大井万紀人、水崎高浩 （専修大学・自然科学研究所）

Onishi and Yoshida: Nucl.Phys. 80 (1966) Onishi and Horibata: PTP 64 (1980)

Continuity of a norm overlap with respect to the Euler angles The Neergård-Wüst method The Pfaffian method

M. Oi and N. Tajima, Phys. Lett. B 606 (2005) K.Hara, A.Hayashi, P. Ring, Nucl. Phys. A 606 (1980) M. Oi, et al., in preparation (2012) --- Limbo-dance method

L.Robledo, Phys.Rev. C 79 (2009) Pfaffian: a polynomial M: anti-symmetric

Bipartite expression

- M. Oi and T. Mizusaki, Phys. Lett. B 707 (2012) T. Mizusaki and M. Oi, Phys. Lett. B 715 (2012) B. Avez and M. Bender, Phys. Rev. C 85 (2012) G. Bertsch and L.M. Robledo, Phys. Rev. Lett. 108 (2012)

- T. Mizusaki and M. Oi, Phys. Lett. B 715 (2012)

- K. Neergård and E. Wüst, Nucl. Phys. A402 (1983) : A polynomial in x

Due to the Onishi formula: : diagonalisation for λ k A necessity to be double-root structure: (…..) 2, or Pair-wise eigenvalues: (λ 1, λ 1 ), (λ 2, λ 2 ), (λ 3, λ 3 ),….

: a general complex matrix ! LINPACK for eigenvalues of a general complex matrix: zgeev.f (based on QR method)

Avez-Bender (PRC85, 2012): “the practical application of the NW technique becomes cumbersome in realistic cases, and has been rarely used in practice.” Schmidt (PPNP52, 2004): VAMPIR “This problem has been first solved by NW, who designed a method to determine the sign of the square root in a unique way. This method is also used in all our numerical applications.”

Robledo (PRC79, 2009 ): “Handling the eigenvalues of non-Hermitian matrices is a difficult task, that increases its complexity if the pairwise degenerate eigenvalues have to be obtained numerically without any symmetry enforcing degeneracy.” (Boson)

E-008 with the NW method E+000 with the Pfaffian Eignevalues at (0,0,9) Norm values

E-010 with the DD NW method E-007 with the NW method E+000 with the Pfaffian Norm overlap Eigenvalues

170 Dy : cranked HFB, P+QQ

: NW(original) : NW(Dim-double) : Pfaffian I max = 60 ℏ For J=20 (cranking).

For the safety of numerical accuracy, -no-prec-div in ifort cannot be switched on, costing computational performance at ~16%. The version of ifort must be 12.1 or higher! 1.Original NW : sec (x1) 2.Pfaffian : sec (x2) 3.Dim-double NW : sec (x3) with i7-875K (OC) for 180 Euler points

○ The Neergård-Wüst method was revisited. ○ The pair structure tends to be slightly lost in the original form, but by means of the “dimension-doubling” formula, the accuracy is slightly improved. ○ These errors do not cause serious problems in angular momentum projection. This is because the errors scatter randomly in the Euler space, unlike the continuity method. ○ Balance between accuracy and comp. performance.

M. Oi and T. Mizusaki, Phys. Lett. B 707 (2012)