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.
Let us consider the systems of nonlinear equations
there exists an
The well-known method for finding the solution to (
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., [
To tackle these disadvantages, a diagonally Newton’s method has been suggested by Leong et al. [
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
It is clear that the only Jacobian information we have is
Our aim here is to build a square matrix, say
In addition, the deviation between
Assume that
Consider the Lagrangian function of (
Multiplying both sides of (
Differentiating
Equating (
Since
Hence, the best possible updating formula for diagonal matrix
Now, we can describe the algorithm for our proposed method as follows.
Choose an initial guess
Compute
Compute
If
If
Let
If
If
Set
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
We can state the following result on the boundedness of
Suppose that
Since
For
After multiplying (
Since
From Assumption
Since
Hence, we obtain
Suppose
From the fact that
Therefore, if we assume that
Hence, by induction,
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 [
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: the number of iteration is at least 200 but no point of CPU time in seconds reaches 200; Insufficient memory to initial the run.
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.
Spares 1 function of Shin et al. [
Trigonometric function of Spedicato [
System of
System of
System of
System of
System of
System of
The numerical results presented in Tables
Numerical results of NM, CN, BM, DQNM, and IDJA methods.
prob | Dim | NM | CN | BM | DQNM | IDJA | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
NI | CPU | NI | CPU | NI | CPU | NI | CPU | NI | CPU | ||
1 | 50 | 7 | 0.046 | 55 | 0.031 | 15 | 0.031 | 14 | 0.016 | 2 | 0.011 |
2 | 50 | 9 | 0.078 | 344 | 0.062 | 15 | 0.031 | 15 | 0.031 | 13 | 0.031 |
3 | 50 | 10 | 0.062 | — | — | — | — | 20 | 0.016 | 10 | 0.016 |
4 | 50 | — | — | — | — | — | — | 19 | 0.031 | 9 | 0.031 |
5 | 50 | 12 | 0.078 | — | — | 42 | 0.031 | 16 | 0.016 | 8 | 0.015 |
6 | 50 | 8 | 0.064 | — | — | 16 | 0.032 | 14 | 0.031 | 7 | 0.014 |
7 | 50 | 8 | 0.094 | — | — | — | — | 25 | 0.031 | 14 | 0.010 |
8 | 50 | 11 | 0.064 | — | — | 11 | 0.0312 | 11 | 0.016 | 9 | 0.016 |
Numerical Results of NM, CN, BM, DQNM, and IDJA methods.
prob | Dim | NM | CN | BM | DQNM | IDJA | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
NI | CPU | NI | CPU | NI | CPU | NI | CPU | NI | CPU | ||
1 | 100 | 7 | 0.156 | 98 | 0.094 | 15 | 0.043 | 14 | 0.016 | 2 | 0.011 |
2 | 100 | 10 | 0.187 | — | — | 18 | 0.062 | 16 | 0.032 | 13 | 0.032 |
3 | 100 | 7 | 0.203 | — | — | 24 | 0.140 | 15 | 0.031 | 7 | 0.015 |
4 | 100 | — | — | — | — | — | — | 13 | 0.031 | 10 | 0.030 |
5 | 100 | 13 | 0.265 | — | — | 53 | 0.109 | 17 | 0.031 | 12 | 0.031 |
6 | 100 | 8 | 0.203 | — | — | 16 | 0.047 | 14 | 0.031 | 7 | 0.017 |
7 | 100 | 8 | 0.185 | — | — | — | — | 26 | 0.031 | 16 | 0.030 |
8 | 100 | 11 | 0.234 | — | — | 11 | 0.094 | 11 | 0.032 | 10 | 0.016 |
Numerical Results of NM, CN, BM, DQNM, and IDJA methods.
prob | Dim | NM | CN | BM | DQNM | IDJA | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
NI | CPU | NI | CPU | NI | CPU | NI | CPU | NI | CPU | ||
1 | 250 | 7 | 0.359 | 100 | 0.109 | 15 | 0.101 | 14 | 0.034 | 2 | 0.032 |
2 | 250 | 11 | 0.640 | — | — | 21 | 0.218 | 18 | 0.032 | 8 | 0.031 |
3 | 250 | 8 | 0.499 | — | — | 29 | 0.250 | 16 | 0.016 | 9 | 0.016 |
4 | 250 | — | — | — | — | — | — | 15 | 0.031 | 10 | 0.032 |
5 | 250 | 14 | 0.827 | — | — | — | — | 19 | 0.031 | 8 | 0.016 |
6 | 250 | 8 | 0.686 | — | — | 24 | 0.250 | 14 | 0.031 | 10 | 0.031 |
7 | 250 | 8 | 0.499 | — | — | — | — | 27 | 0.031 | 14 | 0.031 |
8 | 250 | 11 | 0.484 | — | — | 11 | 0.125 | 11 | 0.031 | 10 | 0.016 |
Numerical results of NM, CN, BM, DQNM, and IDJA methods.
prob | Dim | NM | CN | BM | DQNM | IDJA | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
NI | CPU | NI | CPU | NI | CPU | NI | CPU | NI | CPU | ||
1 | 500 | 7 | 0.796 | 101 | 0.702 | 15 | 0.671 | 14 | 0.016 | 2 | 0.011 |
2 | 500 | 13 | 1.997 | — | — | 23 | 0.972 | 19 | 0.031 | 9 | 0.032 |
3 | 500 | 7 | 1.4352 | — | — | — | — | 17 | 0.031 | 9 | 0.031 |
4 | 500 | — | — | — | — | — | — | 12 | 0.030 | 10 | 0.031 |
5 | 500 | 15 | 2.449 | — | — | — | — | 21 | 0.031 | 9 | 0.031 |
6 | 500 | 8 | 2.184 | — | — | 23 | 0.998 | 14 | 0.032 | 10 | 0.045 |
7 | 500 | 8 | 1.498 | — | — | — | — | 32 | 0.047 | 15 | 0.047 |
8 | 500 | 11 | 1.451 | — | — | 11 | 0.515 | 11 | 0.031 | 9 | 0.031 |
Numerical results of NM, CN, BM, DQNM, and IDJA methods.
prob | Dim | NM | CN | BM | DQNM | IDJA | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
NI | CPU | NI | CPU | NI | CPU | NI | CPU | NI | CPU | ||
1 | 1000 | 7 | 2.730 | 103 | 3.167 | 38 | 9.438 | 14 | 0.016 | 2 | 0.011 |
2 | 1000 | — | — | — | — | 31 | 7.722 | 20 | 0.032 | 8 | 0.043 |
3 | 1000 | 9 | 5.819 | — | — | — | — | 17 | 0.031 | 9 | 0.031 |
4 | 1000 | — | — | — | — | — | — | 11 | 0.064 | 10 | 0.064 |
5 | 1000 | 16 | 8.705 | — | — | — | — | 22 | 0.031 | 10 | 0.031 |
6 | 1000 | 8 | 6.474 | — | — | — | — | 14 | 0.062 | 11 | 0.061 |
7 | 1000 | 8 | 4.321 | — | — | — | — | 38 | 0.062 | 31 | 0.047 |
8 | 1000 | 11 | 4.882 | — | — | 11 | 2.418 | 11 | 0.032 | 10 | 0.031 |
In this paper, we present an improved diagonal quasi-Newton update via new quasi-Cauchy condition for solving large-scale 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 large-scale systems of nonlinear equations.