\infty }_{k = 0 }$ generated by (4.10) satisfies the following.

• 1. $x^{*} \in \hat{X}^{(k)}$    $\forall ~~~ k\ge 0$.

• 2. For $A \,{=}\, \mid I- IGEA (F^{'}(\hat{X}^{(0)})). F^{'}(\hat{X}^{(0)}) \mid$, if the spectral radius $\rho (A)< 1$ then $q(\hat{X}^{(k+1)},x^{*}) \rightarrow 0$ as $k \rightarrow \infty .$

• 3. For all $1\le i, j \le n, L\ge 0, \hat{X} \subseteq \hat{X}^{(0)}$, if $d(\hat{F}^{'}(\hat{X})_{ij})\le L \Vert d(\hat{X})\Vert$ and $S(\hat{X}^{(0)}) \subseteq P(\hat{X}^{(0)}) \subseteq NN(\hat{X}^{(0)}) \subseteq \hat{X}^{(0)}$ then $\{\hat{X}^{(k)}\}^{\infty }_{k = 0 }$ converges to $x^*$ with fourth order rate of convergence given by

  $$\begin{aligned} \Vert d(\hat{X}^{(k+1)})\Vert \le \delta \Vert d(\hat{X}^{(k)})\Vert ^4, ~~\delta \ge 0. \end{aligned}$$

Proof

The proof of (1) and (2) follow similarly as given in Theorem 1. To prove (3), we take diameter on both sides of (4.10) and get

$$\begin{aligned} d(\hat{X}^{(k+1)})\le & {} d(S(\hat{X}^{(k)}))\nonumber \\= & {} d \big ( IGEA (F^{'}(\hat{X}^{(k)}),F(m(\hat{Z}^{(k)}))) \big )\nonumber \\\le & {} d\big (IGEA(F^{'}(\hat{X}^{(k)}))\big ) \mid F(m(\hat{Z}^{(k)}))\mid \nonumber \\\le & {} d\big (IGEA(F^{'}(\hat{X}^{(k)}))\big ) \mid J(m(Z^{(k)}),x^{*}) \mid \mid m(\hat{Z}^{(k)}) - x^{*} \mid \nonumber \\\le & {} d\big (IGEA(F^{'}(\hat{X}^{(k)}))\big ) \mid F^{'}(\hat{X}^{(k)}) \mid d(\hat{Z}^{(k)}) \nonumber \\\le & {} d\big (IGEA(F^{'}(\hat{X}^{(k)}))\big ) \mid F^{'}(\hat{X}^{(0)}) \mid d(\hat{Z}^{(k)}). \end{aligned}$$

(4.11)

From the known result $\Vert d\big (IGEA(F^{'}(\hat{X}^{(k)}))\big )\Vert \le \beta \Vert d(\hat{X}^{(k)})\Vert$ for scaler $\beta \ge 0,$ and $\Vert d(\hat{Z}^{(k)}) \Vert \le \gamma \Vert d(\hat{X}^{(k)})\Vert ^3$ for $\gamma \ge 0$. Thus, from (4.11) using monotonic vector norm, we get $\Vert d(\hat{X}^{(k+1)}) \Vert \le \delta \Vert d(\hat{X}^{(k)})\Vert ^{4}$, where $\delta = \beta \gamma \kappa \Vert \mid F^{'}(\hat{X}^{(0)}) \mid \Vert$ and $\kappa$ is a consequences of the equivalence of all the norms. Hence the fourth order of convergence of the method (4.10) is established. $\square$

5 Numerical examples

In this section, a number of numerical examples are worked out in order to check the applicability and efficiency of the proposed methods PM1 and PM2. All the numerical computations have been performed in INTLAB toolbox developed by Rump [24] on an CPU 3.20GHz with 4GB of RAM running on the windows 7 Professional version 2009 Service Pack 1 on Intel(R) core (TM) i5-3470. The performance measure used are the iterations count and the diameter of the obtained interval vectors. The stopping criteria is used either $\hat{X}^{(k+1)} = \hat{X}^{(k)}$ or $\Vert d(\hat{X}^{(k)})\Vert _{\infty }= \max _{1\le i \le n} d(\hat{X}^{(k)})_{i} \le \epsilon$, where, $\epsilon = 10^{-15}$ to obtain final interval vector $\hat{X}^{*}$ containing the root $x^{*}$.

Example 1

[10] Let $F = (f_{1},f_{2})$, where

$$\begin{aligned} f_{1}(x_{1},x_{2})\equiv & {} x^2_{1}+x^2_{2}-1=0, \\ f_{2}(x_{1},x_{2})\equiv & {} x^2_{1}-x_{2}=0, \end{aligned}$$

be the system of nonlinear equations with   $x^{*} = \{0.786151377757423, 0.618033988749895\}$.

Starting with $\hat{X}^{(0)} = \{[0.7,0.9],[0.5,0.7]\}$, we get final interval vector

$$\begin{aligned} \hat{X}^{*} = \{[0.78615137775742,0.78615137775743], [ 0.61803398874989,0.61803398874990] \} \end{aligned}$$

in three iteration counts by PM1 and in 2 iteration counts by PM2. The diameter for an interval vector $\Vert d(\hat{X}^{(k)})\Vert _{\infty }$ obtained by the methods (3.1), (4.9) and proposed methods are displayed in the Table 1.

Table 1 Comparison of $\Vert d(\hat{X}^{(k)})\Vert _{\infty }$

Example 2

[25] Let $F = (f_{1},f_{2},f_{3})$, where

$$\begin{aligned} f_{1}(x_{1},x_{2},x_{3})\equiv & {} 10x_{1}+sin(x_{1}+x_{2})-1 =0, \\ f_{2}(x_{1},x_{2},x_{3})\equiv & {} 8x_{2}-cos^2(x_{3}-x_{2})-1 =0, \\ f_{3}(x_{1},x_{2},x_{3})\equiv & {} 12x_{3}+sin(x_{3})-1=0, \end{aligned}$$

be the system of nonlinear equations with solution given by   

$$\begin{aligned} x^{*} = \{0.068978349172667, 0.246442418609183, 0.076928911987537\}. \end{aligned}$$

Starting with $\hat{X}^{(0)} = \{[0,1],[0,1],[0,1]\}$, we get the final interval vector

$$\begin{aligned} \hat{X}^{*} \!=\! \{&[0.06897834917266, 0.06897834917267], [ 0.24644241860918,0.24644241860919], \\&[0.07692891198753,0.07692891198754]\}. \end{aligned}$$

in 3 iteration counts by PM1 and PM2. The diameter for an interval vector $\Vert d(\hat{X}^{(k)})\Vert _{\infty }$ obtained by the methods (3.1), (4.9) and proposed methods are displayed in the Table 2.

Table 2 Comparison of $\Vert d(\hat{X}^{(k)})\Vert _{\infty }$

Example 3

[25] Consider the integral equation arising in Chandrasekhar’s work [6]:

$$\begin{aligned} x(s) = f(s)+ \lambda x(s) \int _0^1 \kappa (s,t)(x(t)) dt, \end{aligned}$$

which has many application in radiative transfer theory and the kinetic theory of gases. The abscissas $t_{j}$ and the weights $\varpi _{j}$ are known and given in Table 3 for $n =8$.

Table 3 Abscissas and weights of the Gauss-Legendre quadrature formula for $n=8$

In particular, we consider

$$\begin{aligned} F(x)(s) = x(s)-1-\frac{x(s)}{4} \int _0^1 \frac{s}{s+t} x(t) dt, ~x\in C[0,1],~ s\in [0,1]. \end{aligned}$$

The Gauss-Legendre quadrature formula is used to transform the above problem into a finite dimensional problem which is given by

$$\begin{aligned} \int _0^1 f(t) dt \approx \sum \limits _{j=1}^n \varpi _{j}f(t_{j}), \end{aligned}$$

where $t_{j}$ and $\varpi _{j}$ are computed for $n=8$. Let us denote the approximation of $x(t_{i})$ by $x_{i}~~(i = 1,2,\ldots , 8)$, we obtain the following system of nonlinear equations.

$$\begin{aligned} x_{i}-x_{i} \sum _{j=1}^{8} a_{ij}x_{j}-1 = 0,~~ \mathrm {where} ~~ a_{ij} = \frac{t_{i}\varpi _{j} }{4(t_{i}+t_{j})},~~ i = 1,2,\ldots , 8, \end{aligned}$$

(5.1)

where $t_{j}$ and $\varpi _{j}$ are given in Table 3 for n = 8. From (5.1), we get a system of eight nonlinear equations in eight unknowns. The approximate solution of the above nonlinear system of equations is given by

$$\begin{aligned} x^{*}= & {} \{1.021719731461727, 1.073186381733582, 1.125724893656528,1.169753312169115, \\&\ \ 1.203071751305358, 1.226490874633312, 1.241524600593500, 1.249448516693481 \}. \end{aligned}$$

Starting with $\hat{X}^{(0)} = \{[0,2],\ldots ,[0,2]\},$ we get the final interval vector $\hat{X}^{*}$

$$\begin{aligned} \hat{X}^{*}= & {} \{[1.02171973146172, 1.02171973146173],[1.07318638173358, 1.07318638173359],\\&\ \ [1.12572489365652, 1.12572489365653],[1.16975331216911, 1.16975331216912], \\&\ \ [1.20307175130535, 1.20307175130536],[1.22649087463331, 1.22649087463332], \\&\ \ [1.24152460059349, 1.24152460059351],[1.24944851669348, 1.24944851669349] \}. \end{aligned}$$

in 3 iteration counts by PM1 and in 2 iteration counts by PM2. The diameter for an interval vector $\Vert d(\hat{X}^{(k)})\Vert _{\infty }$ obtained by the methods (3.1), (4.9) and proposed methods are displayed in the Table 4.

Table 4 Comparison of $\Vert d(\hat{X}^{(k)})\Vert _{\infty }$

Example 4

[9] Consider the boundary value problem

$$\begin{aligned} y^{''} = \sin y +y , ~~~ y(0) = 1, ~~~ y(1) = 1. \end{aligned}$$

Divide the interval [0, 1] into m subintervals as

$$\begin{aligned} x_{0} = 0< x_{1}<x_{2} \cdots< x_{m-1}<x_{m} =1, ~~ x_{j+1} = x_{j}+h, ~~ h = \frac{1}{m}. \end{aligned}$$

Define $y_{0} = y(x_{0}) = 1, y_{1} = y(x_{1}),\ldots , y_{m-1} = y(x_{m-1}),y_{m} = y(x_{m})=1$. We discretize the above boundary value problem by using the numerical approximation of second derivative

$$\begin{aligned} y^{''}_{k} = \frac{y_{k-1}-2y_{k}+y_{k+1}}{h^2}, \quad k = 0,1,2,\ldots m-1. \end{aligned}$$

(5.2)

From (), we get a system of $m-1$ nonlinear equations in $m-1$ variables. Taking $m=26$, we get $k =25$. Therefore, we get a system of twenty five nonlinear equations in twenty five unknowns. The approximate root of the above system is given by

$$\begin{aligned} x^{*} = \{&0.028276938174808,0.056637530355823, 0.085165644408382, 0.113945575579911, \\&0.143062259326381, 0.172601483033092,0.202650096146265, 0.233296218130643, \\&0.264629443538373, 0.296741043313635, 0.329724161263333, 0.363674004393813,\\&0.398688025544152, 0.434866096434958, 0.472310668895130, 0.511126921625317,\\&0.551422889404289, 0.593309571143194, 0.636901012644448, 0.682314359331687, \\&0.729669873593532, 0.779090910740754, 0.830703846934660, 0.884637951833381, \\&0.941025198162492 \}. \end{aligned}$$

Starting with $\hat{X}^{(0)}=\{[0,1],\ldots ,[0,1]\}$, we get the final interval vector $\hat{X}^{*}$

$$\begin{aligned} \hat{X}^{*} = \{&\!\![ 0.02827693817480, 0.02827693817481], [ 0.05663753035582, 0.05663753035583], \\&[ 0.08516564440838, 0.08516564440839], [ 0.11394557557991, 0.11394557557992], \\&[ 0.14306225932638, 0.14306225932639], [ 0.17260148303309, 0.17260148303310], \\&[ 0.20265009614626, 0.20265009614627], [ 0.23329621813064, 0.23329621813065], \\&[ 0.26462944353837, 0.26462944353838], [ 0.29674104331363, 0.29674104331364], \\&[ 0.32972416126333, 0.32972416126334], [ 0.36367400439381, 0.36367400439382], \\&[ 0.39868802554415, 0.39868802554416], [ 0.43486609643495, 0.43486609643496], \\&[ 0.47231066889513, 0.47231066889514], [ 0.51112692162531, 0.51112692162532], \\&[ 0.55142288940428, 0.55142288940429], [ 0.59330957114319, 0.59330957114320], \\&[ 0.63690101264444, 0.63690101264445], [ 0.68231435933168, 0.68231435933169], \\&[ 0.72966987359353, 0.72966987359354], [ 0.77909091074075, 0.77909091074076], \\&[ 0.83070384693465, 0.83070384693467], [ 0.88463795183338, 0.88463795183339], \\&[ 0.94102519816249, 0.94102519816250] \}. \end{aligned}$$

in 2 iteration counts by PM1 and PM2. The diameter for an interval vector $\Vert d (\hat{X}^{(k)})\Vert _{\infty }$ obtained by the methods (3.1), (4.9) and proposed methods are displayed in the Table 5.

Table 5 Comparison of $\Vert d(\hat{X}^{(k)})\Vert _{\infty }$

It can be easily seen from the above Tables that the proposed methods either take less or equal number of iterations to converge to the smallest interval containing the root when compared with the methods given by (3.1) and (4.9) respectively. Moreover, the existing methods require the inverse of a matrix for every iteration which takes high computational cost for large nonlinear systems.

6 Conclusions

Higher order multi-step interval iterative methods using the concepts of interval analysis are proposed for solving nonlinear equations in $R^n$. Each method leads to an an interval vector enclosing the approximate solution along with the rigorous error bounds automatically. These methods require solving linear interval systems of equations. Interval extension of Gaussian elimination algorithm is described and used for solving them. Convergence analysis is established to show their third and fourth order of convergence. Numerical examples are worked out to demonstrate the efficiency and applicability of the methods.