a jhb@sddlmZddlZddlmZmZmZmZmZm Z m Z m Z m Z m Z mZmZddlmZmZddlmZddlmZmZddlmZdd lmZmZdd lmZmZdd l m!Z!dd l"m#Z#m$Z$dd lm%Z%gdZ&e'dj(Z(e'dj(Z)dddddddZ*ddZ+d1ddZ,ddZ-d2ddZ.ddZ/ddZ0dd Z1d!d"Z2d#d$Z3d%d&Z4d'd(Z5d)d*Z6d3d+d,Z7d4d-d.Z8d/d0Z9dS)5)productN) dotdiagprod logical_notravel transpose conjugateabsoluteamaxsignisfinitetriu) LinAlgError bandwidth)norm)solveinv)svd)schurrsf2csf) expm_frechet expm_cond)sqrtm)pick_pade_structure pade_UV_calc)single)expmcosmsinmtanmcoshmsinhmtanhmlogmfunmsignmrfractional_matrix_powerrr khatri_raodf)ilr+r*FDcCs8t|}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 ValueErrorAr8>> 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_ssqZ_fractional_matrix_power)r7tscipyr8r8r9r(cs)r(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 -------- >>> 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:rIrJrKZ_logmrHr@rrr print)r7disprMr.errtolerrestr8r8r9r%s7  r%c Cst|}|jdkr6|jdkr6tt|ggS|jdkrHtd|jd|jdkrdtd|jd}t |jdkrt |S|jddd krt|St |j tj s|tj}n|j tjkr|tj}|jd}tj|j|j d }tjd ||f|j d }td d |jddDD]X}||}t|}t|sdttt|||<q&||dddddf<t|\}} | dkr|ddd| ggd| ggd| ggd| ggg9<t||||d} | dkr2|ddks|ddkrt|} t| d| td| dd<tj||ddkrTdndd} t| dddD]} | | } t| d| td| dd<t| d| | d| }|ddkr|td| ddddfdd<n$|td| ddddfdd<qlnt| D]}| | } q"|ddksN|ddkrv|ddkrft| nt| ||<n| ||<q&|S)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]]) rr0z0The input array must be at least two-dimensionalz-Last 2 dimensions of the array must be squarerN)rr)rBcSsg|] }t|qSr8)range).0xr8r8r9 ;zexpm..@zii->i)kg@)r1r2sizendimarrayexpitemrr4minZ empty_likeZ issubdtyperBZinexactastypeZfloat64Zfloat16Zfloat32emptyrranyrrrZeinsumrV _exp_sinchrZtril)r7anZeAZAmindZawZlumsZeAwZdiag_awsdr,Zexp_sd_r8r8r9rsbC        "   D   $ $ &*  ( rcCsXtt|}t|}|dk}||||<t|dd|||<|S)Nr=rS)r1diffrb)rXZ lexp_diffZl_diffZmask_zr8r8r9rhqs  rhcCs@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 -------- >>> 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]]) ??N)r:r1r>rrEr6r8r8r9r{s! 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 -------- >>> 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]]) yrrrsN)r:r1r>rrDr6r8r8r9r s! r 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 -------- >>> 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:rHrrr r6r8r8r9r!s#r!cCs$t|}t|dt|t| S)a  Compute the hyperbolic matrix cosine. This routine uses expm to compute the matrix exponentials. Parameters ---------- A : (N, N) array_like Input array. Returns ------- coshm : (N, N) ndarray Hyperbolic matrix cosine of `A` Examples -------- >>> import numpy as np >>> from scipy.linalg import tanhm, sinhm, coshm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> c = coshm(a) >>> c array([[ 11.24592233, 38.76236492], [ 12.92078831, 50.00828725]]) Verify tanhm(a) = sinhm(a).dot(inv(coshm(a))) >>> t = tanhm(a) >>> s = sinhm(a) >>> t - s.dot(np.linalg.inv(c)) array([[ 2.72004641e-15, 4.55191440e-15], [ 0.00000000e+00, -5.55111512e-16]]) rqr:rHrr6r8r8r9r"s#r"cCs$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 -------- >>> 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]]) rqrtr6r8r8r9r#s#r#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 -------- >>> 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:rHrr"r#r6r8r8r9r$@s#r$c Cs2t|}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|ddrtj}|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 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)ZaxisrNz0funm result may be inaccurate, approximate err =N)r:rrr4rrerBrCabsrVslicerrdrr rHr?r@rAmaxrrrrrr r1infrO)r7funcrPTZrjr.Zmindenpr,jrmZkslvalZdenrGerrr8r8r9r&gs@?   <D(   $ r&cCst|}dd}t||dd\}}dtdtdt|jj}||krL|St|dd}t |}d |}||t |j d} |} t d D]V} t | } d | | } d t| | | } 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+r;) r1rErBrCr?r r@r r )rXrxcr8r8r9 rounded_signs   zsignm..rounded_signr)rPr;r<F)Z compute_uvrqdrz1signm result may be inaccurate, approximate err =N)r:r&r?r@rArBrCrr1r identityr4rVrrrr rO)r7rPrresultrRrQvalsZmax_svrZS0Z prev_errestr,ZiS0ZPpr8r8r9r's0!     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`. 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]]) r0z(The both arrays should be 2-dimensional.rz6The number of columns for both arrays should be equal..N)rS)r1r2r`r5r4ZnewaxisZreshape)ribrr8r8r9r) s1  4r))N)T)T)T): itertoolsrnumpyr1rrrrrrr r r r r rZ scipy.linalgrrZ_miscrZ_basicrrZ _decomp_svdrZ _decomp_schurrrZ _expm_frechetrrZ_matfuncs_sqrtmrZ_matfuncs_expmrrr__all__Zfinfor@r?rAr:rHr(r%rrhrr r!r"r#r$r&r'r)r8r8r8r9s> 8       &. G (('''' i P