-i7SrSSKrSSKJrJrJrJr SSKJr Sr Sr Sr S r S r S rS rS rSrSrSSjrSSjrSrSrSSjrSrSrSrSSjrSSjrSrg)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[(d[R"X5$Sn[[ U55HnX UX-- nM U$)Nr)USE_NAIVE_MATHnpdotrangelenxyresultis S/var/www/html/venv/lib/python3.13/site-packages/scipy/_lib/pyprima/common/linalg.pyinprodrsC >vva| F 3q6]A$+ Mc[R"URS5n[URS5Hn[ XSS2U45X#'M U$)Nr)r zerosshaper rrs r matprod12r%sH XXaggaj !F 1771: 11g&  Mrc[R"URS5n[URS5HnX SS2U4X-- nM U$Nrrr rrr rs r matprod21r,sJ XXaggaj !F 1771: AqD'AD.  Mrc $[R"URSURS45n[URS5HAn[URS5H"nUSS2U4==USS2U4XU4-- ss'M$ MC U$rr)rrrrjs r matprod22r!3s XXqwwqz1771:. /F 1771: qwwqz"A 1a4LAadGa1g- -L# MrcR[(dX-$[UR5S:Xa$[UR5S:Xa [X5$[UR5S:Xa$[UR5S:Xa [ X5$[UR5S:Xa$[UR5S:Xa [ X5$[UR5S:Xa$[UR5S:Xa [ X5$[SURSUR35e)NrzInvalid shapes for x and y: z and )r rrrrrr! ValueError)rrs rmatprodr%;s >s  177|qS\Q.a| QWW s177|q0 QWW s177|q0 QWW s177|q07yaggYOPPrc[(d[R"X5$[R"[ U5[ U545n[ [ U55HnXU-USS2U4'M U$N)r r outerrrr rs routprodr)JsZ >xx~ XXs1vs1v& 'F 3q6]t8F1a4L Mrc $[(d![RRXSS9S$URSnURSn[ XE5n[R "U5nUR5n[US- SS5Hn [XSS2U 45n [[R"U5[R"USS2U 455n [X5(aSXy'MfXU - Xy'XU USS2U 4-- nM U$)N)rcondrr) r r linalglstsqrminrcopyr rabsisminor) AbQRdiagmnrankrrryqyqas rlsqrr<Ss >yyq4033  A  A q9D  A A 4!8R $ AAw RVVAYqAw0 2  ADa=ADaD1QT7N"A% Hrc[(d[R"X5$[R"U5(d [ U5nU$[R"U5(d [ U5nU$[ [R "X/55n[R "[ U5[U5/5nUS[R"[5:aGUS[R"[S- 5:a#[R"[X3-55nU$USS:a6US[R"USUS- USUS- -S-5-nU$SnU$)Nrr@) 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 [(d[RRU5$[R"[ UVs/sHoU-PM sn55nU$s snfr')r r r-normrCrD)rxirs rrIrI}sJ >yy~~a  WWS!,!BR%!,- .F M-sA cp[[U5[R"[U55- 5U:*$r')primasumr1r trilr3tols ristrilrP' CFRWWSV_, - 44rcp[[U5[R"[U55- 5U:*$r')rLr1r triurNs ristriurTrQrc$[(d[RRU5$UR 5nUR Sn[ U5(ayURn[R"X45n[U5H;nSX$U4- X4U4'[USU2SU24USU2U45*X$U4- USU2U4'M= UR$[U5(ac[R"X45n[U5H;nSXU4- X4U4'[USU2SU24USU2U45*XU4- USU2U4'M= U$[U5upRnURn[R"X45n[US- SS5H<nUSS2U4[USS2US-U24X$S-U2U45- X$U4- USS2U4'M> [R"U[S9n[R"SUS- U5Xv'USS2U4RnU$)Nrrr,dtype)r r r-invr0rrPTrr r%rTqrintlinspace)r3r8RBrr5PInvPs rrXrXs >yy}}Q A  A ayy CC HHaV qA!qD'kAdG"1"bqb& 1RaRU844qAw>Abqb!eHss  HHaV qA!qD'kAdG"1"bqb& 1RaRU844qAw>Abqb!eH HQ%a CC HHaV q1ub"%AAw1a!eAg:a%'1* !FF!qD'QAadG&xx%++a1a( agJLL Hrc URSnURSn[R"U5nURn[R"SUS- U[ S9n[ U5GH n[R"[[XFUS-2XaS-245SS9SS9nUS:a.XrU- S- ::a#Xv- nXWXVsXV'XW'XGU/SS24XFU/SS24'[ US- US5Hn[XFXh/45Rn [R"[XFU4XFU45S5XFXh/4'[XFS-US-2Xh/4U 5XFS-US-2Xh/4'[USS2Xh/4U 5USS2Xh/4'M GM URn X:U4$)NrrrVaxisr,)rr eyerYr\r[r argmaxrL primapow2planerotappendr?r%) r3r7r8r5rYr_r krGr]s rrZrZs  A  A q A A AqsAS)A 1X IIhyQqS5!aC%<9B K q5Qa%!)^ FAqtJAD!$VQYvva## | 66Q;x*+ +q6M !''!*%qwwqz"AAadG FI# !''!*%qwwqz"AAdG FI# rc X-$)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 rrfrfs  3Jrc F [(a[U5S:XdS5e[[R"U55(aSnSnGO[ [R "U55(ahS[R"S5- [R"US5-nS[R"S5- [R"US5-nGO.[US5S::a[US5S::aSnSnGO[US5[[US5-::a[R"US5nSnGO[US5[[US5-::aSn[R"US5nGO[ [R"[R"[5[R"U5:[R"U5[R"[S- 5:55(a[U5nUSU- nUSU- nO[US5[US5:a`USUS- n[S[U5[R"SXD--55nU[R"US5-nSU- nXE- nO`USUS- n[S[U5[R"SXD--5/5nU[R"US5-nXE- nSU- n[R "X/U*U//5n[(GasUR"S:Xde[R "[R$"U55(de[USUS- 5[US US -5-S::de[R&"S [R("S S [-55n[+Xg5(de[ [R"[R$"U5[R"U5[R"[S- 5:55(aQ[R,RU5n[[X`-US/- 55[XwU-5::dS5eU$)a( As in MATLAB, planerot(x) returns a 2x2 Givens matrix G for x in R2 so that Y=G@x has Y[1] = 0. Roughly speaking, G = np.array([[x[0]/R, x[1]/R], [-x[1]/R, x[0]/R]]), where R = np.linalg.norm(x). 0. We need to take care of the possibilities of R=0, Inf, NaN, and over/underflow. 1. The G defined above is continuous with respect to X except at 0. Following this definition, G = np.array([[np.sign(x[0]), 0], [0, np.sign(x[0])]]) if x[1] == 0, G = np.array([[0, np.sign(x[1])], [np.sign(x[1]), 0]]) if x[0] == 0 Yet some implementations ignore the signs, leading to discontinuity and numerical instability. 2. Difference from MATLAB: if x contains NaN of consists of only Inf, MATLAB returns a NaN matrix, but we return an identity matrix or a matrix of +/-np.sqrt(2). We intend to keep G always orthogonal. r#zx must be a 2-vectorrrr>)r#r#)rr)rr)rr)rrg|=皙?g.AzG @ X = [||X||, 0])rranyr isnanallisinfrCsignr1r logical_andrrrIrBrArr@maximumminimumisorthr-)rcsrGturjrOs rrgrgsy1v{222{ BHHQK   bhhqk    NRWWQqT] *  NRWWQqT] * ad)q.S1Y!^   ad)sS1Y & GGAaDM  ad)sS1Y &  GGAaDM rwww/"&&);RVVAYQX[^Q^I_=_` a aQA!qA!qA!A$i#ad)#!qt AAs1vrwwq13w/0A 1 AAAA!qt AQAAC 012A 1 AAAA 1&A2q'"#Ayww%vvbkk!n%%%%1T7QtW$%AdGag,=(>>!CCCjj"**VUS["ABa~~~ r~~bkk!nbffQi"'''C-:P.PQ R R q!As13!Q<()SAg->> T@T T> HrcSn[U5U[U5--n[U5SU-[U5--n[R"[U5U:X4:5$)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 rpr#)r1r logical_or)rref sensitivityrefarefbs rr2r2EsW K s8kCF* *D s8a+oA. .D ==ST)4< 88rc 8[R"US5n[(a[R"US5[R"US5:Xde[R"US5[R"US5:Xde[R"US5[R"US5:Xde[U5(aUS:de[U5(aUO_[R"SS[ -[R "[R"US5[R"US55-5n[R"X"[R"[U55-U[R"[U55-/5n[[X55[R"U5- U:*R5=(d< [[X5[R"U5- 5U:*R5nU$)zB This procedure tests whether A = B^{-1} up to the tolerance TOL. rrgMbP?gY@) r sizerrrxrrwrBr1r%rdrs)r3r^rOr8is_invs risinvr[st 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#gamVXV\V\]^V_F_B`ehAh@m@m@oF Mrc [(a[U5(aUS:de[R"US5nU[R"US5:aSnU$[R"[ [ U555(aSnU$[U5(a}[ [URU5[R"U5- 5[R"X[R"[ U55-5:*R5nU$[ [URU5[R"U5- 5S:*R5nU$)z[ This function tests whether the matrix A has orthonormal columns up to the tolerance TOL. rrF) rrr rrrrLr1r%rYrdrwrBrs)r3rOnum_varsis_orths rryryys y 3<<!8O8 wwq!}H"''!Q- N ((8CF# $ $ N 3<<7133?RVVH-==>"**SXZX^X^_bcd_eXfRfBggllnG N 7133?RVVH-==>!CHHJG Nrc[U5S:Xa [S5e[SU55n[SU55nS[-[US5-U-$)a Get a relative tolerance for a set of arrays. Borrowed from COBYQA Parameters ---------- *arrays: tuple Set of `numpy.ndarray` to get the tolerance for. Returns ------- float Relative tolerance for the set of arrays. Raises ------ ValueError If no array is provided. rz$At least one array must be provided.c38# UHoRv M g7fr')r.0rAs r !get_arrays_tol..s.vezzvsc 3# UHEn[R"[R"U[R"U55SS9v MG g7f)?)initialN)r rBr1r@rs rrrs9E rvveBKK./0#>sA Ag$@r)rr$rBr)arraysrweights rget_arrays_tolrs_& 6{a?@@ .v. .D F #:D# & //r)rr')__doc__numpyr constsrrrrrr rrrr!r%r)r<r?rIrPrTrXrZrLrfrgr2rryrrnrrrs44 Q . &55 >24X v9,<@0r