B &]\nG@svddlmZmZmZdddddddd d d d d ddgZddlmZmZmZm Z m Z m Z m Z m Z mZmZmZmZmZddlZddlmZddlmZmZddlmZddlmZddlmZmZddl m!Z!m"Z"ddl#m$Z$e%e&j'Z'e%ej'Z(dddddddZ)ddZ*d+ddZ+ddZ,d,d!d Z-d"dZ.d#dZ/d$dZ0d%dZ1d&dZ2d'dZ3d(dZ4d-d)d Z5d.d*d Z6dS)/)divisionprint_functionabsolute_importexpmcosmsinmtanmcoshmsinhmtanhmlogmfunmsignmsqrtm expm_frechet expm_condfractional_matrix_power) Infdotdiagproduct logical_notravel transpose conjugateabsoluteamaxsignisfinitesingleN)norm)solveinv)triu)svd)schurrsf2csf)rr)r)ilfdFDcCs8t|}t|jdks,|jd|jdkr4td|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. rr z expected square array_like input)npZasarraylenshape ValueError)Ar44lib/python3.7/site-packages/scipy/linalg/matfuncs.py_asarray_square"s "r6cCsVt|rRt|rR|dkr:tdtddt|jj}tj|j d|drR|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. Ng@@g.A)rr g)Zatol) r/Z isrealobj iscomplexobjfepseps_array_precisiondtypecharZallcloseimagreal)r3Btolr4r4r5 _maybe_real:s rAcCs t|}ddl}|jj||S)a Compute the fractional power of a matrix. Proceeds according to the discussion in section (6) of [1]_. Parameters ---------- A : (N, N) array_like Matrix whose fractional power to evaluate. t : float Fractional power. Returns ------- X : (N, N) array_like The fractional power of the matrix. References ---------- .. [1] 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 -------- >>> 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)r6scipy.linalg._matfuncs_inv_ssqlinalg_matfuncs_inv_ssqZ_fractional_matrix_power)r3tscipyr4r4r5r`s(TcCszt|}ddl}|jj|}t||}dt}tt||dt|d}|rnt |r`||krjt d||S||fSdS)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 -------- >>> 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.]]) rNir z0logm result may be inaccurate, approximate err =) r6rBrCrDZ_logmrAr9r!rrprint)r3disprFr,errtolerrestr4r4r5r s6  cCsddl}|jj|S)a Compute the matrix exponential using Pade approximation. Parameters ---------- A : (N, N) array_like or sparse matrix Matrix to be exponentiated. Returns ------- expm : (N, N) ndarray Matrix exponential of `A`. References ---------- .. [1] Awad H. Al-Mohy and Nicholas J. Higham (2009) "A New Scaling and Squaring Algorithm for the Matrix Exponential." SIAM Journal on Matrix Analysis and Applications. 31 (3). pp. 970-989. ISSN 1095-7162 Examples -------- >>> from scipy.linalg import expm, sinm, cosm Matrix version of the formula exp(0) = 1: >>> expm(np.zeros((2,2))) array([[ 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]]) rN)Zscipy.sparse.linalgZsparserCr)r3rFr4r4r5rs,cCs@t|}t|r.dtd|td|Std|jSdS)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 -------- >>> 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]]) g?y?yN)r6r/r7rr>)r3r4r4r5rs  cCs@t|}t|r.dtd|td|Std|jSdS)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 -------- >>> 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]]) yy?yN)r6r/r7rr=)r3r4r4r5r*s  cCs 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 -------- >>> 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]]) )r6rAr"rr)r3r4r4r5rQs"cCs$t|}t|dt|t| S)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 -------- >>> 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]]) g?)r6rAr)r3r4r4r5r ws"cCs$t|}t|dt|t| S)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 -------- >>> 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]]) g?)r6rAr)r3r4r4r5r s"cCs 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 -------- >>> 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]]) )r6rAr"r r )r3r4r4r5r s"c Cs<t|}t|\}}t||\}}|j\}}t|t|}||jj}t|d}x,t d|D]}xt d||dD]} | |} || d| df|| d| df|| d| df} t | | d} t || d| f|| | dft || d| f|| | df} | | } || d| df|| d| df}|dkr\| |} | || d| df<t |t|}qWqdWt t ||t t|}t||}ttdt|jj}|dkr|}t dt|||tt|dd}tttt|ddrt}|r0|d|kr,td||S||fSd S) 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 Examples -------- >>> 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.]]) 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. )rrr g)rr r)Zaxisiz0funm result may be inaccurate, approximate err =N)r6r&r'r1rZastyper;r<absrangeslicerminrrrAr8r9r:maxr!r$rrrrrrG)r3funcrHTZnr,Zmindenpr(jsZkslvalZdenr@errr4r4r5r s@>   <D(   $ cCst|}dd}t||dd\}}dtdtdt|jj}||krL|St|dd}t |}d|}||t |j d} |} x`t d D]T} t | } d| | } dt| | | } tt| | | d }||ks| |krP|} qW|rt|r||kr td || S| |fSd S) 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]) cSsLt|}|jjdkr(dtt|}ndtt|}tt||k|S)Nr*g@@) r/r>r;r<r8rr9rr)xZrxcr4r4r5 rounded_signts   zsignm..rounded_signr)rHg@@)rr )Z compute_uvg?dr z1signm result may be inaccurate, approximate err =N)r6r r8r9r:r;r<r%r/rZidentityr1rLr#rr!rrG)r3rHr[resultrJrIZvalsZmax_svrZZS0Z prev_errestr(ZiS0ZPpr4r4r5rQs0!    )N)T)T)T)7Z __future__rrr__all__Znumpyrrrrrrrrrrrrrr/Zmiscr!Zbasicr"r#Zspecial_matricesr$Z decomp_svdr%Z decomp_schurr&r'Z _expm_frechetrrZ_matfuncs_sqrtmrZfinfofloatr9r8r:r6rArr rrrrr r r r rr4r4r4r5s8   <       &- F0''&&&& h