+ /i7Rt^RIt^RIHtHtHtHt^RIHtRt Rt Rt Rt Rt R tR tR tR tR tRRltRRltRtRtRRltRtRtRtRRltRRltRtR#)z This module provides some basic linear algebra procedures. Translated from Zaikun Zhang's modern-Fortran reference implementation in PRIMA. Dedicated to late Professor M. J. D. Powell FRS (1936--2015). Python translation by Nickolai Belakovski. N) DEBUGGINGEPSREALMAXREALMIN)presentFc\'g\P!W4#^p\\ V44FpW V,W,,, pK! V#)USE_NAIVE_MATHnpdotrangelenxyresultis&& ^/var/www/html/photoedit/myenv/lib/python3.14/site-packages/scipy/_lib/pyprima/common/linalg.pyinprodrsC >vva| F 3q6]A$+ Mc\P!VP^,4p\VP^,4Fp\ WRV3,4W#&K V#)NNN)r zerosshaper rrs&& r matprod12r%sH XXaggaj !F 1771: 11g&  Mrc\P!VP^,4p\VP^,4F!pW RV3,W,,, pK# V#r rr rrr rs&& r matprod21r ,sJ XXaggaj !F 1771: AqD'AD.  Mrc d\P!VP^,VP^,34p\VP^,4FUp\VP^,4F2pVRV3;;,VRV3,WV3,,, uu&K4 KW V#rr)rrrrjs&& r matprod22r#3s XXqwwqz1771:. /F 1771: qwwqz"A 1a4LAadGa1g- -L# Mrcl\'g W,#\VP4^8Xd&\VP4^8Xd \W4#\VP4^8Xd&\VP4^8Xd \ W4#\VP4^8Xd&\VP4^8Xd \ W4#\VP4^8Xd&\VP4^8Xd \ W4#\RVP RVP 24h)rzInvalid shapes for x and y: z and )r rrrrr r# ValueError)rrs&&rmatprodr&;s >s  177|qS\Q.a| QWW s177|q0 QWW s177|q0 QWW s177|q07yaggYOPPrc\'g\P!W4#\P!\ V4\ V434p\ \ V44FpWV,,VRV3&K V#)r)r r outerrrr rs&& routprodr)JsZ >xx~ XXs1vs1v& 'F 3q6]t8F1a4L Mrc \'g)\PPWRR7^,#VP^,pVP^,p\ WE4p\P !V4pVP4p\V^, RR4Fp \WRV 3,4p \\P!V4\P!VRV 3,44p \W4'd^Wy&KkWV ,, Wy&WV ,VRV 3,,, pK V#)N)rcondr) r r linalglstsqrminrcopyr rabsisminor) AbQRdiagmnrankrrryqyqas &&&& rlsqrr<Ss >yyq4033  A  A q9D  A A 4!8R $ AAw RVVAYqAw0 2  ADa=ADaD1QT7N"A% HrcR\'g\P!W4#\P!V4'g\ V4pV#\P!V4'g\ V4pV#\ \P !W.44p\P !\ V4\V4.4pV^,\P!\48dUV^,\P!\R, 48d(\P!\W3,44pV#V^,^8d_V^,\P!V^,V^,, V^,V^,, ,^,4,pV#^pV#)r @) r r hypotisfiniter1arrayr/maxsqrtrrsum)x1x2rrs&& rr?r?js% >xx ;;r?? G H[[__ G H "" # HHc!fc!f% & Q4"'''" "qtbgggck.B'BAC!A H qTAX!rww!QqT AaD1I6:;;A HA Hrc \'g \PPV4#\P!\ VUu.uF qV,NK up44pV#uupiN)r r r-normrCrD)rxirs& rrJrJ}sL >yy~~a  WWS!,!BR%%!,- .F M-sA# cx\\V4\P!\V44, 4V8*#rI)primasumr1r trilr3tols&&ristrilrQ' CFRWWSV_, - 44rcx\\V4\P!\V44, 4V8*#rI)rMr1r triurOs&&ristriurUrRrc\'g \PPV4#VP 4pVP ^,p\ V4'dVPp\P!W34p\V4FSp^W$V3,, W4V3&\VRV1RV13,VRV1V3,4)W$V3,, VRV1V3&KU VP#\V4'd|\P!W34p\V4FSp^WV3,, W4V3&\VRV1RV13,VRV1V3,4)WV3,, VRV1V3&KU V#\V4wrRpVPp\P!W34p\V^, RR4FVpVRV3,\VRV^,V13,W$^,V1V3,4, W$V3,, VRV3&KX \P!V\R7p\P!^V^, V4Wv&VRV3,PpV#)r Nrdtyper,)r r r-invr0rrQTrr r&rUqrintlinspace)r3r8RBrr5PInvPs& rrYrYs >yy}}Q A  A ayy CC HHaV qA!qD'kAdG"1"bqb& 1RaRU844qAw>Abqb!eHss  HHaV qA!qD'kAdG"1"bqb& 1RaRU844qAw>Abqb!eH HQ%a CC HHaV q1ub"%AAw1a!eAg:a%'1* !FF!qD'QAadG&xx%++a1a( agJLL Hrc VP^,pVP^,p\P!V4pVPp\P!^V^, V\ R7p\ V4EFXp\P!\\WFV^,1Wa^,13,4^R7^R7pV^8dDWrV, ^, 8:d0Wv, pWW,WV,uWV&WW&WGV.R3,WFV.R3&\ V^, VR4Fp\WFWh.3,4Pp \P!\WFV3,WFV3,4^4WFWh.3&\WF^,V^,1Wh.3,V 4WF^,V^,1Wh.3&\VRWh.3,V 4VRWh.3&K EK[ VPp W:V3#)r rWaxisrr,)rr eyerZr]r\r argmaxrM primapow2planerotappendr?r&) r3r7r8r5rZr`r"krGr^s & rr[r[s  A  A q A A AqsAS)A 1X IIhyQqS5!aC%<9B K q5Qa%!)^ FAqtJAD!$VQYvva## | 66Q;x*+ +q6M !''!*%qwwqz"AAadG FI# !''!*%qwwqz"AAdG FI# rcW,#)a, Believe it or now, x**2 is not always the same as x*x in Python. In Fortran they appear to be identical. Here's a quick one-line to find an example on your system (well, two liner after importing numpy): list(filter(lambda x: x[1], [(x:=np.random.random(), x**2 - x*x != 0) for _ in range(10000)])) )rs&rrgrgs  3Jrc  \'d\V4^8XgQR4h\\P!V44'd^p^pEMf\ \P !V44'd^\P!^4, \P!V^,4,p^\P!^4, \P!V^,4,pEM\V^,4^8:d\V^,4^8:d^p^pEM\V^,4\\V^,4,8:d"\P!V^,4p^pEM:\V^,4\\V^,4,8:d"^p\P!V^,4pEM\ \P!\P!\4\P!V48\P!V4\P!\R, 4844'd.\V4pV^,V, pV^,V, pEM/\V^,4\V^,48dV^,V^,, p\^\V4\P!^WD,,44pV\P!V^,4,p^V, pWE, pMV^,V^,, p\^\V4\P!^WD,,4.4pV\P!V^,4,pWE, p^V, p\P !W.V)V..4p\'EdVP"R8XgQh\P !\P$!V44'gQh\VR,VR ,, 4\VR ,VR ,,4,^8:gQh\P&!R\P(!RR\,44p\+Wg4'gQh\ \P!\P$!V4\P!V4\P!\R, 4844'd_\P,PV4p\\W`,V^., 44\WwV,48:gQR4hV#) a( As in MATLAB, planerot(x) returns a 2x2 Givens matrix G for x in R2 so that Y=G@x has Y[1] = 0. Roughly speaking, G = np.array([[x[0]/R, x[1]/R], [-x[1]/R, x[0]/R]]), where R = np.linalg.norm(x). 0. We need to take care of the possibilities of R=0, Inf, NaN, and over/underflow. 1. The G defined above is continuous with respect to X except at 0. Following this definition, G = np.array([[np.sign(x[0]), 0], [0, np.sign(x[0])]]) if x[1] == 0, G = np.array([[0, np.sign(x[1])], [np.sign(x[1]), 0]]) if x[0] == 0 Yet some implementations ignore the signs, leading to discontinuity and numerical instability. 2. Difference from MATLAB: if x contains NaN of consists of only Inf, MATLAB returns a NaN matrix, but we return an identity matrix or a matrix of +/-np.sqrt(2). We intend to keep G always orthogonal. zx must be a 2-vectorr>g|=皙?g.AzG @ X = [||X||, 0])rr)r r )rr)r r)rr )rranyr isnanallisinfrCsignr1r logical_andrrrJrBrArr@maximumminimumisorthr-)rcsrGturkrPs& rrhrhsy1v{222{ BHHQK   bhhqk    NRWWQqT] *  NRWWQqT] * ad)q.S1Y!^   ad)sS1Y & GGAaDM  ad)sS1Y &  GGAaDM rwww/"&&);RVVAYQX[^Q^I_=_` a aQA!qA!qA!A$i#ad)#!qt AAs1vrwwq13w/0A 1 AAAA!qt AQAAC 012A 1 AAAA 1&A2q'"#Ayww%vvbkk!n%%%%1T7QtW$%AdGag,=(>>!CCCjj"**VUS["ABa~~~ r~~bkk!nbffQi"'''C-:P.PQ R R q!As13!Q<()SAg->> T@T T> HrcRp\V4V\V4,,p\V4^V,\V4,,p\P!\V4V8W484#)a This function tests whether x is minor compared to ref. It is used by Powell, e.g., in COBYLA. In precise arithmetic, isminor(x, ref) is true if and only if x == 0; in floating point arithmetic, isminor(x, ref) is true if x is 0 or its nonzero value can be attributed to computer rounding errors according to ref. Larger sensitivity means the function is more strict/precise, the value 0.1 being due to Powell. For example: isminor(1e-20, 1e300) -> True, because in floating point arithmetic 1e-20 cannot be added to 1e300 without being rounded to 1e300. isminor(1e300, 1e-20) -> False, because in floating point arithmetic adding 1e300 to 1e-20 dominates the latter number. isminor(3, 4) -> False, because 3 can be added to 4 without being rounded off rq)r1r logical_or)rref sensitivityrefarefbs&& rr2r2EsW K s8kCF* *D s8a+oA. .D ==ST)4< 88rc x\P!V^4p\'d\P!V^4\P!V^48XgQh\P!V^4\P!V^48XgQh\P!V^4\P!V^48XgQh\V4'd V^8gQh\V4'dTMg\P!RR\ ,\P !\P!V^4\P!V^44,4p\P!W"\P!\V44,V\P!\V44,.4p\\W44\P!V4, V8*P4;'gA\\W4\P!V4, 4V8*P4pV#)zB This procedure tests whether A = B^{-1} up to the tolerance TOL. gMbP?gY@) r sizerrrzrryrBr1r&reru)r3r_rPr8is_invs&&& risinvr[sw 1 Aywwq!}1 ---wwq!}1 ---wwq!}1 --- 3<<!8O8 #2::dC#I 277STVW=Z\ZaZabcefZg@h4h#iC &&#RVVCF^+S266#a&>-AB CC71=!BFF1I-#5 : : < o o#gamVXV\V\]^V_F_B`ehAh@m@m@oF Mrc \'d\V4'd V^8gQh\P!V^4pV\P!V^48dRpV#\P!\ \ V444'dRpV#\V4'd\ \VPV4\P!V4, 4\P!W\P!\ V44,48*P4pV#\ \VPV4\P!V4, 4^8*P4pV#)z[ This function tests whether the matrix A has orthonormal columns up to the tolerance TOL. F) rrr rrtrMr1r&rZreryrBru)r3rPnum_varsis_orths&& rr{r{ys y 3<<!8O8 wwq!}H"''!Q- N ((8CF# $ $ N 3<<7133?RVVH-==>"**SXZX^X^_bcd_eXfRfBggllnG N 7133?RVVH-==>!CHHJG Nrc\V4^8Xd \R4h\RV44p\RV44pR\,\VR4,V,#)a Get a relative tolerance for a set of arrays. Borrowed from COBYQA Parameters ---------- *arrays: tuple Set of `numpy.ndarray` to get the tolerance for. Returns ------- float Relative tolerance for the set of arrays. Raises ------ ValueError If no array is provided. z$At least one array must be provided.c38"TFqPxK R#5irI)r.0rAs& r !get_arrays_tol..s.vezzvsc 3"TFLp\P!\P!V\P!V4,4RR7xKN R#5i)?)initialN)r rBr1r@rs& rrrs<E rvveBKK./0#>>sAAg$@r)rr%rBr)arraysrweights* rget_arrays_tolrs_& 6{a?@@ .v. .D F #:D# & //rrrI)__doc__numpyr constsrrrrrr rrr r#r&r)r<r?rJrQrUrYr[rMrgrhr2rr{rrorrrs44 Q . &55 >24X v9,<@0r