Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 875935, 6 pages
http://dx.doi.org/10.1155/2013/875935
Research Article
An Improved Diagonal Jacobian Approximation via a New
Quasi-Cauchy Condition for Solving Large-Scale Systems of
Nonlinear Equations
Mohammed Yusuf Waziri1,2 and Zanariah Abdul Majid1,3
1
Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
Department of Mathematical Sciences, Faculty of Science, Bayero University, Kano PMB 3011, Nigeria
3
Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
2
Correspondence should be addressed to Mohammed Yusuf Waziri; mywaziri@gmail.com
Received 8 August 2012; Revised 14 December 2012; Accepted 15 December 2012
Academic Editor: Turgut Öziş
Copyright © 2013 M. Y. Waziri and Z. Abdul Majid. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
We present a new diagonal quasi-Newton update with an improved diagonal Jacobian approximation for solving large-scale
systems of nonlinear equations. In this approach, the Jacobian approximation is derived based on the quasi-Cauchy condition.
The anticipation has been to further improve the performance of diagonal updating, by modifying the quasi-Cauchy relation so as
to carry some additional information from the functions. The effectiveness of our proposed scheme is appraised through numerical
comparison with some well-known Newton-like methods.
1. Introduction
Let us consider the systems of nonlinear equations
𝐹 (𝑥) = 0,
𝑛
(1)
𝑛
where 𝐹 : 𝑅 → 𝑅 is a nonlinear mapping. Often,
the mapping 𝐹 is assumed to be satisfying the following
assumptions:
(A1) there exists an 𝑥∗ ∈ 𝑅𝑛 s.t 𝐹(𝑥∗ ) = 0;
(A2) 𝐹 is a continuously differentiable mapping in a neighborhood of 𝑥∗ ;
(A3) 𝐹󸀠 (𝑥∗ ) is invertible.
The well-known method for finding the solution to (1) is
the classical Newton’s method which generates a sequence of
iterates {𝑥𝑘 } from a given initial point 𝑥0 via
−1
𝑥𝑘+1 = 𝑥𝑘 − (𝐹󸀠 (𝑥𝑘 )) 𝐹 (𝑥𝑘 ) ,
(2)
where 𝑘 = 0, 1, 2 . . .. The attractive features of this method are
rapid convergence and being easy to implement. Nevertheless, Newton’s method requires the computation of the matrix
entails the first-order derivatives of the systems. In practice,
computations of some functions derivatives are quite costly,
and sometimes they are not available or could not be done
precisely. In this case, Newton’s method cannot be applied
directly.
Moreover, some substantial efforts have been made by
numerous researchers in order to eliminate the well-known
shortcomings of Newton’s method for solving systems of
nonlinear equations, particularly large-scale systems (see,
e.g., [1, 2]). Notwithstanding, most of these modifications
of Newton’s method still have some shortfalls as Newton’s
counterpart. For example, Broyden’s method and Chord
Newton’s method need to store an 𝑛 × 𝑛 matrix, and their
floating points operations, are 𝑂(𝑛2 ), respectively.
To tackle these disadvantages, a diagonally Newton’s
method has been suggested by Leong et al. [3] and showed
that their updating formula is significantly cheaper than
Newton’s method and some of its variants. Based on this
fact, it is pleasing to present an approach which will improve
further the diagonal Jacobian approximation, as well as
reducing the computational cost, floating points operations
and number of iterations. This is what leads to the idea of
2
Journal of Applied Mathematics
this paper. The anticipation has been to further improve the
performance of diagonal updating, by modifying the quasiCauchy relation so as to carry some additional information
from the functions. We organized the paper as follows. In the
next section, we present the details of the proposed method.
Convergence results are present in Section 3. Some numerical
results are reported in Section 4. Finally, conclusions are
made in Section 5.
2. Derivation Process
This section presents a new diagonal quasi-Newton-like
method for solving large-scale systems of nonlinear equations. The quasi-Newton method is an iterative method that
generates a sequence of points {𝑥𝑘 } from a given initial guess
𝑥0 via the following form:
𝑥𝑘+1 = 𝑥𝑘 − 𝛼𝑘 𝐵𝑘 𝐹 (𝑥𝑘 )
𝑘 = 0, 1, 2 . . . ,
(3)
where 𝛼𝑘 is a step length and 𝐵𝑘 is an approximation to the
Jacobian inverse which can be updated at each iteration for
𝑘 = 0, 1, 2 . . .; the updated matrix 𝐵𝑘+1 is chosen in such a
way that it satisfies the secant equation, that is,
𝐵𝑘+1 𝑠𝑘 = 𝑦𝑘 .
(4)
It is clear that the only Jacobian information we have
is 𝑦, and this is only approximation information. To this
end, we incorporate more information from 𝑠𝑘 and 𝐹𝑘 to 𝑦
in order to present a better approximation to the Jacobian
matrix. We consider the modification on 𝑦 presented by Li
and Fukushima [4]:
󵄩 󵄩
𝑦̆ = 𝑦𝑘 + 𝑣𝑘 󵄩󵄩󵄩𝐹𝑘 󵄩󵄩󵄩 𝑠𝑘 ,
(5)
max{−𝑠𝑘𝑇 𝑦𝑘 /‖𝑠𝑘 ‖2 , 0}.
where 𝑣𝑘 = 1 +
Our aim here is to build a square matrix, say 𝐵, using
diagonal updating scheme which is an approximation to the
Jacobian inverse, and we let 𝐵𝑘+1 satisfy the quasi-Cauchy
equation, that is,
𝑦𝑘̆ 𝑇 𝑠𝑘 = 𝑦𝑘̆ 𝑇 𝐵𝑘+1 𝑦𝑘̆ .
(6)
where 𝑉𝑘 = diag((𝑦𝑘̆ (1) )2 , (𝑦𝑘̆ (2) )2 , . . . , (𝑦𝑘̆ (𝑛) )2 ), ∑𝑛𝑖=1 (𝑦𝑘̆ (𝑖) )4 =
tr(𝑉𝑘2 ), and Tr is the trace operation.
Proof. Consider the Lagrangian function of (7):
1 󵄩 󵄩2
𝐿 (Ψ𝑘 , 𝛼) = 󵄩󵄩󵄩Ψ𝑘 󵄩󵄩󵄩𝐹 + 𝛼 (𝑦𝑘̆ 𝑇 Ψ𝑘 𝑦𝑘̆ − 𝑦𝑘̆ 𝑇 𝑠𝑘 + 𝑦𝑘̆ 𝑇 𝐵𝑘 𝑦𝑘̆ ) ,
2
(9)
where 𝛼 is the corresponding Lagrangian multiplier. By
differentiating 𝐿 with respect to each Ψ𝑘(1) , Ψ𝑘(2) , . . . , Ψ𝑘(𝑛) , and
setting them all equal to zero, we obtain
2
Ψ𝑘(𝑖) = −𝛼(𝑦𝑘̆ (𝑖) )
1 󵄩 󵄩2
min 󵄩󵄩󵄩Ψ𝑘 󵄩󵄩󵄩𝐹
2
s.t
(7)
𝑦𝑘̆ 𝑇 (𝐵𝑘 + Ψ𝑘 ) 𝑦𝑘̆ = 𝑦𝑘̆ 𝑇 𝑠𝑘 ,
where ‖ ⋅ ‖𝐹 denotes the Frobenius norm. Hence, the optimal
solution of (7) is given by
𝑇
Ψ𝑘 =
𝑇
(𝑦𝑘̆ 𝑠𝑘 − 𝑦𝑘̆ 𝐵𝑘 𝑦𝑘̆ )
tr (𝑉𝑘2 )
𝑉𝑘 ,
(8)
(10)
Multiplying both sides of (10) by (𝑦𝑘̆ (𝑖) )2 and summing
them all give
𝑛
𝑛
2
4
∑(𝑦𝑘̆ (𝑖) ) Ψ𝑘(𝑖) = −𝛼∑(𝑦𝑘̆ (𝑖) )
𝑖=1
for every 𝑖 = 1, 2, . . . , 𝑛.
𝑖=1
(11)
Differentiating 𝐿 with respect to 𝛼, and since 𝑦𝑘̆ 𝑇 Ψ𝑘 𝑦𝑘̆ =
then we have
∑𝑛𝑖=1 (𝑦𝑘̆ (𝑖) )2 Ψ(𝑖) ,
𝑛
2
∑ ((𝑦𝑘̆ (𝑖) ) Ψ𝑘(𝑖) ) = 𝑦𝑘̆ 𝑇 𝑠𝑘 − 𝑦𝑘̆ 𝑇 𝐵𝑘 𝑦𝑘̆ .
(12)
𝑖=1
Equating (11) and (12) and substituting the relation into
(10), finally we have
Ψ𝑘(𝑖) =
𝑦𝑘̆ 𝑇 𝑠𝑘 − 𝑦𝑘̆ 𝑇 𝐵𝑘 𝑦𝑘̆
∑𝑛𝑖=1
(𝑖) 4
(𝑦𝑘̆ )
2
(𝑦𝑘̆ (𝑖) )
∀𝑖 = 1, 2, . . . , 𝑛.
(13)
Since 𝐵𝑘(𝑖) is a diagonal component of 𝐵𝑘 , 𝑦𝑘̆ (𝑖) is the
𝑖th component of vector 𝑦𝑘̆ , then 𝑉𝑘 = diag((𝑦𝑘̆ (1) )2 , (𝑦𝑘̆ (2) )2 ,
. . . , (𝑦𝑘̆ (𝑛) )2 ) and ∑𝑛𝑖=1 (𝑦𝑘(𝑖) )4 = tr(𝑉𝑘2 ). We further rewrite (13)
as
In addition, the deviation between 𝐵𝑘+1 and 𝐵𝑘 is minimized under some norms; hence, in the following theorem,
we state the resulting update formula for 𝐵𝑘 .
Theorem 1. Assume that 𝐵𝑘+1 be the diagonal update of a
diagonal matrix 𝐵𝑘 . Let us denote the deviation between 𝐵𝑘 and
𝐵𝑘+1 as Ψ𝑘 = 𝐵𝑘+1 − 𝐵𝑘 . Suppose that 𝑦𝑘̆ ≠ 0 which is defined
by (5). Consider the following problem:
∀𝑖 = 1, 2, . . . , 𝑛.
Ψ𝑘 =
(𝑦𝑘̆ 𝑇 𝑠𝑘 − 𝑦𝑘̆ 𝑇 𝐵𝑘 𝑦𝑘̆ )
tr (𝑉𝑘2 )
𝑉𝑘 ,
(14)
which completes the proof.
Hence, the best possible updating formula for diagonal
matrix 𝐵𝑘+1 is given by
𝐵𝑘+1 = 𝐵𝑘 +
(𝑦𝑘̆ 𝑇 𝑠𝑘 − 𝑦𝑘̆ 𝑇 𝐵𝑘 𝑦𝑘̆ )
tr (𝑉𝑘2 )
𝑉𝑘 .
(15)
Now, we can describe the algorithm for our proposed
method as follows.
Algorithm IDJA
Step 1. Choose an initial guess 𝑥0 , 𝜎 ∈ (0, 1), 𝛾 > 1, 𝐵0 = 𝐼𝑛 ,
𝛼0 > 0, and let 𝑘 := 0.
Journal of Applied Mathematics
3
Table 1: Numerical results of NM, CN, BM, DQNM, and IDJA methods.
prob
1
2
3
4
5
6
7
8
NM
Dim
NI
7
9
10
—
12
8
8
11
50
50
50
50
50
50
50
50
CN
CPU
0.046
0.078
0.062
—
0.078
0.064
0.094
0.064
NI
55
344
—
—
—
—
—
—
BM
CPU
0.031
0.062
—
—
—
—
—
—
Step 2. Compute 𝐹(𝑥𝑘 ), and If ‖𝐹(𝑥𝑘 )‖ ≤ 10−8 stop.
Step 3. Compute 𝑑 = −𝐹(𝑥𝑘 )𝐵𝑘 .
Step 4. If ‖𝐹(𝑥𝑘 + 𝛼𝑘 𝑑𝑘 )‖ ≤ 𝜎‖𝐹(𝑥𝑘 )‖, retain 𝛼𝑘 and go to
Step 5. Otherwise set 𝛼𝑘+1 = 𝛼𝑘 /2 and repeat Step 4.
Step 5. If ‖𝐹(𝑥𝑘 + 𝛼𝑘 𝑑𝑘 ) − 𝐹(𝑥𝑘 )‖ ≥ ‖𝐹(𝑥𝑘 + 𝛼𝑘 𝑑𝑘 )‖ − ‖𝐹(𝑥𝑘 )‖,
retain 𝛼𝑘 and go to Step 6. Otherwise set 𝛼𝑘+1 := 𝛼𝑘 × 𝛾 and
repeat Step 5.
Step 6. Let 𝑥𝑘+1 = 𝑥𝑘 + 𝛼𝑘 𝑑𝑘 .
Step 7. If ‖𝑥𝑘+1 − 𝑥𝑘 ‖2 + ‖𝐹(𝑥𝑘 )‖2 ≤ 10−8 stop. Also go to
Step 8.
Step 8. If ‖Δ𝐹𝑘 ‖2 ≥ 𝜖1 where 𝜖1 = 10−4 , compute 𝐵𝑘+1 as
defined by (15); if not, 𝐵𝑘+1 = 𝐵𝑘 .
Step 9. Set 𝑘 := 𝑘 + 1 and go to Step 2.
3. Convergence Result
This section presents local convergence results of the IDJA
methods. To analyze the convergence of these methods, we
will make the following assumptions on nonlinear systems 𝐹.
Assumption 2. (i) 𝐹 is differentiable in an open convex set 𝐸
in R𝑛 .
(ii) There exists 𝑥∗ ∈ 𝐸 such that 𝐹(𝑥∗ ) = 0; 𝐹󸀠 (𝑥) is
continuous for all 𝑥.
(iii) 𝐹󸀠 (𝑥) satisfies the Lipschitz condition of order one
that is there exists a positive constant 𝜇 such that
󵄩
󵄩󵄩 󸀠
󵄩󵄩𝐹 (𝑥) − 𝐹󸀠 (𝑦)󵄩󵄩󵄩 ≤ 𝜇 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩 ,
󵄩
󵄩
(16)
for all 𝑥, 𝑦 ∈ R𝑛 .
(iv) There exist constants 𝑐1 ≤ 𝑐2 such that 𝑐1 ‖𝜔‖2 ≤
𝜔 𝐹 (𝑥)𝜔 ≤ 𝑐2 ‖𝜔‖2 for all 𝑥 ∈ 𝐸 and 𝜔 ∈ R𝑛 .
𝑇 󸀠
We can state the following result on the boundedness
of {‖Ψ𝑘 ‖𝐹 } by assuming that, without loss of generality,
the updating matrix (15) is always used, then we have the
following.
NI
15
15
—
—
42
16
—
11
CPU
0.031
0.031
—
—
0.031
0.032
—
0.0312
NI
14
15
20
19
16
14
25
11
DQNM
CPU
0.016
0.031
0.016
0.031
0.016
0.031
0.031
0.016
IDJA
NI
2
13
10
9
8
7
14
9
CPU
0.011
0.031
0.016
0.031
0.015
0.014
0.010
0.016
Theorem 3. Suppose that {𝑥𝑘 } is generated by Algorithm IDJA
where 𝐵𝑘 is defined by (15). Assume that Assumption 2 holds.
There exists 𝛽 > 0, 𝛿 > 0, 𝛼 > 0 and 𝛾 > 0, such that if 𝑥0 ∈ 𝐸
and 𝐵0 satisfies ‖𝐼 − 𝐵0 𝐹󸀠 (𝑥∗ )‖𝐹 < 𝛿 for all 𝑥𝑘 ∈ 𝐸 then
󵄩󵄩
󵄩
󵄩󵄩𝐼 − 𝐵𝑘 𝐹󸀠 (𝑥∗ )󵄩󵄩󵄩 < 𝛿𝑘 ,
(17)
󵄩
󵄩𝐹
for some constant 𝛿𝑘 > 0, 𝑘 ≥ 0.
Proof. Since ‖𝐵𝑘+1 ‖𝐹 = ‖𝐵𝑘 + Ψ𝑘 ‖𝐹 , it follows that
󵄩󵄩
󵄩
󵄩 󵄩
󵄩 󵄩
󵄩󵄩𝐵𝑘+1 󵄩󵄩󵄩𝐹 ≤ 󵄩󵄩󵄩𝐵𝑘 󵄩󵄩󵄩𝐹 + 󵄩󵄩󵄩Ψ𝑘 󵄩󵄩󵄩𝐹 .
For 𝑘 = 0 and assuming 𝐵0 = 𝐼, we have
󵄨 𝑇
󵄨󵄨
𝑇
󵄨󵄨 (𝑖) 󵄨󵄨 󵄨󵄨󵄨󵄨 𝑦0̆ 𝑠0 − 𝑦0̆ 𝐵0 𝑦0̆
(𝑖) 2 󵄨󵄨󵄨
̆
)
(
𝑦
󵄨󵄨
󵄨󵄨󵄨Ψ0 󵄨󵄨󵄨 = 󵄨󵄨󵄨
0
󵄨󵄨
tr (𝑉02 )
󵄨󵄨
󵄨
󵄨󵄨 𝑇
󵄨󵄨
𝑇
󵄨󵄨𝑦0̆ 𝑠0 − 𝑦0̆ 𝐵0 𝑦0̆ 󵄨󵄨 (max) 2
󵄨 (𝑦̆
≤ 󵄨
),
0
tr (𝑉02 )
(18)
(19)
where (𝑦0̆ (max) )2 is the largest element among (𝑦0̆ (𝑖) )2 , 𝑖 =
1, 2, . . . , 𝑛.
After multiplying (19) by (𝑦0̆ (max) )2 /(𝑦0̆ (max) )2 and substituting tr(𝑉02 ) = ∑𝑛𝑖=1 (𝑦0̆ (𝑖) )4 , we have
󵄨󵄨󵄨𝑦̆ 𝑇 𝑠 − 𝑦̆ 𝑇 𝐵 𝑦̆ 󵄨󵄨󵄨
4
󵄨 0 0
󵄨󵄨 (𝑖) 󵄨󵄨
0
0 0 󵄨󵄨
󵄨󵄨Ψ0 󵄨󵄨 ≤ 󵄨
(𝑦0̆ (max) ) .
(20)
󵄨 󵄨
(max) 2 𝑛
(𝑖) 4
(𝑦0̆
) ∑𝑖=1 (𝑦0̆ )
Since (𝑦0̆ (max) )4 / ∑𝑛𝑖=1 (𝑦0̆ (𝑖) )4 ≤ 1, then (20) turns into
󵄨󵄨 𝑇 󸀠
󵄨󵄨
𝑇
󵄨󵄨 (𝑖) 󵄨󵄨 󵄨󵄨󵄨𝑦0̆ 𝐹 (𝑥) 𝑦0̆ − 𝑦0̆ 𝐵0 𝑦0̆ 󵄨󵄨󵄨
.
(21)
󵄨󵄨󵄨Ψ0 󵄨󵄨󵄨 ≤
2
(𝑦0̆ (max) )
From Assumption 2 and 𝐵0 = 𝐼, (21) becomes
𝑇
󵄨󵄨 (𝑖) 󵄨󵄨 |𝑐 − 1| (𝑦0̆ 𝑦0̆ )
󵄨󵄨Ψ0 󵄨󵄨 ≤
,
2
󵄨 󵄨
(𝑦0̆ (max) )
(22)
where 𝑐 = max{|𝑐1 |, |𝑐2 |}.
Since (𝑦0̆ (𝑖) )2 ≤ (𝑦0̆ (max) )2 for 𝑖 = 1, . . . , 𝑛, it follows that
2
(max)
)
󵄨󵄨 (𝑖) 󵄨󵄨 𝑛 |𝑐 − 1| (𝑦0̆
󵄨󵄨Ψ0 󵄨󵄨 ≤
.
󵄨 󵄨
(max) 2
(𝑦0̆
)
(23)
4
Journal of Applied Mathematics
Hence, we obtain
󵄩󵄩 󵄩󵄩
3/2
󵄩󵄩Ψ0 󵄩󵄩𝐹 ≤ 𝑛 |𝑐 − 1| .
Suppose 𝛼 = 𝑛3/2 |𝑐 − 1|, then
󵄩󵄩 󵄩󵄩
󵄩󵄩Ψ0 󵄩󵄩𝐹 ≤ 𝛼.
From the fact that ‖𝐵0 ‖𝐹 = √𝑛, it follows that
󵄩󵄩 󵄩󵄩
󵄩󵄩𝐵1 󵄩󵄩𝐹 ≤ 𝛽,
Problem 3. System of 𝑛 nonlinear equations
(24)
𝑓𝑖 (𝑥) = sin (1 − 𝑥𝑖 )
𝑛
(25)
× ∑ 𝑥𝑖2 + 2𝑥𝑛−1
𝑖=1
1
1
− 3𝑥𝑛−2 − 𝑥𝑛−4 + 𝑥𝑛−5 − 𝑥𝑖 ln (9 + 𝑥𝑖 )
2
2
(26)
where 𝛽 = √𝑛 + 𝛼 > 0.
Therefore, if we assume that ‖𝐼 − 𝐵0 𝐹󸀠 (𝑥∗ )‖𝐹 < 𝛿, then
󵄩󵄩
󵄩
󵄩
󵄩
󵄩󵄩𝐼 − 𝐵1 𝐹󸀠 (𝑥∗ )󵄩󵄩󵄩 = 󵄩󵄩󵄩𝐼 − (𝐵0 + Ψ0 ) 𝐹󸀠 (𝑥∗ )󵄩󵄩󵄩
󵄩
󵄩𝐹 󵄩
󵄩𝐹
󵄩󵄩
󵄩󵄩
󵄩
󸀠
∗ 󵄩
󵄩
≤ 󵄩󵄩󵄩𝐼 − 𝐵0 𝐹 (𝑥 )󵄩󵄩󵄩𝐹 + 󵄩󵄩󵄩Ψ0 𝐹󸀠 (𝑥∗ )󵄩󵄩󵄩󵄩𝐹
󵄩
󵄩
󵄩
󵄩 󵄩 󵄩
≤ 󵄩󵄩󵄩󵄩𝐼 − 𝐵0 𝐹󸀠 (𝑥∗ )󵄩󵄩󵄩󵄩𝐹 + 󵄩󵄩󵄩Ψ0 󵄩󵄩󵄩𝐹 󵄩󵄩󵄩󵄩𝐹󸀠 (𝑥∗ )󵄩󵄩󵄩󵄩𝐹 ;
(27)
−
9
exp (1 − 𝑥𝑛 ) + 2
2
𝑖 = 1, 2, . . . , 𝑛, 𝑥0 = (0, 0, . . . , 0)𝑇 .
(31)
Problem 4. System of 𝑛 nonlinear equations
𝑓𝑖 (𝑥) = 𝑥𝑖2 − 4 exp (sin (4 − 𝑥𝑖2 ))
2
therefore, ‖𝐼 − 𝐵1 𝐹󸀠 (𝑥∗ )‖𝐹 < 𝛿 + 𝛼𝜙 = 𝛿1 .
Hence, by induction, ‖𝐼 − 𝐵𝑘 𝐹󸀠 (𝑥∗ )‖𝐹 < 𝛿𝑘 for all 𝑘.
+
4. Numerical Results
(28)
The identity matrix has been chosen as an initial approximate Jacobian inverse.
We further design the codes to terminates whenever one
of the following happens:
(i) the number of iteration is at least 200 but no point of
𝑥𝑘 that satisfies (28) is obtained;
(ii) CPU time in seconds reaches 200;
(iii) Insufficient memory to initial the run.
𝑛
𝑓𝑖 (𝑥) = (∑𝑥𝑖 ) (𝑥𝑖 − 2) + (cos 𝑥𝑖 − 2) − 1
𝑖=1
(29)
𝑖 = 1, . . . , 𝑛, 𝑥0 = (1, 1, 1, . . . , 1) .
Problem 6. System of 𝑛 nonlinear equations
𝑛
𝑓𝑖 (𝑥) = ∑𝑥𝑖2 − (sin (𝑥𝑖 ) − 𝑥𝑖4 + sin 𝑥𝑖2 )
𝑖=1
(34)
𝑖 = 1, . . . , 𝑛, 𝑥0 = (.5, .5, .5, . . . , .5) .
Problem 7. System of 𝑛 nonlinear equations
𝑛
𝑓𝑗 (𝑥) = (∑ 𝑥𝑖2 − 1) (𝑥𝑗 − 1)
𝑖=1
𝑛
+𝑥𝑗 (∑ (𝑥𝑖 − 1)) − 𝑛 + 1
𝑖=1
(35)
𝑓𝑛 (𝑥) = (∑ 𝑥𝑖2 − 1) (𝑥𝑛 − 1) + (cos 𝑥𝑛 − 1) − 1
𝑖=1
𝑗 = 2, . . . , 𝑛 − 1, 𝑥0 = (.5, .5, .5, . . .) .
Problem 2. Trigonometric function of Spedicato [6]
𝑛
Problem 8. System of 𝑛 nonlinear equations
𝑓𝑖 (𝑥) = 𝑛 − ∑ cos 𝑥𝑗 + 𝑖 (1 − cos 𝑥𝑖 ) − sin 𝑥𝑖 ,
1 1
1
𝑖 = 1, . . . , 𝑛 , 𝑥0 = ( , , . . . , ) .
𝑛 𝑛
𝑛
(33)
𝑛
Problem 1. Spares 1 function of Shin et al. [5]:
𝑗=1
(32)
Problem 5. System of 𝑛 nonlinear equations
The performance of these methods are compared in terms
of number of iterations and CPU time in seconds. In the
following, some details on the benchmarks test problems are
presented.
𝑓𝑖 (𝑥) = 𝑥𝑖2 − 1 𝑖 = 1, 2, . . . , 𝑛, 𝑥0 = (5, 5, . . . , 5) .
2𝑛 − ∑𝑛𝑖=1 𝑥𝑖
cos 𝑥𝑖
𝑖 = 1, . . . , 𝑛, 𝑥0 = (2.8, 2.8, 2.8, . . . , 2.8) .
In this section, the performance of IDJA method has been
presented, when compared with Broyden’s method (BM),
Chord Newton’s method (CN), Newton’s method (NM), and
(DQNM) method proposed by [3], respectively. The codes are
written in MATLAB 7.4 with a double precision computer; the
stopping condition used is
󵄩
󵄩󵄩 󵄩󵄩 󵄩󵄩
−8
󵄩󵄩𝑠𝑘 󵄩󵄩 + 󵄩󵄩𝐹 (𝑥𝑘 )󵄩󵄩󵄩 ≤ 10 .
2
+ sin (4 − 𝑥𝑖 ) + 𝑖(𝑥𝑛 − 𝑥𝑖 )
(30)
𝑓𝑖 (𝑥) = (1 − 𝑥𝑖2 ) + 𝑥𝑖 + 𝑥𝑖2 𝑥𝑛−2 𝑥𝑛−1 𝑥𝑛 − 2
𝑖 = 1, 2, . . . , 𝑛, 𝑥0 = (.5, .5, . . . , .5) .
(36)
Journal of Applied Mathematics
5
Table 2: Numerical Results of NM, CN, BM, DQNM, and IDJA methods.
prob
Dim
1
2
3
4
5
6
7
8
100
100
100
100
100
100
100
100
NM
NI
7
10
7
—
13
8
8
11
CN
CPU
0.156
0.187
0.203
—
0.265
0.203
0.185
0.234
NI
98
—
—
—
—
—
—
—
BM
CPU
0.094
—
—
—
—
—
—
—
NI
15
18
24
—
53
16
—
11
CPU
0.043
0.062
0.140
—
0.109
0.047
—
0.094
NI
14
16
15
13
17
14
26
11
DQNM
CPU
0.016
0.032
0.031
0.031
0.031
0.031
0.031
0.032
IDJA
NI
2
13
7
10
12
7
16
10
CPU
0.011
0.032
0.015
0.030
0.031
0.017
0.030
0.016
Table 3: Numerical Results of NM, CN, BM, DQNM, and IDJA methods.
prob
Dim
1
2
3
4
5
6
7
8
250
250
250
250
250
250
250
250
NM
NI
7
11
8
—
14
8
8
11
CN
CPU
0.359
0.640
0.499
—
0.827
0.686
0.499
0.484
NI
100
—
—
—
—
—
—
—
BM
CPU
0.109
—
—
—
—
—
—
—
NI
15
21
29
—
—
24
—
11
CPU
0.101
0.218
0.250
—
—
0.250
—
0.125
NI
14
18
16
15
19
14
27
11
DQNM
CPU
0.034
0.032
0.016
0.031
0.031
0.031
0.031
0.031
IDJA
NI
2
8
9
10
8
10
14
10
CPU
0.032
0.031
0.016
0.032
0.016
0.031
0.031
0.016
Table 4: Numerical results of NM, CN, BM, DQNM, and IDJA methods.
prob
Dim
1
2
3
4
5
6
7
8
500
500
500
500
500
500
500
500
NM
NI
7
13
7
—
15
8
8
11
CN
CPU
0.796
1.997
1.4352
—
2.449
2.184
1.498
1.451
NI
101
—
—
—
—
—
—
—
BM
CPU
0.702
—
—
—
—
—
—
—
NI
15
23
—
—
—
23
—
11
CPU
0.671
0.972
—
—
—
0.998
—
0.515
NI
14
19
17
12
21
14
32
11
DQNM
CPU
0.016
0.031
0.031
0.030
0.031
0.032
0.047
0.031
IDJA
NI
2
9
9
10
9
10
15
9
CPU
0.011
0.032
0.031
0.031
0.031
0.045
0.047
0.031
Table 5: Numerical results of NM, CN, BM, DQNM, and IDJA methods.
prob
Dim
1
2
3
4
5
6
7
8
1000
1000
1000
1000
1000
1000
1000
1000
NM
NI
7
—
9
—
16
8
8
11
CN
CPU
2.730
—
5.819
—
8.705
6.474
4.321
4.882
NI
103
—
—
—
—
—
—
—
BM
CPU
3.167
—
—
—
—
—
—
—
NI
38
31
—
—
—
—
—
11
CPU
9.438
7.722
—
—
—
—
—
2.418
NI
14
20
17
11
22
14
38
11
DQNM
CPU
0.016
0.032
0.031
0.064
0.031
0.062
0.062
0.032
IDJA
NI
2
8
9
10
10
11
31
10
CPU
0.011
0.043
0.031
0.064
0.031
0.061
0.047
0.031
6
The numerical results presented in Tables 1, 2, 3, 4, and
5 demonstrate clearly the proposed method (IDJA) shows
good improvements, when compared with NM, CN, BM, and
DQNM, respectively. In addition, it is worth mentioning, the
IDJA method does not require more storage locations than
classic diagonal quasi-Newton’s methods. One can observe
from the tables that the proposed method (IDJA) is faster
than DQNM methods and required little time to solve the
problems when compared to the other Newton-like methods
and still keeping memory requirement and CPU time in
seconds to only 𝑂(𝑛).
5. Conclusions
In this paper, we present an improved diagonal quasi-Newton
update via new quasi-Cauchy condition for solving largescale Systems of nonlinear equations (IDJA). The Jacobian
inverse approximation is derived based on the quasi-Cauchy
condition. The anticipation has been to further improve the
diagonal Jacobian, by modifying the quasi-Cauchy relation so
as to carry some additional information from the functions.
It is also worth mentioning that the method is capable of
significantly reducing the execution time (CPU time), as
compared to NM, CN, BM, and DQNM methods while
maintaining good accuracy of the numerical solution to some
extent. Another fact that makes the IDJA method appealing
is that throughout the numerical experiments it never fails to
converge. Hence, we can claim that our method (IDJA) is a
good alternative to Newton-type methods for solving largescale systems of nonlinear equations.
References
[1] J. E. Dennis, Jr. and R. B. Schnabel, Numerical Methods for
Unconstrained Optimization and Nonlinear Equations, Prentice
Hall, Englewood Cliffs, NJ, USA, 1983.
[2] C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, vol. 16, SIAM, Philadelphia, Pa, USA, 1995.
[3] W. J. Leong, M. A. Hassan, and M. Waziri Yusuf, “A matrixfree quasi-Newton method for solving large-scale nonlinear
systems,” Computers & Mathematics with Applications, vol. 62,
no. 5, pp. 2354–2363, 2011.
[4] D.-H. Li and M. Fukushima, “A modified BFGS method and
its global convergence in nonconvex minimization,” Journal of
Computational and Applied Mathematics, vol. 129, no. 1-2, pp.
15–35, 2001.
[5] B.-C. Shin, M. T. Darvishi, and C.-H. Kim, “A comparison
of the Newton-Krylov method with high order Newton-like
methods to solve nonlinear systems,” Applied Mathematics and
Computation, vol. 217, no. 7, pp. 3190–3198, 2010.
[6] E. Spedicato, “Cumputational experience with quasi-Newton
algorithms for minimization problems of moderatetly large
size,” Tech. Rep. CISE-N-175 3, pp. 10–41, 1975.
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