ftj^YXddlZddlmZddlmZddlZddlmZmZddlm Z ddl m Z m Z m Z mZddlmZmZmZdd lmZdd lmZmZdd lmZdd lmZmZmZmZdd lmZgdZd=dZ dZ!dZ" d>dZ#e dd d>dZ$dZ%dZ&dZ'dZ(dZ)dZ*dZ+e dd d?dZ, d@d Z- dAd!Z.e d"d#d$dd%Z/e dd dBd&Z0dCd'Z1e d(d)dd*Zd+Z2 dDd.Z3dEddd/d0Z4dEd1Z5e dd dFd2Z6d3e6_7e dddddd4d5Z8e d dGd6Z9e d7 dHd8Z: dCd9Z;dId:Z<e d)d(d;d<Z=dS)JN)warn)product) atleast_1d atleast_2d)_apply_over_batch)get_lapack_funcs_normalize_lapack_dtype_normalize_lapack_dtype1_compute_lwork) LinAlgError _datacopied LinAlgWarning)_asarray_validated)_decomp _decomp_svd)levinson)find_det_from_lu bandwidth issymmetric ishermitian)_batched_linalg) solvesolve_triangular solveh_banded solve_bandedsolve_toeplitzsolve_circulantinvdetlstsqpinvpinvhmatrix_balancematmul_toeplitzc|dkrtd| dd|ks|dkrtd|dS|d}||kstd|dd td dSdS) zB Check arguments during the different steps of the solution phase rz$LAPACK reported an illegal value in z -th argument.zMatrix is singular.NEzIll-conditioned matrix (rcond=z.6gz): result may not be accurate. stacklevel) ValueErrorr rr)ninfolamchrcondr's Q/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/scipy/linalg/_basic.py _solve_checkr1s axxTTTTUUU TUaZZ/000 } c A QJJ +e <+++ q * * * * * * JcR|jd}t|\}}||cxkrdkrnnd}nq|dkrd}nh|dkrd}n_|dkr|dkr |dkrd}nJtj|jtjrt |rd}nt|rd }nd }|||fS) Nrdiagonallower triangularupper triangularrr( tridiagonal hermitian symmetricgeneral)shapernp issubdtypedtypecomplexfloatingrr)ar,n_belown_abovekinds r0_find_matrix_structurerD/s  A ||GW'Q A! A! A'Q,,1q55 qw 2 3 3 A Q ' !!r2c ggg}}}t|D]x\}}|dr|||ddkr!|d|d|d|dr!|d|d|dy|rtd |d |r&td d |d |r5t jdd |d tddSdS)zVFormat/emit errors/warnings from a lowlevel batched routine. See inv, solve. is_singular lapack_inforzslice z emits lapack info=is_ill_conditionedz has rcond = r/z%A singular matrix detected: slice(s) z are singular.zInternal LAPACK errors: ,.z$An ill-conditioned matrix detected: r(r)N) enumerateappendr r+joinwarningsrr)err_lstsingular lapack_errill_condidcts r0_format_emit_errors_warningsrUEsl &(R(jHG$$EE3 }   OOA    }  ! !   QqQQS=OQQ R R R # $ E OOCQCCS\CC D D D  LH L L L   MKCHHZ4H4HKKKLLL } H388H3E3E H H H         r2FTc|dvrt||||||||Sdddddddd d d d d d d |d }|d krt|dtjt ||} tjt ||} t| | \} } t| |\} }| j dkrtd| j | j d| j dkrtd| j tj | j tj r|rtd| jds&| j jdksd}| } | jds&| j jdksd}| } | j dk} | r | dddf} | jdk} | j d| j dkr!| std| j d| j tj| j dd| j dd} tj| | | j ddz} tj| | | j ddz} | jdks | jdkr tj| }| r|d}|S| r8| dkrt/d| | z }| r|dn|S|o| j dko | jd}|o| j dko | jd}t1j| | |||||\}}|rt5|| r|d}|S)a& Solve the equation ``a @ x = b`` for ``x``, where `a` is a square matrix. If the data matrix is known to be a particular type then supplying the corresponding string to ``assume_a`` key chooses the dedicated solver. The available options are ============================= ================================ diagonal 'diagonal' tridiagonal 'tridiagonal' banded 'banded' upper triangular 'upper triangular' lower triangular 'lower triangular' symmetric 'symmetric' (or 'sym') hermitian 'hermitian' (or 'her') symmetric positive definite 'positive definite' (or 'pos') general 'general' (or 'gen') ============================= ================================ Array argument(s) of this function may have additional "batch" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see :ref:`linalg_batch` for details. Parameters ---------- a : array_like, shape (..., N, N) Square left-hand side matrix or a batch of matrices. b : (..., N, NRHS) array_like Input data for the right hand side or a batch of right-hand sides. lower : bool, default: False Ignored unless ``assume_a`` is one of ``'sym'``, ``'her'``, or ``'pos'``. If True, the calculation uses only the data in the lower triangle of `a`; entries above the diagonal are ignored. If False (default), the calculation uses only the data in the upper triangle of `a`; entries below the diagonal are ignored. overwrite_a : bool, default: False Allow overwriting data in `a` (may enhance performance). overwrite_b : bool, default: False Allow overwriting data in `b` (may enhance performance). check_finite : bool, default: True 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. assume_a : str, optional Valid entries are described above. If omitted or ``None``, checks are performed to identify structure so the appropriate solver can be called. transposed : bool, default: False If True, solve ``a.T @ x == b``. Raises `NotImplementedError` for complex `a`. Returns ------- x : ndarray, shape (N, NRHS) or (..., N) The solution array. Raises ------ ValueError If size mismatches detected or input a is not square. LinAlgError If the computation fails because of matrix singularity. LinAlgWarning If an ill-conditioned input a is detected. NotImplementedError If transposed is True and input a is a complex matrix. Notes ----- If the input b matrix is a 1-D array with N elements, when supplied together with an NxN input a, it is assumed as a valid column vector despite the apparent size mismatch. This is compatible with the numpy.dot() behavior and the returned result is still 1-D array. The general, symmetric, Hermitian and positive definite solutions are obtained via calling ?GETRF/?GETRS, ?SYSV, ?HESV, and ?POTRF/?POTRS routines of LAPACK respectively. The datatype of the arrays define which solver is called regardless of the values. In other words, even when the complex array entries have precisely zero imaginary parts, the complex solver will be called based on the data type of the array. Examples -------- Given `a` and `b`, solve for `x`: >>> import numpy as np >>> a = np.array([[3, 2, 0], [1, -1, 0], [0, 5, 1]]) >>> b = np.array([2, 4, -1]) >>> from scipy.linalg import solve >>> x = solve(a, b) >>> x array([ 2., -2., 9.]) >>> a @ x == b array([ True, True, True], dtype=bool) Batches of matrices are supported, with and without structure detection: >>> a = np.arange(12).reshape(3, 2, 2) # a batch of 3 2x2 matrices >>> A = a.transpose(0, 2, 1) @ a # A is a batch of 3 positive definite matrices >>> b = np.ones(2) >>> solve(A, b) # this automatically detects that A is pos.def. array([[ 1. , -0.5], [ 3. , -2.5], [ 5. , -4.5]]) >>> solve(A, b, assume_a='pos') # bypass structucture detection array([[ 1. , -0.5], [ 3. , -2.5], [ 5. , -4.5]]) )banded)lower overwrite_a overwrite_b check_finiteassume_a transposedr e) Nr:genr4r7r6r5pospositive definitesymr9herr8unknown% is not a recognized matrix structurer[&Expected at least ndim=2, got a1.ndim=%Expected square matrix, got a1.shape=Wscipy.linalg.solve can currently not solve a^T x = b or a^H x = b for complex matrices.ALIGNED=TrNzincompatible shapes: a1.shape=z and b1.shape=).rzA singular matrix detected. F_CONTIGUOUS)solve0getr+r<rrr_ensure_dtype_cdszr ndimr;r=r>r?NotImplementedErrorflags byteordercopysizebroadcast_shapes broadcast_to empty_likeitemr r_solverU)r@brXrYrZr[r\r] structurea1b1b_is_1D a_is_scalar batch_shapexoutrOs r0rrcsf: q;K%Z    Q###   c(IIHKKKLLL )!,GGG H HB )!,GGG H HB B ' 'FB-b+>>OB  w{{D"'DDEEE x|rx|##CCCDDD }RXr122;z;!#:;; ; HY 28#5#<#< WWYY HY 28#5#<#< WWYYglG 4['Q,K x|rx|##K#LLLLLMMM%bhssmRXcrc]CCK [28BCC=8 9 9B [28BCC=8 9 9B w!||rw!|| M"    & A/ 7799>>;<< <2g%.s6{{3.O27a<Obh~6NKO27a<Obh~6NK!' B 5*k;JAw.$W--- fI Hr2)r@rn)r1|2c H d}tt|d} tt|d} t| | \} } | jd} |pt | |}|pt | |}| jd| jdkrt d| | jdkr | dkr | jdkst d| jdkr]ttj d| j tj d| j j } tj | | S| jdkr!| dkr | d d d f} n | d d d f} d }|d vrt |d |d vrtj| sd}d\} }|t!| \}} }| j jdvrt%dd}nt%dd}|r(d}d}tj| rt'dnd}d}|dkrt)| |}n|dkrt+|| |}n|dkrR| t-| n| |f\} }|r | j|| fn| | |f\}} }t1| ||}t3| ||||}n(|dvrt5||| |}nt7|| |}dtj}}|dvrmt%d| | f\}}}|| |\}}}t;| ||||| ||\}}t;| |||||\}}n0|d vrat%d | | f\}}}t=|| |} || | | |||!\}}}}t;| ||||||"\}}n|d#vrat%d$| | f\}!}"}#t=|#| |} |"| | | |||!\}}}}t;| ||!||||"\}}nf|dkrltj| }$| j|$z j}tj |$}%|%!}&|&dkr|&n|&|%"z }n|dkr|r| jn| } tj| d%tj| dtj| d})}(}'t%d&| | f\}*}+},|*|'|(|)\}'}(})}-}.}t;| ||+|'|(|)|-|.| |'\}}t;| ||,|'|(|)|-|.|\}}n0|dkrt%d(| | f\}/}0tj#d| z|zdz|jdf|/j }||| d d d f<|/| ||| d |)\}}1}}t;| ||0| |||1|\}}n|dvrT|d*k}tI| | |||+\}}t;| |t%d,| | f}2|2| |rd-nd./\}}nHt%d0| | f\}3}4|4| | |||1\}}}t;| ||3||\}}t;| ||||r|%}|S)2a Solve the equation ``a @ x = b`` for ``x``, where `a` is a square matrix. If the data matrix is known to be a particular type then supplying the corresponding string to ``assume_a`` key chooses the dedicated solver. The available options are ============================= ================================ diagonal 'diagonal' tridiagonal 'tridiagonal' banded 'banded' upper triangular 'upper triangular' lower triangular 'lower triangular' symmetric 'symmetric' (or 'sym') hermitian 'hermitian' (or 'her') symmetric positive definite 'positive definite' (or 'pos') general 'general' (or 'gen') ============================= ================================ Parameters ---------- a : (N, N) array_like Square input data b : (N, NRHS) array_like Input data for the right hand side. lower : bool, default: False Ignored unless ``assume_a`` is one of ``'sym'``, ``'her'``, or ``'pos'``. If True, the calculation uses only the data in the lower triangle of `a`; entries above the diagonal are ignored. If False (default), the calculation uses only the data in the upper triangle of `a`; entries below the diagonal are ignored. overwrite_a : bool, default: False Allow overwriting data in `a` (may enhance performance). overwrite_b : bool, default: False Allow overwriting data in `b` (may enhance performance). check_finite : bool, default: True 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. assume_a : str, optional Valid entries are described above. If omitted or ``None``, checks are performed to identify structure so the appropriate solver can be called. transposed : bool, default: False If True, solve ``a.T @ x == b``. Raises `NotImplementedError` for complex `a`. Returns ------- x : (N, NRHS) ndarray The solution array. Raises ------ ValueError If size mismatches detected or input a is not square. LinAlgError If the computation fails because of matrix singularity. LinAlgWarning If an ill-conditioned input a is detected. NotImplementedError If transposed is True and input a is a complex matrix. Notes ----- If the input b matrix is a 1-D array with N elements, when supplied together with an NxN input a, it is assumed as a valid column vector despite the apparent size mismatch. This is compatible with the numpy.dot() behavior and the returned result is still 1-D array. The general, symmetric, Hermitian and positive definite solutions are obtained via calling ?GESV, ?SYSV, ?HESV, and ?POSV routines of LAPACK respectively. The datatype of the arrays define which solver is called regardless of the values. In other words, even when the complex array entries have precisely zero imaginary parts, the complex solver will be called based on the data type of the array. Examples -------- Given `a` and `b`, solve for `x`: >>> import numpy as np >>> a = np.array([[3, 2, 0], [1, -1, 0], [0, 5, 1]]) >>> b = np.array([2, 4, -1]) >>> from scipy import linalg >>> x = linalg.solve(a, b) >>> x array([ 2., -2., 9.]) >>> np.dot(a, x) == b array([ True, True, True], dtype=bool) Frmrrz$Input a needs to be a square matrix.z2Input b has to have same number of rows as input arnr>NT>NrfrjrgrirWr:r4r8r9r7r5r6rhrl>rjr8r9NNfFr.fdIrr1r4r7rW>r5r6>rfr:)gecongetrfgetrs)rY)transrZ)norm)heconhesv hesv_lwork)lworkrXrYrZ)rX>rir9)syconsysv sysv_lworkr^)gttrfgttrsgtcon)rZ)gbsvgbcon overwrite_abrZr5)rXrZrtrconLU)uplo)poconposv)rXrYrZ)&rrrrxr;rr+r~rr<eyer>onesrry iscomplexobjrDcharr rz_matrix_norm_diagonal_matrix_norm_tridiagonalrT _to_banded_matrix_norm_banded_matrix_norm_triangular_matrix_norm_generalinfr1r diagabsminmaxzeros_solve_triangularravel)5r@rrXrYrZr[r\r]rrrr,dtrArBr.rranorma2abr-r/rrrluipvtrrrhesv_lwrrrsysv_lwdiag_a abs_diag_adiag_mindlrdu_gttrf_gttrs_gtcondu2ipivrrpiv_trconrrs5 r0rvrv3syHG &qu=== > >B &qu=== > >B B ' 'FB  A3R!3!3K3R!3!3K x{bhqk!!?@@@BHQKQ27a<<'(( ( w!|| 26!28,,,bgarx.H.H.H I I O}Rr**** w!|| 66D!!!GBBAAAtGBTTTHKKKLLL'''0C0C'!GW%;B%?%?"'7  x} 444 444  ?2   ?%'>?? ? ?  :%b,77 ] " "(r<@@ X  ,3O9R==='7AS>>4QeBD111 tt ) ) )/1?ACR J JtWw511 DRu',-8-8:::D!T QeBe5999 tt ) ) )/1?ACR J JtWw511 DRu',-8-8:::D!T QeBe5999 tt Z   TF] VF^^ >>##$MMx*..:J:J/J ] " " 'RTTRGBOORWR^^RWR^^rA!12MPRTVx!X!X%+VB2%6%6"Ar3dQ&QCr{KKK4QfRBT599 tt X  &'82r(CC e Xqy7*Q. >r2cp|jd}||zdz}tj||f|j}tj|||<t d|dzD]!}tj|||||z |df<"t d|dzD]#}tj|| |||zd| f<$|S)Nrrr)r;r<rr>rrange)rArBr@r,rowsrrSs r0rros  A W q D 4)17 + + +B'!**BwK 1gk " ",, gamm7Q;? 1gk " "..!wq1"~~7Q;!  Ir2ctj|tjtjs d|DStjtj}tjjdkr|r tjn tjn7tjjdkr|r tj n tj fd|DS)Nc3TK|]#}|tjV$dSr)astyper<float64).0arrays r0 z%_ensure_dtype_cdsz..s0==U RZ((======r2 @c3FK|]}|dVdS)Fr}N)r)rrr>s r0rz%_ensure_dtype_cdsz..s3 @ @ELLUL + + @ @ @ @ @ @r2) r< result_typer=inexactr?finfobits complex64float32 complex128r)arrayscomplexr>s @r0rxrx{s NF #E = + +>==f====mE2#566G xr!! '7 RZ %  # #!(8 bj @ @ @ @ @ @ @@r2cjt||}t||}t|jdks|jd|jdkrtd|jd|jdkr td|jd|jd|jdkr]t t jd|j t j d|j j} t j || S|pt||}t||||||\} } | S) a Solve the equation ``a @ x = b`` for ``x``, where `a` is a triangular matrix. Parameters ---------- a : (M, M) array_like A triangular matrix b : (M,) or (M, N) array_like Right-hand side matrix in ``a x = b`` lower : bool, optional Use only data contained in the lower triangle of `a`. Default is to use upper triangle. trans : {0, 1, 2, 'N', 'T', 'C'}, optional Type of system to solve: ======== ========= trans system ======== ========= 0 or 'N' a x = b 1 or 'T' a^T x = b 2 or 'C' a^H x = b ======== ========= unit_diagonal : bool, optional If True, diagonal elements of `a` are assumed to be 1 and will not be referenced. overwrite_b : bool, optional Allow overwriting data in `b` (may enhance performance) 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. Returns ------- x : (M,) or (M, N) ndarray Solution to the system ``a x = b``. Shape of return matches `b`. Raises ------ LinAlgError If `a` is singular Notes ----- .. versionadded:: 0.9.0 Examples -------- Solve the lower triangular system a x = b, where:: [3 0 0 0] [4] a = [2 1 0 0] b = [2] [1 0 1 0] [4] [1 1 1 1] [2] >>> import numpy as np >>> from scipy.linalg import solve_triangular >>> a = np.array([[3, 0, 0, 0], [2, 1, 0, 0], [1, 0, 1, 0], [1, 1, 1, 1]]) >>> b = np.array([4, 2, 4, 2]) >>> x = solve_triangular(a, b, lower=True) >>> x array([ 1.33333333, -0.66666667, 2.66666667, -1.33333333]) >>> a.dot(x) # Check the result array([ 4., 2., 4., 2.]) rmrnrrzexpected square matrixz shapes of a  and b  are incompatibler) rlenr;r+r~rr<rr>rrrr) r@rrrX unit_diagonalrZr[rr dt_nonemptyr_s r0rrs2L AL 9 9 9B AL 9 9 9B 28}}RXa[BHQK771222 x{bhqk!!TTTTTTUUU w!||& F1BH % % %rwq'A'A'A   }R{33333R!3!3K RUE=+ N NDAq Hr2cTdddd||}td||f\}|jjs|dkr|||||||\}}n||j||| | |\}}|dkr||fS|dkrt d|dz t d| d ) Nrrrn)NrC)trtrs)rZrXrunitdiagz/singular matrix: resolution failed at diagonal illegal value in z-th argument of internal trtrs)rwr r{ f_contiguousrr r+) rrrrXrrZrrr-s r0rrs!! $ $ ( ( 6 6E j2r( 3 3FE xA %BKu#m===44%bkU"'i-AAA4 qyy$w axxTDQRFTTUUU N$NNN O OOr2c 8|\}}t|||||||S)a Solve the equation ``a @ x = b`` for ``x``, where ``a`` is the banded matrix defined by `ab`. The matrix a is stored in `ab` using the matrix diagonal ordered form:: ab[u + i - j, j] == a[i,j] Example of `ab` (shape of a is (6,6), `u` =1, `l` =2):: * a01 a12 a23 a34 a45 a00 a11 a22 a33 a44 a55 a10 a21 a32 a43 a54 * a20 a31 a42 a53 * * The documentation is written assuming array arguments are of specified "core" shapes. However, array argument(s) of this function may have additional "batch" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see :ref:`linalg_batch` for details. Parameters ---------- (l, u) : (integer, integer) Number of non-zero lower and upper diagonals ab : (`l` + `u` + 1, M) array_like Banded matrix b : (M,) or (M, K) array_like Right-hand side overwrite_ab : bool, optional Discard data in `ab` (may enhance performance) overwrite_b : bool, optional Discard data in `b` (may enhance performance) 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. Returns ------- x : (M,) or (M, K) ndarray The solution to the system a x = b. Returned shape depends on the shape of `b`. Examples -------- Solve the banded system a x = b, where:: [5 2 -1 0 0] [0] [1 4 2 -1 0] [1] a = [0 1 3 2 -1] b = [2] [0 0 1 2 2] [2] [0 0 0 1 1] [3] There is one nonzero diagonal below the main diagonal (l = 1), and two above (u = 2). The diagonal banded form of the matrix is:: [* * -1 -1 -1] ab = [* 2 2 2 2] [5 4 3 2 1] [1 1 1 1 *] >>> import numpy as np >>> from scipy.linalg import solve_banded >>> ab = np.array([[0, 0, -1, -1, -1], ... [0, 2, 2, 2, 2], ... [5, 4, 3, 2, 1], ... [1, 1, 1, 1, 0]]) >>> b = np.array([0, 1, 2, 2, 3]) >>> x = solve_banded((1, 2), ab, b) >>> x array([-2.37288136, 3.93220339, -4. , 4.3559322 , -1.3559322 ]) )rrZr[) _solve_banded)l_and_urrrrZr[nlowernuppers r0rrs9VVV Q\%0| M M MMr2)rr)rr)rrnc t||d}t||d}|jd|jdkrtd||zdz|jdkr'td||zdzd|jdd |jdkr]t t jd|j t jd|j j} t j || S|pt||}|jddkr&t j || } | ||dfz} | S||cxkrdkrfnnc|pt||}td ||f\} |dddf} |dddf} |d ddf}| || | |||||\}} } }}nftd||f\}t j d |z|zdz|jdf|j }|||dddf<|||||d|\}}}}|dkr|S|dkrtdtd| d)NT)r[ as_inexactr^r&shapes of ab and b are not compatible.rzCinvalid values for the number of lower and upper diagonals: l+u+1 (z) does not equal ab.shape[0] ()rr)gtsvrn)rrzsingular matrixr z"-th argument of internal gbsv/gtsv)rr;r+r~rr<rr>rrrrr rr )rrrrrrZr[rrrb2rrrrrrr-rrrrs r0r r Ks B\d K K KB ALT J J JB x|rx{""ABBB bhqk)) R!# R RCE8A; R R R    w!|| 26!28,,,bgarx.H.H.H I I O}Rr****3R!3!3K x|q XbK 1 1 1 bm  1#:{2r':':  RH55 122Y q!!!tH 3B3Z"d2q"b, #/>>QAtt!RH55 Xqx&(1,bhqk:$* M M M677AAA:4BT,7999CD qyy axx+,,, R$RRR S SSr2ct||}t||}|jd|jdkrtd|jdkr]t t jd|jt jd|jj}t j ||S|pt||}|pt||}|jddkrytd||f\} |r|dd d fj } |dd df} n/|dd d fj } |ddd f } | | | ||||\} } } }n(td ||f\}|||||| \}} }|dkrt|d |dkrtd | d| S)a Solve the equation ``a @ x = b`` for ``x``, where ``a`` is the Hermitian positive-definite banded matrix defined by `ab`. Uses Thomas' Algorithm, which is more efficient than standard LU factorization, but should only be used for Hermitian positive-definite matrices. The matrix ``a`` is stored in `ab` either in lower diagonal or upper diagonal ordered form: ab[u + i - j, j] == a[i,j] (if upper form; i <= j) ab[ i - j, j] == a[i,j] (if lower form; i >= j) Example of `ab` (shape of ``a`` is (6, 6), number of upper diagonals, ``u`` =2):: upper form: * * a02 a13 a24 a35 * a01 a12 a23 a34 a45 a00 a11 a22 a33 a44 a55 lower form: a00 a11 a22 a33 a44 a55 a10 a21 a32 a43 a54 * a20 a31 a42 a53 * * Cells marked with * are not used. Parameters ---------- ab : (``u`` + 1, M) array_like Banded matrix b : (M,) or (M, K) array_like Right-hand side overwrite_ab : bool, optional Discard data in `ab` (may enhance performance) overwrite_b : bool, optional Discard data in `b` (may enhance performance) lower : bool, optional Is the matrix in the lower form. (Default is upper form) 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. Returns ------- x : (M,) or (M, K) ndarray The solution to the system ``a x = b``. Shape of return matches shape of `b`. Notes ----- In the case of a non-positive definite matrix ``a``, the solver `solve_banded` may be used. Examples -------- Solve the banded system ``A x = b``, where:: [ 4 2 -1 0 0 0] [1] [ 2 5 2 -1 0 0] [2] A = [-1 2 6 2 -1 0] b = [2] [ 0 -1 2 7 2 -1] [3] [ 0 0 -1 2 8 2] [3] [ 0 0 0 -1 2 9] [3] >>> import numpy as np >>> from scipy.linalg import solveh_banded ``ab`` contains the main diagonal and the nonzero diagonals below the main diagonal. That is, we use the lower form: >>> ab = np.array([[ 4, 5, 6, 7, 8, 9], ... [ 2, 2, 2, 2, 2, 0], ... [-1, -1, -1, -1, 0, 0]]) >>> b = np.array([1, 2, 2, 3, 3, 3]) >>> x = solveh_banded(ab, b, lower=True) >>> x array([ 0.03431373, 0.45938375, 0.05602241, 0.47759104, 0.17577031, 0.34733894]) Solve the Hermitian banded system ``H x = b``, where:: [ 8 2-1j 0 0 ] [ 1 ] H = [2+1j 5 1j 0 ] b = [1+1j] [ 0 -1j 9 -2-1j] [1-2j] [ 0 0 -2+1j 6 ] [ 0 ] In this example, we put the upper diagonals in the array ``hb``: >>> hb = np.array([[0, 2-1j, 1j, -2-1j], ... [8, 5, 9, 6 ]]) >>> b = np.array([1, 1+1j, 1-2j, 0]) >>> x = solveh_banded(hb, b) >>> x array([ 0.07318536-0.02939412j, 0.11877624+0.17696461j, 0.10077984-0.23035393j, -0.00479904-0.09358128j]) rmr^rrrrrn)ptsvN)pbsv)rXrrZz&th leading minor not positive definiter zth argument of internal pbsv)rr;r+r~rr<rr>rrrr realconjr )rrrrZrXr[rrrrrerrr-rcs r0rr~sR B\ : : :B AL 9 9 9B x|rx{""ABBB w!|| 26!28,,,bgarx.H.H.H I I O}Rr****3R!3!3K6;r2#6#6L x{a RH55  !1aaa4 A1crc6 AA1aaa4 A1abb5   AaB l)++2q$$!RH55T"bL&1333 1d axxTIIIJJJ axxPdUPPPQQQ Hr2ct|tr|n|tj|f\}}t ||||S)a Solve the equation ``T @ x = b`` for ``x``, where ``T`` is a Toeplitz matrix defined by `c_or_cr`. The Toeplitz matrix has constant diagonals, with ``c`` as its first column and ``r`` as its first row. If ``r`` is not given, ``r == conjugate(c)`` is assumed. The documentation is written assuming array arguments are of specified "core" shapes. However, array argument(s) of this function may have additional "batch" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see :ref:`linalg_batch` for details. Parameters ---------- c_or_cr : array_like or tuple of (array_like, array_like) The vector ``c``, or a tuple of arrays (``c``, ``r``). If not supplied, ``r = conjugate(c)`` is assumed; in this case, if c[0] is real, the Toeplitz matrix is Hermitian. r[0] is ignored; the first row of the Toeplitz matrix is ``[c[0], r[1:]]``. b : (M,) or (M, K) array_like Right-hand side in ``T x = b``. 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 (result entirely NaNs) if the inputs do contain infinities or NaNs. Returns ------- x : (M,) or (M, K) ndarray The solution to the system ``T @ x = b``. Shape of return matches shape of `b`. See Also -------- toeplitz : Toeplitz matrix Notes ----- The solution is computed using Levinson-Durbin recursion, which is faster than generic least-squares methods, but can be less numerically stable. Examples -------- Solve the Toeplitz system ``T @ x = b``, where:: [ 1 -1 -2 -3] [1] T = [ 3 1 -1 -2] b = [2] [ 6 3 1 -1] [2] [10 6 3 1] [5] To specify the Toeplitz matrix, only the first column and the first row are needed. >>> import numpy as np >>> c = np.array([1, 3, 6, 10]) # First column of T >>> r = np.array([1, -1, -2, -3]) # First row of T >>> b = np.array([1, 2, 2, 5]) >>> from scipy.linalg import solve_toeplitz, toeplitz >>> x = solve_toeplitz((c, r), b) >>> x array([ 1.66666667, -1. , -2.66666667, 2.33333333]) Check the result by creating the full Toeplitz matrix and multiplying it by ``x``. We should get `b`. >>> T = toeplitz(c, r) >>> T.dot(x) array([ 1., 2., 2., 5.]) ) isinstancetupler< conjugate_solve_toeplitz)c_or_crrr[rrs r0rr sDV!%00 V77w W@U@U6VDAq 1aL 1 11r2)rr)r#rct||f|d\}}}}jdkrtjStj|ddd|ft djdkr&ttj\}}nCtj fdtj dD}|j |}|S)NT) keep_b_shaperr^z)illegal value, `b` is a required argumentrc tg|]4}ttjdd|fd5S)Nr)rr<ascontiguousarray)rrSrvalss r0 z#_solve_toeplitz..lsO999!"&dB,@111a4,I,IJJ1M999r2) _validate_args_for_toeplitz_opsr~r<r concatenater+ryrr' column_stackrr;reshape) rr#rr[r>b_shaperrr(s ` @r0r!r!Zs= A1R"W:q/ * *DyDEEEv{{b21556611 O99999&+AGAJ&7&7999 : : AIw  Hr2c|}|dkr ||jz }d|cxkr |jkrnn |j|Std|d)Nr'zaxis' entry is out of bounds)ryr;r+)anamer@axisaxs r0 _get_axis_lenr4sse B Avv af Bwr{ <<<< = ==r2raiser^cBtj|}td||}tj|}td||}||kr td|jd|jd|jdkr]t tjd|jtj d|jj} tj || Stj tj ||d d } tj | } |z| d |ztjtjjz}|jd kr||jd z}ntj|}| |k} tj| } | r|dkrt)dd| | <tj tj ||d d }|| z }| r#tj|t,| z}d||<tj |d }tj|stj|s|j}|d krtj |d |}|S)aSolve the equation ``C @ x = b`` for ``x``, where ``C`` is a circulant matrix defined by `c`. `C` is the circulant matrix associated with the vector `c`. The system is solved by doing division in Fourier space. The calculation is:: x = ifft(fft(b) / fft(c)) where `fft` and `ifft` are the fast Fourier transform and its inverse, respectively. For a large vector `c`, this is *much* faster than solving the system with the full circulant matrix. Parameters ---------- c : array_like The coefficients of the circulant matrix. b : array_like Right-hand side matrix in ``a x = b``. singular : str, optional This argument controls how a near singular circulant matrix is handled. If `singular` is "raise" and the circulant matrix is near singular, a `LinAlgError` is raised. If `singular` is "lstsq", the least squares solution is returned. Default is "raise". tol : float, optional If any eigenvalue of the circulant matrix has an absolute value that is less than or equal to `tol`, the matrix is considered to be near singular. If not given, `tol` is set to:: tol = abs_eigs.max() * abs_eigs.size * np.finfo(np.float64).eps where `abs_eigs` is the array of absolute values of the eigenvalues of the circulant matrix. caxis : int When `c` has dimension greater than 1, it is viewed as a collection of circulant vectors. In this case, `caxis` is the axis of `c` that holds the vectors of circulant coefficients. baxis : int When `b` has dimension greater than 1, it is viewed as a collection of vectors. In this case, `baxis` is the axis of `b` that holds the right-hand side vectors. outaxis : int When `c` or `b` are multidimensional, the value returned by `solve_circulant` is multidimensional. In this case, `outaxis` is the axis of the result that holds the solution vectors. Returns ------- x : ndarray Solution to the system ``C x = b``. Raises ------ LinAlgError If the circulant matrix associated with `c` is near singular. See Also -------- circulant : circulant matrix Notes ----- For a 1-D vector `c` with length `m`, and an array `b` with shape ``(m, ...)``, solve_circulant(c, b) returns the same result as solve(circulant(c), b) where `solve` and `circulant` are from `scipy.linalg`. .. versionadded:: 0.16.0 Examples -------- >>> import numpy as np >>> from scipy.linalg import solve_circulant, solve, circulant, lstsq >>> c = np.array([2, 2, 4]) >>> b = np.array([1, 2, 3]) >>> solve_circulant(c, b) array([ 0.75, -0.25, 0.25]) Compare that result to solving the system with `scipy.linalg.solve`: >>> solve(circulant(c), b) array([ 0.75, -0.25, 0.25]) A singular example: >>> c = np.array([1, 1, 0, 0]) >>> b = np.array([1, 2, 3, 4]) Calling ``solve_circulant(c, b)`` will raise a `LinAlgError`. For the least square solution, use the option ``singular='lstsq'``: >>> solve_circulant(c, b, singular='lstsq') array([ 0.25, 1.25, 2.25, 1.25]) Compare to `scipy.linalg.lstsq`: >>> x, resid, rnk, s = lstsq(circulant(c), b) >>> x array([ 0.25, 1.25, 2.25, 1.25]) A broadcasting example: Suppose we have the vectors of two circulant matrices stored in an array with shape (2, 5), and three `b` vectors stored in an array with shape (3, 5). For example, >>> c = np.array([[1.5, 2, 3, 0, 0], [1, 1, 4, 3, 2]]) >>> b = np.arange(15).reshape(-1, 5) We want to solve all combinations of circulant matrices and `b` vectors, with the result stored in an array with shape (2, 3, 5). When we disregard the axes of `c` and `b` that hold the vectors of coefficients, the shapes of the collections are (2,) and (3,), respectively, which are not compatible for broadcasting. To have a broadcast result with shape (2, 3), we add a trivial dimension to `c`: ``c[:, np.newaxis, :]`` has shape (2, 1, 5). The last dimension holds the coefficients of the circulant matrices, so when we call `solve_circulant`, we can use the default ``caxis=-1``. The coefficients of the `b` vectors are in the last dimension of the array `b`, so we use ``baxis=-1``. If we use the default `outaxis`, the result will have shape (5, 2, 3), so we'll use ``outaxis=-1`` to put the solution vectors in the last dimension. >>> x = solve_circulant(c[:, np.newaxis, :], b, baxis=-1, outaxis=-1) >>> x.shape (2, 3, 5) >>> np.set_printoptions(precision=3) # For compact output of numbers. >>> x array([[[-0.118, 0.22 , 1.277, -0.142, 0.302], [ 0.651, 0.989, 2.046, 0.627, 1.072], [ 1.42 , 1.758, 2.816, 1.396, 1.841]], [[ 0.401, 0.304, 0.694, -0.867, 0.377], [ 0.856, 0.758, 1.149, -0.412, 0.831], [ 1.31 , 1.213, 1.603, 0.042, 1.286]]]) Check by solving one pair of `c` and `b` vectors (cf. ``x[1, 1, :]``): >>> solve_circulant(c[1], b[1, :]) array([ 0.856, 0.758, 1.149, -0.412, 0.831]) rrz Shapes of c rrrr(rr^r2N)rr5znear singular circulant matrix.r)r<rr4r+r;r~raranger>rrfftmoveaxisrrrrepsr-anyr ones_likeboolifftrr)rrrPtolcaxisbaxisoutaxisncnbrfcabs_fc near_zerosis_near_singularfbqmaskrs r0rr|skl aA sAu % %B aA sAu % %B RxxRRRRRRSSS v{{ RYq888WQag666888= }Qb)))) BK5"--B 7 7B VBZZF {jjbj!!B&"*)=)=)AA 9??++ci$.//CC-$$C3Jvj)) w  ?@@ @BzN BK5"--B 7 7B RA |AT***Z7$  AB A OA  "/!"4"4 F"}} K2w ' ' Hr2)r\rXc t||}|jdkrtd|j|jd|jdkrtd|j|jdkrCt t jd|jj}t j ||St||\}}|j d s&|jj d ksd }| }|o|jdko |j d }dddd dddddd |}tj||||\}} | rt!| |S)a. Compute the inverse of a matrix. If the data matrix is known to be a particular type then supplying the corresponding string to ``assume_a`` key chooses the dedicated solver. The available options are ============================= ================================ general 'general' (or 'gen') diagonal 'diagonal' upper triangular 'upper triangular' lower triangular 'lower triangular' symmetric positive definite 'pos' symmetric 'sym' Hermitian 'her' ============================= ================================ For the 'pos' option, only the triangle of the input matrix specified in the `lower` argument is used, and the other triangle is not referenced. Likewise, an explicit `assume_a='diagonal'` means that off-diagonal elements are not referenced. Array argument(s) of this function may have additional "batch" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see :ref:`linalg_batch` for details. Parameters ---------- a : array_like, shape (..., M, M) Square matrix (or a batch of matrices) to be inverted. overwrite_a : bool, optional Discard data in `a` (may improve performance). Default is 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. assume_a : str, optional Valid entries are described above. If omitted or ``None``, checks are performed to identify structure so the appropriate solver can be called. lower : bool, optional Ignored unless `assume_a` is one of 'sym', 'her', or 'pos'. If True, the calculation uses only the data in the lower triangle of `a`; entries above the diagonal are ignored. If False (default), the calculation uses only the data in the upper triangle of `a`; entries below the diagonal are ignored. Returns ------- ainv : ndarray Inverse of the matrix `a`. Raises ------ LinAlgError If `a` is singular. ValueError If `a` is not square, or not 2D. Examples -------- >>> import numpy as np >>> from scipy import linalg >>> a = np.array([[1., 2.], [3., 4.]]) >>> linalg.inv(a) array([[-2. , 1. ], [ 1.5, -0.5]]) >>> np.dot(a, linalg.inv(a)) array([[ 1., 0.], [ 0., 1.]]) Notes ----- The input array ``a`` may represent a single matrix or a collection (a.k.a. a "batch") of square matrices. For example, if ``a.shape == (4, 3, 2, 2)``, it is interpreted as a ``(4, 3)``-shaped batch of :math:`2\times 2` matrices. This routine checks the condition number of the `a` matrix and emits a `LinAlgWarning` for ill-conditioned inputs. rmrnror^rprqrrrsrtTrur_rarbrcrdre) Nr:rfr4r6r5rgrirj)rryr+r;r~rr<rr>rr r{r|r}r_invrU) r@rYr[r\rXrrrinv_arOs r0rrIszd AL 9 9 9B w{{D"'DDEEE x|rx|##CCCDDD w!|| *** + + 1}Rr****.b+>>OB HY 28#5#<#< WWYYO27a<Obh~6NKQ   I%)"ieLLNE7.$W--- Lr2cV|rtj|ntj|}|jdkrt d|jd|jdkrt d|jdt ||\}}t|jdkr_|jj dvr tj n tj }|jdkr |d Stj |jd d| S|jdd d kre|d }|jdkr|d}|jj dvr|S|jj dkr| dn| dSt||s|s|d}|jdr |jds|d}|jdkrKt#|}tj|rtj |ntj |S|jj }|dvr|rdnd}tj|jd d|}t+d|jd dDD]}t#||||<|S)a Compute the determinant of a matrix The determinant is a scalar that is a function of the associated square matrix coefficients. The determinant value is zero for singular matrices. Array argument(s) of this function may have additional "batch" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see :ref:`linalg_batch` for details. Parameters ---------- a : (..., M, M) array_like Input array to compute determinants for. overwrite_a : bool, optional Allow overwriting data in a (may enhance performance). 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. Returns ------- det : (...) float or complex Determinant of `a`. For stacked arrays, a scalar is returned for each (m, m) slice in the last two dimensions of the input. For example, an input of shape (p, q, m, m) will produce a result of shape (p, q). If all dimensions are 1 a scalar is returned regardless of ndim. Notes ----- The determinant is computed by performing an LU factorization of the input with LAPACK routine 'getrf', and then calculating the product of diagonal entries of the U factor. Even if the input array is single precision (float32 or complex64), the result will be returned in double precision (float64 or complex128) to prevent overflows. Examples -------- >>> import numpy as np >>> from scipy import linalg >>> a = np.array([[1,2,3], [4,5,6], [7,8,9]]) # A singular matrix >>> linalg.det(a) 0.0 >>> b = np.array([[0,2,3], [4,5,6], [7,8,9]]) >>> linalg.det(b) 3.0 >>> # An array with the shape (3, 2, 2, 2) >>> c = np.array([[[[1., 2.], [3., 4.]], ... [[5., 6.], [7., 8.]]], ... [[[9., 10.], [11., 12.]], ... [[13., 14.], [15., 16.]]], ... [[[17., 18.], [19., 20.]], ... [[21., 22.], [23., 24.]]]]) >>> linalg.det(c) # The resulting shape is (3, 2) array([[-2., -2.], [-2., -2.], [-2., -2.]]) >>> linalg.det(c[0, 0]) # Confirm the (0, 0) slice, [[1, 2], [3, 4]] -2.0 rnz1The input array must be at least two-dimensional.r^rpzALast 2 dimensions of the array must be square but received shape rJrFD?N)r;r>)rr).rrr8dDrrDr)order C_CONTIGUOUS WRITEABLErrc,g|]}t|Sr8)r)rrs r0r)zdet..@s999aq999r2)r<rasarrayryr+r;r rr>rrrrrrr}r{r isrealobjisloweremptyr)r@rYr[rdtypr dtype_charinds r0r r sF%1 C a bjmmB w{{LMMM x|rx|##<02<<<== =/r;??OB  BH~X]$66rzzBM 7a<<499 7#2#d;;; ; x}  ] 7a<<BB 8=D I!##!5!5ryy~~~299S>>I r1  $ $s##B H^ $ +)> WW3W   w!||r""#%<#4#4L 3"-:L:LMJT&..009SSc (28CRC= 3 3 3C9928CRC=999:--#BsG,,C Jr2c ^t||}t||}t|jdkrtd|j\} } t|jdkr|jd} nd} | |jdkr!td| d|jdd| dks| dkrt j| f|jdd zt j|| } | dkr%tj|d dz} nt j d } | | dt j d fS|}| tj }|d vrtd|dt||df||f\}}|j jdkrdnd}| | kret|jdkr)t j| | f|j }||d | d d f<n"t j| |j }||d | <|}|pt||}|pt||}|t j|j j}|dvr2|dkr,t%|| | | |}|||||||\}} }}}}nc|dkr]|r-t%|| | | |\}}||||||dd\} }}}n.t%|| | | |\}}}|||||||dd\} }}}|dkrt'd|dkrtd| d|t jg| j }| | krE| d | }|| kr3t jt j| | d dzd }|} | |||fS|dkrt%|| | | |}t j|jddftj }||||||dd\}} }}}|dkrtd| d| | kr | d | }|} | t jg| j |d fSd S)a  Compute least-squares solution to the equation ``a @ x = b``. Compute a vector x such that the 2-norm ``|b - A x|`` is minimized. Parameters ---------- a : (M, N) array_like Left-hand side array b : (M,) or (M, K) array_like Right hand side array cond : float, optional Cutoff for 'small' singular values; used to determine effective rank of a. Singular values smaller than ``cond * largest_singular_value`` are considered zero. overwrite_a : bool, optional Discard data in `a` (may enhance performance). Default is False. overwrite_b : bool, optional Discard data in `b` (may enhance performance). Default is 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. lapack_driver : str, optional Which LAPACK driver is used to solve the least-squares problem. Options are ``'gelsd'``, ``'gelsy'``, ``'gelss'``. Default (``'gelsd'``) is a good choice. However, ``'gelsy'`` can be slightly faster on many problems. ``'gelss'`` was used historically. It is generally slow but uses less memory. .. versionadded:: 0.17.0 Returns ------- x : (N,) or (N, K) ndarray Least-squares solution. residues : (K,) ndarray or float Square of the 2-norm for each column in ``b - a x``, if ``M > N`` and ``rank(A) == n`` (returns a scalar if ``b`` is 1-D). Otherwise a (0,)-shaped array is returned. rank : int Effective rank of `a`. s : (min(M, N),) ndarray or None Singular values of `a`. The condition number of ``a`` is ``s[0] / s[-1]``. Raises ------ LinAlgError If computation does not converge. ValueError When parameters are not compatible. See Also -------- scipy.optimize.nnls : linear least squares with non-negativity constraint Notes ----- When ``'gelsy'`` is used as a driver, `residues` is set to a (0,)-shaped array and `s` is always ``None``. Examples -------- >>> import numpy as np >>> from scipy.linalg import lstsq >>> import matplotlib.pyplot as plt Suppose we have the following data: >>> x = np.array([1, 2.5, 3.5, 4, 5, 7, 8.5]) >>> y = np.array([0.3, 1.1, 1.5, 2.0, 3.2, 6.6, 8.6]) We want to fit a quadratic polynomial of the form ``y = a + b*x**2`` to this data. We first form the "design matrix" M, with a constant column of 1s and a column containing ``x**2``: >>> M = x[:, np.newaxis]**[0, 2] >>> M array([[ 1. , 1. ], [ 1. , 6.25], [ 1. , 12.25], [ 1. , 16. ], [ 1. , 25. ], [ 1. , 49. ], [ 1. , 72.25]]) We want to find the least-squares solution to ``M.dot(p) = y``, where ``p`` is a vector with length 2 that holds the parameters ``a`` and ``b``. >>> p, res, rnk, s = lstsq(M, y) >>> p array([ 0.20925829, 0.12013861]) Plot the data and the fitted curve. >>> plt.plot(x, y, 'o', label='data') >>> xx = np.linspace(0, 9, 101) >>> yy = p[0] + p[1]*xx**2 >>> plt.plot(xx, yy, label='least squares fit, $y = a + bx^2$') >>> plt.xlabel('x') >>> plt.ylabel('y') >>> plt.legend(framealpha=1, shadow=True) >>> plt.grid(alpha=0.25) >>> plt.show() rmrnzInput array a should be 2Drrz=Shape mismatch: a and b should have the same number of rows (z != z).Nrr7)r)gelsdgelsygelsszLAPACK driver "z" is not found_lworkrTF)rdrbrd)rYrZrbz,SVD did not converge in Linear Least Squaresr z-th argument of internal rcz-th argument of internal gelsy)rrr;r+r<r common_typelinalgrr]r!default_lapack_driverr r>rCrrr<r r rZsumrint32r)r@rcondrYrZr[ lapack_driverrrmr,nrhsrresiduesdriver lapack_func lapack_lwork real_datarrvsrankworkr-iworkrworkresidsx1jptvjs r0r!r!Fs/` AL 9 9 9B AL 9 9 9B 28}}5666 8DAq 28}}x{BHQK=&'==-/Xa[===>> >Avva HaTBHQRRL(r20F0F G G G 66y~~bq~1114HHx~~H(Arx~~-- F ~, 000A6AAABBB 0&4:1B1B1B2D24b!;!;K%*/366UI1uu rx==A  1d);+<===BBrr111uII!;#4555BBrrF 3R!3!3K3R!3!3K |x )**. ### W  "<AtTBBE(3 BD%@K@K)M)M)M %Aq!T4w   C-lAq$MM u#.;r2u/4dE5$J$J 1dDD'5\1a594'A'A#ue#.;r2ueU/3UE$C$C 1dD !88LMM M !88STESSMSS Bag... q552A2Bqyyqu q 0q999A&$!! 7  |Q4>>x!a(999)k"b$*/??1at !88V$VVVWW W q552A2BA"(2qw''t33  r2rb)atolrtol return_rankr[c6t||}tj|dd\}}}|jj}t j|d} |dn|}|.t|jt j |j zn|}|dks|dkrtd|| |zz} t j || k} |ddd| f}||d| z}||d| z j} |r| | fS| S)aI Compute the (Moore-Penrose) pseudo-inverse of a matrix. Calculate a generalized inverse of a matrix using its singular-value decomposition ``U @ S @ V`` in the economy mode and picking up only the columns/rows that are associated with significant singular values. If ``s`` is the maximum singular value of ``a``, then the significance cut-off value is determined by ``atol + rtol * s``. Any singular value below this value is assumed insignificant. Parameters ---------- a : (M, N) array_like Matrix to be pseudo-inverted. atol : float, optional Absolute threshold term, default value is 0. .. versionadded:: 1.7.0 rtol : float, optional Relative threshold term, default value is ``max(M, N) * eps`` where ``eps`` is the machine precision value of the datatype of ``a``. .. versionadded:: 1.7.0 return_rank : bool, optional If True, return the effective rank of the matrix. 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. Returns ------- B : (N, M) ndarray The pseudo-inverse of matrix `a`. rank : int The effective rank of the matrix. Returned if `return_rank` is True. Raises ------ LinAlgError If SVD computation does not converge. See Also -------- pinvh : Moore-Penrose pseudoinverse of a hermitian matrix. Notes ----- If ``A`` is invertible then the Moore-Penrose pseudoinverse is exactly the inverse of ``A`` [1]_. If ``A`` is not invertible then the Moore-Penrose pseudoinverse computes the ``x`` solution to ``Ax = b`` such that ``||Ax - b||`` is minimized [1]_. References ---------- .. [1] Penrose, R. (1956). On best approximate solutions of linear matrix equations. Mathematical Proceedings of the Cambridge Philosophical Society, 52(1), 17-19. doi:10.1017/S0305004100030929 Examples -------- Given an ``m x n`` matrix ``A`` and an ``n x m`` matrix ``B`` the four Moore-Penrose conditions are: 1. ``ABA = A`` (``B`` is a generalized inverse of ``A``), 2. ``BAB = B`` (``A`` is a generalized inverse of ``B``), 3. ``(AB)* = AB`` (``AB`` is hermitian), 4. ``(BA)* = BA`` (``BA`` is hermitian) [1]_. Here, ``A*`` denotes the conjugate transpose. The Moore-Penrose pseudoinverse is a unique ``B`` that satisfies all four of these conditions and exists for any ``A``. Note that, unlike the standard matrix inverse, ``A`` does not have to be a square matrix or have linearly independent columns/rows. As an example, we can calculate the Moore-Penrose pseudoinverse of a random non-square matrix and verify it satisfies the four conditions. >>> import numpy as np >>> from scipy import linalg >>> rng = np.random.default_rng() >>> A = rng.standard_normal((9, 6)) >>> B = linalg.pinv(A) >>> np.allclose(A @ B @ A, A) # Condition 1 True >>> np.allclose(B @ A @ B, B) # Condition 2 True >>> np.allclose((A @ B).conj().T, A @ B) # Condition 3 True >>> np.allclose((B @ A).conj().T, B @ A) # Condition 4 True rmF) full_matricesr[initialN&atol and rtol values must be positive.)rrsvdr>rrXr<rr;rr<r+rirr) r@r~rrr[uruvhtmaxSvalrvBs r0r"r"s)H 1<888AqEJJJHAq"  A 6!R D224D.2l3qw<<"(1++/ ) )D r tbyyABBB  C 6!c'??D !!!UdU( A5D5MA RY A$wr2cXt||}tj||dd\}}|jj}t jt j|d} |dn|}|.t|j t j |j zn|}|dks|dkrtd|| |zz} t|| k} d || z } |dd| f}|| z| jz} |r| t| fS| S) au Compute the (Moore-Penrose) pseudo-inverse of a Hermitian matrix. Calculate a generalized inverse of a complex Hermitian/real symmetric matrix using its eigenvalue decomposition and including all eigenvalues with 'large' absolute value. Parameters ---------- a : (N, N) array_like Real symmetric or complex hermetian matrix to be pseudo-inverted atol : float, optional Absolute threshold term, default value is 0. .. versionadded:: 1.7.0 rtol : float, optional Relative threshold term, default value is ``N * eps`` where ``eps`` is the machine precision value of the datatype of ``a``. .. versionadded:: 1.7.0 lower : bool, optional Whether the pertinent array data is taken from the lower or upper triangle of `a`. (Default: lower) return_rank : bool, optional If True, return the effective rank of the matrix. 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. Returns ------- B : (N, N) ndarray The pseudo-inverse of matrix `a`. rank : int The effective rank of the matrix. Returned if `return_rank` is True. Raises ------ LinAlgError If eigenvalue algorithm does not converge. See Also -------- pinv : Moore-Penrose pseudoinverse of a matrix. Examples -------- For a more detailed example see `pinv`. >>> import numpy as np >>> from scipy.linalg import pinvh >>> rng = np.random.default_rng() >>> a = rng.standard_normal((9, 6)) >>> a = np.dot(a, a.T) >>> B = pinvh(a) >>> np.allclose(a, a @ B @ a) True >>> np.allclose(B, B @ a @ B) True rmFev)rXr[rprrNrrS)rreighr>rrXr<rrr;rr<r+rrr)r@r~rrXrr[rurrrr above_cutoff psigma_diagrs r0r#r#s+J 1<888A <U4 H H HDAq  A 6"&))R ( ( (D224D.2l3qw<<"(1++/ ) )D r tbyyABBB  CFFSLL,'K !!!\/A [AFFHHJ&A#k""""r2)Arnctjt|d}tj|jst d|jdkrttjd|j \}}tj ||j }|rJtj |t|}tj t|} ||| ffS|tj ||j fStd|f} | |||| \}} } } }|dkrt d | d tj | t}| | | d z|| | d z<| t"d d z } |jd}tj |} | |krIt%| | d zddddd D]$\}}||z |kr| ||z |g| |||z g<%| dkr3t%| d| D]\}}||kr | ||g| ||g<|r||| ffStj | }tj ||| <|tj||ddffS)a# Compute a diagonal similarity transformation for row/column balancing. The balancing tries to equalize the row and column 1-norms by applying a similarity transformation such that the magnitude variation of the matrix entries is reflected to the scaling matrices. Moreover, if enabled, the matrix is first permuted to isolate the upper triangular parts of the matrix and, again if scaling is also enabled, only the remaining subblocks are subjected to scaling. Parameters ---------- A : (n, n) array_like Square data matrix for the balancing. permute : bool, optional The selector to define whether permutation of A is also performed prior to scaling. scale : bool, optional The selector to turn on and off the scaling. If False, the matrix will not be scaled. separate : bool, optional This switches from returning a full matrix of the transformation to a tuple of two separate 1-D permutation and scaling arrays. overwrite_a : bool, optional This is passed to xGEBAL directly. Essentially, overwrites the result to the data. It might increase the space efficiency. See LAPACK manual for details. This is False by default. Returns ------- B : (n, n) ndarray Balanced matrix T : (n, n) ndarray A possibly permuted diagonal matrix whose nonzero entries are integer powers of 2 to avoid numerical truncation errors. scale, perm : (n,) ndarray If ``separate`` keyword is set to True then instead of the array ``T`` above, the scaling and the permutation vectors are given separately as a tuple without allocating the full array ``T``. Notes ----- The balanced matrix satisfies the following equality .. math:: B = T^{-1} A T The scaling coefficients are approximated to the nearest power of 2 to avoid round-off errors. This algorithm is particularly useful for eigenvalue and matrix decompositions and in many cases it is already called by various LAPACK routines. The algorithm is based on the well-known technique of [1]_ and has been modified to account for special cases. See [2]_ for details which have been implemented since LAPACK v3.5.0. Before this version there are corner cases where balancing can actually worsen the conditioning. See [3]_ for such examples. The code is a wrapper around LAPACK's xGEBAL routine family for matrix balancing. .. versionadded:: 0.19.0 References ---------- .. [1] B.N. Parlett and C. Reinsch, "Balancing a Matrix for Calculation of Eigenvalues and Eigenvectors", Numerische Mathematik, Vol.13(4), 1969, :doi:`10.1007/BF02165404` .. [2] R. James, J. Langou, B.R. Lowery, "On matrix balancing and eigenvector computation", 2014, :arxiv:`1401.5766` .. [3] D.S. Watkins. A case where balancing is harmful. Electron. Trans. Numer. Anal, Vol.23, 2006. Examples -------- >>> import numpy as np >>> from scipy import linalg >>> x = np.array([[1,2,0], [9,1,0.01], [1,2,10*np.pi]]) >>> y, permscale = linalg.matrix_balance(x) >>> np.abs(x).sum(axis=0) / np.abs(x).sum(axis=1) array([ 3.66666667, 0.4995005 , 0.91312162]) >>> np.abs(y).sum(axis=0) / np.abs(y).sum(axis=1) array([ 1.2 , 1.27041742, 0.92658316]) # may vary >>> permscale # only powers of 2 (0.5 == 2^(-1)) array([[ 0.5, 0. , 0. ], # may vary [ 0. , 1. , 0. ], [ 0. , 0. , 1. ]]) Trmz/The data matrix for balancing should be square.rrnrr;gebal)scalepermuterYzHxGEBAL exited with the internal error "illegal value in argument number z8.". See LAPACK documentation for the xGEBAL error codes.rFrNr^)r<rrequalr;r+r~r$rr>rr>rr9r floatrintrKr)rrrseparaterYb_nt_nrscalingpermrlohipsr-r,r`riperms r0r$r$sF (>>>??A 8QW LJKKK v{{!"&!'":":":;;S M!39 - - -  &l1CFF333G9SVV$$Dwo% %"-33333 g - -E%,7999Ar2r4 axxL?CeLLLMM M l2U+++G"RT'{GBr!tG 3U # #a 'B  A 9Q<s r0rz2_validate_args_for_toeplitz_ops..s2==arz!5)))======r2rr^) rrrr r+r;r<rrrryr-r~) r"rr[r%enforce_squarerr#r. is_not_squareis_cmplxr>s @r0r*r*sV'5!!1 q| < < < q| < < < w\ B B B KKMMy:;;;1<888AgGGAJ!'!*,M5=5QWQZ171:-E-E3444q!!MR_Q%7%7M2?1;M;MH% 5BMM2:E====Aq!9===GAq!v{{<{ IIb!   1 IIagaj "" : : aE7 ""r2c ddlm}m}m}m}t |t r|n|t j|f\}} t| |||||||| S)a|Efficient Toeplitz Matrix-Matrix Multiplication using FFT This function returns the matrix multiplication between a Toeplitz matrix and a dense matrix. The Toeplitz matrix has constant diagonals, with c as its first column and r as its first row. If r is not given, ``r == conjugate(c)`` is assumed. The documentation is written assuming array arguments are of specified "core" shapes. However, array argument(s) of this function may have additional "batch" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see :ref:`linalg_batch` for details. Parameters ---------- c_or_cr : array_like or tuple of (array_like, array_like) The vector ``c``, or a tuple of arrays (``c``, ``r``). If not supplied, ``r = conjugate(c)`` is assumed; in this case, if c[0] is real, the Toeplitz matrix is Hermitian. r[0] is ignored; the first row of the Toeplitz matrix is ``[c[0], r[1:]]``. x : (M,) or (M, K) array_like Matrix with which to multiply. 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 (result entirely NaNs) if the inputs do contain infinities or NaNs. workers : int, optional To pass to scipy.fft.fft and ifft. Maximum number of workers to use for parallel computation. If negative, the value wraps around from ``os.cpu_count()``. See scipy.fft.fft for more details. Returns ------- T @ x : (M,) or (M, K) ndarray The result of the matrix multiplication ``T @ x``. Shape of return matches shape of `x`. See Also -------- toeplitz : Toeplitz matrix solve_toeplitz : Solve a Toeplitz system using Levinson Recursion Notes ----- The Toeplitz matrix is embedded in a circulant matrix and the FFT is used to efficiently calculate the matrix-matrix product. Because the computation is based on the FFT, integer inputs will result in floating point outputs. This is unlike NumPy's `matmul`, which preserves the data type of the input. This is partly based on the implementation that can be found in [1]_, licensed under the MIT license. More information about the method can be found in reference [2]_. References [3]_ and [4]_ have more reference implementations in Python. .. versionadded:: 1.6.0 References ---------- .. [1] Jacob R Gardner, Geoff Pleiss, David Bindel, Kilian Q Weinberger, Andrew Gordon Wilson, "GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration" with contributions from Max Balandat and Ruihan Wu. Available online: https://github.com/cornellius-gp/gpytorch .. [2] J. Demmel, P. Koev, and X. Li, "A Brief Survey of Direct Linear Solvers". In Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, editors. Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia, 2000. Available at: http://www.netlib.org/utk/people/JackDongarra/etemplates/node384.html .. [3] R. Scheibler, E. Bezzam, I. Dokmanic, Pyroomacoustics: A Python package for audio room simulations and array processing algorithms, Proc. IEEE ICASSP, Calgary, CA, 2018. https://github.com/LCAV/pyroomacoustics/blob/pypi-release/ pyroomacoustics/adaptive/util.py .. [4] Marano S, Edwards B, Ferrari G and Fah D (2017), "Fitting Earthquake Spectra: Colored Noise and Incomplete Data", Bulletin of the Seismological Society of America., January, 2017. Vol. 107(1), pp. 276-291. Examples -------- Multiply the Toeplitz matrix T with matrix x:: [ 1 -1 -2 -3] [1 10] T = [ 3 1 -1 -2] x = [2 11] [ 6 3 1 -1] [2 11] [10 6 3 1] [5 19] To specify the Toeplitz matrix, only the first column and the first row are needed. >>> import numpy as np >>> c = np.array([1, 3, 6, 10]) # First column of T >>> r = np.array([1, -1, -2, -3]) # First row of T >>> x = np.array([[1, 10], [2, 11], [2, 11], [5, 19]]) >>> from scipy.linalg import toeplitz, matmul_toeplitz >>> matmul_toeplitz((c, r), x) array([[-20., -80.], [ -7., -8.], [ 9., 85.], [ 33., 218.]]) Check the result by creating the full Toeplitz matrix and multiplying it by ``x``. >>> toeplitz(c, r) @ x array([[-20, -80], [ -7, -8], [ 9, 85], [ 33, 218]]) The full matrix is never formed explicitly, so this routine is suitable for very large Toeplitz matrices. >>> n = 1000000 >>> matmul_toeplitz([1] + [0]*(n-1), np.ones(n)) array([1., 1., 1., ..., 1., 1., 1.], shape=(1000000,)) rn)r:r@rfftirfft) r:r@rrrrr<r _matmul_toepltiz) r"rr[workersr:r@rrrr#s r0r%r%sr~-,,,,,,,,,,, %00 V77w W@U@U6VDAq Aq!WlCtU S SSr2)rrc t||f||dd\}}}} } |j\} } t|} t|}| |zdz }t| dkr| fn| | f}|jdkrt j||St j||dddf}t j|st j|rO||d|dd}|||d|}|||zd|d| ddf}nO||d|dd}|||d|}|||zd|| d| ddf}|j|S) NF)r%rrrrr^)r2r)r,r2r)r2rr,) r*r;rr~r<rr+rr-)r#rrrr[r:r@rrr>x_shaper,rmT_nrowsT_ncolsp return_shape embedded_colfft_matfft_x mat_times_xs r0rrX s=q!faKPMRTTTAq!UG 7DAq!ffG!ffG'AA!$W!2!2G::! L v{{}Ql3333>1a1Rj/22L |$$ ?(:(: ?#lG<<<DDRKKAG444d75=q#*,,,,4WHaaaK9 $|!W===EEb!LLQ!!W555eGEM$+q2222:7(AAA+?  ;  --r2r)FFFTNF)rFFFT)rFFF)FFT)FFFT)T)r5Nr^rr)FT)NFFTN)NNTFT)TTFF)FN)>rNr itertoolsrnumpyr<rrscipy._lib._utilrlapackr r r r _miscr rrrrrr!r_cythonized_array_utilsrrrrr__all__r1rDrUrrvrrrrrrrxrrrr rrr4rrr r!rhr"r#r$r*r%rr8r2r0rsi((((((((......;:::::::::''''''""""""""%%%%%%333333333333 A A A * * * * """,   <*/9=M M M M `8\***/9=P P P +*P f   LLL ###     A A A8\**?D59X X X +*X xCH"'PPPP(BG"MMMMMMMM`=-LII/T/TJI/Td8\**FK#I I I +*I XL2L2L2L2^8X|44  54 0>>>15/0I I I I ZzTzzzzz~zzzz|8\**:?+/H4H4H4+*H4V&8tTxxxxxv8;@ZZZZz89>$W)W)W)W)v48F#F#F#F#RBTBTBTBTJ8X|44..54...r2