Table 1 Results of the numerical contrast experiments 1–11

1

Wang and Liang (2005)

6.7780

(2.00003, 1.66665)

44

0.18

\(10^{-4}\)

Jiao (2009)

6.77778

(2.0, 1.666667)

58

<1

\(10^{-8}\)

TPRM

6.77778

(2.0000, 1.6667)

1

0.027

\(10^{-8}\)

2

Jiao (2009)

−4.0

(2.0, 1.0, 3.0)

43

–

\(10^{-8}\)

TPRM

−4.0

(2.0000, 1.0000, 3.0000)

1

0.054

\(10^{-8}\)

3

Gao et al. (2010)

10.0042

(2.0003, 7.9999)

27

10.83

\(10^{-3}\)

Chen and Jiao (2009)

10.00009

(1.999998, 7.9999988)

41

0.02

\(10^{-5}\)

TPRM

10.0

(2.0000,8.0000)

2

0.0407

\(10^{-8}\)

4

Gao et al. (2010)

0.0000

(0.0002, 0.0001, 0, 0, 0, 0, 0, 0, 0, 0, 0)

36

16.03

\(10^{-3}\)

TPRM

0.0000

\((0.00, 2.3453, 0.0000, 6.4121, 1.9434, 0.00, 2.4858, 8.4448, 6.9770, 5.8001, 5.1340)\)

13

1.2758

\(10^{-8}\)

5

Thoai (1991)

0.8902

(1.314792, 1.39555, 0, 0.42329)

6

0.1880

\(10^{-6}\)

TPRM

0.89019

(1.3148, 0.1396, 0.0000, 0.4233)

1

0.00355

\(10^{-8}\)

6

Shen et al. (2008)

11.475

(0.61824, 0.0000)

29

0.01

\(10^{-3}\)

TPRM

4.0000

(0.0000, 0.0000)

1

0.022

\(10^{-8}\)

7

Chen and Jiao (2009)

−15.000

(2.0, 1)

1657

120.58

\(10^{-6}\)

TPRM

−15.0000

(2.0000, 1.0000)

110

57.224

\(10^{-8}\)

8

Shen and Jiao (2006)

0.0000

(2.00, 1.00)

24

–

\(10^{-3}\)

Jiao and Liu (2015)

0.00000003

(2.0000061, 1.0)

16

0.018

\(10^{-8}\)

TPRM

0.0000

(2.0000, 1.0000)

1

0.05406

\(10^{-8}\)

9

Shen and Jiao (2006)

1.1771

(1.17709, 2.1772)

434

1

\(10^{-3}\)

Jiao and Liu (2015)

1.17708

(1.17709, 2.1772)

189

0.226

\(10^{-6}\)

TPRM

1.1770

(1.177088, 2.17718)

3

0.66936

\(10^{-8}\)

10

Jiao and Liu (2015)

3.0000

(0.0000,4.0000)

25

0.750

\(10^{-8}\)

TPRM

3.0000

(0.0000,4.0000)

1

0.02456

\(10^{-8}\)

11

TPRM

−25.0000

(0.0000, −5.0000)

47

22.64563

\(10^{-8}\)