Singularities of Møller-Plesset series: example "a16"

[n₁, n₂, n₃] approximant is defined as a solution of the quadratic equation
A(z)f² + B(z)f + C(z) = 0
with polynomial coefficients A(z), B(z) and C(z) of degree n₃, n₂ and n₁ respectively.

Square-root singularities are determined as zeroes of the discriminant
D(z) = B²(z) - 4A(z)C(z).
The weight c of the singularity z_c is defined so that
f ~ c(1 - z/z_c)^1/2 at z -> z_c.
The weight is calculated by formula
c = 1/2[-z(D/A²)']^1/2
where r. h. s. of the above equation is evaluated at z = z_c.

Table 1. Singularities with their weights for the quadratic approximant [1, 0, 0]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.2927

0.202
Singularities of quadratic [1, 0, 0] approximant

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Table 2. Singularities with their weights for the quadratic approximant [1, 1, 0]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.2872

0.2

2

287370.4764

94.7 i

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Table 3. Singularities with their weights for the quadratic approximant [1, 1, 1]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5145

0.376
Singularities of quadratic [1, 1, 1] approximant

2

-8.5766

0.877

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Table 4. Singularities with their weights for the quadratic approximant [2, 1, 1]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.4728

0.338
Singularities of quadratic [2, 1, 1] approximant

2

-4.1308

0.426

3

-6.9349

0.314 i

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Table 5. Singularities with their weights for the quadratic approximant [2, 2, 1]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.4469

0.294
Singularities of quadratic [2, 2, 1] approximant

2

-6.0145

1.91

3

-10.0817

0.705 i

4

168.9953

2.18 i

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Table 6. Singularities with their weights for the quadratic approximant [2, 2, 2]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5145

0.502
Singularities of quadratic [2, 2, 2] approximant

2

3.1178

0.617 i

3

4.4759

11.

4

-5.4372

0.36

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Table 7. Singularities with their weights for the quadratic approximant [3, 2, 2]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.3384 + 0.2217 i

0.00495 - 0.144 i
Singularities of quadratic [3, 2, 2] approximant

2

1.3384 - 0.2217 i

0.00495 + 0.144 i

3

1.3613

0.1

4

-4.9916

0.328

5

-20.3182

0.323 i

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Table 8. Singularities with their weights for the quadratic approximant [3, 3, 2]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5711 + 0.3143 i

0.278 - 0.223 i
Singularities of quadratic [3, 3, 2] approximant

2

1.5711 - 0.3143 i

0.278 + 0.223 i

3

2.1419

0.501

4

-4.2286

0.157

5

-17.6661

0.22 i

6

45.1719

0.471 i

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Table 9. Singularities with their weights for the quadratic approximant [3, 3, 3]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5988 + 0.3244 i

0.321 - 0.174 i
Singularities of quadratic [3, 3, 3] approximant

2

1.5988 - 0.3244 i

0.321 + 0.174 i

3

2.3332

0.828

4

-4.266

0.152

5

11.4301

0.49 i

6

-224.8279

0.111 i

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Table 10. Singularities with their weights for the quadratic approximant [4, 3, 3]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.6049 + 0.3342 i

0.319 - 0.128 i
Singularities of quadratic [4, 3, 3] approximant

2

1.6049 - 0.3342 i

0.319 + 0.128 i

3

2.402

1.11

4

-4.159

0.127

5

13.8183

0.448 i

6

-0.5204 + 17.4617 i

0.0733 + 0.274 i

7

-0.5204 - 17.4617 i

0.0733 - 0.274 i

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Table 11. Singularities with their weights for the quadratic approximant [4, 4, 3]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.6043 + 0.3399 i

0.314 - 0.0993 i
Singularities of quadratic [4, 4, 3] approximant

2

1.6043 - 0.3399 i

0.314 + 0.0993 i

3

2.3637

1.07

4

-3.1665

0.03

5

-3.544

0.0319 i

6

-6.5087 + 2.4242 i

0.062 - 0.198 i

7

-6.5087 - 2.4242 i

0.062 + 0.198 i

8

63.3988

0.602 i

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Table 12. Singularities with their weights for the quadratic approximant [4, 4, 4]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.6 + 0.3118 i

0.315 - 0.249 i
Singularities of quadratic [4, 4, 4] approximant

2

1.6 - 0.3118 i

0.315 + 0.249 i

3

-1.9444 + 0.0014 i

0.00402 + 0.00403 i

4

-1.9444 - 0.0014 i

0.00402 - 0.00403 i

5

2.3916

0.817

6

-4.0796

0.0955

7

6.0118

0.794 i

8

14.2009

7.06

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Table 13. Singularities with their weights for the quadratic approximant [5, 4, 4]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

-0.2487 + 0.e-5 i

2.81e-6 + 2.81e-6 i
Singularities of quadratic [5, 4, 4] approximant

2

-0.2487 - 0.e-5 i

2.81e-6 - 2.81e-6 i

3

1.5974 + 0.3079 i

0.287 - 0.272 i

4

1.5974 - 0.3079 i

0.287 + 0.272 i

5

2.4255

0.888

6

-4.0841

0.105

7

6.6919

0.556 i

8

10.9588 + 16.2174 i

0.154 + 0.424 i

9

10.9588 - 16.2174 i

0.154 - 0.424 i

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Table 14. Singularities with their weights for the quadratic approximant [5, 5, 4]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.2265 + 0.0048 i

0.0213 - 0.0215 i
Singularities of quadratic [5, 5, 4] approximant

2

1.2265 - 0.0048 i

0.0213 + 0.0215 i

3

1.6418 + 0.3752 i

0.32 + 0.12 i

4

1.6418 - 0.3752 i

0.32 - 0.12 i

5

2.3689

1.49

6

-3.1548

0.043

7

-3.3424

0.0446 i

8

-5.5501

123.

9

-9.8609

0.22 i

10

82.8276

0.694 i

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Table 15. Singularities with their weights for the quadratic approximant [5, 5, 5]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.2934 + 0.0093 i

0.03 - 0.0305 i
Singularities of quadratic [5, 5, 5] approximant

2

1.2934 - 0.0093 i

0.03 + 0.0305 i

3

1.6559 + 0.3853 i

0.293 + 0.18 i

4

1.6559 - 0.3853 i

0.293 - 0.18 i

5

2.3776

1.81

6

-3.5435

0.0491

7

-4.7537

0.0539 i

8

-4.982 + 3.2699 i

0.0712 - 0.112 i

9

-4.982 - 3.2699 i

0.0712 + 0.112 i

10

-22.3706

4.41

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Table 16. Singularities with their weights for the quadratic approximant [6, 5, 5]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.3937 + 0.0365 i

0.0338 - 0.0298 i
Singularities of quadratic [6, 5, 5] approximant

2

1.3937 - 0.0365 i

0.0338 + 0.0298 i

3

1.7322 + 0.4588 i

0.0791 - 0.279 i

4

1.7322 - 0.4588 i

0.0791 + 0.279 i

5

2.079

1.29

6

-3.0593 + 0.0721 i

0.017 + 0.0172 i

7

-3.0593 - 0.0721 i

0.017 - 0.0172 i

8

-4.3937

0.0964

9

4.814

36.5 i

10

7.3289

0.63

11

-11.1419

0.48 i

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Table 17. Singularities with their weights for the quadratic approximant [6, 6, 5]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.4408 + 0.065 i

0.0492 - 0.0375 i
Singularities of quadratic [6, 6, 5] approximant

2

1.4408 - 0.065 i

0.0492 + 0.0375 i

3

1.6892 + 0.5332 i

0.0325 - 0.148 i

4

1.6892 - 0.5332 i

0.0325 + 0.148 i

5

2.0631 + 0.2204 i

0.735 + 0.0419 i

6

2.0631 - 0.2204 i

0.735 - 0.0419 i

7

3.0388

134.

8

-3.2432

0.208

9

-3.3141

0.205 i

10

-5.164

0.922

11

-9.9841

0.29 i

12

156.5158

1.06 i

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Table 18. Singularities with their weights for the quadratic approximant [6, 6, 6]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5753 + 0.1598 i

0.923 - 3.46 i
Singularities of quadratic [6, 6, 6] approximant

2

1.5753 - 0.1598 i

0.923 + 3.46 i

3

1.5774 + 0.4361 i

0.159 - 0.0536 i

4

1.5774 - 0.4361 i

0.159 + 0.0536 i

5

1.6759 + 0.172 i

0.0187 - 0.195 i

6

1.6759 - 0.172 i

0.0187 + 0.195 i

7

1.9546

0.172

8

-3.1416

0.0334

9

-3.3477

0.0359 i

10

-5.219

5.52

11

17.7145

0.469 i

12

-17.843

0.176 i

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Table 19. Singularities with their weights for the quadratic approximant [7, 6, 6]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.4616

0.0494
Singularities of quadratic [7, 6, 6] approximant

2

1.4751 + 0.3218 i

0.0262 + 0.00637 i

3

1.4751 - 0.3218 i

0.0262 - 0.00637 i

4

1.4641 + 0.4295 i

0.0134 - 0.0205 i

5

1.4641 - 0.4295 i

0.0134 + 0.0205 i

6

1.9438 + 0.5333 i

0.0706 + 0.118 i

7

1.9438 - 0.5333 i

0.0706 - 0.118 i

8

-3.1049

0.0132

9

-3.5762

0.016 i

10

-4.1153 + 7.9739 i

0.000498 - 0.0628 i

11

-4.1153 - 7.9739 i

0.000498 + 0.0628 i

12

-10.8104 + 2.4534 i

0.0385 + 0.0328 i

13

-10.8104 - 2.4534 i

0.0385 - 0.0328 i

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Table 20. Singularities with their weights for the quadratic approximant [7, 7, 6]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.4253 + 0.2636 i

0.0302 - 0.0184 i
Singularities of quadratic [7, 7, 6] approximant

2

1.4253 - 0.2636 i

0.0302 + 0.0184 i

3

1.485 + 0.3406 i

0.0159 + 0.0437 i

4

1.485 - 0.3406 i

0.0159 - 0.0437 i

5

1.5298

0.0716

6

1.8707 + 0.4191 i

0.214 + 0.147 i

7

1.8707 - 0.4191 i

0.214 - 0.147 i

8

-3.1509

0.0308

9

-3.3832

0.0335 i

10

-5.3578

28.1

11

6.8793

1.09 i

12

9.9577

6.69

13

-13.295

0.199 i

14

153.6699

0.97 i

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Table 1. Singularities with their weights for the quadratic approximant [1, 0, 0] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.2927	0.202

Table 2. Singularities with their weights for the quadratic approximant [1, 1, 0] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.2872	0.2
2	287370.4764	94.7 i

Table 3. Singularities with their weights for the quadratic approximant [1, 1, 1] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5145	0.376
2	-8.5766	0.877

Table 4. Singularities with their weights for the quadratic approximant [2, 1, 1] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.4728	0.338
2	-4.1308	0.426
3	-6.9349	0.314 i

Table 5. Singularities with their weights for the quadratic approximant [2, 2, 1] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.4469	0.294
2	-6.0145	1.91
3	-10.0817	0.705 i
4	168.9953	2.18 i

Table 6. Singularities with their weights for the quadratic approximant [2, 2, 2] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5145	0.502
2	3.1178	0.617 i
3	4.4759	11.
4	-5.4372	0.36

Table 7. Singularities with their weights for the quadratic approximant [3, 2, 2] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.3384 + 0.2217 i	0.00495 - 0.144 i
2	1.3384 - 0.2217 i	0.00495 + 0.144 i
3	1.3613	0.1
4	-4.9916	0.328
5	-20.3182	0.323 i

Table 8. Singularities with their weights for the quadratic approximant [3, 3, 2] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5711 + 0.3143 i	0.278 - 0.223 i
2	1.5711 - 0.3143 i	0.278 + 0.223 i
3	2.1419	0.501
4	-4.2286	0.157
5	-17.6661	0.22 i
6	45.1719	0.471 i

Table 9. Singularities with their weights for the quadratic approximant [3, 3, 3] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5988 + 0.3244 i	0.321 - 0.174 i
2	1.5988 - 0.3244 i	0.321 + 0.174 i
3	2.3332	0.828
4	-4.266	0.152
5	11.4301	0.49 i
6	-224.8279	0.111 i

Table 10. Singularities with their weights for the quadratic approximant [4, 3, 3] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.6049 + 0.3342 i	0.319 - 0.128 i
2	1.6049 - 0.3342 i	0.319 + 0.128 i
3	2.402	1.11
4	-4.159	0.127
5	13.8183	0.448 i
6	-0.5204 + 17.4617 i	0.0733 + 0.274 i
7	-0.5204 - 17.4617 i	0.0733 - 0.274 i

Table 11. Singularities with their weights for the quadratic approximant [4, 4, 3] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.6043 + 0.3399 i	0.314 - 0.0993 i
2	1.6043 - 0.3399 i	0.314 + 0.0993 i
3	2.3637	1.07
4	-3.1665	0.03
5	-3.544	0.0319 i
6	-6.5087 + 2.4242 i	0.062 - 0.198 i
7	-6.5087 - 2.4242 i	0.062 + 0.198 i
8	63.3988	0.602 i

Table 12. Singularities with their weights for the quadratic approximant [4, 4, 4] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.6 + 0.3118 i	0.315 - 0.249 i
2	1.6 - 0.3118 i	0.315 + 0.249 i
3	-1.9444 + 0.0014 i	0.00402 + 0.00403 i
4	-1.9444 - 0.0014 i	0.00402 - 0.00403 i
5	2.3916	0.817
6	-4.0796	0.0955
7	6.0118	0.794 i
8	14.2009	7.06

Table 13. Singularities with their weights for the quadratic approximant [5, 4, 4] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	-0.2487 + 0.e-5 i	2.81e-6 + 2.81e-6 i
2	-0.2487 - 0.e-5 i	2.81e-6 - 2.81e-6 i
3	1.5974 + 0.3079 i	0.287 - 0.272 i
4	1.5974 - 0.3079 i	0.287 + 0.272 i
5	2.4255	0.888
6	-4.0841	0.105
7	6.6919	0.556 i
8	10.9588 + 16.2174 i	0.154 + 0.424 i
9	10.9588 - 16.2174 i	0.154 - 0.424 i

Table 14. Singularities with their weights for the quadratic approximant [5, 5, 4] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.2265 + 0.0048 i	0.0213 - 0.0215 i
2	1.2265 - 0.0048 i	0.0213 + 0.0215 i
3	1.6418 + 0.3752 i	0.32 + 0.12 i
4	1.6418 - 0.3752 i	0.32 - 0.12 i
5	2.3689	1.49
6	-3.1548	0.043
7	-3.3424	0.0446 i
8	-5.5501	123.
9	-9.8609	0.22 i
10	82.8276	0.694 i

Table 15. Singularities with their weights for the quadratic approximant [5, 5, 5] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.2934 + 0.0093 i	0.03 - 0.0305 i
2	1.2934 - 0.0093 i	0.03 + 0.0305 i
3	1.6559 + 0.3853 i	0.293 + 0.18 i
4	1.6559 - 0.3853 i	0.293 - 0.18 i
5	2.3776	1.81
6	-3.5435	0.0491
7	-4.7537	0.0539 i
8	-4.982 + 3.2699 i	0.0712 - 0.112 i
9	-4.982 - 3.2699 i	0.0712 + 0.112 i
10	-22.3706	4.41

Table 16. Singularities with their weights for the quadratic approximant [6, 5, 5] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.3937 + 0.0365 i	0.0338 - 0.0298 i
2	1.3937 - 0.0365 i	0.0338 + 0.0298 i
3	1.7322 + 0.4588 i	0.0791 - 0.279 i
4	1.7322 - 0.4588 i	0.0791 + 0.279 i
5	2.079	1.29
6	-3.0593 + 0.0721 i	0.017 + 0.0172 i
7	-3.0593 - 0.0721 i	0.017 - 0.0172 i
8	-4.3937	0.0964
9	4.814	36.5 i
10	7.3289	0.63
11	-11.1419	0.48 i

Table 17. Singularities with their weights for the quadratic approximant [6, 6, 5] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.4408 + 0.065 i	0.0492 - 0.0375 i
2	1.4408 - 0.065 i	0.0492 + 0.0375 i
3	1.6892 + 0.5332 i	0.0325 - 0.148 i
4	1.6892 - 0.5332 i	0.0325 + 0.148 i
5	2.0631 + 0.2204 i	0.735 + 0.0419 i
6	2.0631 - 0.2204 i	0.735 - 0.0419 i
7	3.0388	134.
8	-3.2432	0.208
9	-3.3141	0.205 i
10	-5.164	0.922
11	-9.9841	0.29 i
12	156.5158	1.06 i

Table 18. Singularities with their weights for the quadratic approximant [6, 6, 6] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5753 + 0.1598 i	0.923 - 3.46 i
2	1.5753 - 0.1598 i	0.923 + 3.46 i
3	1.5774 + 0.4361 i	0.159 - 0.0536 i
4	1.5774 - 0.4361 i	0.159 + 0.0536 i
5	1.6759 + 0.172 i	0.0187 - 0.195 i
6	1.6759 - 0.172 i	0.0187 + 0.195 i
7	1.9546	0.172
8	-3.1416	0.0334
9	-3.3477	0.0359 i
10	-5.219	5.52
11	17.7145	0.469 i
12	-17.843	0.176 i

Table 19. Singularities with their weights for the quadratic approximant [7, 6, 6] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.4616	0.0494
2	1.4751 + 0.3218 i	0.0262 + 0.00637 i
3	1.4751 - 0.3218 i	0.0262 - 0.00637 i
4	1.4641 + 0.4295 i	0.0134 - 0.0205 i
5	1.4641 - 0.4295 i	0.0134 + 0.0205 i
6	1.9438 + 0.5333 i	0.0706 + 0.118 i
7	1.9438 - 0.5333 i	0.0706 - 0.118 i
8	-3.1049	0.0132
9	-3.5762	0.016 i
10	-4.1153 + 7.9739 i	0.000498 - 0.0628 i
11	-4.1153 - 7.9739 i	0.000498 + 0.0628 i
12	-10.8104 + 2.4534 i	0.0385 + 0.0328 i
13	-10.8104 - 2.4534 i	0.0385 - 0.0328 i

Table 20. Singularities with their weights for the quadratic approximant [7, 7, 6] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.4253 + 0.2636 i	0.0302 - 0.0184 i
2	1.4253 - 0.2636 i	0.0302 + 0.0184 i
3	1.485 + 0.3406 i	0.0159 + 0.0437 i
4	1.485 - 0.3406 i	0.0159 - 0.0437 i
5	1.5298	0.0716
6	1.8707 + 0.4191 i	0.214 + 0.147 i
7	1.8707 - 0.4191 i	0.214 - 0.147 i
8	-3.1509	0.0308
9	-3.3832	0.0335 i
10	-5.3578	28.1
11	6.8793	1.09 i
12	9.9577	6.69
13	-13.295	0.199 i
14	153.6699	0.97 i