A Rank-one Fitting Method with Descent Direction for Solving Symmetric Nonlinear Equations *

In this paper, a rank-one updated method for solving symmetric nonlinear equations is proposed. This method possesses some features: 1) The updated matrix is positive definite whatever line search technique is used; 2) The search direction is descent for the norm function; 3) The global convergence of the given method is established under reasonable conditions. Numerical results show that the presented method is interesting .


Introduction
Consider the following system of nonlinear equations: where F:R n →R n is continuously differentiable and the Jacobian ▽F(x) of F(x) is symmetric for all x∈R n .Let θ(x) be the norm function defined by The following iterative method is used for solving (1) where x k is the current iterative point, d k is a search direction, and a k is a positive step-size.
It is well known that there are many methods [1][2][3][4][5][6][7][8][9] for the unconstrained optimization problems min ( )( ) where the BFGS method is one of the most effective quasi-Newton methods [10][11][12][13][14][15][16][17].These years, lots of modified BFGS methods (see [18][19][20][21][22][23]) have been proposed for UP.Different from their techniques, Xu [24] presented a rank-one fitting algorithm for UP and the numerical examples are very interesting.Motivated by their idea, we give a new rank-one fitting algorithm for (1) which possesses the global convergence, the method can ensure that the updated matrices are positive definite without carrying out any line search, the search direction is descent for the normal function, and the numerical results is more competitive than those of the BFGS method for the test problem.
For nonlinear equations, the global convergence is due to Griewank [25] for Broyden's rank one method.Fan [1], Yuan [26], Yuan, Lu andWei [27], and Zhang [28] presented the trust region algorithms for nonlinear equations.Zhu [29] gave a family of nonmonotone backtracking inexact quasi-Newton algorithms for solving smooth nonlinear equations.In particular, a Gauss-Newton-based BFGS method is proposed by Li and Fukushima [30] for solving symmetric nonlinear equations, and the modified methods [31,32] are studied.
The line search rules play an important role for solving the optimization problems.In the following, we briefly review some line search technique to obtain the stepsize a k .
Brown and Saad [33] proposed the following line search method: where k [29] gave the nonmonotone line search technique: and M is a nonnegative integer.Yuan and Lu [32] presented a new backtracking inexact technique to obtain the stepsize a k : where δ∈(0,1) is a constant, and d k is a solution of the system of linear Equation ( 9).Li and Fukashima [11] give a line search technique to determine a positive step-size a k satisfying where δ1 andδ2 are positive constants, and {ε k } is a positive sequence such that The Formula (7) means that {F(x k )} is approximately norm descent when k is sufficiently large.Gu, Li, Qi, and Zhou [14] presented a descent line search technique as follows whereδ1 andδ2 are positive constants.In this paper, we also use the Formula (8) as line search to find the step-size a k : The search direction d k : play a main role in line search methods for solving optimization problems too, and d k : is a solution of the system of linear equation where B k is often generated by BFGS update formula where Is there another way to determine the update formula?Accordingly the search direction d k is determined by the way.In this paper, the updated matrix B k is generated by the following rank-one updated formula where, as k = 0, B 0 is the given symmetric positive definite matrix, Then we use the following formula to get the search direction, 1 ( ) B k follows ( 11), a k-1 is the steplength used at the previous iteration, and the Equation ( 14) is inspired by [34].
Throughout the paper, we use these notations: . is the Euclidean norm, and F(x k ) and F(x k+1 ) are replaced by F k and F k-1 , respectively.This paper is organized as follows.In the next section, the algorithm is stated.The global convergence convergence is established in Section 3. The numerical results are reported in Section 4.

Rank-One Updated Algorithm (ROUA).
Step 0: Choose an initial point Step 1: Otherwise, solving linear Equation ( 13) to get d k ; Step 2: Find a k is the largest number of {1,r,r 2 ,…} such that (8); Step 3: Let the next iterative point be x k+1 = x k +a k d k ; Step 4: Update B k+1 and H k+1 by the Formulas ( 11) and ( 12) respectively; Step 5: In this paper, we also give the normal BFGS method for solving (1), the algorithm which has the same conditions to ROUA is stated as follows.
Remark 1. a) By the Step 0 of ROUA, there should exist constants λ 1 ≥λ0＞0 such that b) By the Step 4 of ROUA, it is easy to deduce that the updated matrices are symmetric

Convergence Analysis
This section will establish the global convergence for ROUA.
Let Ω be the level set defined by In order to establish the global convergence of ROUA, the following assumptions are needed [30,34,35].
2) The Jacobian of F is symmetric, bounded and uniformly nonsingular on Ω 1 , i.e., there exist constants M≥m＞0 such that In particular, for all x∈Ω 1 , we have where x * stand for the unique solution of (1) in Ω 1 .
Lemma 3.1 Let Assumption A hold.Consider ROUA.
Then for any d∈R n , then there exist constants m 0 such that 2 0 , i.e., the matrix B k is positive for all k.
Proof.By ROUA, we know that the initial matrix B0 is symmetric positive, and then we have (15).Using (11), for k≥1, we have Let m 0 =λ0.Then we get (22).The proof is complete.
Since B k is positive definite, then d k which is determined by (13) has the unique solution.The following lemma can found in [34], here we also give the process of this proof.Lemma 3.2 Let Assumption A hold.If x k is not a sta-tionary point of (2), then there exists a constant a'＞0 depending on k such that when a k -1 ∈(0, a'), the unique solution d(a k -1 ) of ( 13) such that 1 ( ) ( ) 0 Moreover, inequality Proof.By ( 14), we can deduce that From ( 13), we get Since x k is not a stationary point of (2), we have ▽ F(x k )F(x k )≠0.By ▽F(x k ) is symmetric and B k is positive.We obtain (24).
However, the right hand side of ( 25) is O(a k -1 ).Thus, inequality (25) holds for all a k -1＞0 sufficiently small.The proof is complete.
The above lemma shows that line search technique (8) is reasonable, and the given algorithm is well defined.Lemma 3.2 also shows that the sequence {θ(x k )} is strictly decreasing.By Lemma 3.2, it is not difficult to get the following lemma.Proof.By the line search (8), we get Summing these inequalities (30) for k from 0 to ∞ we obtain ( 28) and ( 29).Then we complete the proof of this Lemma., and which mean that the updated matrices are all positive by ROUA.
Proof.By the updated Formula (11), we have By ( 28), we know that 2 0 Accordingly, we get (28).By ( 32), (31), and the Remark 1(b), we can deduce that the updated matrices are all symmetric and positive.Consider we obtain (32) immediately.So, we complete the lemma.By (32), (31), and (34), we have Now we establish the global convergence theorem of ROUA.

Theorem 3.1 Let Assumption A hold and
Therefore, we only discuss the case of (38).In this case, for all k sufficiently large and By Lemma 3.3, we know that {x k }∈Ω is bounded, considering (35), it is easy to deduce that {q k (a k-1 ) and {d k } are bounded.Let {x k } and {d k (a)} converge to x * and dx * , respectively.Then we have Let both sides of (40) be divided by a k ' and take limits as k→∞ we obtain * * ( ) 0 By ( 31) and ( 13), we have As k→∞ taking limits in both of (43) yields

Lemma 3 . 5
Let Assumption A hold.Consider ROUA.Then { } k B converges, for all k and any d∈R n then there exist constants m 0 and M 0 such that 2 0


be generated by ROUA.Then the sequence {x k } converges to the unique solution x * of (1