5[g\_ddlmZddlZddlmZmZmZmZmZm Z m Z m Z m Z m Z mZmZddlmZmZddlmZddlmZmZddlmZdd lmZmZdd lmZmZdd l m!Z!dd l"m#Z#m$Z$gd Z%ejLdjNZ'ejLdjNZ(dddddddZ)dZ*d dZ+dZ,d!dZ-dZ.dZ/dZ0dZ1dZ2dZ3dZ4dZ5d!dZ6d!dZ7dZ8y)")productN) dotdiagprod logical_notravel transpose conjugateabsoluteamaxsignisfinitetriu) LinAlgError bandwidth)norm)solveinv)svd)schurrsf2csf) expm_frechet expm_cond)sqrtm)pick_pade_structure pade_UV_calc)expmcosmsinmtanmcoshmsinhmtanhmlogmfunmsignmrfractional_matrix_powerrr khatri_raodf)ilr+r*FDctj|}t|jdk7s|jd|jdk7r t d|S)a Wraps asarray with the extra requirement that the input be a square matrix. The motivation is that the matfuncs module has real functions that have been lifted to square matrix functions. Parameters ---------- A : array_like A square matrix. Returns ------- out : ndarray An ndarray copy or view or other representation of A. rrz expected square array_like input)npasarraylenshape ValueErrorAs Q/var/www/html/bid-api/venv/lib/python3.12/site-packages/scipy/linalg/_matfuncs.py_asarray_squarer:"sI$ 1 A 177|qAGGAJ!''!*4;<< Hctj|rvtj|ra|1tdztdzdt |j j}tj|jd|r |j}|S)a( Return either B or the real part of B, depending on properties of A and B. The motivation is that B has been computed as a complicated function of A, and B may be perturbed by negligible imaginary components. If A is real and B is complex with small imaginary components, then return a real copy of B. The assumption in that case would be that the imaginary components of B are numerical artifacts. Parameters ---------- A : ndarray Input array whose type is to be checked as real vs. complex. B : ndarray Array to be returned, possibly without its imaginary part. tol : float Absolute tolerance. Returns ------- out : real or complex array Either the input array B or only the real part of the input array B. @@g.Arr)atol) r2 isrealobj iscomplexobjfepseps_array_precisiondtypecharallcloseimagreal)r8Btols r9 _maybe_realrM:sf4 ||A2??1- ;3h3s7+,>> import numpy as np >>> from scipy.linalg import fractional_matrix_power >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> b = fractional_matrix_power(a, 0.5) >>> b array([[ 0.75592895, 1.13389342], [ 0.37796447, 1.88982237]]) >>> np.dot(b, b) # Verify square root array([[ 1., 3.], [ 1., 4.]]) rN)r:scipy.linalg._matfuncs_inv_ssqlinalg_matfuncs_inv_ssq_fractional_matrix_power)r8tscipys r9r(r(`s/R A) << ) ) B B1a HHr;c&t|}ddl}|jjj |}t ||}dt z}tt||z dt|dz }|rt|r||k\r td||S||fS)a Compute matrix logarithm. The matrix logarithm is the inverse of expm: expm(logm(`A`)) == `A` Parameters ---------- A : (N, N) array_like Matrix whose logarithm to evaluate disp : bool, optional Print warning if error in the result is estimated large instead of returning estimated error. (Default: True) Returns ------- logm : (N, N) ndarray Matrix logarithm of `A` errest : float (if disp == False) 1-norm of the estimated error, ||err||_1 / ||A||_1 References ---------- .. [1] Awad H. Al-Mohy and Nicholas J. Higham (2012) "Improved Inverse Scaling and Squaring Algorithms for the Matrix Logarithm." SIAM Journal on Scientific Computing, 34 (4). C152-C169. ISSN 1095-7197 .. [2] Nicholas J. Higham (2008) "Functions of Matrices: Theory and Computation" ISBN 978-0-898716-46-7 .. [3] Nicholas J. Higham and Lijing lin (2011) "A Schur-Pade Algorithm for Fractional Powers of a Matrix." SIAM Journal on Matrix Analysis and Applications, 32 (3). pp. 1056-1078. ISSN 0895-4798 Examples -------- >>> import numpy as np >>> from scipy.linalg import logm, expm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> b = logm(a) >>> b array([[-1.02571087, 2.05142174], [ 0.68380725, 1.02571087]]) >>> expm(b) # Verify expm(logm(a)) returns a array([[ 1., 3.], [ 1., 4.]]) rNrz0logm result may be inaccurate, approximate err =) r:rOrPrQ_logmrMrDrrrprint)r8disprTr.errtolerrests r9r%r%sn A) &&,,Q/AAqA #XF $q'!)Q $q!* ,F 6V#3 Df M&yr;c tj|}|jdk(rG|jdkr8tjtj |j ggS|jdkr td|jd|jdk7r td|jd}t|jdk(rtj|S|jddd k(rtj |Stj|jtjs |jtj}n<|jtj k(r|jtj"}|jd}tj$|j|j }tj$d ||f|j }t'|jddDcgc] }t)|c}D]P}||}t+|}t-|s?tj.tj tj.|||<^||dddddf<t1|\} } | dk7r*|dd xxxd| zggd | zggd | zggd| zgggzccct3||| |d} | dk7rF|ddk(s |ddk(r tj.|} tj | d| zztj4d| ddtj.||ddk(rdnd} t)| dz ddD]}| | z} tj | d| zztj4d| ddt7| d| zz| d| zzz}|ddk(r#|tj4d| ddddfdd|tj4d| ddddfddnt)| D]}| | z} |ddk(s|ddk(r7|ddk(rtj8| ntj:| ||<L| ||<S|Scc}w)aCompute the matrix exponential of an array. Parameters ---------- A : ndarray Input with last two dimensions are square ``(..., n, n)``. Returns ------- eA : ndarray The resulting matrix exponential with the same shape of ``A`` Notes ----- Implements the algorithm given in [1], which is essentially a Pade approximation with a variable order that is decided based on the array data. For input with size ``n``, the memory usage is in the worst case in the order of ``8*(n**2)``. If the input data is not of single and double precision of real and complex dtypes, it is copied to a new array. For cases ``n >= 400``, the exact 1-norm computation cost, breaks even with 1-norm estimation and from that point on the estimation scheme given in [2] is used to decide on the approximation order. References ---------- .. [1] Awad H. Al-Mohy and Nicholas J. Higham, (2009), "A New Scaling and Squaring Algorithm for the Matrix Exponential", SIAM J. Matrix Anal. Appl. 31(3):970-989, :doi:`10.1137/09074721X` .. [2] Nicholas J. Higham and Francoise Tisseur (2000), "A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra." SIAM J. Matrix Anal. Appl. 21(4):1185-1201, :doi:`10.1137/S0895479899356080` Examples -------- >>> import numpy as np >>> from scipy.linalg import expm, sinm, cosm Matrix version of the formula exp(0) = 1: >>> expm(np.zeros((3, 2, 2))) array([[[1., 0.], [0., 1.]], [[1., 0.], [0., 1.]], [[1., 0.], [0., 1.]]]) Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta)) applied to a matrix: >>> a = np.array([[1.0, 2.0], [-1.0, 3.0]]) >>> expm(1j*a) array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) >>> cosm(a) + 1j*sinm(a) array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) rr1z0The input array must be at least two-dimensionalz-Last 2 dimensions of the array must be squarerN)rr)rF@zii->i)kg@)r2r3sizendimarrayexpitemrr5min empty_like issubdtyperFinexactastypefloat64float16float32emptyrrangeranyrrreinsum _exp_sinchrtril)r8aneAAmxindawlumseAwdiag_awsdr,exp_sd_s r9rrsF 1 Avv{qvvzxx"&&*+,--vvzLMMwwr{aggbk!IJJ  A AGG}}}Q wwrs|vvvay =="** - HHRZZ  BJJ  HHRZZ   A !'' )B 1a)177 +B1773B<8D '3qr3B3w<8;>D '3ssABw<8;,qA)C" qEQJBqEQJ&(eqjbggclbggclBsGBsGc:f Ig9sQ:ctjtj|}tj|}|dk(}||xx||zcc<tj|dd|||<|S)Nr?r])r2diffrg)r{ lexp_diffl_diffmask_zs r9rurunsiq "I WWQZF r\F vg&&/)q"vf~.If r;ct|}tj|r dtd|ztd|zzzStd|zjS)a! Compute the matrix cosine. This routine uses expm to compute the matrix exponentials. Parameters ---------- A : (N, N) array_like Input array Returns ------- cosm : (N, N) ndarray Matrix cosine of A Examples -------- >>> import numpy as np >>> from scipy.linalg import expm, sinm, cosm Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta)) applied to a matrix: >>> a = np.array([[1.0, 2.0], [-1.0, 3.0]]) >>> expm(1j*a) array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) >>> cosm(a) + 1j*sinm(a) array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) ??)r:r2rBrrJr7s r9rrxsMB A qDAJc!e,--BqDzr;ct|}tj|r dtd|ztd|zz zStd|zjS)a  Compute the matrix sine. This routine uses expm to compute the matrix exponentials. Parameters ---------- A : (N, N) array_like Input array. Returns ------- sinm : (N, N) ndarray Matrix sine of `A` Examples -------- >>> import numpy as np >>> from scipy.linalg import expm, sinm, cosm Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta)) applied to a matrix: >>> a = np.array([[1.0, 2.0], [-1.0, 3.0]]) >>> expm(1j*a) array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) >>> cosm(a) + 1j*sinm(a) array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) yrr)r:r2rBrrIr7s r9r r sMB A qd2a4j4A;.//BqDzr;c ht|}t|tt|t |S)a Compute the matrix tangent. This routine uses expm to compute the matrix exponentials. Parameters ---------- A : (N, N) array_like Input array. Returns ------- tanm : (N, N) ndarray Matrix tangent of `A` Examples -------- >>> import numpy as np >>> from scipy.linalg import tanm, sinm, cosm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> t = tanm(a) >>> t array([[ -2.00876993, -8.41880636], [ -2.80626879, -10.42757629]]) Verify tanm(a) = sinm(a).dot(inv(cosm(a))) >>> s = sinm(a) >>> c = cosm(a) >>> s.dot(np.linalg.inv(c)) array([[ -2.00876993, -8.41880636], [ -2.80626879, -10.42757629]]) )r:rMrrr r7s r9r!r!s+F A q%Qa1 22r;cbt|}t|dt|t| zzS)a  Compute the hyperbolic matrix cosine. This routine uses expm to compute the matrix exponentials. Parameters ---------- A : (N, N) array_like Input array. Returns ------- coshm : (N, N) ndarray Hyperbolic matrix cosine of `A` Examples -------- >>> import numpy as np >>> from scipy.linalg import tanhm, sinhm, coshm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> c = coshm(a) >>> c array([[ 11.24592233, 38.76236492], [ 12.92078831, 50.00828725]]) Verify tanhm(a) = sinhm(a).dot(inv(coshm(a))) >>> t = tanhm(a) >>> s = sinhm(a) >>> t - s.dot(np.linalg.inv(c)) array([[ 2.72004641e-15, 4.55191440e-15], [ 0.00000000e+00, -5.55111512e-16]]) rr:rMrr7s r9r"r"0F A q#a48!34 55r;cbt|}t|dt|t| z zS)a Compute the hyperbolic matrix sine. This routine uses expm to compute the matrix exponentials. Parameters ---------- A : (N, N) array_like Input array. Returns ------- sinhm : (N, N) ndarray Hyperbolic matrix sine of `A` Examples -------- >>> import numpy as np >>> from scipy.linalg import tanhm, sinhm, coshm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> s = sinhm(a) >>> s array([[ 10.57300653, 39.28826594], [ 13.09608865, 49.86127247]]) Verify tanhm(a) = sinhm(a).dot(inv(coshm(a))) >>> t = tanhm(a) >>> c = coshm(a) >>> t - s.dot(np.linalg.inv(c)) array([[ 2.72004641e-15, 4.55191440e-15], [ 0.00000000e+00, -5.55111512e-16]]) rrr7s r9r#r#rr;c ht|}t|tt|t |S)a Compute the hyperbolic matrix tangent. This routine uses expm to compute the matrix exponentials. Parameters ---------- A : (N, N) array_like Input array Returns ------- tanhm : (N, N) ndarray Hyperbolic matrix tangent of `A` Examples -------- >>> import numpy as np >>> from scipy.linalg import tanhm, sinhm, coshm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> t = tanhm(a) >>> t array([[ 0.3428582 , 0.51987926], [ 0.17329309, 0.86273746]]) Verify tanhm(a) = sinhm(a).dot(inv(coshm(a))) >>> s = sinhm(a) >>> c = coshm(a) >>> t - s.dot(np.linalg.inv(c)) array([[ 2.72004641e-15, 4.55191440e-15], [ 0.00000000e+00, -5.55111512e-16]]) )r:rMrr"r#r7s r9r$r$=s+F A q%a%(3 44r;c t|}t|\}}t||\}}|j\}}t |t |}|j |j j}t|d}td|D]}td||z dzD]} | |z} || dz | dz f|| dz | dz f|| dz | dz fz z} t| | dz } t|| dz | f|| | dz ft|| dz | f|| | dz fz } | | z} || dz | dz f|| dz | dz fz }|dk7r| |z } | || dz | dz f<t|t|}tt||tt|}t||}t t"dt$|j j}|dk(r|}tdt'|||z t)t+|ddz}t-t/t1t3|drt4j6}|r|d|zkDr t9d||S||fS) a Evaluate a matrix function specified by a callable. Returns the value of matrix-valued function ``f`` at `A`. The function ``f`` is an extension of the scalar-valued function `func` to matrices. Parameters ---------- A : (N, N) array_like Matrix at which to evaluate the function func : callable Callable object that evaluates a scalar function f. Must be vectorized (eg. using vectorize). disp : bool, optional Print warning if error in the result is estimated large instead of returning estimated error. (Default: True) Returns ------- funm : (N, N) ndarray Value of the matrix function specified by func evaluated at `A` errest : float (if disp == False) 1-norm of the estimated error, ||err||_1 / ||A||_1 Notes ----- This function implements the general algorithm based on Schur decomposition (Algorithm 9.1.1. in [1]_). If the input matrix is known to be diagonalizable, then relying on the eigendecomposition is likely to be faster. For example, if your matrix is Hermitian, you can do >>> from scipy.linalg import eigh >>> def funm_herm(a, func, check_finite=False): ... w, v = eigh(a, check_finite=check_finite) ... ## if you further know that your matrix is positive semidefinite, ... ## you can optionally guard against precision errors by doing ... # w = np.maximum(w, 0) ... w = func(w) ... return (v * w).dot(v.conj().T) References ---------- .. [1] Gene H. Golub, Charles F. van Loan, Matrix Computations 4th ed. Examples -------- >>> import numpy as np >>> from scipy.linalg import funm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> funm(a, lambda x: x*x) array([[ 4., 15.], [ 5., 19.]]) >>> a.dot(a) array([[ 4., 15.], [ 5., 19.]]) )rrrr?r>r)axisrVz0funm result may be inaccurate, approximate err =)r:rrr5rrmrFrGabsrrslicerrir r rMrCrDrEmaxrrrrrrr2infrX)r8funcrYTZrxr.mindenpr,jrkslvaldenrLerrs r9r&r&dsa~ A 8DAq 1a=DAq 77DAq T$q']A A 4\F1a[q!A#a%AAA!A#qs( q1ac{QqsAaCx[89A1Q3-Ca!Sk1S!A#X;/#a!Sk1S!A#X;2OOCCAAaC1H+!A#qs( +CczGAac1Q3hKS*F! C1Iy1./AAqAs ,QWW\\: ;C } aS3v:tDAJ'::; Dc J#v r;cRt|}d}t||d\}}dtzdtzdt|j j }||kr|St|d}tj|}d|z }||tj|jdzz} |} td D]N} t| } d| | zz} dt| | | zz} tt| | | z d }||ks| |k(rn|} P|rt!|r||k\r t#d || S| |fS) a' Matrix sign function. Extension of the scalar sign(x) to matrices. Parameters ---------- A : (N, N) array_like Matrix at which to evaluate the sign function disp : bool, optional Print warning if error in the result is estimated large instead of returning estimated error. (Default: True) Returns ------- signm : (N, N) ndarray Value of the sign function at `A` errest : float (if disp == False) 1-norm of the estimated error, ||err||_1 / ||A||_1 Examples -------- >>> from scipy.linalg import signm, eigvals >>> a = [[1,2,3], [1,2,1], [1,1,1]] >>> eigvals(a) array([ 4.12488542+0.j, -0.76155718+0.j, 0.63667176+0.j]) >>> eigvals(signm(a)) array([-1.+0.j, 1.+0.j, 1.+0.j]) ctj|}|jjdk(rdtzt |z}ndt zt |z}tt||kD|zS)Nr+r=) r2rJrFrGrCr rDr r )r{rxcs r9 rounded_signzsignm..rounded_signs[ WWQZ 88==C Da ACQAXb\A%+,,r;r)rYr=r>F) compute_uvrdrz1signm result may be inaccurate, approximate err =)r:r&rCrDrErFrGrr2r identityr5rrrrrrrX)r8rYrresultr[rZvalsmax_svrS0 prev_errestr,iS0Pps r9r'r's8B A-!\2NFFTc#g &'7 8I8I'J KF   qU #D WWT]F F A Qr{{1771:& & &BK 3Z"g "s(^ #b"+b. !c"bk"na( F?kV3   6V#3 Ev N 6zr;cvtj|}tj|}|jdk(r|jdk(s td|jd|jdk(s td|j dk(s|j dk(rG|jd|jdz}|jd}tj |||fS|dddtjddf|dtjddddfz}|jd |jddzS) a Khatri-rao product A column-wise Kronecker product of two matrices Parameters ---------- a : (n, k) array_like Input array b : (m, k) array_like Input array Returns ------- c: (n*m, k) ndarray Khatri-rao product of `a` and `b`. See Also -------- kron : Kronecker product Notes ----- The mathematical definition of the Khatri-Rao product is: .. math:: (A_{ij} \bigotimes B_{ij})_{ij} which is the Kronecker product of every column of A and B, e.g.:: c = np.vstack([np.kron(a[:, k], b[:, k]) for k in range(b.shape[1])]).T Examples -------- >>> import numpy as np >>> from scipy import linalg >>> a = np.array([[1, 2, 3], [4, 5, 6]]) >>> b = np.array([[3, 4, 5], [6, 7, 8], [2, 3, 9]]) >>> linalg.khatri_rao(a, b) array([[ 3, 8, 15], [ 6, 14, 24], [ 2, 6, 27], [12, 20, 30], [24, 35, 48], [ 8, 15, 54]]) r1z(The both arrays should be 2-dimensional.rz6The number of columns for both arrays should be equal.r)r5.N)r]) r2r3rer6r5rdrjnewaxisreshape)rwbrrxrs r9r)r)s b 1 A 1 A FFaKAFFaKCDD 771: #,- - vv{affk GGAJ # GGAJ}}Qq!f-- #q"**a  1S"**a%:#;;A 99UQWWQR[( ))r;)N)T)9 itertoolsrnumpyr2rrrrrr r r r r rr scipy.linalgrr_miscr_basicrr _decomp_svdr _decomp_schurrr _expm_frechetrr_matfuncs_sqrtmr_matfuncs_expmrr__all__finforDrCrEr:rMr(r%rrurr r!r"r#r$r&r'r)r;r9rsDDDD0)2"= &bhhsmrxx}C 0 L+I\DNVr%P%P$3N$6N$6N$5NfRM`C*r;