ó ]™xTc@ s|ddlmZddlTddlZddlZddlmZddlmZddl m Z ddl m Z de _d „Zd efd „ƒYZd „Zd „Zd„Zddd„Zed„Zdefd„ƒYZdefd„ƒYZddd$d„Zdddd„Zddd„Zdddd„Zdefd „ƒYZd!„Z ddd"„Z!d#„Z"dS(%i’’’’(tdivision(t*N(tmatrix(tflex(trational(tmaths Fits a gaussian function to a list of points. Parameters ---------- x : scitbx.array_family.flex.double y : scitbx.array_family.flex.double Returns ------- scitbx.math.gaussian_fit_1d_analytical Examples -------- >>> from scitbx.array_family import flex >>> from scitbx.math import gaussian_fit_1d_analytical >>> x, y = flex.double(), flex.double() >>> x.append(-1) ; y.append(.1) >>> x.append(0) ; y.append(1) >>> x.append(1) ; y.append(.1) >>> fit = gaussian_fit_1d_analytical(x, y) >>> print fit.a, fit.b 1.0 2.30258509299 cC stjj|ƒS(N(Rtmediant dispersion(tdata((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pytmedian_statistics#stline_given_pointscB s#eZd„Zd„Zd„ZRS(cC s\g|D]}tj|ƒ^q|_|jd|jd|_t|jjƒƒ|_dS(Nii(Rtcoltpointstdeltatfloattnorm_sqt delta_norm_sq(tselfR tpoint((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyt__init__(s%cC sL|jdkr$||jdjƒS|jj||jdƒjƒ|jS(sAhttp://mathworld.wolfram.com/Point-LineDistance3-Dimensional.htmli(RR RR tcross(RR((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyt distance_sq-s cC sA|jd|j|jƒ |jjƒ}|jd|j|S(Ni(R tdotR R(RRtt((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pytperp_xyz4s+(t__name__t __module__RRR(((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyR &s  cC s2tt|d|d|ddddƒdƒS(Niiigš?g@gšæ(tmaxtmin(tr((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyt*r3_rotation_cos_rotation_angle_from_matrix8s(cC sYt|ƒ}|dkr=tj|ƒ}|tj|ƒ}nd}d|||jƒS(s— Chirikjian, G. Stochastic models, information theory, and Lie groups. Volume 2, pp 39-40. Applied and Numerical Harmonic Analysis, Birkhauser gš?gą?(RRtacostsint transpose(Rt cos_thetatthetata1((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pytr3_rotation_matrix_logarithm?s   cG sq|std‚nddl}ddl}t|jg|D]}|jjt|ƒƒ^q:ƒ}t|j ƒƒS(sż Calculates an approximation of the Riemannian (geometric) mean through quaternion averaging Arithmetic and geometric solutions for average rigid-body rotation (2010). I Sharf, A Wolf & M.B. Rubin, Mechanism and Machine Theory, 45, 1239-1251 sAverage of empty sequencei’’’’N( t TypeErrortoperatort scitbx.matrixtreducetaddRR t%r3_rotation_matrix_as_unit_quaterniont%r3_rotation_unit_quaternion_as_matrixt normalize(tmatricesR'tscitbxtetaverage((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyt1r3_rotation_average_rotation_matrix_from_matricesQs    +idg-Cėā6?c C s|std‚n|dks*|dkr6td‚nddlm}ddl}||}dt|ƒ}|d}x‚t|ƒD]t}|jƒ} |t|j g|D]} |j j d| | ƒ^q؃} | j ƒ|krē|S|| j ƒ9}qƒWtd ‚dS( s, Calculates an approximation of the Riemannian (geometric) mean by iteratively averaging the rotation matrix logarithms Nonparametric Second-order Theory of Error Propagation on Motion Groups (2008). Y. Wang & G. S. Chirikjian, The International Journal of Robotics Research 27, 1258-1273 sAverage of empty sequenceisInvalid iteration parametersi’’’’(tso3_lie_algebraNgš?RsIteration limit exceeded(R&t ValueErrort scitbx.mathR3R'tlentrangetinverseR)R*telementtfrom_rotation_matrixRt exponentialt RuntimeError( R.tmaxitert convergenceR3R'tconv_sqtnormtcurrenttiR8tmtlog_diff((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyt,r3_rotation_average_rotation_via_lie_algebrahs&       -c C s`|r-g|D]}|tjd^q }ng|D]}tj|ƒ^q4}g|D]}tj|ƒ^qV}tj|d|d|d|d|d|d |d|d|d|d|d|d|d|d|d|d|d|d |d|d|d|d|d|d|d |d|d|d|df ƒS(Ni“iii(RtpitcosR Rtsqr(tanglestdegtatcts((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyteuler_angles_as_matrixs'""'('(terf_verificationcB seZdd„Zd„ZRS(g»½×Łß|Ū=cC s||_d|_dS(Ni(t tolerancet max_delta(RRP((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyR”s cC s||ƒ}t|ƒt|ƒkr t|ƒjƒjdƒ}t|ƒjƒjdƒ}t|ƒt|ƒksxt‚t|ƒdkr§|d|dks§t‚ntt|dƒt|dƒƒ}|j|kr ||_||jkr |G|G|G|GHq q ndS(NR0iii( tstrtlowertsplitR6tAssertionErrortabsRRQRP(Rtftxtexpected_resulttresulttmetmexR ((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyt__call__„s $ (RRRR](((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyROŸs tminimum_covering_sphere_ndcB s5eZdddd„Zd„Zd„Zd„ZRS(iicC sst|ƒdks$|dks$t‚|dkr9d}nd|_t|ƒdkrj||_||_dSt|ƒdkrœt|dƒ|_||_dSt|djƒ}dt|ƒ}tj gt t|ƒƒD] }|^qŲƒ}xjtj dg|ƒ} x.t |j|ƒD]\}} | || 7} qWtj g|D]} t | | ƒ^qDƒ} d} x5t |j| jƒD]\}} | || d7} qWtj| ƒ} | jƒ}|| ||krŌPng}x4t |j| jƒD]\}} |j|| ƒqšWtj |ƒ}|jƒ}|dks>t‚||}|jd7_qš| j|_||_dS(Nigķµ ÷Ę°>igš?i(R6RUtNonet _n_iterationst_centert_radiusttupletelemsRtrowtxrangeR tzipRVRtsqrtRtappendtsum(RR tepsilontradius_if_one_or_no_pointstcenter_if_no_pointstn_dimtwRBtweightsRXRtradiitsigmaRttautw_rtsum_w_r((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyRµsJ$     .," "   cC s|jS(N(R`(R((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyt n_iterationsßscC s|jS(N(Ra(R((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pytcenterāscC s|jS(N(Rb(R((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pytradiusåsN(iii(RRR_RRvRwRx(((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyR^³s  &  iic C sU|dkrd}nt|tjƒr0t}nt}|d|d|d|d|ƒS(Ngķµ ÷Ę°>R RkRlRm(R_t isinstanceRt vec3_doubletminimum_covering_sphere_3dR^(R RkRlRmtimpl((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pytminimum_covering_spherečs  cC s=|dk r'|jƒdks't‚ntj||||ƒS(Ni(R_tndRUtextt!row_echelon_back_substitution_int(trow_echelon_formtvtsolutiontindependent_flags((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyR€ųs cC sL|dk st‚|dk r9|jƒdks9t‚ntj|||ƒS(Ni(R_RUR~Rt#row_echelon_back_substitution_float(RR‚Rƒ((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyR…s  gź-™—q=i cC st|tjƒr!|jƒ}nt|tjƒr]|jƒdksKt‚tj|ƒ}n|d krrd}n,t |j ƒƒtj tj |ƒƒ|}t ||ƒ}||}|j ƒ}|j ƒ}t d|d|d|ƒ} | jdkr | jdtj| jƒd|ƒ} nš|jƒs2t‚|j ƒ}tjjj|ƒ} | jƒ} | jƒ} | j|ƒj| ƒ}| j|ƒ}| j}t d|d|d|ƒ} | j|ksĖt‚| jdtj| jdƒd|ƒ} | d k r| j| ƒ} n| S( sAssumes a is symmetric, without checking to avoid overhead. Special case of generalized_inverse(a) * b taking advantage of the fact that a is real and symmetric. Opportunistic algorithm: first assumes that a has full rank. If this is true, solves a*x=b for x using simple back-substitution. Only if a is rank-deficient: To obtain the x with minimum norm, transforms a to a basis formed by its eigenvectors, solves a*x=b in this basis, then transforms x back to the original basis system. Returns None if a*x=b has no solution. iita_worktb_workt min_abs_pivott free_valuesRktmax_rankN(RyRtrectas_flex_double_matrixt n_columnsRURtdoubleR_RtallRVt deep_copytrow_echelon_full_pivotingtnullitytback_substitutiontis_square_matrixR/tlinalgt eigensystemtreal_symmetrictvectorstmatrix_transposetmatrix_multiplytrank(RKtbtrelative_min_abs_pivottabsolute_min_abs_pivott back_substitution_epsilon_factorRˆRktawtbwtefRXtesRLtctRŠ((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyt)solve_a_x_eq_b_min_norm_given_a_sym_b_col s@  ,          tfinite_difference_computationcB s eZdZd„Zd„ZRS(s° Computing the derivative of a scalar function f(x, ...) wrt x, where x may be a vector, involves computing f(x + h, ...) where h is suitably chosen a delta. So as to keep numerical accuracy, it is best to follow a well known recipe to choose h knowing its direction. When using that value, a finite difference computation typically reaches a precision of self.precision Reference: chap 5.7 of Numerical Recipes in C cC s.tjtjjjƒ}tj|ƒ|_dS(N(RRhR/tdouble_numeric_limitsRkt precision(RR0((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyRSscC s<||j}}|t|ƒ}||}|||}|S(sĀ Compute the value h in the given direction that is most suitable for use in finite difference computation around the given point x. The vector 'direction' may not be a unit-vector. (RØRV(RRt directionRXRMtuth((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyt best_deltaWs  (RRt__doc__RR¬(((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyR¦Hs  c C stj}tj}|tjd}|tjd}|d|||ƒ||ƒ}|d|||ƒ||ƒ} |d|||ƒ} || | fS(sj Returns a point on sphere of radius r: see definition of polar (sperical) coordinate system in 3D. i“iii(RR RGRF( Rts_degtt_degRwR RGRMRRXtytz((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pytpoint_on_spherebs  ""c C sA||gjdƒdks!t‚|d|d|d|d|d|dg}tj|gƒ}|dk r‡|dksÜt‚nU|dks™t‚tj|dd|dd|ddƒ}t||ƒd}|dt|dƒ}tjƒ}x-t |dƒD]} |j || |ƒqW|j |ƒ|S(Niii( tcountR_RURRzRRhtintRRftextendRi( tstarttendtntsteptvecRt vec_lengthtdrR RB((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pytequally_spaced_points_on_vectorps!3 /  cC s%t|ƒ}t|ƒ}|drS||d}||dd }||d}n>||d||ddd}||d }||d}t|ƒ}|drĘ||d}||d}nD||d||ddd}||d||ddd}|d||||dfS(sŗ Returns the Tukey five number summary (min, lower hinge, median, upper hinge, max) for a sequence of observations. This function gives the same results as R's fivenum function. iiii’’’’(tsortedR6(RtsortsRøtmedRStuppertlhingetuhinge((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pytfive_number_summarys    "  ""(iii(#t __future__Rtscitbx.math.exttscitbx.linalg.eigensystemR/tscitbx.math.gaussianRtscitbx.array_familyRtboostRtstdlibRtgaussian_fit_1d_analyticalR­R tobjectR RR%R2REtFalseRNROR^R_R}R€R…R„R¦R²R½RÄ(((sB/public2/ftp/pub/ccp4/6.5/ccp4-6.5/lib/py2/scitbx/math/__init__.pyts@        % 7    8