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� d�c}� � �> � d dl mZ d dlZd dlmZmZ ddlmZ ddlm Z m Z ddl m Z m Z mZ ddlmZ dd lmZmZ dd lmZ g d �Zd"d �Z d#d�Z d$d�Z d%d�Z d&d�Zd'd�Zd� Z d(d�Z d)d�Z d)d�Z d*d�Z d e _! d+d �Z" d,d �Z# d-d �Z$ d'd �Z%d.d!�Z&dS )/� )�warnN)� atleast_1d� atleast_2d� )�get_flinalg_funcs)�get_lapack_funcs�_compute_lwork)� LinAlgError� _datacopied� LinAlgWarning)�_asarray_validated)�_decomp� _decomp_svd)�levinson) �solve�solve_triangular� solveh_banded� solve_banded�solve_toeplitz�solve_circulant�inv�det�lstsq�pinv�pinvh�matrix_balance�matmul_toeplitzc � � |dk r#t d� | � � � � �d|k rt d� � �|�dS |d� � }||k r,t d� |� � t d�� � dS dS ) zB Check arguments during the different steps of the solution phase r z3LAPACK reported an illegal value in {}-th argument.zMatrix is singular.N�EzBIll-conditioned matrix (rcond={:.6g}): result may not be accurate.� �� stacklevel)� ValueError�formatr r r )�n�info�lamch�rcondr s �5/usr/lib/python3/dist-packages/scipy/linalg/_basic.py� _solve_checkr* s� � � �a�x�x�� � #�V�T�E�]�]�,� ,� ,� �T����/�0�0�0� �}��� ��c� � �A� �q�y�y� � +�+1�6�%�=�=� �q� *� *� *� *� *� *� �y� FT�genc �b � d} t t | |�� � � � } t t ||�� � � � } | j d } |pt | | � � }|pt | |� � }| j d | j d k rt d� � �| | j d k r | dk r | j dk st d� � �| j dk r&t j | � � � � � S | j dk r!| dk r | ddd�f } n | dd�df } d} |rd } t | t d � � � d }|d vr"t d� |� � � � �|dk rt j | � � sd}| j j dv rt# dd�� � }nt# dd�� � }t# d| f� � }|r(d}d}t j | � � rt% d� � �nd}d} ||| � � }|dk rmt# d| | f� � \ }}} || |� � � \ }}}t' | |� � |||| ||� � � \ }}t' | |� � ||||� � � \ }}�n|dk r^t# d | | f� � \ }} } t) | | |� � } | | | | |||� � � \ }}}}t' | |� � ||||� � \ }}n�|dk r^t# d!| | f� � \ } } }!t) |!| |� � } | | | | |||� � � \ }}}}t' | |� � | |||� � \ }}nHt# d"| | f� � \ }"}# |#| | |||�#� � \ }}}t' | |� � |"||� � \ }}t' | |||� � | r|� � � }|S )$a� Solves the linear equation set ``a @ x == b`` for the unknown ``x`` for square `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 =================== ======== generic matrix 'gen' symmetric 'sym' hermitian 'her' positive definite 'pos' =================== ======== If omitted, ``'gen'`` is the default structure. 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. Parameters ---------- a : (N, N) array_like Square input data b : (N, NRHS) array_like Input data for the right hand side. sym_pos : bool, default: False, deprecated Assume `a` is symmetric and positive definite. .. deprecated:: 0.19.0 This keyword is deprecated and should be replaced by using ``assume_a = 'pos'``. `sym_pos` will be removed in SciPy 1.11.0. lower : bool, default: False Ignored if ``assume_a == 'gen'`` (the default). 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, {'gen', 'sym', 'her', 'pos'} Valid entries are explained above. 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 matrix is singular. 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 generic, symmetric, Hermitian and positive definite solutions are obtained via calling ?GESV, ?SYSV, ?HESV, and ?POSV routines of LAPACK respectively. 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) F�� check_finiter r z$Input a needs to be a square matrix.z2Input b has to have same number of rows as input aNTz�The 'sym_pos' keyword is deprecated and should be replaced by using 'assume_a = "pos"'. 'sym_pos' will be removed in SciPy 1.11.0.� r! �pos)r, �sym�herr1 z'{} is not a recognized matrix structurer3 r2 �fFr' �f��dtype�d�lange�IzWscipy.linalg.solve can currently not solve a^T x = b or a^H x = b for complex matrices.�1r, )�gecon�getrf�getrs�� overwrite_a)�trans� overwrite_b)�norm)�hecon�hesv� hesv_lwork)�lwork�lowerr@ rB )�sycon�sysv� sysv_lwork)�pocon�posv)rH r@ rB )r r r �shaper r# �size�np�asfortranarray�copy�ndimr �DeprecationWarningr$ � iscomplexobjr7 �charr �NotImplementedErrorr* r �ravel)$�a�b�sym_posrH r@ rB r/ �assume_a� transposed�b_is_1D�a1�b1r% �messager' r9 rA rC �anormr< r= r> �lu�ipvtr& �xr( rD rE �hesv_lwrG rI rJ �sysv_lwrL rM s$ r) r r ( s� � �J �G� �&�q�|�D�D�D� E� E�B� �&�q�|�D�D�D� E� E�B� ��� �A� �3��R��!3�!3�K� �3��R��!3�!3�K� �x��{�b�h�q�k�!�!��?�@�@�@��B�H�Q�K����Q���2�7�a�<�<� � '� (� (� (� �w�!�|�|�� ������+�+�+� �w�!�|�|� ��6�6��D�!�!�!�G��B�B��A�A�A�t�G��B��� � �/�� �W�(�Q�7�7�7�7����3�3�3�� � "�F�8� ,� ,�.� .� .� �5���� ��!4�!4���� �x�}�� � � ���4�4�4��� ���4�4�4�� �W�r�e� ,� ,�E� � ����� �?�2� � � ?�%� '>� ?� ?� ?� ?� ���� �E�$��O�O�E� �5��� .�/J�02�B�x� 9� 9���u�e� ��r�{�;�;�;���D�$��Q�� � � ��%��D�"� #��>� >� >���4��Q�� � � ��e�B��D�1�1�1� ��t�t� �U� � � /� 1?�AC�R�� J� J� ��t�W� �w��5�1�1�� �D��R�u�',�-8�-8� :� :� :���D�!�T� �Q�� � � ��e�B��e�,�,� ��t�t� �U� � � /� 1?�AC�R�� J� J� ��t�W� �w��5�1�1�� �D��R�u�',�-8�-8� :� :� :���D�!�T� �Q�� � � ��e�B��e�,�,� ��t�t� '�'8�(*�B�x�1� 1� ��t��d�2�r��'2�'2�4� 4� 4� ��A�t� �Q�� � � ��e�B��&�&� ��t���D�%��'�'�'�� � �G�G�I�I�� �Hr+ c �� � t | |�� � }t ||�� � }t |j � � dk s |j d |j d k rt d� � �|j d |j d k r-t d� |j |j � � � � �|pt ||� � }dddd�� ||� � }t d||f� � \ } |j j s|dk r | ||||||� � � \ } } n | |j ||| | |� � � \ } } | dk r| S | dk rt d | dz z � � �t d | z � � �) a� Solve the equation `a x = b` for `x`, assuming 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.]) r. r0 r r �expected square matrixz(shapes of a {} and b {} are incompatible)�N�T�C)�trtrs)rB rH rA �unitdiagz1singular matrix: resolution failed at diagonal %dz0illegal value in %dth argument of internal trtrs) r �lenrN r# r$ r �getr �flags� f_contiguousrk r ) rY rZ rA rH � unit_diagonalrB r/ r_ r` rm re r& s r) r r s� � �J �A�L� 9� 9� 9�B� �A�L� 9� 9� 9�B� �2�8�}�}����R�X�a�[�B�H�Q�K� 7� 7��1�2�2�2� �x��{�b�h�q�k�!�!��C� �&���2�8�4�4�6� 6� 6� �3��R��!3�!3�K��!�!� $� $� (� (��� 6� 6�E� �j�2�r�(� 3� 3�F�E� �x� � A��� � ��%��B�K�u� #�m�=� =� =���4�4� �%���b�k�U��"'�i�-�A� A� A���4� �q�y�y��� �a�x�x��M� ��6�#� $� $� $� �G��e� � � � r+ c �� � t ||d�� � }t ||d�� � }|j d |j d k rt d� � �| \ }} || z dz |j d k r%t d|| z dz |j d fz � � �|pt ||� � }|j d dk r$t j || �� � } | |d z } | S || cxk rdk rfn nc|pt ||� � }t d ||f� � \ } |ddd �f } |dd d �f } |d d d�f } | || | |||||� � \ }} } }}nft d ||f� � \ }t j d |z | z dz |j d f|j �� � }|||d �d d �f< ||| ||d|�� � \ }}}}|dk r|S |dk rt d� � �t d| z � � �)an Solve the equation a x = b for x, assuming a is banded matrix. 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 * * 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 ]) T)r/ � as_inexact���r �&shapes of ab and b are not compatible.r zfinvalid values for the number of lower and upper diagonals: l+u+1 (%d) does not equal ab.shape[0] (%d)�rR )r r )�gtsvNr0 )�gbsvr6 )� overwrite_abrB �singular matrixz5illegal value in %d-th argument of internal gbsv/gtsv) r rN r# r rP �arrayr �zerosr7 r )�l_and_u�abrZ r{ rB r/ r_ r` �nlower�nupper�b2ry �dur8 �dl�du2re r& rz �a2rc �pivs r) r r m sx � �L �B�\�d� K� K� K�B� �A�L�T� J� J� J�B� �x��|�r�x��{�"�"��A�B�B�B� ��V�V� �����b�h�q�k�)�)�� �#)�F�?�Q�#6���� �"D�E� F� F� F� �3��R��!3�!3�K� �x��|�q��� �X�b�K�� 1� 1� 1�� �b��h���� � �� � � � �1� � � � � �#�:�{�2�r�':�':� � ��R��H�5�5��� ��1�2�2��Y�� �q�!�!�!�t�H�� ��3�B�3��Z�� "�d�2�q�"�b�,� �#/�� >� >���Q��A�t�t� !��R��H�5�5��� �X�q��x�&�(�1�,�b�h�q�k�:�$�*� M� M� M����6�7�7�A�A�A�:�� �4����B�T�,7� 9� 9� 9���C��D� �q�y�y��� �a�x�x��+�,�,�,� � !�$(�5�)� *� *� *r+ c � � t | |�� � }t ||�� � }|j d |j d k rt d� � �|pt ||� � }|pt || � � }|j d dk ryt d||f� � \ }|r |ddd�f j } |ddd�f } n/|ddd�f j } |ddd�f � � � } || | ||||� � \ } } } } n(t d ||f� � \ } ||||||� � � \ }} } | dk rt d | z � � �| dk rt d | z � � �| S ) a� Solve equation a x = b. a is Hermitian positive-definite banded matrix. 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]) r. rv r rw r0 )�ptsvNr )�pbsv)rH r{ rB z(%dth leading minor not positive definitez/illegal value in %dth argument of internal pbsv)r rN r# r r �real�conjr )r� rZ r{ rB rH r/ r_ r` r� r8 �er� re r& r� �cs r) r r � s� � �N �B�\� :� :� :�B� �A�L� 9� 9� 9�B� �x��|�r�x��{�"�"��A�B�B�B� �3��R��!3�!3�K� �6�;�r�2�#6�#6�L� �x��{�a��� ��R��H�5�5��� � !��1�a�a�a�4�� �A��1�c�r�c�6� �A�A��1�a�a�a�4�� �A��1�a�b�b�5� ��� � �A� ��a��B� �l� )�+� +���2�q�$�$� !��R��H�5�5����T�"�b��L�&1�3� 3� 3� ��1�d� �a�x�x��D�t�K�L�L�L� �a�x�x�� �#'�%�(� )� )� )� �Hr+ c � �� � t | �|d�� � \ }}�}}t j |ddd� |f� � � ��t d� � ��j dk r&t � t j �� � � � \ }}nCt j �� fd�t �j d � � D � � � � } |j |� }|S ) a Solve a Toeplitz system using Levinson Recursion 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. Parameters ---------- c_or_cr : array_like or tuple of (array_like, array_like) The vector ``c``, or a tuple of arrays (``c``, ``r``). Whatever the actual shape of ``c``, it will be converted to a 1-D array. 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:]]``. Whatever the actual shape of ``r``, it will be converted to a 1-D array. 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.]) T)� keep_b_shaperv r Nz)illegal value, `b` is a required argumentr c �t �� g | ]4}t �t j �d d �|f � � � � d ��5S )Nr )r rP �ascontiguousarray)�.0�irZ �valss ��r) � <listcomp>z"solve_toeplitz.<locals>.<listcomp>� sO �� � 9� 9� 9�!"� &�d�B�,@��1�1�1�a�4��,I�,I� J� J�1� M� 9� 9� 9r+ ) � _validate_args_for_toeplitz_opsrP � concatenater# rS r r� � column_stack�rangerN �reshape) �c_or_crrZ r/ �rr� r7 �b_shapere �_r� s ` @r) r r _ s� ��� �P >���L�t� 5� 5� 5��A�q�!�U�G� �>�1�R��"�W�:�q�/� *� *�D��y��D�E�E�E��v��{�{���b� 2�1� 5� 5�6�6���1�1� �O� 9� 9� 9� 9� 9�&+�A�G�A�J�&7�&7� 9� 9� 9� :� :�� �A�I�w� �� �Hr+ c � � |}|dk r ||j z }d|cxk r |j k rn n |j | S t d| �d�� � �)Nr �'z axis' entry is out of bounds)rS rN r# )�anamerY �axis�axs r) � _get_axis_lenr� � sb � � �B� �A�v�v� �a�f� ���B�������������w�r�{�� �*�%�%�%�A� B� B�Br+ �raiserv c �p � t j | � � } t d| |� � }t j |� � }t d||� � }||k r-t d� | j |j � � � � �t j � t j | |d� � d�� � } t j | � � } |�l| � d�� � |z t j t j � � j z }|j dk r|j dz |_ nt j |� � }| |k } t j | � � } | r|d k rt d � � �d | | < t j � t j ||d� � d�� � } | | z }| r#t j |t � � � | z }d ||< t j � |d�� � }t j | � � st j |� � s|j }|dk rt j |d|� � }|S )a� Solve C x = b for x, where C is a circulant matrix. `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]) r� rZ z(Shapes of c {} and b {} are incompatiblerv �r� N� )r r� z near singular circulant matrix.r r6 r )rP r r� r# r$ rN �fft�moveaxis�abs�max�finfo�float64�eps�anyr � ones_like�bool�ifftrU r� )r� rZ �singular�tol�caxis�baxis�outaxis�nc�nb�fc�abs_fc� near_zeros�is_near_singular�fb�q�maskre s r) r r � s � �j � �a���A� �s�A�u� %� %�B� � �a���A� �s�A�u� %� %�B� �R�x�x��C� �&���!�'�2�2�4� 4� 4� ����B�K��5�"�-�-�B�� 7� 7�B� �V�B�Z�Z�F� �{��j�j�b�j�!�!�B�&���"�*�)=�)=�)A�A�� �9��?�?�� �D�(�C�I�I��-��$�$�C��3� �J��v�j�)�)��� � �w� � � � ?�@�@� @� �B�z�N� ����B�K��5�"�-�-�B�� 7� 7�B� �R��A�� � �|�A�T�*�*�*�Z�7����$�� �� � �A�B� � � �A� �O�A� � � �"�/�!�"4�"4� � �F���"�}�}� �K��2�w� '� '�� �Hr+ c � � t | |�� � }t |j � � dk s |j d |j d k rt d� � �|pt || � � }t d|f� � \ }}} |||�� � \ }}} | dk r?t ||j d � � } t d| z � � } |||| d� � � \ } } | dk rt d � � �| dk rt d | z � � �| S ) a� Compute the inverse of a matrix. Parameters ---------- a : array_like Square matrix 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. 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.]]) r. r0 r r ri )r= �getri� getri_lworkr? g)\��(�?)rG � overwrite_lur| z7illegal value in %d-th argument of internal getrf|getri) r ro rN r# r r r �intr ) rY r@ r/ r_ r= r� r� rc r� r& rG �inv_as r) r r � s6 � �P �A�L� 9� 9� 9�B� �2�8�}�}����R�X�a�[�B�H�Q�K� 7� 7��1�2�2�2� �3��R��!3�!3�K� !1� 2A�24��!8� !8� �E�5�+� �E�"�+�6�6�6�M�B��T� �q�y�y� �{�B�H�Q�K�8�8�� �D�5�L�!�!���e�B��5�q�A�A�A� ��t� �a�x�x��+�,�,�,� �a�x�x�� '�*.��/� 0� 0� 0� �Lr+ c �J � t | |�� � }t |j � � dk s |j d |j d k rt d� � �|pt || � � }t d|f� � \ } |||�� � \ }}|dk rt d| z � � �|S ) a� Compute the determinant of a matrix The determinant of a square matrix is a value derived arithmetically from the coefficients of the matrix. The determinant for a 3x3 matrix, for example, is computed as follows:: a b c d e f = A g h i det(A) = a*e*i + b*f*g + c*d*h - c*e*g - b*d*i - a*f*h Parameters ---------- a : (M, M) array_like A square matrix. 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`. Notes ----- The determinant is computed via LU factorization, LAPACK routine z/dgetrf. Examples -------- >>> import numpy as np >>> from scipy import linalg >>> a = np.array([[1,2,3], [4,5,6], [7,8,9]]) >>> linalg.det(a) 0.0 >>> a = np.array([[0,2,3], [4,5,6], [7,8,9]]) >>> linalg.det(a) 3.0 r. r0 r r ri )r r? z5illegal value in %d-th argument of internal det.getrf)r ro rN r# r r )rY r@ r/ r_ �fdet�a_detr& s r) r r � s� � �^ �A�L� 9� 9� 9�B� �2�8�}�}����R�X�a�[�B�H�Q�K� 7� 7��1�2�2�2� �3��R��!3�!3�K� �h��� .� .�E�D��$�r�{�3�3�3�K�E�4� �a�x�x�� %�(,�u�-� .� .� .� �Lr+ c �r � t | |�� � }t ||�� � }t |j � � dk rt d� � �|j \ } } t |j � � dk r|j d } nd} | |j d k r.t d� | |j d � � � � �| dk s| dk r�t j | f|j dd� z t j ||� � �� � } | dk r%t j � |d� � � dz } nt j d � � } | | dt j d � � fS |}|� t j }|d vrt d |z � � �t |d |z f||f� � \ }}|j j dk rdnd}| | k ret |j � � dk r)t j | | f|j �� � }||d| �dd�f< n"t j | |j �� � }||d| �< |}|pt! || � � }|pt! ||� � }|� t j |j � � j }|dv �r1|dk r,t' || | | |� � } |||||||�� � \ }} }}}}nc|dk r]|r-t' || | | |� � \ }} ||||||dd� � \ } }}}n.t' || | | |� � \ }}} |||||||dd� � \ } }}}|dk rt) d� � �|dk rt d| |fz � � �t j g | j �� � }| | k rE| d| � } || k r3t j t j | | d� � � dz d� � � }| } | |||fS |dk r�t' || | | |� � }t j |j d dft j �� � } |||| ||dd� � \ }} } }}|dk rt d| z � � �| | k r | d| � } | } | t j g | j � � |dfS dS )a Compute least-squares solution to equation Ax = 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 ``ndim(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() r. r0 zInput array a should be 2Dr r zGShape mismatch: a and b should have the same number of rows ({} != {}).Nr6 r� )r )�gelsd�gelsy�gelssz LAPACK driver "%s" is not foundz%s_lworkr5 TF)r� r� r� )r@ rB r� z,SVD did not converge in Linear Least Squaresz.illegal value in %d-th argument of internal %sr� z1illegal value in %d-th argument of internal gelsy)r ro rN r# r$ rP r~ � common_type�linalgrC �emptyr �default_lapack_driverr r7 �kindr r� r� r r �asarray�sumr� �int32r} ) rY rZ �condr@ rB r/ � lapack_driverr_ r` �mr% �nrhsre �residues�driver� lapack_func� lapack_lwork� real_datar� rG �v�s�rank�workr& �iwork�rwork�resids�x1�jptv�js r) r r s � �^ �A�L� 9� 9� 9�B� �A�L� 9� 9� 9�B� �2�8�}�}�����5�6�6�6� �8�D�A�q� �2�8�}�}�����x��{������B�H�Q�K���� 0�06��q�"�(�1�+�0F�0F�H� H� H��A�v�v��a��� �H�a�T�B�H�Q�R�R�L�(���r�2�0F�0F� G� G� G�� ��6�6��y�~�~�b�q�~�1�1�1�4�H�H��x��~�~�H��(�A�r�x��~�~�-�-� �F� �~��,�� �0�0�0��:�V�C�D�D�D� 0�&�1;�f�1D�2F�24�b��!;� !;� �K� � %�*�/�3�6�6���U�I��1�u�u� �r�x�=�=�A� � ���1�d�)�;�+<�=�=�=�B��B�r��r�1�1�1�u�I�I���!�;�#4�5�5�5�B��B�r��r�F� �� �3��R��!3�!3�K� �3��R��!3�!3�K� �|��x� �)�*�*�.�� �#�#�#� �W� � �"�<��A�t�T�B�B�E�(3� �B��D�%�@K�@K�)M� )M� )M� %�A�q�!�T�4��� �w� � �� C� -�l�A�q�$�� M� M� ��u�#.�;�r�2�u�/4�d�E�5�$J� $J� ��1�d�D�D� '5�\�1�a�59�4�'A� 'A�#��u�e�#.�;�r�2�u�e�U�/3�U�E�$C� $C� ��1�d�D� �!�8�8� � L�M�M� M� �!�8�8� � M�!%��}� 5� 6� 7� 7� 7���B�a�g�.�.�.�� �q�5�5��2�A�2��B��q�y�y�����q����u� � �q� 0�q�9�9�9���A��&�$��!�!� �7� � � �|�Q��4��>�>���x���!��a�(���9�9�9�� )�k�"�b�$��*/��� ?� ?���1�a��t� �!�8�8� � %�(,�u� -� .� .� .� �q�5�5��2�A�2��B��A��"�(�2�q�w�'�'��t�3�3� � r+ r� c � � t | |�� � } t j | dd�� � \ }}} |j j � � � } t j |� � } |s|rt dt d�� � |s|r |�|�|p|}d}|�dn|}|�.t | j � � t j | � � j z n|}|dk s|dk rt d � � �|| |z z } t j || k � � } |dd�d| �f }||d| � z }|| d| � z � � � j }|r|| fS |S ) a 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. cond, rcond : float, optional In older versions, these values were meant to be used as ``atol`` with ``rtol=0``. If both were given ``rcond`` overwrote ``cond`` and hence the code was not correct. Thus using these are strongly discouraged and the tolerances above are recommended instead. In fact, if provided, atol, rtol takes precedence over these keywords. .. versionchanged:: 1.7.0 Deprecated in favor of ``rtol`` and ``atol`` parameters above and will be removed in future versions of SciPy. .. versionchanged:: 1.3.0 Previously the default cutoff value was just ``eps*f`` where ``f`` was ``1e3`` for single precision and ``1e6`` for double precision. 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. Examples -------- >>> 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, a @ B @ a) True >>> np.allclose(B, B @ a @ B) True r. F)� full_matricesr/ z�Use of the "cond" and "rcond" keywords are deprecated and will be removed in future versions of SciPy. Use "atol" and "rtol" keywords insteadr0 r! N� �&atol and rtol values must be positive.)r r �svdr7 rV rH rP r� r rT rN r� r� r# r� r� rk )rY �atol�rtol� return_rankr/ r� r( �ur� �vh�t�maxS�valr� �Bs r) r r � s � �V �1�<�8�8�8�A���q��E�J�J�J�H�A�q�"� �� ��� � �A� �6�!�9�9�D� � J�� J� � '�(:�q� J� J� J� J� � �� �D�L�t�|��}������ �2�2�4�D�.2�l�3�q�w�<�<�"�(�1�+�+�/� )� )��D� �r� � �t�b�y�y��A�B�B�B� ���� �C� �6�!�c�'�?�?�D� �!�!�!�U�d�U�(� �A���5�D�5��M�A� �R����Y� � � � � � �A�� ��$�w���r+ c �R � t | |�� � } t j | |d�� � \ }}|j j � � � }t j t j |� � � � } |�dn|}|�.t | j � � t j |� � j z n|}|dk s|dk rt d� � �|| |z z } t |� � | k } d|| z } |dd�| f }|| z |� � � j z } |r| t | � � fS | S )a� 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. Examples -------- >>> 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 r. F)rH r/ Nr� r� g �?)r r �eighr7 rV rH rP r� r� rN r� r� r# r� rk ro )rY r� r� rH r� r/ r� r� r� r� r� � above_cutoff� psigma_diagr� s r) r r O s$ � �z �1�<�8�8�8�A� �<���U� ;� ;� ;�D�A�q� �� ��� � �A� �6�"�&��)�)� � �D�� �2�2�4�D�.2�l�3�q�w�<�<�"�(�1�+�+�/� )� )��D� �r� � �t�b�y�y��A�B�B�B� ���� �C���F�F�S�L�L���,��'�K� �!�!�!�\�/��A� �[� �A�F�F�H�H�J�&�A�� ��#�k�"�"�"�"��r+ c � � t j t | d�� � � � } t j | j � st d� � �t d| f� � } || |||�� � \ }}}} } | dk r#t d� | � � � � �t j | t �� � } | ||d z � | ||d z �< | � t d � � � d z } | j d } t j | � � } || k rIt | |d z d � d d d � d � � D ]$\ }}| |z |k r�| | |z |g | || |z g< �%|dk r3t | d |� � � D ]\ }}||k r� | ||g | ||g< � |r|| | ffS t j | � � }t j | � � || < |t j | � � |d d �f fS )a4 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. 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. 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 ----- 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. ]]) Tr. z/The data matrix for balancing should be square.�gebal)�scale�permuter@ r z�xGEBAL exited with the internal error "illegal value in argument number {}.". See LAPACK documentation for the xGEBAL error codes.r6 r Frx Nrv )rP r r �equalrN r# r r$ r� �float�astyper� �arange� enumerate� empty_like�diag)�Ar r �separater@ r� r� �lo�hi�psr& �scalingr% �perm�indre �iperms r) r r � s1 � �F � �(���>�>�>�?�?�A� �8�Q�W� � L��J�K�K�K� �g��� -� -�E� �%����,7�9� 9� 9��A�r�2�r�4� �a�x�x�� � #�F�D�5�M�M�+� +� +� �l�2�U�+�+�+�G��"�R��T�'�{�G�B�r�!�t�G�� ���3�U�� #� #�a� '�B� ��� �A� �9�Q�<�<�D� �A�v�v� ��2�a�4�5�5� �$�$�B�$���3�3� 0� 0�F�C����u��z�z�� #�Q�s�U�A�J� /�D�!�Q�s�U�� � � �A�v�v� ��3�B�3��(�(� ,� ,�F�C���a�x�x�� !�3��(�^�D�!�S��N�N�� "��7�D�/�!�!� �M�$� � �E��)�A�,�,�E�$�K� �b�g�g� � �u�a�a�a�x�(� (�(r+ c � � � t | t � � rL| \ }}t ||�� � � � � }t ||�� � � � � }n7t | |�� � � � � }|� � � }|�t d� � �t ||�� � }|j }|j d |j d k }|r|s |j d |j d k rt d� � �t j |� � p't j |� � pt j |� � } | r t j n t j � � fd�|||fD � � \ }}}|j dk r|s|� dd� � }n,|j dk r!|� |j d d� � }|||� |fS ) a� Validate arguments and format inputs for toeplitz functions Parameters ---------- c_or_cr : array_like or tuple of (array_like, array_like) The vector ``c``, or a tuple of arrays (``c``, ``r``). Whatever the actual shape of ``c``, it will be converted to a 1-D array. 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:]]``. Whatever the actual shape of ``r``, it will be converted to a 1-D array. b : (M,) or (M, K) array_like Right-hand side in ``T x = b``. check_finite : bool 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. keep_b_shape : bool Whether to convert a (M,) dimensional b into a (M, 1) dimensional matrix. enforce_square : bool, optional If True (default), this verifies that the Toeplitz matrix is square. Returns ------- r : array 1d array corresponding to the first row of the Toeplitz matrix. c: array 1d array corresponding to the first column of the Toeplitz matrix. b: array (M,), (M, 1) or (M, K) dimensional array, post validation, corresponding to ``b``. dtype: numpy datatype ``dtype`` stores the datatype of ``r``, ``c`` and ``b``. If any of ``r``, ``c`` or ``b`` are complex, ``dtype`` is ``np.complex128``, otherwise, it is ``np.float``. b_shape: tuple Shape of ``b`` after passing it through ``_asarray_validated``. r. Nz `b` must be an array, not None.r zIncompatible dimensions.c 3 �D �K � | ]}t j |�� � � V � �dS )r6 N)rP r� )r� r� r7 s �r) � <genexpr>z2_validate_args_for_toeplitz_ops.<locals>.<genexpr>v s2 �� � � �=�=�a�r�z�!�5�)�)�)�=�=�=�=�=�=r+ r rv ) � isinstance�tupler rX � conjugater# rN rP rU � complex128�doublerS r� ) r� rZ r/ r� �enforce_squarer� r� r� � is_not_square�is_cmplxr7 s @r) r� r� 7 s� �� �V �'�5�!�!� ����1� �q�|� <� <� <� B� B� D� D�� �q�|� <� <� <� B� B� D� D��� �w�\� B� B� B� H� H� J� J�� �K�K�M�M���y��:�;�;�;��1�<�8�8�8�A��g�G��G�A�J�!�'�!�*�,�M�� 5�=� 5�Q�W�Q�Z�1�7�1�:�-E�-E��3�4�4�4���q�!�!�M�R�_�Q�%7�%7�M�2�?�1�;M�;M�H� %� 4�B�M�M�2�9�E�=�=�=�=�A�q�!�9�=�=�=�G�A�q�!��v��{�{�<�{� �I�I�b�!� � ��� ��1��� �I�I�a�g�a�j�"� %� %�� �a��E�7� "�"r+ c � � ddl m }m}m}m} t | ||dd�� � \ }} }} } |j \ } } t | � � }t |� � }||z dz }t j | |ddd� f� � }t j |� � st j |� � rO ||d|�� � � dd� � } |||d|� � � } |||z d|�� � d |�d d �f }nO ||d|�� � � dd� � } |||d|� � � } |||z d||� � � d |�d d �f }t | � � dk r|fn|| f} |j |� 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. Parameters ---------- c_or_cr : array_like or tuple of (array_like, array_like) The vector ``c``, or a tuple of arrays (``c``, ``r``). Whatever the actual shape of ``c``, it will be converted to a 1-D array. 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:]]``. Whatever the actual shape of ``r``, it will be converted to a 1-D array. 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.]) r0 )r� r� �rfft�irfftF)r� r r rv r )r� �workers)r% r� r N)r� r r% ) r� r� r r r� rN ro rP r� rU r� )r� re r/ r r� r� r r r� r� r7 �x_shaper% r� �T_nrows�T_ncols�p� embedded_col�fft_mat�fft_x� mat_times_x� return_shapes r) r r � s� � �x -�,�,�,�,�,�,�,�,�,�,�,� =���L�u�U� L� L� L��A�q�!�U�G� �7�D�A�q��!�f�f�G��!�f�f�G��'��A� �A��>�1�a��1�R��j�/�2�2�L� ��|�$�$� ?����(:�(:� ?��#�l��G�<�<�<�D�D�R��K�K����A���G�4�4�4���d�7�5�=�q�#*�,� ,� ,�,4�W�H�a�a�a�K�9� � � �$�|�!�W�=�=�=�E�E�b�!�L�L����Q�!�!�W�5�5�5���e�G�E�M��$+�q�2� 2� 2�2:�7�(�A�A�A�+�?� � "%�W� � ��!2�!2�G�:�:��!� �L� �;� � � -�-r+ )NN)FFFFTr, F)r FFFT)FFT)FFFT)T)r� Nrv r r )FT)NFFTN)NNFTNN)NNTFT)TTFF)FN)'�warningsr �numpyrP r r � _flinalg_pyr �lapackr r �_miscr r r r r � r �_solve_toeplitzr �__all__r* r r r r r r� r r r r r� r r r r� r r� r+ r) �<module>r2 s� �� � � � � � � � � � � (� (� (� (� (� (� (� (� *� *� *� *� *� *� 4� 4� 4� 4� 4� 4� 4� 4� :� :� :� :� :� :� :� :� :� :� '� '� '� '� '� '� "� "� "� "� "� "� "� "� %� %� %� %� %� %� A� A� A�� *� *� *� *�" 9>�9>��a � a � a � a �H @E�59�^ � ^ � ^ � ^ �B BG� "�i*� i*� i*� i*�X GL� #�C � C � C � C �LX � X � X � X �vC� C� C� 15�/0�C � C � C � C �NI� I� I� I�\8� 8� 8� 8�x ;@�+/�H4� H4� H4� H4�V &�� � CG� �j� j� j� j�Z <A��S� S� S� S�l :?� $�O)� O)� O)� O)�f 48�F#� F#� F#� F#�RW.� W.� W.� W.� W.� W.r+
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