o vThYb@sddlZddlmZddlZddlmZmZmZmZm Z m Z m Z m Z m Z mZmZmZddlmZmZddlmZddlmZmZddlmZdd lmZmZdd lmZm Z dd l!m"Z"dd l#m$Z$m%Z%dd l&m'Z'gdZ(e)dj*Z*e)dj*Z+dddddddZ,ddZ-d1ddZ.ddZ/d2ddZ0ddZ1ddZ2dd Z3d!d"Z4d#d$Z5d%d&Z6d'd(Z7d)d*Z8d2d+d,Z9d2d-d.Z:d/d0Z;dS)3N)product) dotdiagprod logical_notravel transpose conjugateabsoluteamaxsignisfinitetriu) LinAlgError bandwidth)norm)solveinv)svd)schurrsf2csf) expm_frechet expm_cond)sqrtm)pick_pade_structure pade_UV_calc) _funm_loops)expmcosmsinmtanmcoshmsinhmtanhmlogmfunmsignmrfractional_matrix_powerrr khatri_raodf)ilr+r*FDcCs8t|}t|jdks|jd|jdkrtd|S)a Wraps asarray with the extra requirement that the input be a square matrix. The motivation is that the matfuncs module has real functions that have been lifted to square matrix functions. Parameters ---------- A : array_like A square matrix. Returns ------- out : ndarray An ndarray copy or view or other representation of A. rrz expected square array_like input)npasarraylenshape ValueErrorAr8W/home/air/segue/gemini/back/venv/lib/python3.10/site-packages/scipy/linalg/_matfuncs.py_asarray_square$s "r:cCsVt|r)t|r)|durtdtddt|jj}tj|j d|dr)|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)atol) r1 isrealobj iscomplexobjfepseps_array_precisiondtypecharallcloseimagreal)r7Btolr8r8r9 _maybe_real<s rKcCs 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 -------- >>> 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)r7tscipyr8r8r9r(bs)r(TcCst|}ddl}|jj|}t||}dt}tjdddt t ||dtjt |d|j dj d}Wdn1sBwY|r`t |rQ||kr^d |}tj|td d |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 Emit 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.]]) rNignore)divideinvalidrrDr8z1logm result may be inaccurate, approximate err = r0) stacklevel)r1r2rLrMrN_logmrKrBerrstaterrrDrHr warningswarnRuntimeWarning)r7disprQr.errtolerrestmessager8r8r9r%s 7 0 r%c Cst|}|jdkr|jdkrtt|ggS|jdkr$td|jd|jdkr2tdt |jdkrKt tj d|j dj }tj ||dS|jdd d krYt|St|j tjsh|tj}n |j tjkrt|tj}|jd}tj|j|j d}tjd ||f|j d}td d |jd dDD]&}||}t|}t|sttt|||<q||dd d d d f<t|\} } | dkrtd| dt|| } | dkr| dkrtd| dtd| d|d} | dkr|ddks |ddkrt|} t| d| td| d d <tj||ddkr,dndd}t| dddD]V}| | } t| d| td| d d <t| d| |d| }|ddkr{|td| dd d dfd d <q8|td| d ddd fd d <q8n t| D]}| | } q|ddks|ddkr|ddkrt | nt!| ||<q| ||<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 squarerrVN)rrcSsg|]}t|qSr8)range).0xr8r8r9 =szexpm..znscipy.linalg.expm could not allocate sufficient memory while trying to compute the Pade structure (error code z).izkscipy.linalg.expm could not allocate sufficient memory while trying to compute the exponential (error code z^scipy.linalg.expm got an internal LAPACK error during the exponential computation (error code )zii->i)kg@)"r1r2sizendimarrayexpitemrr4minreyerD empty_like issubdtypeinexactastypefloat64float16float32emptyrranyrr MemoryErrorr RuntimeErroreinsumrd _exp_sinchrtril)r7arDneAAmindawlumsinfoeAwdiag_awsdr,exp_sd_r8r8r9rsx C     "      $ $ (( ( rcCsXtt|}t|}|dk}||||<t|dd|||<|S)Nr=ra)r1diffrm)rf lexp_diffl_diffmask_zr8r8r9r}s  r}cCs<t|}t|rdtd|td|Std|jS)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:r1r@rrHr6r8r8r9r! rcCs<t|}t|rdtd|td|Std|jS)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:r1r@rrGr6r8r8r9r rr cC 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:rKrrr r6r8r8r9r!#r!cCs$t|}t|dt|t| S)a  Compute the hyperbolic matrix cosine. This routine uses expm to compute the matrix exponentials. Parameters ---------- A : (N, N) array_like Input array. Returns ------- coshm : (N, N) ndarray Hyperbolic matrix cosine of `A` Examples -------- >>> import numpy as np >>> from scipy.linalg import tanhm, sinhm, coshm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> c = coshm(a) >>> c array([[ 11.24592233, 38.76236492], [ 12.92078831, 50.00828725]]) Verify tanhm(a) = sinhm(a).dot(inv(coshm(a))) >>> t = tanhm(a) >>> s = sinhm(a) >>> t - s.dot(np.linalg.inv(c)) array([[ 2.72004641e-15, 4.55191440e-15], [ 0.00000000e+00, -5.55111512e-16]]) rr:rKrr6r8r8r9r"#r"cCs$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]]) rrr6r8r8r9r#(rr#cCr)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:rKrr"r#r6r8r8r9r$Orr$c Cst|}t|\}}t||\}}|j\}}t|t|}||jj}t|d}t ||||\}}t t ||t t |}t ||}ttdt|jj}|dkrV|}tdt|||tt|dd} tttt|ddrwtj} |r| d|krtd| |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.]]) )rrr<r=rr)axisrRz0funm result may be inaccurate, approximate err =)r:rrr4rrtrDrEabsrrrr rKrArBrCromaxrrrrrr r1infprint) r7funcr]TZrr.mindenrJerrr8r8r9r&vs*?    $  r&cCst|}dd}t||dd\}}dtdtdt|jj}||kr&|St|dd}t |}d |}||t |j d} |} t d D]+} t | } d | | } d t| | | } tt| | | d }||ksn| |krpn|} qG|rt|r}||krtd || 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]) cSsLt|}|jjdkrdtt|}ndtt|}tt||k|S)Nr+r;) r1rHrDrErAr rBr r )rfrxcr8r8r9 rounded_signs  zsignm..rounded_signr)r]r;r<F) compute_uvrdrz1signm result may be inaccurate, approximate err =)r:r&rArBrCrDrErr1r identityr4rdrrrr r)r7r]rresultr_r^valsmax_svrS0 prev_errestr,iS0Ppr8r8r9r's0!    r'cCst|}t|}|jdkr|jdkstd|jd|jdks&td|jdks0|jdkrH|jd|jd}|jd}tj|||fdS|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 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.r)r4.N)ra) r1r2rkr5r4rjrqnewaxisreshape)rbrrrr8r8r9r)$s -  4r))N)T)sB 8       & .I( ((''' ' ^ P