Cpj.SrSSKrSSKJr S/rSSjrSr\S5r\S5r S r S r S r S r SS jrSrSrSrg)zSparse block 1-norm estimator. N)aslinearoperator onenormestc[U5nURSURS:wa [S5eURSnX:a[R"[U5R [R "U555nURXU4:wa"[SS[UR5-5e[U5RSS9nURU4:wa"[SS[UR5-5e[R"U5n[XX5n USS2U4n Xxn O[XRX5uppn U(dU(aU 4nU(aX4- nU(aX4- nU$U $)a Compute a lower bound of the 1-norm of a sparse array. Parameters ---------- A : ndarray or other linear operator A linear operator that can be transposed and that can produce matrix products. t : int, optional A positive parameter controlling the tradeoff between accuracy versus time and memory usage. Larger values take longer and use more memory but give more accurate output. itmax : int, optional Use at most this many iterations. compute_v : bool, optional Request a norm-maximizing linear operator input vector if True. compute_w : bool, optional Request a norm-maximizing linear operator output vector if True. Returns ------- est : float An underestimate of the 1-norm of the sparse array. v : ndarray, optional The vector such that ||Av||_1 == est*||v||_1. It can be thought of as an input to the linear operator that gives an output with particularly large norm. w : ndarray, optional The vector Av which has relatively large 1-norm. It can be thought of as an output of the linear operator that is relatively large in norm compared to the input. Notes ----- This is algorithm 2.4 of [1]_. In [2]_ it is described as follows. "This algorithm typically requires the evaluation of about 4t matrix-vector products and almost invariably produces a norm estimate (which is, in fact, a lower bound on the norm) correct to within a factor 3." .. versionadded:: 0.13.0 References ---------- .. [1] Nicholas J. Higham and Francoise Tisseur (2000), "A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra." SIAM J. Matrix Anal. Appl. Vol. 21, No. 4, pp. 1185-1201. .. [2] Awad H. Al-Mohy and Nicholas J. Higham (2009), "A new scaling and squaring algorithm for the matrix exponential." SIAM J. Matrix Anal. Appl. Vol. 31, No. 3, pp. 970-989. Examples -------- >>> import numpy as np >>> from scipy.sparse import csc_array >>> from scipy.sparse.linalg import onenormest >>> A = csc_array([[1., 0., 0.], [5., 8., 2.], [0., -1., 0.]], dtype=float) >>> A.toarray() array([[ 1., 0., 0.], [ 5., 8., 2.], [ 0., -1., 0.]]) >>> onenormest(A) 9.0 >>> np.linalg.norm(A.toarray(), ord=1) 9.0 rz1expected the operator to act like a square matrixzinternal error: zunexpected shape axisN)rshape ValueErrornpasarraymatmatidentity Exceptionstrabssumargmaxelementary_vector_onenormest_coreH)Atitmax compute_v compute_wn A_explicit col_abs_sumsargmax_jvwestnmults nresamplesresults [/var/www/html/pdf-tiff/venv/lib/python3.13/site-packages/scipy/sparse/linalg/_onenormest.pyrr sdT AwwqzQWWQZLMM  AvZZ 0 3 : :2;;q> JK   v %.'#j.>.>*??A A:***2   ! &.'#l.@.@*AAC C99\* a * q({ #$(8CC(J%:I  dNF  dNF  c^^SmUU4SjnU$)z Decorator for an elementwise function, to apply it blockwise along first dimension, to avoid excessive memory usage in temporaries. c>>URST:aT"U5$T"UST5n[R"URS4URSS-URS9nXST&A[ TURST5HnT"XUT-5X#UT-&M U$)Nrrdtype)r r zerosr,range)xy0yj block_sizefuncs r&wrapper%_blocked_elementwise..wrappers 771: "7Na n%B!''!*!"5RXXFAkzN:qwwqz:>$(Qz\):$;AjL!?Hr')r4r5r3s` @r&_blocked_elementwiser8ys J  Nr'cfUR5nSXS:H'U[R"U5-nU$)a! This should do the right thing for both real and complex matrices. From Higham and Tisseur: "Everything in this section remains valid for complex matrices provided that sign(A) is redefined as the matrix (aij / |aij|) (and sign(0) = 1) transposes are replaced by conjugate transposes." rr)copyr r)XYs r& sign_round_upr=s0 AA1fINA Hr'cT[R"[R"U5SS9$)Nrr)r maxr)r;s r&_max_abs_axis1r@s 66"&&)! $$r'c SnSn[SURSU5H;n[R"[R"XX1-5SS9nUcUnM7X$- nM= U$)Nr)rr)r.r r rr)r;r3rr2r1s r&_sum_abs_axis0rCs_J A 1aggaj* - FF266!al+,1 5 9A FA . Hr'c@[R"U[S9nSX!'U$)Nr+r)r r-float)rir s r&rrs % A AD Hr'cURS:wdURUR:wa [S5eURSn[R"X5U:H$)Nrz2expected conformant vectors with entries in {-1,1}r)ndimr r r dot)r r!rs r&vectors_are_parallelrJsJ vv{agg(MNN  A 66!<1 r'cx^URH)m[U4SjUR55(aM) g g)Nc3<># UHn[TU5v M g7fNrJ.0r!r s r& ;every_col_of_X_is_parallel_to_a_col_of_Y..s;s!'1--sFT)Tany)r;r<r s @r&(every_col_of_X_is_parallel_to_a_col_of_YrVs. SS;qss;;; r'c^^TRup4TSS2U4m[UU4Sj[U555(agUb%[U4SjUR55(agg)Nc3J># UHn[TTSS2U45v M g7frMrN)rPr2r;r s r&rQ*column_needs_resampling..s$ >X 1QT7 + +Xs #Tc3<># UHn[TU5v M g7frMrNrOs r&rQrYs73a#Aq))3rSF)r rUr.rT)rFr;r<rrr s ` @r&column_needs_resamplingr[sU 77DA !Q$A >U1X >>>} 71337 7 7 r'cv[RRSSURSS9S-S- USS2U4'g)Nr)sizer)r randomrandintr )rFr;s r&resample_columnras7ii11771:6q81>8W"56 Q{ | !X&A 1f ((r')r]FFrM)__doc__numpyr scipy.sparse.linalgr__all__rr8r=r@rCrrJrVr[rarfrr7r'r&rsx0 .k\*    %%    =(O)r'