a a'M@sRgdZddlmZmZmZmZmZmZmZm Z m Z m Z m Z m Z mZddlZddlmZddlmZmZddlmZddlmZdd lmZmZdd lmZmZdd lm Z e!e"j#Z#e!ej#Z$ddddddd Z%d dZ&d*ddZ'ddZ(d+ddZ)ddZ*ddZ+ddZ,ddZ-ddZ.d d!Z/d"d#Z0d,d$d%Z1d-d&d'Z2d(d)Z3dS).)expmcosmsinmtanmcoshmsinhmtanhmlogmfunmsignmsqrtm expm_frechet expm_condfractional_matrix_power khatri_rao) Infdotdiagprod logical_notravel transpose conjugateabsoluteamaxsignisfinitesingleN)norm)solveinv)triu)svd)schurrsf2csf)r r )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. rrz expected square array_like input)npasarraylenshape ValueErrorAr4e/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/scipy/linalg/matfuncs.py_asarray_square!s "r6cCsVt|rRt|rR|dur: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. N@@g.Arr)Zatol) r-Z isrealobj iscomplexobjfepseps_array_precisiondtypecharZallcloseimagreal)r3Btolr4r4r5 _maybe_real9s rDcCs 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(rTcCszt|}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.]]) rNrz0logm result may be inaccurate, approximate err =) r6rErFrGZ_logmrDr<rrrprint)r3disprIr*errtolerrestr4r4r5rs6  rcCsddl}|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.linalgsparserFr)r3rIr4r4r5rs,rcCs@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]]) ??N)r6r-r:rrAr2r4r4r5rs  rcCs@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]]) yrQrRN)r6r-r:rr@r2r4r4r5r)s  rcCs 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]]) )r6rDr rrr2r4r4r5rPs"rcCs$t|}t|dt|t| S)a Compute the hyperbolic matrix cosine. This routine uses expm to compute the matrix exponentials. Parameters ---------- A : (N, N) array_like Input array. Returns ------- coshm : (N, N) ndarray Hyperbolic matrix cosine of `A` Examples -------- >>> from scipy.linalg import tanhm, sinhm, coshm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> c = coshm(a) >>> c array([[ 11.24592233, 38.76236492], [ 12.92078831, 50.00828725]]) Verify tanhm(a) = sinhm(a).dot(inv(coshm(a))) >>> t = tanhm(a) >>> s = sinhm(a) >>> t - s.dot(np.linalg.inv(c)) array([[ 2.72004641e-15, 4.55191440e-15], [ 0.00000000e+00, -5.55111512e-16]]) rPr6rDrr2r4r4r5rvs"rcCs$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]]) rPrSr2r4r4r5rs"rcCs 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]]) )r6rDr rrr2r4r4r5rs"rc Cs0t|}t|\}}t||\}}|j\}}t|t|}||jj}t|d}t d|D]}t d||dD]} | |} || d| df|| d| df|| d| df} t | | d} t || d| f|| | dft || d| f|| | df} | | } || d| df|| d| df}|dkrT| |} | || d| df<t |t|}qxq`t t ||t t|}t||}ttdt|jj}|dkr|}t dt|||tt|dd}tttt|ddrt}|r$|d|kr td||S||fSd S) a{ Evaluate a matrix function specified by a callable. Returns the value of matrix-valued function ``f`` at `A`. The function ``f`` is an extension of the scalar-valued function `func` to matrices. Parameters ---------- A : (N, N) array_like Matrix at which to evaluate the function func : callable Callable object that evaluates a scalar function f. Must be vectorized (eg. using vectorize). disp : bool, optional Print warning if error in the result is estimated large instead of returning estimated error. (Default: True) Returns ------- funm : (N, N) ndarray Value of the matrix function specified by func evaluated at `A` errest : float (if disp == False) 1-norm of the estimated error, ||err||_1 / ||A||_1 Examples -------- >>> from scipy.linalg import funm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> funm(a, lambda x: x*x) array([[ 4., 15.], [ 5., 19.]]) >>> a.dot(a) array([[ 4., 15.], [ 5., 19.]]) Notes ----- This function implements the general algorithm based on Schur decomposition (Algorithm 9.1.1. in [1]_). If the input matrix is known to be diagonalizable, then relying on the eigendecomposition is likely to be faster. For example, if your matrix is Hermitian, you can do >>> from scipy.linalg import eigh >>> def funm_herm(a, func, check_finite=False): ... w, v = eigh(a, check_finite=check_finite) ... ## if you further know that your matrix is positive semidefinite, ... ## you can optionally guard against precision errors by doing ... # w = np.maximum(w, 0) ... w = func(w) ... return (v * w).dot(v.conj().T) References ---------- .. [1] Gene H. Golub, Charles F. van Loan, Matrix Computations 4th ed. )rrrr9r8r)ZaxisrJz0funm result may be inaccurate, approximate err =N)r6r$r%r0rZastyper>r?absrangeslicerminrrrDr;r<r=maxrr"rrrrrrK)r3funcrLTZnr*Zmindenpr&jsZkslvalZdenrCerrr4r4r5r s@>   <D(   $ r cCst|}dd}t||dd\}}dtdtdt|jj}||krL|St|dd}t |}d|}||t |j d} |} t d D]V} t | } d| | } dt| | | } tt| | | d }||ks| |krq|} q|r t|r||krtd || 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(r7) r-rAr>r?r;rr<rr)xrxcr4r4r5 rounded_signss   zsignm..rounded_signr)rLr7r8)Z compute_uvrPdrz1signm result may be inaccurate, approximate err =N)r6r r;r<r=r>r?r#r-ridentityr0rUr!rrrrK)r3rLreresultrNrMvalsZmax_svrdZS0Z prev_errestr&ZiS0ZPpr4r4r5r Ps0!     r cCst|}t|}|jdkr(|jdks0td|jd|jdksLtd|dddtjddf|dtjddddf}|d|jddS)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`. 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 See Also -------- kron : Kronecker product Examples -------- >>> 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]]) r,z(The both arrays should be 2-dimensional.rz6The number of columns for both arrays should be equal..N))r-r.ndimr1r0ZnewaxisZreshape)abrdr4r4r5rs0  4r)N)T)T)T)4__all__Znumpyrrrrrrrrrrrrrr-miscrbasicr r!Zspecial_matricesr"Z decomp_svdr#Z decomp_schurr$r%Z _expm_frechetr r Z_matfuncs_sqrtmr Zfinfofloatr<r;r=r6rDrrrrrrrrrr r rr4r4r4r5s4<       &- F0''&&&& h P