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The function takes the coefficients of quadratic equation and returns the real roots. .. math:: coeffs[0]x^2 + coeffs[1]x + coeffs[2] = 0 Args: coeffs : The coefficients of quadratic equation :`(B, 3)` Returns: A tensor of shape `(B, 2)` containing the real roots to the quadratic equation. Example: >>> coeffs = torch.tensor([[1., 4., 4.]]) >>> roots = solve_quadratic(coeffs) .. note:: In cases where a quadratic polynomial has only one real root, the output will be in the format [real_root, 0]. And for the complex roots should be represented as 0. This is done to maintain a consistent output shape for all cases. ÚBÚ3Nrééégà?©ÚdeviceÚdtype) rrÚshaperrÚtorchÚanyÚsqrtÚ bitwise_and) r ÚaÚbÚcÚdeltaÚ mask_negativeÚ mask_zeroÚinv_2aÚ solutionsÚ sqrt_deltaÚmask©r"ú]/home/ubuntu/.local/lib/python3.10/site-packages/kornia/geometry/solvers/polynomial_solver.pyÚsolve_quadratics$   ""r$c.Cs8t|ddgƒtjtj|j|jd}|dd…df}|dd…df}|dd…df}|dd…df}tt|ƒdf|j|jd}|dk}|dk}|dk} ||@| @} ||@| @} t  | ¡rot || dd…fƒ|| dd…f<t  | ¡r‚tjd |j|jd|| df<d ||} | ||} | | }| ||}| ||}d||d }d | |d |d| |d }|||}|||}d | }t |ƒ}t |ƒ}t |ƒ}|||<|||<|||<|dk|dk@}|dk|dk@}t  |¡rt  d||d ¡||}|||df<|dk|dk@}|dk|dk@}t  |¡r;t || || || gdd||<|dk|dk@}|dk|dk@}t  |¡r¥t ||t || ¡¡} t || ¡}!d|!t| dƒ||}"d|!t| d|dƒ||}#d|!t| d|dƒ||}$t |"|#|$gdd||<t|ƒ}%||%|<|%dk|dk@}&|dk|dk@}'t  |'¡rt|ƒ}(t|ƒ})t |¡}*|*dk}+t  |+¡rt  |*|+t ||+¡d ¡|(|+<|dk},t  |,¡rý|(|, |(|,<||+ |(|+|)|+<|(|'|)|'||'}-|-||&df<|S)aSolve given cubic equation. The function takes the coefficients of cubic equation and returns the real roots. .. math:: coeffs[0]x^3 + coeffs[1]x^2 + coeffs[2]x + coeffs[3] = 0 Args: coeffs : The coefficients cubic equation : `(B, 4)` Returns: A tensor of shape `(B, 3)` containing the real roots to the cubic equation. Example: >>> coeffs = torch.tensor([[32., 3., -11., -6.]]) >>> roots = solve_cubic(coeffs) .. note:: In cases where a cubic polynomial has only one or two real roots, the output for the non-real roots should be represented as 0. Thus, the output for a single real root should be in the format [real_root, 0, 0], and for two real roots, it should be [real_root_1, real_root_2, 0]. r Ú4rNrr réçð?é éé6gUUUUUUÕ?©Údimg@rg¼‰Ø—²Òœ<)rrÚtensorÚmathÚpirrrÚlenrr$rÚpowrÚacosrrrÚabs).r Ú_PIrrrÚdrÚ mask_a_zeroÚ mask_b_zeroÚ mask_c_zeroÚmask_first_orderÚmask_second_orderÚinv_aÚb_aÚb_a2Úc_aÚd_aÚQÚRÚQ3ÚDÚb_a_3Úa_Q_zeroÚa_R_zeroÚa_D_zeroÚ mask_Q_zeroÚmask_Q_zero_solutionsÚ x0_Q_zeroÚ mask_QR_zeroÚmask_QR_zero_solutionsÚ mask_D_zeroÚmask_D_zero_solutionsÚ theta_D_zeroÚ sqrt_Q_D_zeroÚ x0_D_zeroÚ x1_D_zeroÚ x2_D_zeroÚ a_D_positiveÚmask_D_positive_solutionÚmask_D_positiveÚADÚBDÚR_absÚmask_R_positiveÚmask_R_positive_Ú x0_D_positiver"r"r#Ú solve_cubicZsŒ   $         ÿ $$    "  r]éé r )rr)r r )rr )rr)ér)r&r&)é r&)ér)érb)r(rb)érc)é rc)r_rd)é r`)ér`)ér(rrcCsJtj|jks tj|jkrtj|j|jda| d¡| d¡jddtS)aEMultiply two polynomials of the first order [@nister2004efficient]. Args: a: a first order polynomial for variables :math:`(x,y,z,1)`. b: a first order polynomial for variables :math:`(x,y,z,1)`. Returns: degree 2 poly with the order :math:`(x^2, x*y, x*z, x, y^2, y*z, y, z^2, z, 1)`. rrr éþÿÿÿ©Ú start_dim)ÚT_deg1rrÚtoÚ unsqueezeÚflatten)rrr"r"r#Úmultiply_deg_one_polyös  rpé(é)ér )r r)rr)rbr&)r^r&)rr)r`r)r&rb)rarb)érc)érc)érd)érd)rcr`)r(r`)rrr`)rdr()rer()ér()r_r_)ér_)rfrf)rgrf)é rf)rhra)é$ra)ére)ére)érg)érg)é!rg)r)rh)é%rh)ér^)érs)é"rs)é#rt)é&rt)é'rvcCsRtj|jks tj|jkrtj|j|jda| d¡| d¡}|jdd}|tS)a¿Multiply two polynomials a and b of degrees two and one [@nister2004efficient]. Args: a: a second degree poly for variables :math:`(x^2, x*y, x*z, x, y^2, y*z, y, z^2, z, 1)`. b: a first degree poly for variables :math:`(x y z 1)`. 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[@nister2004efficient]. 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