^{1, 2}

^{1, 3}

^{1}

^{2}

^{3}

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

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.