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As far as design matrix depends only on vector of knots, `k` and `x` it is useful to make tests for different boundary conditions (and as following different vectors of knots). c.[RRS5n[R"UR U5S-S- 5nUR U5S-S- nUS:XaUSUS'[ XEXS9n[R "[UR5U- S- 5n[URXq5"U5n[R"UURU5R5n [XR-US S 9 [XS S 9 g ) zA To avoid repetition of code the following function is provided. r(rrr8rr5bc_typer/rxryN)rLrSrr random_sampler eyelenr3r design_matrixtoarrayrr4) rVr5rWrr|rrr4 des_matr_def des_matr_csrs rXrun_design_matrix_testsHTestBSpline.test_design_matrix_bc_types..run_design_matrix_testss))''-C))!,r1B67A!!!$r)B.A*$u!%aaADs466{Q*+A"466103L"00151244;GI  L66115 A LU CrorEr<clampednaturalrr: not-a-knotrrrN)r)rTr_rVr5bcs rXtest_design_matrix_bc_types'TestBSpline.test_design_matrix_bc_typessg D*  (B #A" -)q!QA #A, 7  q!A #A* 5ror)FTrdegreerc [RRS5nURSUS--5n[R"U5[R "U5peUn[R [R"US- US- U5[R"XVSUS--5[R"US-US-U54n[R"[U5U- S- 5n [XXq5n [U "U5[R"XHXq5R55 [R"US- US- US-US-/5nU(d;[R "["5 [R"XHXq5 SSS5 g[U "U5[R"XHXq5R55 g!,(df  g=f)z;Test that design_matrix(x) is equivalent to BSpline(..)(x).rrhr/r:rN)rLrSrrXaminamaxrrirYrZr rr[r\arrayrcr rJ) rTrrhrr|xminxmaxr5r3r4bsplines rX'test_design_matrix_same_as_BSpline_call3TestBSpline.test_design_matrix_same_as_BSpline_call%sxii##D)   bFQJ/ 0WWQZd  EE"++dQhq&9++d!vz*:;++dQhq&9: ; FF3q6A:> "!/ AJ--aACKKM HHdRi4#:tbyA Bz*%%aA;+*  %%aA;CCE +*s :G Gc[RRS5nSnSn[R"UR U5S-S- 5nUR U5S-S- n[ XEUS9n[ SS5HTnUSUnUSUn [R"UURU5R5n [XR-U S S 9 MV g) Nrrhr<rUrr5r/r=rxry) rLrSrrrXr rr r[r3r\rr4) rTrrVr5r|rrr%xcycr^s rXtest_design_matrix_x_shapes'TestBSpline.test_design_matrix_x_shapesAsii##D)   GGC%%a(2-2 3   a 2 % *!!!,q!A2AB2AB"00151244;GI  L6612E B roct/SQn[R"SUS5R5n[U/SQ/SS9 g)N)r1r1r1r?r@rArArAr?r<)g?gmਪ?gK}\UU?r>rxry)r r[r\r)rTr3des_matrs rXtest_design_matrix_t_shapes'TestBSpline.test_design_matrix_t_shapesRs7 ,((Q2::<;<" $roc[RRS5nSnSn[R"UR U5S-S- 5nUR U5S-S- n[ XEUS9n[ [5 [R"XFRSSS2U5 SSS5 Sn/S Qn/S Qn[ [5 [R"XGU5 SSS5 g!,(df  NH=f!,(df  g=f) Nrrhr<rUrrsr8r:)r>r1r?r@rA@r) rLrSrrrXr rHrJr r[r3)rTrrVr5r|rrr3s rXtest_design_matrix_asserts&TestBSpline.test_design_matrix_assertsZsii##D)   GGC%%a(2-2 3   a 2 % *!!!,: &  ! !!VVDbD\1 5'  $  : &  ! !! *' & ' & ' &s?(C)C:) C7: DrW)rcrbrrdc[RRS5n[R"URS55nURS5nUS:XaUSUS'[ X4US9n[ R "XQS9n[R"SSS5n[U"U5U"U5SS 9 [X4US9n[URURSS 9 g) Nrrrrr8rWr/r\ry) rLrSrrrr from_power_basisrirr r4) rTrWrr|rcbrrkbspl_news rXtest_from_power_basis!TestBSpline.test_from_power_basislsii##D) GGCJJrN # JJrN j aDAbE w /''< [[Ar "2Ru5%aG< 7roc[RRS5n[R"URS55nURS5URS5S--nUS:XaUSUS'[ X4US9n[ R "XQS9n[X4RUS9n[X4RUS9n[URURSUR--SS 9 g) Nrrr0rrr8rr\ry) rLrSrrrr rr rrrr4) rTrWrr|rrr bspl_new_real bspl_new_imags rXtest_from_power_basis_complex)TestBSpline.test_from_power_basis_complex|sii##D) GGCJJrN # JJrNSZZ^b0 0 j aDAbE w /''<*1ffgF *1ffgF  "}2F FUSroc[R"/SQ5n[R"/SQ5n[R"[ XSS9SS9n[ UR /SQSS9 g) a5 For x = [0, 1, 2, 3, 4] and y = [1, 1, 1, 1, 1] the coefficients of Cubic Spline in the power basis: $[[0, 0, 0, 0, 0],\$ $[0, 0, 0, 0, 0],\$ $[0, 0, 0, 0, 0],\$ $[1, 1, 1, 1, 1]]$ It could be shown explicitly that coefficients of the interpolating function in B-spline basis are c = [1, 1, 1, 1, 1, 1, 1] r;)r/r/r/r/r/rcr)r1r/r/r/r/r/r/r\ryN)rLrlr rrrr4)rTr|rrs rXtest_from_power_basis_exmp&TestBSpline.test_from_power_basis_exmpsM HH_ % HH_ %'' A)(L09; 7eDrocR[R"SS/5n[R"S/5nURSS9 URSS9 [R"SSS5nURSS9 [ XSS9n[ U"U5[R "U5S-5 g)Nrr/r@Fwriterhr2)rLrlsetflagsrir rrr)rTr3r4rkrWs rXtest_read_onlyTestBSpline.test_read_onlys HHaV  HHcUO     [[Ar " %  a ""r||B/#56roc<^[TS9nU4Sjn[SX25 g)Nrwcj>URup nTRX#X#*S- S5nU"U5 g)Nr/i')rbri)rrWr3r5rkrUs rX worker_fn/TestBSpline.test_concurrency..worker_fns3eeGA!QT1RT7E2B bErorh)rar&)rTrUrWrs ` rXtest_concurrencyTestBSpline.test_concurrencys 2 &  I1rocygwinz'threading.get_native_id not implemented)rr c f[5n[R"SSS5nU"U5n[R"5n[R "[ URSUS355SURRURRS9nURUSS&[R "[ URSUS355SURRURRS9nURUSS&XblXrl [U"U5U5 g) Nrr/rhr3z.datzw+)moderGrr4) rarLri threading get_native_idmemmapstrjoinr3rGrr4r)rTtmpdirrWrkrltidt_mmc_mms rX test_memmapTestBSpline.test_memmaps ! [[Ar "R5%%'yyV[[1SE78t ssyy ;##QyyV[[1SE78t ssyy ;##Q"x(ror9N)9r:r;r<r=rYrermrsrrskip_xp_backendsrrrrrrrrrrrrrrrrr r rrr'r-r4rArcmark parametrizerrJrQrfrprvrzr~rrrrrxfailsysplatformrdrr>r9rorXr,r,1s!F )>D @l+OPk*KLgk2 /3MQ /A ;1 .E&W? ( 6? FG4A M :# F 7+BZ 1!1 [[VU2q\2%3%:D(6T [[],EF [[XuQx01G4C"$+$ [[Y)CD 8D 8 [[Y)CD TD TE& 7 2 [[  8 )  )ror,cF\rSrSr\R R S/SQ5S5r\R R S/SQ5S5rSr \R R S /S Q5S 5r \R R S S S/5S5r Sr Sr Srg ) TestInsertixval)r>r1rCr=@@c hURSURS9nURU5S-n[X4SS9nURUR UR 4n[[X56nURU5n[URURUR5SS9 [URURURSUR *S- 5SS9 XS:waUOUSSn URU S USSUSS--45n [U"U 5U"U 5SS9 URU5S-n [X:SS9n [X2R!XJ4SS 9SS9n U RU5n [U RURSS9 [U RUR!URU5RU RU5R4SS 9SS9 XS:waUOUSSn URU S USSUSS--45n [U "U 5U "U 5SS9 g) NrrFr<rsr\ryr/r8rr)rPrQsinr _t_cr5r r insert_knotrr3rRr4concatcosr)rTrrUr|rrNrbspl_1fspl_1rky1spl_y1spl_yyspl_yy1s rX test_insertTestInsert.test_inserts IIarzzI * FF1IqL +vvsvvsuu%&+,%FHH!5EBFHHYsuufQh,?!@uMB%-QQsV YYC123B01 2Br7VVAY\#AQ/#Axxax'@AF$$T* 5777 II HHcood+--v/A/A$/G/I/IJQRH S B%-QQsV YYC123B01 2r GBKer:)r1r<)rr)r=r:)rr:c \URSURS9nURU5S-n[XESS9n[ [ XR URUR4US96nURX5n[URURUR5SS9 [URURURSUR*S- 5SS9 XS :waUOUSS n URU S USSUSS --45n [U"U 5U"U 5SS9 g) NrrFr<rsmr\ryr/r8r)rPrQrr r rrrr5rrr3rRr4r) rTrrrUr|rrNrrrks rXtest_insert_multiTestInsert.test_insert_multis IIarzzI * FF1IqL +&'>!DE(FHH!5EBFHHYsuufQh,?!@uMB%-QQsV YYC123B01 2Br7roc [RRS5nSup4UR[R"UR X4-S-S955nURUR USS4S95n[ XVU5nURUR XTS-XT*S- S95nURU5n URUR XTS-XT*S- SS 95n [U"U 5U "U 5S S 9 g) N90rDr/rr<r:)lowhighrrrrr\ry) rLrS default_rngrRruniformr rr) rTrUrrVr5r3r4rNxvrrks rXtest_insert_randomTestInsert.test_insert_randomsii##E* JJrwws{{A{67 8 JJs{{Aq {2 3aA ZZ A#Qr!tW = ># ZZ A#Qr!tW2 F GBr7ror) r皙?r?rA@@@gffffff@rc URSURS9nURU5S-n[X4SS9upVn[ UR XV45upV[ XVUSS9nURU5n [XRSS9upn[U RUR U 5S S 9 [U RSU*S - UR U SU*S - 5S S 9 [RRS 5R!S SSS9n UR U 5n [U "U 5UR [#XX455S S 9 g)NrrFr<rsrrTperr\ryr/rrr)r)rPrQrrr{rRr rrrbrr3r4rLrSrrr)rTrrUr|rr3r4r5rNrtfcfrrks rXtest_insert_periodicTestInsert.test_insert_periodics IIarzzI * FF1IqL#a2::v&aA:6#2wwD1 Be<!A 2er!t9(=EJ %%d+333K ZZ b 2::eE;.G#HuUrorNrcRURSURS9S-[R-nUR U5UR U5pT[ X4SU--SS9nXl[ X4SS9nXl[ X5SS9nXlSn URU 5n URU 5n URU 5n [U RU RSS 9 [U RU RSS 9 [U RU RSU R--SS 9 g) NrrFr:r0r<rs @r\ry) rPrQrLpirrr rrrr3r4) rTrrUr|y_rey_imrNspl_respl_imrrspl_1respl_1ims rX test_complexTestInsert.test_complex0s IIarzzI *Q . 6VVAYq d 2d7Na8%#Aq1(#Aq1( #$$R($$R(77R \!9FrocSnSn[R"S/US--/SQ-S/US---5n[R"[U5U- S- 5n[ X4USS9n[ [ 5 [XXB4S S 9 SSS5 [ [ 5 URU5 SSS5 g!,(df  N8=f!,(df  g=f) Nrr<rr/r:r<r=rrrrTr) rLrlonesrZr rHrJrr)rTrr5r3r4rNs rX+test_insert_periodic_too_few_internal_knots6TestInsert.test_insert_periodic_too_few_internal_knotsGs  HHaS!A#Y-QqS 9 : GGCFQJN #aA:6 : & 21yd +': & OOB ' &' &' &s0 B/C/ B= CcSn[R"S/US--/SQ-S/US---5n[R"[U5U- S- 5n[ X#U5n[ [ 5 URS5 SSS5 [ [ 5 URS5 SSS5 [ [ 5 URSSS9 SSS5 g!,(df  N`=f!,(df  NH=f!,(df  g=f) Nr<rr/rrr8rr)rLrlrrZr rHrJr)rTr5r3r4rNs rXtest_insert_no_extrap TestInsert.test_insert_no_extrap[s  HHaS!A#Y-QqS 9 : GGCFQJN #aA : & OOB ': & OOA ': & OOAO #' & ' &' &' &s$/CC,C= C), C:= D r9)r:r;r<r=rcrrrrrrrrrr>r9rorXrrs [[V%AB!=C!=H [[C 8 8 8 [[T$8VV" [[]T:,>?G@G, ( $rorc SSjnSHRn[US9n[[U55H.up4U"XC5 [SUS-5HnU"XCUSS5 M M0 MT g)Nc URupVn[R"U5n[RUSS- SUSSUSS--USS-4n[ [ XXg4U5U"X5X4SUSUR 3S9 g) Nrrrr/r8zder = z k = )r^r_err_msg)rbrLuniquerrrr5) rWrrr^r_r3r4r5r|s rX check_splev,test_knots_multiplicity..check_splevos%%a IIaL EE!A$s(C122A/2s: ;aQC0!A)se6!##.G Iror/r:r<r=rrsr/r)rrxrx)ra enumerate_make_multiplesr)rr5rWrrIrs rXtest_knots_multiplicityrks^I ! $q12EA  Q!}B3u5%3rocJUS:XaX2Us=::a X2S-:aS$ S$X2U-X2:XaSnO$XU- X2U-X2- - [XS- X#5-nX2U-S-X2S-:XaSnXE-$X2U-S-U- X2U-S-X2S-- - [XS- US-U5-nXE-$)zk Naive way to compute B-spline basis functions. Useful only for testing! computes B(x; t[i],..., t[i+k+1]) rr/r1r>_naive_Br|r5r%r3c1c2s rXrrs  Avda(!aC&(s1(c11v~ A$hQ3!$ '(1c1*@ @1Qx1qS6  G!AhlQs1uXA#. /(1c1Q32J J Groc ^^^^^TTT:XaTmOURTT5S- mTTTs=::a TTS-::de eTT:aTTRST- :de[UUUUU4Sj[STS-555$)z5 Naive B-spline evaluation. Useful only for testing! r/rc3X># UHnTTU- [TTTU- T5-v M! g7fNr).0rr4r%r5r3r|s rX _naive_eval..s-F 1q1vAqsA.. s'*) searchsortedrsumr)r|r3r4r5rUr%s```` @rXrrs AaDy  OOAq !A % Q41 !A#    6a!''!*q.(( ( FFa1 F FFroc^^^^TRSTS-- nUTS-:deTRSU:deTTTs=::a TU::de e[UUUU4Sj[U555$)z'Naive B-spline evaluation, another way.rr/c3L># UHnTU[TTUT5-v M g7frrrr%r4r5r3r|s rXr _naive_eval_2..s%=Hqqthq!Q**H!$)rrr)r|r3r4r5rUrVs```` rXrrsr  acA !8O8 771:?? Q41 !    =E!H= ==roc[U5US-- nXCS-:deURSU:deSn[U5HCn[R"XXc-S-SS9"U5nXRU[ R "U5-- nME U$)Nr/rr>r:Fr)rZrrr rrL nan_to_num)r|r3r4r5rVsr%rWs rXrrs A!A#A !8O8 771:?? A 1X  ! !!ac!e*% @ C qTBMM!$ $$ Hroc[R"U5n[R"XS:US:-US:US:-US:US:*-/SSS/5nURU5$)z*A linear B-spline function B(x | 0, 1, 2).rr:r/cgNr>r9r|s rXB_012..sbrocU$rr9rs rXrrsAroc SU- $Nr?r9rs rXrrsAro)rL atleast_1d piecewiserR)r|rUresults rXrzrzsp aA \\!1uQ/!VA.!VQ/1)+~FHF ::f roc[R"U5nUS:US:US:-US:/nUS:Xa SSS/nOUS:Xa SSS /nO[S U35e[R"XU5nU$) z0A quadratic B-spline function B(x | 0, 1, 2, 3).r/r:rcX-S- $rr9rs rXrB_0123..s 13r6rocSUS- S-- $)Ng?rr:r9rs rXrrs41T6A+-rocSU- S-S- $)Nr@r:r9rs rXrrsBqD19q=rocgNr1r9rs rXrr2rocg)Ngr9rs rXrrs3rocgr#r9rs rXrrr$roznever be here: der=)rLrrJr)r|rcondsfuncspiecess rXrrs aA UQUq1u%q1u -E ax!-(* .se455 \\!E *F Mro#r<c&[RRS5n[R"URX-S-55nURU5nUR U5UR U5pT[ R "XEU5$)N{r/)rLrSrrrRr construct_fast)rVr5rUrr3r4s rXrarasj ))   $C  13q5!"A 1 A ::a="**Q-q  ! !! **roc#v# URURp!URR5nUSUSS&USUS'[ X1U5v URR5nUSUSUS-&[ X1U5v URR5nUSX2*S- S&[ X1U5v g7f) zIncrease knot multiplicity.rrNr/r8)r4r5r3copyr )rWr4r5t1s rXrrs 33q B2Br"I VBrF "  B!uBt!H "  B2Br!tuI " sB7B9cV\rSrSrSrSrSrSrSrSr Sr S r S r S r S rS rg) TestInteropicn[R"SS[R-S5n[R"U5n[ X5nUR UR UR4UlXUsUl Ul Ul [R"SS[R-S5Ul [RUR UR UR 4n[R"XD45Ul[!UR URUR5Ulg)NrrArr1)rLrirrr r3r4r5rbrkrrWxnewc_dstackrr b2)rTrkrrWrs rX setup_methodTestInterop.setup_methods [[BruuHb ) VVBZ r &CCacc?#%1 $&KK2bee8R0 UU133QSS= !))RH%!##tww,roc URURURp2n[[ X5U"U5SSS9 [[ XR 5U"U5SSS9 [[ R"UVs/sHn[ XB5PM sn5U"U5SSS9 [[SS9 [ X5 SSS5 [[SURR55S-nURRU5nURXcR 4n[[ R"[ X55U"U5RU5SSS9 gs snf!,(df  N=f)Nr\r]zCalling splev.. with BSplinematchr/r)r7rWr:rrrbrLrRrHrJrErr4r2 transposer3r5)rTr7rWr:r|rFrrbs rX test_splevTestInterop.test_splevsii d$e% 9dEE*$e% 9 #>AE!K#>?$e% 9 :-K L $OM5BDDII& '$ . 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A!%! R"@Aq5!{{17A!eH!{{17AeH B  1 A /ra@IIOOA1 ?#!ror9N)0r:r;r<r=rLrirrkrrrrrrrcrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rr>r9rorXrrs RBEE "B B88;; [[S,/*0*  [[S,/ *0 *? I  [[S"45 -6 -7" > 7 [[S,/ ?0 ?2 = ,;, T [[j) E* E>& ;5. 5?"*@ [[S,/&0&  " 130j 9?rorcURUR:XdeURURU-S-:XdeURn[R"XD4[RS9n[ U5HOnXnXrU:XaUnO[R "X'5S- n[ R"X#Xx5n XXhU- US-24'MQ [R"XQ5n U $)zAssemble an spline order k with knots t to interpolate y(x) using full matrices. Not-a-knot BC only. This routine is here for testing only (even though it's functional). r/rF) rrLrmrQrrr*evaluate_all_bsplslr) r|rr3r5rVArrleftbbr4s rXrrs 66QVV   66QVVaZ!^ ## # A !rzz*A 1Xt Q4<D??1+a/D ' 'd 9 !!VDF]  A Hroc:[[RXU45upnURnURU- S- n[R"XE4[R S9n[ U5HOnXnXU:XaUn O[R"X(5S- n [R"X#X5n XXyU- U S-24'MQ [R"URU5n [R"URU5n [R"X5n XU 44$)z,Make the least-square spline, full matrices.r/rF)r{rLrRrrmrQrrr*rdotTrr)r|rr3r5rrVrrrrrr?Yr4s rXmake_lsq_full_matrixr$s"**qQi(GA! 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HHUOQXXe_ahhuo6 eafn # WR[u=roc[R"SSS5nUSSS2nUSSS2n[US5n[X4USUS9 g)Nr8r/rr<rR)rLrirr)rTr%rkr|rr3s rXrTestLSQ.test_sliced_inputzsI[[Q $ ssG ssG AqMa1V4roc [R"S5R[5nUS-n[ US5n[R [R [R *4HnXSS'[[[X#XAS9 M g)N r:r<r8r+) rLrPrSrXrrNrOrHrJr)rTr%r|rr3rs rXtest_checkfiniteTestLSQ.test_checkfinitesd IIbM  ' qD AqM&&"&&266'*AbE *oqQ N+rocURURURpCnURSS9 URSS9 URSS9 [ X#XAS9 g)NFr)r|rr3r%)r|rr3rrrTs rXrTestLSQ.test_read_onlysO&&$&&$&&a      !A5ror5r/cURURp2[[R"USUSS5U5n[ X#XASS9n[ X#XASS9nUSSUSS-S- n[ U"U5U"U5S S 9 g) Nrr8rr&rRr'r/r?r\ry)r|rrrLrirr)rTr5r|rr3 spl_norm_eqspl_qrrks rXtest_qr_vs_norm_eqTestLSQ.test_qr_vs_norm_eqs~vvtvv1 BKK!aeQ/ 3%aA9E  qd;eafn # B%@rocT[R"URS5n[R"URS5n[ URURUR SSS9n[ XUR SSS9nUSSUSS-S- n[ U"U5U"U5SS 9 g) Nr:r<r'rRr/r8r?r\ry)rLrepeatr|rrr3r)rTr|rrspl_2rks rXtest_duplicatesTestLSQ.test_duplicatess IIdffa  IIdffa !DIdff$?eafn #b 5959ror9N)&r:r;r<r=rLrSrrrVr5rr|rrrir3parametrize_lsq_methodsr1r7r>rrErIrrOrr\rrbrrcrrrrrirnr>r9rorXr)r)sv ))   %C DAq  1 A 1 A AaD!B%+Q/A ! !   =:: 0 05 M MM,55 > >55OO66 [[S$uQ{"34A5A:ror)c4\rSrSrSrSr\S5rSrSr g) PackedMatrixia_A simplified CSR format for when non-zeros in each row are consecutive. Assuming that each row of an `(m, nc)` matrix 1) only has `nz` non-zeros, and 2) these non-zeros are consecutive, we only store an `(m, nz)` matrix of non-zeros and a 1D array of row offsets. This way, a row `i` of the original matrix A is ``A[i, offset[i]: offset[i] + nz]``. cXlX lX0lURS:XdeURS:XdeURSURS:Xdeg)Nr:r/r)rrncr2r)rTrrrts rX__init__PackedMatrix.__init__sQ vv{{{{awwqzV\\!_,,,rocLURRSUR4$Nr)rrrtrs rXrPackedMatrix.shapesvv||A''rocn[R"UR5nURRSn[ URS5H_n[ UR URU- U5nURUSU24XURUURUU-24'Ma U$)Nr/r)rLrmrrrminrtr)rToutnelemr%nels rXtodensePackedMatrix.todenseshhtzz" Qsyy|$Adgg A.6C:>&&DSD/C4;;q>$++a.3"666 7% ro)rrtrN) r:r;r<r=__doc__rupropertyrrr>r9rorXrrrrs%-((rorrc  SSKJn URnURnURnUR upxUR SU:XdeUR 5n UR 5n [X'5Hn X[n [X5Hn U [X5:a M'U"XS4XS45upnUXS4'[SU5H&n[XXU4XU45uXU4'XUS- 4'M( SXS4'[U R S5H#n[XXU4XU45uXU4'XU4'M% M M [[U R S55n[U [R"U[RS9U5nUU 4$)zbThis is a python counterpart of the `_qr_reduce` routine, declared in interpolate/src/__fitpack.h r)dlartgr/r>r8rF)scipy.linalg.lapackrrrrtrr2rr{fprotarrrrLrlint64)a_prstartrowrrrrtrnzRrr%oirr4rrroffsR_ps rX _qr_reduce_pyrs+ A ZZF B GGEA 771:?? A B 8  YrACJQ!tWa1g.GA!AdG1b\%+A!qD'1T7%C"Q$ac6"AeH288A;'%+A"T(B!tH%E"a4"T(( * aggaj! "D q"((4rxx8" =C 7Nroc0X-X--nU*U-X--nXE4$)z@Givens rotate [a, b]. [aa] = [ c s] @ [a] [bb] [-s c] [b] r9)r4rrrWr0rs rXrrs) #)B AB 6MrocURnURup4URnURSURS:Xde[R"USU5nXS- X%S- S4- XeS- S4'[ US- SS5HGn[ XEU- 5nX'SU2S4XgS-Xx-2S4-RSS9n XU - X'S4- XgS4'MI U$)zxBacksubsitution solve upper triangular banded `R @ c = y.` `R` is in the "packed" format: `R[i, :]` is `a[i, i:i+k+1]` rNr/.r:r8r)rrrtrLrrr{r) rrrrrrtr4r%r~summs rXfpbackrs A GGEA B 771: ## # afAT7Q!tQwZ'AdCiL 2a4R "dmQsUD.!Ac!%in$55:::BdTka1g-S& ! Hroc8\rSrSrSrSrSrSrSrSr Sr g ) TestGivensQRicrSn[R"U[S9nUS-SSU-- -n[X25nX4XR4$)Nr<rFr/)rLrPrXr)rTrVr5r|rr3s rX_get_xytTestGivensQR._get_xyts@  IIau % qD1ac7N  Qzroc VSnURU5up#pE[R"X$U5n[R"UR 55upxUR U-n URupXRSU- S- :XdeURSSUS-2n [R"U [RS9n URRXS-5n [XU 5nUSS2S4n[R "XX5 [#[R$"UR 5U-UR 5U- 5[R&"U5SS9 [#[R$"[)XSS2S4- 5[)XSS2S4-55[R&"U 5SS9 [R*"X5n[R,"UR.UR0X/XE[R2"U5U5un n[#UUSS2S4SS9 g) Nrhrr/rFr\rygvIh%L=-a=)rr r[rr'rr"rindicesrLascontiguousarrayrdatarrrr* qr_reducerminimumrr:rrrrtrr)rTrVr|rr3r5a_csrqrqTyrrtrrry_c_fullc_bandedrs rX test_vs_fullTestGivensQR.test_vs_full#s ]]1% a%%aA.uuU]]_%ccAg WWQZ!^a''''w!A#w'%%fBHH= JJ  qA# & B ' q$wZ1b-  199;?#$99;?457]]15EE S 3s1X~#6#&s1X~#689;s9KRW Y!!!addA1aRTU!QAU;rocSnURU5up#pE[R"X$U5nURupxXRSU- S- :XdeURSSUS-2n [ R "U [ RS9n URRXuS-5n [XU5n USS2S4n [X5up[R"XX5 [U RU RSS9 [!U R"U R"SS9 U R$U R$:Xde[XSS9 g) Nrhrr/rFr\ryF) check_dtype)rr r[rrrLrrrrrrrr*rrrrrrt)rTrVr|rr3r5rrrtrrrrRRrs rXtest_py_vs_compiled TestGivensQR.test_py_vs_compiledFs ]]1% a%%aA. WWQZ!^a''''w!A#w'%%fBHH= JJ  qA# & B ' q$wZq%1b.acc. 188?uu}}U+rocSnURU5up#pE[R"SUS-[S9n[R "X$XV5upxn UR Sn [R"X$U5n XSS2S4-R5n U RRXS-45n U RSSUS-2R[R5n[X}SS9 [!X5 XR SU- S- :Xdeg)Nrhr/rFrr\ry)rrLrPrXr* data_matrixrr r[tocsrrrrrSrrr)rTrVr|rr3r5r5rrrtrra_wA_offset_s rXtest_data_matrixTestGivensQR.test_data_matrixas ]]1% a IIa1E * ,,Q18 2 GGAJ%%aA.Dz!((* XX  qA#h '++g1g&--bhh7E*(WWQZ!^a''''roc SnURU5up#pE[RX3S-4n[R"X$U[R "U55upgn[ XgU5n [R"XgX5 [X5n [R"XhX#XE[R "U5U5un n [XSS9 g)Nrhr:rxry) rrLr8r*rrrrrrrr) rTrVr|rr3r5rrrtrr4rrs rX test_fpbackTestGivensQR.test_fpbackrs ]]1% a EE!T'N ,,Q1bll1oF 2 B '1b, 1L??1!bll1oqIAqE*roc SnURU5up#pE[R"US-4[S9nUHMn[ [ R "XEXqUS-5U5 [ [ R "XEXqSU-5U5 MO g)Nrhr/rFr:)rrLrmrXrr*r)rTrVr|rr3r5 zero_arrayrs rXtest_evaluate_all_bspl#TestGivensQR.test_evaluate_all_bspls{ ]]1% aXXq1uhe4 D **1!a%@* N **1!A#>  Lror9N) r:r;r<r=rrrrrrr>r9rorXrrs"!rr/r8rz)``x`` and ``y`` length must be at least 5r>)rLrSrrrXrnormalrHrJrrr2rPrrcr )rTrrVr|rx_duplexception_messages rXtest_invalid_input&TestSmoothingSpline.test_invalid_inputsii##D)  GGC%%a(1,q0 1 qD266!a%= 1a4 '#**Ra*@ @: & !!qrU +' : & !!AB% +' : & !!))Aq/1 5': & !!DbD'1 -'1Iq : & !& ,' IIaL GGAJG ]]:-> ? !! '@ ?+' & & & & &' & ' &@ ?sHG;G "G2H H9 H% G G/2 H H H"% H3c[R"[S55nUSnUSnUSnSSS5 [WW5"U5n[ UWSSSS9 g!,(df  N.=f) a} Data is generated in the following way: >>> np.random.seed(1234) >>> n = 100 >>> x = np.sort(np.random.random_sample(n) * 4 - 2) >>> y = np.sin(x) + np.random.normal(scale=.5, size=n) >>> np.savetxt('x.csv', x) >>> np.savetxt('y.csv', y) We obtain the result of performing the GCV smoothing splines package (by Woltring, gcvspl) on the sample data points using its version for Octave (https://github.com/srkuberski/gcvspl). In order to use this implementation, one should clone the repository and open the folder in Octave. In Octave, we load up ``x`` and ``y`` (generated from Python code above): >>> x = csvread('x.csv'); >>> y = csvread('y.csv'); Then, in order to access the implementation, we compile gcvspl files in Octave: >>> mex gcvsplmex.c gcvspl.c >>> mex spldermex.c gcvspl.c The first function computes the vector of unknowns from the dataset (x, y) while the second one evaluates the spline in certain points with known vector of coefficients. >>> c = gcvsplmex( x, y, 2 ); >>> y0 = spldermex( x, c, 2, x, 0 ); If we want to compare the results of the gcvspl code, we can save ``y0`` in csv file: >>> csvwrite('y0.csv', y0); z gcvspl.npzr|ry_GCVSPLN-C6?F)r^r_r)rLloadrrr)rTrr|rry_comprs rXtest_compare_with_GCVSPL,TestSmoothingSpline.test_compare_with_GCVSPLsgRWWY|, -S AS AJ'H . 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IIbM qD : & QR8 9' & &s A  Ac URSURS9nURU[R-S- 5nSn[ [ X#USS95n/SQ/SQ/SQ/S Q/S Q/nUVs/sHoqRXqRS9PM nn[U5[U5:Xde[XV5Hup[XS S 9 M [X#USS9un n [US URU5S S 9 gs snf)aY To generate the `wanted` list below apply the following diff and rerun the test. The stdout will contain successive iterations of the `t` array. $ git diff scipy/interpolate/fitpack/fpcurf.f diff --git a/scipy/interpolate/fitpack/fpcurf.f b/scipy/interpolate/fitpack/fpcurf.f index 1afb1900f1..d817e51ad8 100644 --- a/scipy/interpolate/fitpack/fpcurf.f +++ b/scipy/interpolate/fitpack/fpcurf.f @@ -216,6 +216,9 @@ c t(j+k) <= x(i) <= t(j+k+1) and store it in fpint(j),j=1,2,...nrint. do 190 l=1,nplus c add a new knot. call fpknot(x,m,t,n,fpint,nrdata,nrint,nest,1) + print*, l, nest, ': ', t + print*, "n, nmax = ", n, nmax + c if n=nmax we locate the knots as for interpolation. if(n.eq.nmax) go to 10 c test whether we cannot further increase the number of knots. rrFr<rr;)r>r>r>r>rrrr) r>r>r>r>rArrrr r>r>r>r>r?rArrrr) r>r>r>r>r?rArrrrr) r>r>r>r>r?r@rAr}rrrrr\ryr8N) rPrQrrLrrr"rRrZrnrr) rTrUr|rr5r@wantedwantr3rrs rX test_s_switchTestGenerateKnots.test_s_switch s. 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