9=eaddlmZddlZddlmZmZmZmZmZm Z m Z m Z m Z m Z mZmZmZddlmZddlmZmZddlmZddlmZmZdd lmZdd lmZm Z dd l!m"Z"m#Z#dd l$m%Z%dd l&m'Z'm(Z(gdZ)ej*dj+Z+ej*dj+Z,dddddddZ-dZ.d"dZ/dZ0d#dZ1dZ2dZ3dZ4dZ5dZ6dZ7dZ8dZ9d#dZ:d#d Z;d!Z>> 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)r:tscipys r;r*r*as9R A)))) < ) B B1a H HHr=TcVt|}ddl}|jj|}t ||}dt z}tt||z dt|dz }|r't|r||krtd||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<rQrRrS_logmrOrFrr rprint)r:disprVr0errtolerrests r;r'r'sn A)))) &,,Q//AAqA #XF $q''!)A  a *F  N6V#3#3 Df M M M&yr=c  tj|}|jdkrE|jdkr:tjtj|ggS|jdkrtd|jd|jdkrtd|jd}t|jdkrtj |S|jddd krtj|Stj |j tj s|t}n4|j tjkr|tj}|jddd krQ|d dgdgf|d dgdgf|d dgdgf|d dgdgff\}}}}t%||z dzd |z|zzd z }tj||zd z }||z d z } tj|} tj| } |dk} tj|| || z | | <tj|j|j } || | | zzz| d dgdgf<||z| z| d dgdgf<||z| z| d dgdgf<|| | | zz z| d dgdgf<tj|r| jS| S|jd}tj|j|j } tjd||f|j }t3d|jddDD]}||}t5|}t7|s= 400``, the exact 1-norm computation cost, breaks even with 1-norm estimation and from that point on the estimation scheme given in [2] is used to decide on the approximation order. References ---------- .. [1] Awad H. Al-Mohy and Nicholas J. Higham, (2009), "A New Scaling and Squaring Algorithm for the Matrix Exponential", SIAM J. Matrix Anal. Appl. 31(3):970-989, :doi:`10.1137/09074721X` .. [2] Nicholas J. Higham and Francoise Tisseur (2000), "A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra." SIAM J. Matrix Anal. Appl. 21(4):1185-1201, :doi:`10.1137/S0895479899356080` Examples -------- >>> import numpy as np >>> from scipy.linalg import expm, sinm, cosm Matrix version of the formula exp(0) = 1: >>> expm(np.zeros((3, 2, 2))) array([[[1., 0.], [0., 1.]], [[1., 0.], [0., 1.]], [[1., 0.], [0., 1.]]]) Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta)) applied to a matrix: >>> a = np.array([[1.0, 2.0], [-1.0, 3.0]]) >>> expm(1j*a) array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) >>> cosm(a) + 1j*sinm(a) array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) rr3z0The input array must be at least two-dimensionalz-Last 2 dimensions of the array must be squarerN)rr)r3r3.g@)rHc,g|]}t|S)range).0xs r; zexpm..Ns888aq888r=@zii->i)k)$r4r5sizendimarrayexpitemrr7min empty_like issubdtyperHinexactastypefloatfloat16float32csqrtcosh ones_likesinhemptyrCrLrranyrrreinsumre _exp_sinchrtril)r:ana1a2a3a4mueApD2AmD2coshMusinchMumaskeAAmindawlumseAwdiag_awsdr.exp_sd_s r;r r sF 1 Av{{qvzzx"&**+,---vzzLMMMwr{agbk!!IJJJ  A AG}}Q wrss|vvayy ="* - -! HHUOO BJ   HHRZ  wrss|vC!qcM*C!qcM*C!qcM*C!qcM*,BBBrEA:"R' ( ( +2rz""R|,v&&Qw4))BtH4 Xqwrx 0 0 0!Vd7l%:;3aS=!BJ03aS=!BJ03aS=!Vd7l%:;3aS= <?? 7N   A !' ) ) )B 1a)17 + + +B88173B3<888911 sV r]]2ww gbfRWR[[1122BsG 1aaa7 "2&&1 66 rrFFFaR {a1"gYK28*qb |L LFFFRAe 661 1 '"++-/VGa1"g4E-F-F '3''*WRA!22;;;qsB++ E EA)C24"r(8J1K1KBIgs++AAA.'28(<==a1"gNF!uzz>D '3qrr3B3w<88;;>D '3ssABBw<88;; Eq$$A)CC qEQJJBqEQJJ&(eqjjbgclllbgcllBsGGBsGG Ir=ctjtj|}tj|}|dk}||xx||zcc<tj|dd|||<|S)NrAr_)r4diffro)rg lexp_diffl_diffmask_zs r;rrsyq ""I WQZZF 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<r4rDr rLr9s r;r!r!sZB A qDAJJc!e,--BqDzzr=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<r4rDr rKr9s r;r"r"sZB A qd2a4jj4A;;.//BqDzzr=c t|}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<rOrr!r"r9s r;r#r#s8F A q%Qa11 2 22r=ct|}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<rOr r9s r;r$r$:F A q#a488!34 5 55r=ct|}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]]) rrr9s r;r%r%,rr=c t|}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<rOrr$r%r9s r;r&r&Ss8F A q%a%((33 4 44r=c t|}t|\}}t||\}}|j\}}t |t |}||jj}t|d}td|D]}td||z dzD]} | |z} || dz | dz f|| dz | dz f|| dz | dz fz z} t| | dz } t|| dz | f|| | dz ft|| dz | f|| | dz fz } | | z} || dz | dz f|| dz | dz fz }|dkr| |z } | || dz | dz f<t|t|}tt||tt|}t||}t t"dt$|jj}|dkr|}tdt'|||z t)t+|ddz}t-t/t1t3|drt4}|r|d|zkrt7d||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.]]) )rrrrAr@r)axisrXz0funm result may be inaccurate, approximate err =)r<rrr7rrurHrIabsreslicerrqr r rOrErFrGmaxrrrr rrrrZ)r:funcr[TZrr0mindenpr.jrkslvaldenrNerrs r;r(r(zs~ A 88DAq 1Q<> Dc J J J#v r=ct|}d}t||d\}}dtzdtzdt|jj}||kr|St|d}tj |}d|z }||tj |j dzz} |} td D]`} t| } d| | zz} dt| | | zz} tt| | | z d }||ks| |krn|} a|r't!|r||krt#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|}|jjdkrdtzt |z}ndt zt |z}tt||k|zS)Nr-r?) r4rLrHrIrEr rFrr )rgrxcs r; rounded_signzsignm..rounded_signse WQZZ 8=C  Da AACQAXb\\A%+,,,r=r)r[r?r@F) compute_uvrdrz1signm result may be inaccurate, approximate err =)r<r(rErFrGrHrIrr4r identityr7rerrrrrZ)r:r[rresultr]r\valsmax_svrS0 prev_errestr.iS0Pps r;r)r)s~B A---!\222NFFDCG $ $%5fl6G%H IF   qU # # #D WT]]F F A Qr{171:&& & &BK 3ZZ"gg "s(^ #b**R- c"RjjmA&& F??kV33 E   O6V#3#3 Ev N N N 6zr=ctj|}tj|}|jdkr |jdkstd|jd|jdkstd|dddtjddf|dtjddddfz}|d|jddzS)a Khatri-rao product A column-wise Kronecker product of two matrices Parameters ---------- a : (n, k) array_like Input array b : (m, k) array_like Input array Returns ------- c: (n*m, k) ndarray Khatri-rao product of `a` and `b`. See Also -------- kron : Kronecker product Notes ----- The mathematical definition of the Khatri-Rao product is: .. math:: (A_{ij} \bigotimes B_{ij})_{ij} which is the Kronecker product of every column of A and B, e.g.:: c = np.vstack([np.kron(a[:, k], b[:, k]) for k in range(b.shape[1])]).T Examples -------- >>> import numpy as np >>> from scipy import linalg >>> a = np.array([[1, 2, 3], [4, 5, 6]]) >>> b = np.array([[3, 4, 5], [6, 7, 8], [2, 3, 9]]) >>> linalg.khatri_rao(a, b) array([[ 3, 8, 15], [ 6, 14, 24], [ 2, 6, 27], [12, 20, 30], [24, 35, 48], [ 8, 15, 54]]) r3z(The both arrays should be 2-dimensional.rz6The number of columns for both arrays should be equal..N)r_)r4r5rmr8r7newaxisreshape)rbrs r;r+r+3sb 1 A 1 A FaKKAFaKKCDDD 71: # #,-- - #qqq"*aaa  1S"*aaa%:#;;A 99UQWQRR[( ) ))r=)N)T)= itertoolsrnumpyr4rrrrrr r r r r rrrnumpy.lib.scimathrry scipy.linalgrr_miscr_basicrr _decomp_svdr _decomp_schurrr _expm_frechetrr_matfuncs_sqrtmr_matfuncs_expmrr__all__finforFrErGr<rOr*r'r rr!r"r#r$r%r&r(r)r+rdr=r;rsDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD++++++0///////))))))))22222222""""""======== & & &bhsmmrx}}CC   0    L+I+I+I\DDDDNkkk\%%%P%%%P$3$3$3N$6$6$6N$6$6$6N$5$5$5NffffRMMMM`=*=*=*=*=*r=