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Theory and Modern Applications

Table 1 Computational details of Example 2 for sequence \(\{q^{k}\}\)

From: On q-BFGS algorithm for unconstrained optimization problems

k

\(q^{k}\)

\(x=(2, 3)^{T}\)

f(x)

\(f(q^{k}x)\)

\(g_{q^{k}}(1)\)

\(g_{q^{k}}(2)\)

1

\((0.3200, 0.3200)^{T}\)

8.4877

7.3482

4.0387

0.5585

2

\((0.9200, 0.9200)^{T}\)

8.4877

8.4043

6.8282

0.3474

3

\((0.9060, 0.9060)^{T}\)

8.4877

8.3889

6.7358

0.3501

4

\((0.9535, 0.9535)^{T}\)

8.4877

8.4401

7.0561

0.3413

5

\((0.9669, 0.9669)^{T}\)

8.4877

8.4540

7.1500

0.3390

6

\((0.9765, 0.9765)^{T}\)

8.4877

8.4639

7.2183

0.3373

7

\((0.9823, 0.9823)^{T}\)

8.4877

8.4698

7.2598

0.3363

8

\((0.9862, 0.9862)^{T}\)

8.4877

8.4738

7.2881

0.3357

9

\((0.9890, 0.9890)^{T}\)

8.4877

8.4766

7.3080

0.3352

10

\((0.9910, 0.9910)^{T} \)

8.4877

8.4786

7.3226

0.3349

11

\((0.9925, 0.9925)^{T}\)

8.4877

8.4801

7.3336

0.3346

12

\((0.9936, 0.9936)^{T}\)

8.4877

8.4813

7.3420

0.3344

13

\((0.9945, 0.9945)^{T}\)

8.4877

8.4822

7.3487

0.3343

14

\((0.9952, 0.9952)^{T}\)

8.4877

8.4829

7.3541

0.3341

15

\((0.9958, 0.9958)^{T}\)

8.4877

8.4835

7.3584

0.3340

16

\((0.9963, 0.9963)^{T}\)

8.4877

8.4840

7.3620

0.3339

17

\((0.9967, 0.9967)^{T}\)

8.4877

8.4844

7.3650

0.3339

18

\((0.9971, 0.9971)^{T}\)

8.4877

8.4847

7.3675

0.3338

19

\((0.9974, 0.9974)^{T}\)

8.4877

8.4850

7.3697

0.3338

20

\((0.9976, 0.9976)^{T}\)

8.4877

8.4853

7.3715

0.3337

21

\((0.9978, 0.9978)^{T}\)

8.4877

8.4855

7.3731

0.3337

22

\((0.9980, 0.9980)^{T}\)

8.4877

8.4857

7.3745

0.3337

23

\((0.9982, 0.9982)^{T}\)

8.4877

8.4859

7.3757

0.3336

24

\((0.9983, 0.9983)^{T}\)

8.4877

8.4860

7.3768

0.3336

25

\((0.9985, 0.9985)^{T}\)

8.4877

8.4861

7.3777

0.3336

26

\((0.9986, 0.9986)^{T}\)

8.4877

8.4862

7.3785

0.3336

27

\((0.9987, 0.9987)^{T}\)

8.4877

8.4863

7.3793

0.3336

28

\((0.9988, 0.9988)^{T}\)

8.4877

8.4864

7.3800

0.3335

29

\((0.9989, 0.9989)^{T}\)

8.4877

8.4865

7.3806

0.3335

30

\((0.9989, 0.9989)^{T}\)

8.4877

8.4866

7.3811

0.3335