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IA~IAHaH)AAHHcЉH9H5KH9L5u;L9uL9v„nfHUH;V@HEH~H9AHAt HDM D^ DD@@8A  H}8A  Hv8DA4A`DDE9HtL$ HeL$ L=2:L9 DL$(DD$LLHT$IHp!H=QE1!Wf.HD$8H0:I9B LHL$8HLLLL$L$cL$LL$t"HM9LHH5#H81H|$@HtHx HH$H|$HHtHx HHH|$PHtHx HH!H|$XHtHx HHH|$`HtHx HHH|$hHtHx HHH|$pHtHx HHH|$xHtHx HHt`H=OE1 HL[]A\A]A^A_DIEHJH?PHcHmcYRuHM>%4*1E111LNAtAHNLL$XHL$$tHD$LnHD$PAEtAEL&Ll$HA$tA$Ld$@HHMH,7H$HMH5`7Ht$D$(tHD$D$(HD$xg1Lv0AtALt$pH^(tH\$hHn EtEHl$`1E11E11H~8tH|$xL94,H5DH9L9uL9v„LML;NHUHFH9@H@t HD] DV DD8A  H}8A  Hv8Ѓ  DA9It $IH $cHLLI؋D$0PDL$(HT$ AY^IH* H|$@HtHx HH!H|$HHtHx HH H|$PHtHx HHH|$XHtHx HHH|$`HtHx HHH|$hHtHx HHH|$pHtHx HHcH|$xHIH>HH1֖'L=4L9H5BL9L9vL%@FA$tA$H-EH ?HDŽ$IT$H$H9 H4H9 HXH LGM~"1LLL9- I9$ HI9uHB8IHH$1LHHI$xHI$2 HtHӏHExHHE^ %UD$IHL OAAUH OH2HxH5:IH81踙XZD$(=HHH)HHHL×HcЉH9Hu 蚗HuH2H5GH8菕zHKHD$8II tH5H-BEtEH\$hHl$`H:L=-2AtALt$pL|$hLM H51H4$D$tH$H|$xLd$@HD$pIHH51Ht$$tHD$HD$xIHD$PD$(Ll$HLL$XHD$HV1H$HL$ ,L$ HH)H;D$H;<$L9L$,H|$ ΖH|$ L$,Hx HHS@IHH HMHAUL LH50H4$D$tH$HD$pIAL=0AtAL|$hLH-@EtEHl$`虒菒腒{qg%]9AAAQHH HcЉH9QAAAQHH HHcЉH9,LHL$8HLLLL$L$YLL$L$HHH|D$,L$ 踑D$,L$ ZL=d/L9uL9uHL$ L$ HI H;D$L;4${M9rL $螔 $Ix HIiqLH$tRH$1HHHH$HH5tHHL)HD@HEHHI9PH0L;HuAHt2LL$ H,LL$H_L%?A$tA$H-r>H 9HDŽ$IT$H$H9EHB.H95HXHgLGM~"1LLL9I9HI9uHB8IH H$1LHHI$xHI$ HtH艉HExHHE  HD$PLl$HLL$XHD$H,H$H$I9H 0LL}WLN(H8A@IE*H}(LU8A@IDIHIL $豎 $#aH$1ɺL蘑HDD+H$1ɺLbHtDDH+H@`H2HH"LIHH@H;+DIFHwJH)AFHHcЉH9ukI3HI&LL$ 菍L$ HHH)HHt|HtWLHcЉH9tHH*H5H8IXHIKL>AFAVHH HcЉH9=AFAVHH HHcЉH9It$FtdH^E1 uMl$H=qHLH茏HI$HI$LaH9ID$0"H7HHH9t.HuH5*H9tHHH9tHuH9It$FH^E1 uMl$H=HLH豎HI$HI$L膋DGD;HV(H8A@HE HU(H}8A@HEIt$FJH9ID$0!ǍHt'I$iHI$[LNH$(H5H8蕋~Ht'I$jHI$\L觊OH'H5|H8LHHH9t.HuH5_(H9tHHH9tHuH9It$F}rHHPtZH'HѾHH81IF蒌H HZ'H5` H8胊H?'H5H81ItH@`HHHLIHH@H;U'IGHwJH)AGHHcЉH9ugIeHIXLL$ L$ CHHH)HHtmHtHL茋HcЉH9tHtH]&H5H8f{IHt؃iAGAWHH HcЉH9LAGAWHH HHcЉH9*H@`HBHH2LIH!H@H; &LLL$ ILL$ IHILωL$ ԇL$ HP H3%HѾHD H81躌IG24HIH$H5H8%.HPLL$ t~H$HѾH H81NLL$ IHIuLh觉HZHo$H5uH8蘇HQ$H5 H81(LL$ H2$H5 H81 I\HIOLyBHHtakes no arguments%.200s() %s (%zd given)takes exactly one argumentBad call flags for CyFunctiontakes no keyword arguments%.200s() %s_cython_3_1_6LRNUcannot import name %S__pyx_capi____loader__loader__file__origin__package__parent__path__submodule_search_locationsneeds an argumentan integer is requiredkeywords must be stringsTMissing type objectfloat divisionCname '%U' is not definedbuiltinscython_runtime__builtins__scipy.linalg._decomp_updatedoes not matchnumpydtypeflatiterbroadcastndarraygenericnumberunsignedintegerinexactcomplexfloatingflexiblecharacterufuncscipy.linalg.cython_blascaxpyccopycgemmcgemvcgerucscalcswapctrmmdaxpydcopydgemmdgemvdgerdnrm2drotdscaldswapdtrmmdznrm2saxpyscnrm2scopysgemmsgemvsgersnrm2srotsscalsswapstrmmzaxpyzcopyzgemmzgemvzgeruzscalzswapztrmmscipy.linalg.cython_lapackcgeqrfclarfclarfgclartgcrotcunmqrdgeqrfdlarfdlarfgdlartgdormqrsgeqrfslarfslarfgslartgsormqrzgeqrfzlarfzlarfgzlartgzrotzunmqrnumpy._core._multiarray_umathnumpy.core._multiarray_umath_ARRAY_API_ARRAY_API is NULL pointernumpy.import_arrayat leastat mostqr_deleteqr_update_form_qTuexactlyqr_insert__reduce____module____dictoffset____vectorcalloffset____weaklistoffset__func_doc__doc__func_name__name____qualname__func_dict__dict__func_globals__globals__func_closure__closure__func_code__code__func_defaults__defaults____kwdefaults____annotations___is_coroutineC function %.200s.%.200s has wrong signature (expected %.500s, got %.500s)%.200s does not export expected C function %.200sInterpreter change detected - this module can only be loaded into one interpreter per process.Shared Cython type %.200s is not a type objectShared Cython type %.200s has the wrong size, try recompiling__int__ returned non-int (type %.200s). The ability to return an instance of a strict subclass of int is deprecated, and may be removed in a future version of Python.__int__ returned non-int (type %.200s)value too large to convert to intunbound method %.200S() needs an argument%.200s.%.200s is not a type object%.200s.%.200s size changed, may indicate binary incompatibility. Expected %zd from C header, got %zd from PyObject%.200s() keywords must be strings%s() got multiple values for keyword argument '%U' while calling a Python objectNULL result without error in PyObject_Calljoin() result is too long for a Python string%s() got an unexpected keyword argument '%U'Cannot convert %.200s to %.200s../../tmp/build-env-99e64tom/lib/python3.12/site-packages/numpy/__init__.cython-30.pxd__annotations__ must be set to a dict object__name__ must be set to a string object__qualname__ must be set to a string object__defaults__ must be set to a tuple objectchanges to cyfunction.__defaults__ will not currently affect the values used in function calls__kwdefaults__ must be set to a dict objectchanges to cyfunction.__kwdefaults__ will not currently affect the values used in function callsfunction's dictionary may not be deletedsetting function's dictionary to a non-dictcalling %R should have returned an instance of BaseException, not %Rraise: exception class must be a subclass of BaseExceptionscipy.linalg._decomp_update.reorthscipy.linalg._decomp_update.reorthxscipy/linalg/_decomp_update.pyxModule '_decomp_update' has already been imported. Re-initialisation is not supported.compile time Python version %d.%d of module '%.100s' %s runtime version %d.%dvoid (int *, __pyx_t_float_complex *, __pyx_t_float_complex *, int *, __pyx_t_float_complex *, int *)void (int *, __pyx_t_float_complex *, int *, __pyx_t_float_complex *, int *)void (char *, char *, int *, int *, int *, __pyx_t_float_complex *, __pyx_t_float_complex *, int *, __pyx_t_float_complex *, int *, __pyx_t_float_complex *, __pyx_t_float_complex *, int *)void (char *, int *, int *, __pyx_t_float_complex *, __pyx_t_float_complex *, int *, __pyx_t_float_complex *, int *, __pyx_t_float_complex *, __pyx_t_float_complex *, int *)void (int *, int *, __pyx_t_float_complex *, __pyx_t_float_complex *, int *, __pyx_t_float_complex *, int *, __pyx_t_float_complex *, int *)void (int *, __pyx_t_float_complex *, __pyx_t_float_complex *, int *)void (char *, char *, char *, char *, int *, int *, __pyx_t_float_complex *, __pyx_t_float_complex *, int *, __pyx_t_float_complex *, int *)void (int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *)void (int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *)void (char *, char *, int *, int *, int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *)void (char *, int *, int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *)void (int *, int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *)__pyx_t_5scipy_6linalg_11cython_blas_d (int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *)void (int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, __pyx_t_5scipy_6linalg_11cython_blas_d *)void (int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *)void (char *, char *, char *, char *, int *, int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *, __pyx_t_5scipy_6linalg_11cython_blas_d *, int *)__pyx_t_5scipy_6linalg_11cython_blas_d (int *, __pyx_t_double_complex *, int *)void (int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *)__pyx_t_5scipy_6linalg_11cython_blas_s (int *, __pyx_t_float_complex *, int *)void (int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *)void (char *, char *, int *, int *, int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *)void (char *, int *, int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *)void (int *, int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *)__pyx_t_5scipy_6linalg_11cython_blas_s (int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *)void (int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, __pyx_t_5scipy_6linalg_11cython_blas_s *)void (int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *)void (char *, char *, char *, char *, int *, int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *, __pyx_t_5scipy_6linalg_11cython_blas_s *, int *)void (int *, __pyx_t_double_complex *, __pyx_t_double_complex *, int *, __pyx_t_double_complex *, int *)void (int *, __pyx_t_double_complex *, int *, __pyx_t_double_complex *, int *)void (char *, char *, int *, int *, int *, __pyx_t_double_complex *, __pyx_t_double_complex *, int *, __pyx_t_double_complex *, int *, __pyx_t_double_complex *, __pyx_t_double_complex *, int *)void (char *, int *, int *, __pyx_t_double_complex *, __pyx_t_double_complex *, int *, __pyx_t_double_complex *, int *, __pyx_t_double_complex *, __pyx_t_double_complex *, int *)void (int *, int *, __pyx_t_double_complex *, __pyx_t_double_complex *, int *, __pyx_t_double_complex *, int *, __pyx_t_double_complex *, int *)void (int *, __pyx_t_double_complex *, __pyx_t_double_complex *, int *)void (char *, char *, char *, char *, int *, int *, __pyx_t_double_complex *, __pyx_t_double_complex *, int *, __pyx_t_double_complex *, int *)void (int *, int *, __pyx_t_float_complex *, int *, __pyx_t_float_complex *, __pyx_t_float_complex *, int *, int *)void (char *, int *, int *, __pyx_t_float_complex *, int *, __pyx_t_float_complex *, __pyx_t_float_complex *, int *, __pyx_t_float_complex *)void (int *, __pyx_t_float_complex *, __pyx_t_float_complex *, int *, __pyx_t_float_complex *)void (__pyx_t_float_complex *, __pyx_t_float_complex *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, __pyx_t_float_complex *, __pyx_t_float_complex *)void (int *, __pyx_t_float_complex *, int *, __pyx_t_float_complex *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, __pyx_t_float_complex *)void (char *, char *, int *, int *, int *, __pyx_t_float_complex *, int *, __pyx_t_float_complex *, __pyx_t_float_complex *, int *, __pyx_t_float_complex *, int *, int *)void (int *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, int *, int *)void (char *, int *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_d *)void (int *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_d *)void (__pyx_t_5scipy_6linalg_13cython_lapack_d *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, __pyx_t_5scipy_6linalg_13cython_lapack_d *)void (char *, char *, int *, int *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, int *, int *)void (int *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, int *, int *)void (char *, int *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_s *)void (int *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_s *)void (__pyx_t_5scipy_6linalg_13cython_lapack_s *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, __pyx_t_5scipy_6linalg_13cython_lapack_s *)void (char *, char *, int *, int *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_s *, int *, int *)void (int *, int *, __pyx_t_double_complex *, int *, __pyx_t_double_complex *, __pyx_t_double_complex *, int *, int *)void (char *, int *, int *, __pyx_t_double_complex *, int *, __pyx_t_double_complex *, __pyx_t_double_complex *, int *, __pyx_t_double_complex *)void (int *, __pyx_t_double_complex *, __pyx_t_double_complex *, int *, __pyx_t_double_complex *)void (__pyx_t_double_complex *, __pyx_t_double_complex *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, __pyx_t_double_complex *, __pyx_t_double_complex *)void (int *, __pyx_t_double_complex *, int *, __pyx_t_double_complex *, int *, __pyx_t_5scipy_6linalg_13cython_lapack_d *, __pyx_t_double_complex *)void (char *, char *, int *, int *, int *, __pyx_t_double_complex *, int *, __pyx_t_double_complex *, __pyx_t_double_complex *, int *, __pyx_t_double_complex *, int *, int *)_ARRAY_API is not PyCapsule objectmodule compiled against ABI version 0x%x but this version of numpy is 0x%xmodule was compiled against NumPy C-API version 0x%x (NumPy 1.23) but the running NumPy has C-API version 0x%x. Check the section C-API incompatibility at the Troubleshooting ImportError section at https://numpy.org/devdocs/user/troubleshooting-importerror.html#c-api-incompatibility for indications on how to solve this problem.FATAL: module compiled as unknown endianFATAL: module compiled as little endian, but detected different endianness at runtimeinit scipy.linalg._decomp_update'%.200s' object is not subscriptablecannot fit '%.200s' into an index-sized integerscipy.linalg._decomp_update.validate_arrayscipy.linalg._decomp_update.validate_qrtoo many values to unpack (expected %zd)need more than %zd value%.1s to unpack'NoneType' object is not iterablescipy.linalg._decomp_update.qr_deletescipy.linalg._decomp_update.qr_insert_row%.200s() takes %.8s %zd positional argument%.1s (%zd given)scipy.linalg._decomp_update.form_qTuscipy.linalg._decomp_update.qr_insert_colscipy.linalg._decomp_update.qr_updatescipy.linalg._decomp_update._form_qTuscipy.linalg._decomp_update.qr_insert_cython_3_1_6.cython_function_or_method_cython_3_1_6._common_types_metatype<+ўͤԤweStcQ?- %ݰ.F9hs27N2 qr_update(Q, R, u, v, overwrite_qruv=False, check_finite=True) Rank-k QR update If ``A = Q R`` is the QR factorization of ``A``, return the QR factorization of ``A + u v**T`` for real ``A`` or ``A + u v**H`` for complex ``A``. Parameters ---------- Q : (M, M) or (M, N) array_like Unitary/orthogonal matrix from the qr decomposition of A. R : (M, N) or (N, N) array_like Upper triangular matrix from the qr decomposition of A. u : (M,) or (M, k) array_like Left update vector v : (N,) or (N, k) array_like Right update vector overwrite_qruv : bool, optional If True, consume Q, R, u, and v, if possible, while performing the update, otherwise make copies as necessary. Defaults to False. check_finite : bool, optional Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Default is True. Returns ------- Q1 : ndarray Updated unitary/orthogonal factor R1 : ndarray Updated upper triangular factor See Also -------- qr, qr_multiply, qr_delete, qr_insert Notes ----- This routine does not guarantee that the diagonal entries of `R1` are real or positive. .. versionadded:: 0.16.0 References ---------- .. [1] Golub, G. H. & Van Loan, C. F. Matrix Computations, 3rd Ed. (Johns Hopkins University Press, 1996). .. [2] Daniel, J. W., Gragg, W. B., Kaufman, L. & Stewart, G. W. Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization. Math. Comput. 30, 772-795 (1976). .. [3] Reichel, L. & Gragg, W. B. Algorithm 686: FORTRAN Subroutines for Updating the QR Decomposition. ACM Trans. Math. Softw. 16, 369-377 (1990). Examples -------- >>> import numpy as np >>> from scipy import linalg >>> a = np.array([[ 3., -2., -2.], ... [ 6., -9., -3.], ... [ -3., 10., 1.], ... [ 6., -7., 4.], ... [ 7., 8., -6.]]) >>> q, r = linalg.qr(a) Given this q, r decomposition, perform a rank 1 update. >>> u = np.array([7., -2., 4., 3., 5.]) >>> v = np.array([1., 3., -5.]) >>> q_up, r_up = linalg.qr_update(q, r, u, v, False) >>> q_up array([[ 0.54073807, 0.18645997, 0.81707661, -0.02136616, 0.06902409], # may vary (signs) [ 0.21629523, -0.63257324, 0.06567893, 0.34125904, -0.65749222], [ 0.05407381, 0.64757787, -0.12781284, -0.20031219, -0.72198188], [ 0.48666426, -0.30466718, -0.27487277, -0.77079214, 0.0256951 ], [ 0.64888568, 0.23001 , -0.4859845 , 0.49883891, 0.20253783]]) >>> r_up array([[ 18.49324201, 24.11691794, -44.98940746], # may vary (signs) [ 0. , 31.95894662, -27.40998201], [ 0. , 0. , -9.25451794], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]]) The update is equivalent, but faster than the following. >>> a_up = a + np.outer(u, v) >>> q_direct, r_direct = linalg.qr(a_up) Check that we have equivalent results: >>> np.allclose(np.dot(q_up, r_up), a_up) True And the updated Q is still unitary: >>> np.allclose(np.dot(q_up.T, q_up), np.eye(5)) True Updating economic (reduced, thin) decompositions is also possible: >>> qe, re = linalg.qr(a, mode='economic') >>> qe_up, re_up = linalg.qr_update(qe, re, u, v, False) >>> qe_up array([[ 0.54073807, 0.18645997, 0.81707661], # may vary (signs) [ 0.21629523, -0.63257324, 0.06567893], [ 0.05407381, 0.64757787, -0.12781284], [ 0.48666426, -0.30466718, -0.27487277], [ 0.64888568, 0.23001 , -0.4859845 ]]) >>> re_up array([[ 18.49324201, 24.11691794, -44.98940746], # may vary (signs) [ 0. , 31.95894662, -27.40998201], [ 0. , 0. , -9.25451794]]) >>> np.allclose(np.dot(qe_up, re_up), a_up) True >>> np.allclose(np.dot(qe_up.T, qe_up), np.eye(3)) True Similarly to the above, perform a rank 2 update. >>> u2 = np.array([[ 7., -1,], ... [-2., 4.], ... [ 4., 2.], ... [ 3., -6.], ... [ 5., 3.]]) >>> v2 = np.array([[ 1., 2.], ... [ 3., 4.], ... [-5., 2]]) >>> q_up2, r_up2 = linalg.qr_update(q, r, u2, v2, False) >>> q_up2 array([[-0.33626508, -0.03477253, 0.61956287, -0.64352987, -0.29618884], # may vary (signs) [-0.50439762, 0.58319694, -0.43010077, -0.33395279, 0.33008064], [-0.21016568, -0.63123106, 0.0582249 , -0.13675572, 0.73163206], [ 0.12609941, 0.49694436, 0.64590024, 0.31191919, 0.47187344], [-0.75659643, -0.11517748, 0.10284903, 0.5986227 , -0.21299983]]) >>> r_up2 array([[-23.79075451, -41.1084062 , 24.71548348], # may vary (signs) [ 0. , -33.83931057, 11.02226551], [ 0. , 0. , 48.91476811], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]]) This update is also a valid qr decomposition of ``A + U V**T``. >>> a_up2 = a + np.dot(u2, v2.T) >>> np.allclose(a_up2, np.dot(q_up2, r_up2)) True >>> np.allclose(np.dot(q_up2.T, q_up2), np.eye(5)) Trueqr_insert(Q, R, u, k, which='row', rcond=None, overwrite_qru=False, check_finite=True) QR update on row or column insertions If ``A = Q R`` is the QR factorization of ``A``, return the QR factorization of ``A`` where rows or columns have been inserted starting at row or column ``k``. Parameters ---------- Q : (M, M) array_like Unitary/orthogonal matrix from the QR decomposition of A. R : (M, N) array_like Upper triangular matrix from the QR decomposition of A. u : (N,), (p, N), (M,), or (M, p) array_like Rows or columns to insert k : int Index before which `u` is to be inserted. which: {'row', 'col'}, optional Determines if rows or columns will be inserted, defaults to 'row' rcond : float Lower bound on the reciprocal condition number of ``Q`` augmented with ``u/||u||`` Only used when updating economic mode (thin, (M,N) (N,N)) decompositions. If None, machine precision is used. Defaults to None. overwrite_qru : bool, optional If True, consume Q, R, and u, if possible, while performing the update, otherwise make copies as necessary. Defaults to False. check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Default is True. Returns ------- Q1 : ndarray Updated unitary/orthogonal factor R1 : ndarray Updated upper triangular factor Raises ------ LinAlgError : If updating a (M,N) (N,N) factorization and the reciprocal condition number of Q augmented with ``u/||u||`` is smaller than rcond. See Also -------- qr, qr_multiply, qr_delete, qr_update Notes ----- This routine does not guarantee that the diagonal entries of ``R1`` are positive. .. versionadded:: 0.16.0 References ---------- .. [1] Golub, G. H. & Van Loan, C. F. Matrix Computations, 3rd Ed. (Johns Hopkins University Press, 1996). .. [2] Daniel, J. W., Gragg, W. B., Kaufman, L. & Stewart, G. W. Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization. Math. Comput. 30, 772-795 (1976). .. [3] Reichel, L. & Gragg, W. B. Algorithm 686: FORTRAN Subroutines for Updating the QR Decomposition. ACM Trans. Math. Softw. 16, 369-377 (1990). Examples -------- >>> import numpy as np >>> from scipy import linalg >>> a = np.array([[ 3., -2., -2.], ... [ 6., -7., 4.], ... [ 7., 8., -6.]]) >>> q, r = linalg.qr(a) Given this QR decomposition, update q and r when 2 rows are inserted. >>> u = np.array([[ 6., -9., -3.], ... [ -3., 10., 1.]]) >>> q1, r1 = linalg.qr_insert(q, r, u, 2, 'row') >>> q1 array([[-0.25445668, 0.02246245, 0.18146236, -0.72798806, 0.60979671], # may vary (signs) [-0.50891336, 0.23226178, -0.82836478, -0.02837033, -0.00828114], [-0.50891336, 0.35715302, 0.38937158, 0.58110733, 0.35235345], [ 0.25445668, -0.52202743, -0.32165498, 0.36263239, 0.65404509], [-0.59373225, -0.73856549, 0.16065817, -0.0063658 , -0.27595554]]) >>> r1 array([[-11.78982612, 6.44623587, 3.81685018], # may vary (signs) [ 0. , -16.01393278, 3.72202865], [ 0. , 0. , -6.13010256], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]]) The update is equivalent, but faster than the following. >>> a1 = np.insert(a, 2, u, 0) >>> a1 array([[ 3., -2., -2.], [ 6., -7., 4.], [ 6., -9., -3.], [ -3., 10., 1.], [ 7., 8., -6.]]) >>> q_direct, r_direct = linalg.qr(a1) Check that we have equivalent results: >>> np.dot(q1, r1) array([[ 3., -2., -2.], [ 6., -7., 4.], [ 6., -9., -3.], [ -3., 10., 1.], [ 7., 8., -6.]]) >>> np.allclose(np.dot(q1, r1), a1) True And the updated Q is still unitary: >>> np.allclose(np.dot(q1.T, q1), np.eye(5)) Trueqr_delete(Q, R, k, int p=1, which='row', overwrite_qr=False, check_finite=True) QR downdate on row or column deletions If ``A = Q R`` is the QR factorization of ``A``, return the QR factorization of ``A`` where ``p`` rows or columns have been removed starting at row or column ``k``. Parameters ---------- Q : (M, M) or (M, N) array_like Unitary/orthogonal matrix from QR decomposition. R : (M, N) or (N, N) array_like Upper triangular matrix from QR decomposition. k : int Index of the first row or column to delete. p : int, optional Number of rows or columns to delete, defaults to 1. which: {'row', 'col'}, optional Determines if rows or columns will be deleted, defaults to 'row' overwrite_qr : bool, optional If True, consume Q and R, overwriting their contents with their downdated versions, and returning appropriately sized views. Defaults to False. check_finite : bool, optional Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Default is True. Returns ------- Q1 : ndarray Updated unitary/orthogonal factor R1 : ndarray Updated upper triangular factor See Also -------- qr, qr_multiply, qr_insert, qr_update Notes ----- This routine does not guarantee that the diagonal entries of ``R1`` are positive. .. versionadded:: 0.16.0 References ---------- .. [1] Golub, G. H. & Van Loan, C. F. Matrix Computations, 3rd Ed. (Johns Hopkins University Press, 1996). .. [2] Daniel, J. W., Gragg, W. B., Kaufman, L. & Stewart, G. W. Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization. Math. Comput. 30, 772-795 (1976). .. [3] Reichel, L. & Gragg, W. B. Algorithm 686: FORTRAN Subroutines for Updating the QR Decomposition. ACM Trans. Math. Softw. 16, 369-377 (1990). Examples -------- >>> import numpy as np >>> from scipy import linalg >>> a = np.array([[ 3., -2., -2.], ... [ 6., -9., -3.], ... [ -3., 10., 1.], ... [ 6., -7., 4.], ... [ 7., 8., -6.]]) >>> q, r = linalg.qr(a) Given this QR decomposition, update q and r when 2 rows are removed. >>> q1, r1 = linalg.qr_delete(q, r, 2, 2, 'row', False) >>> q1 array([[ 0.30942637, 0.15347579, 0.93845645], # may vary (signs) [ 0.61885275, 0.71680171, -0.32127338], [ 0.72199487, -0.68017681, -0.12681844]]) >>> r1 array([[ 9.69535971, -0.4125685 , -6.80738023], # may vary (signs) [ 0. , -12.19958144, 1.62370412], [ 0. , 0. , -0.15218213]]) The update is equivalent, but faster than the following. >>> a1 = np.delete(a, slice(2,4), 0) >>> a1 array([[ 3., -2., -2.], [ 6., -9., -3.], [ 7., 8., -6.]]) >>> q_direct, r_direct = linalg.qr(a1) Check that we have equivalent results: >>> np.dot(q1, r1) array([[ 3., -2., -2.], [ 6., -9., -3.], [ 7., 8., -6.]]) >>> np.allclose(np.dot(q1, r1), a1) True And the updated Q is still unitary: >>> np.allclose(np.dot(q1.T, q1), np.eye(3)) True this function only exists to expose the cdef version below for testing purposes. Here we perform minimal input validation to ensure that the inputs meet the requirements below. Unable to allocate memory for array'u' must have the same type as Q and Ru and v must have the same number of dimensionsq must be either F or C contiguousq and u must be a blas compatible type: f d F or Dnumpy._core.umath failed to importnumpy._core.multiarray failed to importarray must not contain infs or NaNsUpdate rank larger than np.dot(Q, R).Unable to allocate memory for array.The {0}th argument to ?ormqr/?unmqr was invalidThe {0}th argument to ?geqrf was invalidShape of u is incorrect, should be 1 <= u.ndim <= 2Second dimension of u and v must be the sameReorthogonalization Failed, unable to perform row deletion.Q and R must have the same dtypeQ and R do not have compatible shapes. Expected (M,M) (M,N) or (M,N) (N,N) but found Input array too large for use with BLASwnAS7.j-q -q3ajuIV;c& yl#YfAjas!as! ^1AWAXZqgQe1 AS9F! 1'which' must be either 'row' or 'col''u' should be either (N,) or (p,N) when inserting rows. Found %s.'u' should be either (M,) or (M,p) when inserting columns. Found %s.u and v must have the same type as Q and Rscipy/linalg/_decomp_update.pyx'rcond' is not used when updating full, (M,M) (M,N) decompositions and must be None.q and u must have the same type. Routines for updating QR decompositions .. versionadded:: 0.16.0 Rank-k QR update If ``A = Q R`` is the QR factorization of ``A``, return the QR factorization of ``A + u v**T`` for real ``A`` or ``A + u v**H`` for complex ``A``. Parameters ---------- Q : (M, M) or (M, N) array_like Unitary/orthogonal matrix from the qr decomposition of A. R : (M, N) or (N, N) array_like Upper triangular matrix from the qr decomposition of A. u : (M,) or (M, k) array_like Left update vector v : (N,) or (N, k) array_like Right update vector overwrite_qruv : bool, optional If True, consume Q, R, u, and v, if possible, while performing the update, otherwise make copies as necessary. Defaults to False. check_finite : bool, optional Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Default is True. Returns ------- Q1 : ndarray Updated unitary/orthogonal factor R1 : ndarray Updated upper triangular factor See Also -------- qr, qr_multiply, qr_delete, qr_insert Notes ----- This routine does not guarantee that the diagonal entries of `R1` are real or positive. .. versionadded:: 0.16.0 References ---------- .. [1] Golub, G. H. & Van Loan, C. F. Matrix Computations, 3rd Ed. (Johns Hopkins University Press, 1996). .. [2] Daniel, J. W., Gragg, W. B., Kaufman, L. & Stewart, G. W. Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization. Math. Comput. 30, 772-795 (1976). .. [3] Reichel, L. & Gragg, W. B. Algorithm 686: FORTRAN Subroutines for Updating the QR Decomposition. ACM Trans. Math. Softw. 16, 369-377 (1990). Examples -------- >>> import numpy as np >>> from scipy import linalg >>> a = np.array([[ 3., -2., -2.], ... [ 6., -9., -3.], ... [ -3., 10., 1.], ... [ 6., -7., 4.], ... [ 7., 8., -6.]]) >>> q, r = linalg.qr(a) Given this q, r decomposition, perform a rank 1 update. >>> u = np.array([7., -2., 4., 3., 5.]) >>> v = np.array([1., 3., -5.]) >>> q_up, r_up = linalg.qr_update(q, r, u, v, False) >>> q_up array([[ 0.54073807, 0.18645997, 0.81707661, -0.02136616, 0.06902409], # may vary (signs) [ 0.21629523, -0.63257324, 0.06567893, 0.34125904, -0.65749222], [ 0.05407381, 0.64757787, -0.12781284, -0.20031219, -0.72198188], [ 0.48666426, -0.30466718, -0.27487277, -0.77079214, 0.0256951 ], [ 0.64888568, 0.23001 , -0.4859845 , 0.49883891, 0.20253783]]) >>> r_up array([[ 18.49324201, 24.11691794, -44.98940746], # may vary (signs) [ 0. , 31.95894662, -27.40998201], [ 0. , 0. , -9.25451794], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]]) The update is equivalent, but faster than the following. >>> a_up = a + np.outer(u, v) >>> q_direct, r_direct = linalg.qr(a_up) Check that we have equivalent results: >>> np.allclose(np.dot(q_up, r_up), a_up) True And the updated Q is still unitary: >>> np.allclose(np.dot(q_up.T, q_up), np.eye(5)) True Updating economic (reduced, thin) decompositions is also possible: >>> qe, re = linalg.qr(a, mode='economic') >>> qe_up, re_up = linalg.qr_update(qe, re, u, v, False) >>> qe_up array([[ 0.54073807, 0.18645997, 0.81707661], # may vary (signs) [ 0.21629523, -0.63257324, 0.06567893], [ 0.05407381, 0.64757787, -0.12781284], [ 0.48666426, -0.30466718, -0.27487277], [ 0.64888568, 0.23001 , -0.4859845 ]]) >>> re_up array([[ 18.49324201, 24.11691794, -44.98940746], # may vary (signs) [ 0. , 31.95894662, -27.40998201], [ 0. , 0. , -9.25451794]]) >>> np.allclose(np.dot(qe_up, re_up), a_up) True >>> np.allclose(np.dot(qe_up.T, qe_up), np.eye(3)) True Similarly to the above, perform a rank 2 update. >>> u2 = np.array([[ 7., -1,], ... [-2., 4.], ... [ 4., 2.], ... [ 3., -6.], ... [ 5., 3.]]) >>> v2 = np.array([[ 1., 2.], ... [ 3., 4.], ... [-5., 2]]) >>> q_up2, r_up2 = linalg.qr_update(q, r, u2, v2, False) >>> q_up2 array([[-0.33626508, -0.03477253, 0.61956287, -0.64352987, -0.29618884], # may vary (signs) [-0.50439762, 0.58319694, -0.43010077, -0.33395279, 0.33008064], [-0.21016568, -0.63123106, 0.0582249 , -0.13675572, 0.73163206], [ 0.12609941, 0.49694436, 0.64590024, 0.31191919, 0.47187344], [-0.75659643, -0.11517748, 0.10284903, 0.5986227 , -0.21299983]]) >>> r_up2 array([[-23.79075451, -41.1084062 , 24.71548348], # may vary (signs) [ 0. , -33.83931057, 11.02226551], [ 0. , 0. , 48.91476811], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]]) This update is also a valid qr decomposition of ``A + U V**T``. >>> a_up2 = a + np.dot(u2, v2.T) >>> np.allclose(a_up2, np.dot(q_up2, r_up2)) True >>> np.allclose(np.dot(q_up2.T, q_up2), np.eye(5)) True  QZ!! q4AvS D #S ;as#Q +;1 53e1 *AQ 3b ! 3b2Rs#S *AQ 1 rAr!4s"ErQauBfCr w+1A&at9Aq%QgQqq q3a!Ba 7!4q 7!4q9F!-QcHF$a d$gQfA-QcIV4q!t4wafA-Qc4DF$a(d$gQ-Qc4EV4q)t4wa uCqr''2Qack!j 7!4q 7!4q9F!'q3hfA d$afA'q3ivQ!t4qfA'q3.>fA(d$a'q3.?vQ)t4q 2Qe61A s! 3b JcK{!3c0a JcK{!3c$K/?q Eq 53e1 *AQ 3b ! 3b2Rs#S *AQwat1wat1 3c9F!!#S86 d!fA!#S9F!!t1fA!#S+;6(d!!#S+>> import numpy as np >>> from scipy import linalg >>> a = np.array([[ 3., -2., -2.], ... [ 6., -7., 4.], ... [ 7., 8., -6.]]) >>> q, r = linalg.qr(a) Given this QR decomposition, update q and r when 2 rows are inserted. >>> u = np.array([[ 6., -9., -3.], ... [ -3., 10., 1.]]) >>> q1, r1 = linalg.qr_insert(q, r, u, 2, 'row') >>> q1 array([[-0.25445668, 0.02246245, 0.18146236, -0.72798806, 0.60979671], # may vary (signs) [-0.50891336, 0.23226178, -0.82836478, -0.02837033, -0.00828114], [-0.50891336, 0.35715302, 0.38937158, 0.58110733, 0.35235345], [ 0.25445668, -0.52202743, -0.32165498, 0.36263239, 0.65404509], [-0.59373225, -0.73856549, 0.16065817, -0.0063658 , -0.27595554]]) >>> r1 array([[-11.78982612, 6.44623587, 3.81685018], # may vary (signs) [ 0. , -16.01393278, 3.72202865], [ 0. , 0. , -6.13010256], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ]]) The update is equivalent, but faster than the following. >>> a1 = np.insert(a, 2, u, 0) >>> a1 array([[ 3., -2., -2.], [ 6., -7., 4.], [ 6., -9., -3.], [ -3., 10., 1.], [ 7., 8., -6.]]) >>> q_direct, r_direct = linalg.qr(a1) Check that we have equivalent results: >>> np.dot(q1, r1) array([[ 3., -2., -2.], [ 6., -7., 4.], [ 6., -9., -3.], [ -3., 10., 1.], [ 7., 8., -6.]]) >>> np.allclose(np.dot(q1, r1), a1) True And the updated Q is still unitary: >>> np.allclose(np.dot(q1.T, q1), np.eye(5)) True QR downdate on row or column deletions If ``A = Q R`` is the QR factorization of ``A``, return the QR factorization of ``A`` where ``p`` rows or columns have been removed starting at row or column ``k``. Parameters ---------- Q : (M, M) or (M, N) array_like Unitary/orthogonal matrix from QR decomposition. R : (M, N) or (N, N) array_like Upper triangular matrix from QR decomposition. k : int Index of the first row or column to delete. p : int, optional Number of rows or columns to delete, defaults to 1. which: {'row', 'col'}, optional Determines if rows or columns will be deleted, defaults to 'row' overwrite_qr : bool, optional If True, consume Q and R, overwriting their contents with their downdated versions, and returning appropriately sized views. Defaults to False. check_finite : bool, optional Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Default is True. Returns ------- Q1 : ndarray Updated unitary/orthogonal factor R1 : ndarray Updated upper triangular factor See Also -------- qr, qr_multiply, qr_insert, qr_update Notes ----- This routine does not guarantee that the diagonal entries of ``R1`` are positive. .. versionadded:: 0.16.0 References ---------- .. [1] Golub, G. H. & Van Loan, C. F. Matrix Computations, 3rd Ed. (Johns Hopkins University Press, 1996). .. [2] Daniel, J. W., Gragg, W. B., Kaufman, L. & Stewart, G. W. Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization. Math. Comput. 30, 772-795 (1976). .. [3] Reichel, L. & Gragg, W. B. Algorithm 686: FORTRAN Subroutines for Updating the QR Decomposition. ACM Trans. Math. Softw. 16, 369-377 (1990). Examples -------- >>> import numpy as np >>> from scipy import linalg >>> a = np.array([[ 3., -2., -2.], ... [ 6., -9., -3.], ... [ -3., 10., 1.], ... [ 6., -7., 4.], ... [ 7., 8., -6.]]) >>> q, r = linalg.qr(a) Given this QR decomposition, update q and r when 2 rows are removed. >>> q1, r1 = linalg.qr_delete(q, r, 2, 2, 'row', False) >>> q1 array([[ 0.30942637, 0.15347579, 0.93845645], # may vary (signs) [ 0.61885275, 0.71680171, -0.32127338], [ 0.72199487, -0.68017681, -0.12681844]]) >>> r1 array([[ 9.69535971, -0.4125685 , -6.80738023], # may vary (signs) [ 0. , -12.19958144, 1.62370412], [ 0. , 0. , -0.15218213]]) The update is equivalent, but faster than the following. >>> a1 = np.delete(a, slice(2,4), 0) >>> a1 array([[ 3., -2., -2.], [ 6., -9., -3.], [ 7., 8., -6.]]) >>> q_direct, r_direct = linalg.qr(a1) Check that we have equivalent results: >>> np.dot(q1, r1) array([[ 3., -2., -2.], [ 6., -9., -3.], [ 7., 8., -6.]]) >>> np.allclose(np.dot(q1, r1), a1) True And the updated Q is still unitary: >>> np.allclose(np.dot(q1.T, q1), np.eye(3)) True Only arrays with dtypes float32, float64, complex64, and complex128 are supported.One of the columns of u lies in the span of Q. 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