U Gv f~<@sdZddlZddlmZdgZd!ddZd d Zed d Zed dZ ddZ ddZ ddZ ddZ d"ddZddZddZddZdd ZdS)#zSparse block 1-norm estimator. N)aslinearoperator onenormestFcCs&t|}|jd|jdkr$td|jd}||krtt|t|}|j||fkrrtddt|jt |j dd}|j|fkrtddt|jt |}t ||} |dd|f} ||} nt ||j||\} } } } } |s|r| f}|r || f7}|r|| f7}|S| SdS)a Compute a lower bound of the 1-norm of a sparse matrix. 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 matrix. 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_matrix >>> from scipy.sparse.linalg import onenormest >>> A = csc_matrix([[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 ZaxisN)rshape ValueErrornpasarraymatmatidentity Exceptionstrabssumargmaxelementary_vector_onenormest_coreH)AtitmaxZ compute_vZ compute_wnZ A_explicitZ col_abs_sumsZargmax_jvwestnmults nresamplesresultr C/tmp/pip-unpacked-wheel-96ln3f52/scipy/sparse/linalg/_onenormest.pyr s8J          csdfdd}|S)z Decorator for an elementwise function, to apply it blockwise along first dimension, to avoid excessive memory usage in temporaries. cs|jdkr|S|d}tj|jdf|jdd|jd}||d<~t|jdD]$}||||||<ql|SdS)Nrrdtype)rr zerosr$range)xZy0yj block_sizefuncr r!wrappers& "z%_blocked_elementwise..wrapperr )r,r-r r*r!_blocked_elementwiseys r.cCs&|}d||dk<|t|}|S)a9 This should do the right thing for both real and complex matrices. From Higham and Tisseur: "Everything in this section remains valid for complex matrices provided that sign(A) is redefined as the matrix (aij / |aij|) (and sign(0) = 1) transposes are replaced by conjugate transposes." rr)copyr rXYr r r! sign_round_ups  r3cCstjt|ddS)Nrr)r maxr)r1r r r!_max_abs_axis1sr5cCsZd}d}td|jd|D]:}tjt||||dd}|dkrL|}q||7}q|S)Nr"rr)r&rr rr)r1r+rr)r(r r r!_sum_abs_axis0s  r7cCstj|td}d||<|S)Nr#r)r r%float)rirr r r!rsrcCs8|jdks|j|jkrtd|jd}t|||kS)Nrz2expected conformant vectors with entries in {-1,1}r)ndimrr r dot)rrrr r r!vectors_are_parallels r<cs.|jD]"tfdd|jDsdSqdS)Nc3s|]}t|VqdSNr<.0rrr r! sz;every_col_of_X_is_parallel_to_a_col_of_Y..FT)Tanyr0r rAr!(every_col_of_X_is_parallel_to_a_col_of_Ys rEcsbj\}}dd|ftfddt|Dr:dS|dk r^tfdd|jDr^dSdS)Nc3s$|]}tdd|fVqdSr=r>)r@r)r1rr r!rBsz*column_needs_resampling..Tc3s|]}t|VqdSr=r>r?rAr r!rBsF)rrDr&rC)r9r1r2rrr rFr!column_needs_resamplings rGcCs0tjjdd|jdddd|dd|f<dS)Nrrsizer)r randomrandintr)r9r1r r r!resample_columnsrLcCst||p||kSr=)r Zallclose)abr r r!less_than_or_closesrOcCst|}t|}|jd}t||f}|dkrbtjjdd||dfddd|ddddf<|t|}d}d}d} t|} t| |} t | } t | } | | ddd} t | }t| |}t|}| dkrtt|t|dd| f|dd| frqt|dddd|} || }t|D] }t|| ||dd|f<qD| dkrt|d|dstdt|d| dstd| dkrt|D]"}t| |||std q| }|}| d7} q| | fS) a" This is Algorithm 2.2. Parameters ---------- A : ndarray or other linear operator A linear operator that can produce matrix products. AT : ndarray or other linear operator The transpose of A. t : int, optional A positive parameter controlling the tradeoff between accuracy versus time and memory usage. Returns ------- g : sequence A non-negative decreasing vector such that g[j] is a lower bound for the 1-norm of the column of A of jth largest 1-norm. The first entry of this vector is therefore a lower bound on the 1-norm of the linear operator A. This sequence has length t. ind : sequence The ith entry of ind is the index of the column A whose 1-norm is given by g[i]. This sequence of indices has length t, and its entries are chosen from range(n), possibly with repetition, where n is the order of the operator A. Notes ----- This algorithm is mainly for testing. It uses the 'ind' array in a way that is similar to its usage in algorithm 2.4. This algorithm 2.2 may be easier to test, so it gives a chance of uncovering bugs related to indexing which could have propagated less noticeably to algorithm 2.4. rrrrHNzinvariant (2.2) is violatedzinvariant (2.3) is violated)rrr onesrJrKr8r&r r r7rsortr3r5rOr4r;argsortrr)rATrA_linear_operatorAT_linear_operatorrr1Zg_prevZh_prevkindr2gbest_jSZhr)r r r!_algorithm_2_2sN' 2   0      r_cCst|}t|}|dkr td|dkr0td|jd}||krJtdd}d}tj||ftd} |dkrtd|D]} t| | qvt|D]"} t| | rt| | |d7}qq| t|} tj dtj d} d} tj ||ftd} d}d}t | | }|d7}t |}t|}t|}|| ks4|dkrV|dkrF||}|dd|f}|dkrr|| krr| }q|} | }||krqt|} ~t| |rq|dkrt|D]*} t| | |rt| | |d7}qq~t | | }|d7}t|}~|dkr$t|||kr$qt|ddd d|t| }~|dkrt|d|| rvqt|| }t||||f}t|D] }t|||| dd|f<q|d|t|d|| }t| |f} |d7}qt||}|||||fS) a Compute a lower bound of the 1-norm of a sparse matrix. Parameters ---------- A : ndarray or other linear operator A linear operator that can produce matrix products. AT : ndarray or other linear operator The transpose of A. t : int, optional A positive parameter controlling the tradeoff between accuracy versus time and memory usage. itmax : int, optional Use at most this many iterations. Returns ------- est : float An underestimate of the 1-norm of the sparse matrix. 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. nmults : int, optional The number of matrix products that were computed. nresamples : int, optional The number of times a parallel column was observed, necessitating a re-randomization of the column. Notes ----- This is algorithm 2.4. rz$at least two iterations are requiredrzat least one column is requiredrz't should be smaller than the order of Ar#NrP)rr rr rRr8r&rLrGr%Zintpr r r7r4rr3rEr5rTlenr/Zin1dallZ concatenater)rrUrrrVrWrrrr1r9Zind_histZest_oldr\rXrYr2Zmagsrr[Zind_bestrZS_oldr]r^seenr)Znew_indrr r r!rDs)               (   "  r)rrFF)N)__doc__Znumpyr Zscipy.sparse.linalgr__all__rr.r3r5r7rr<rErGrLrOr_rr r r r!s$  n     g