gYbddlZddlmZddlZddlmZmZmZmZm 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(ej)dj*Z*ej)dj*Z+dddddddZ,dZ-d"dZ.dZ/d#dZ0dZ1dZ2dZ3dZ4dZ5dZ6dZ7dZ8d#dZ9d#d Z:d!Z;dS)$N)product) dotdiagprod logical_notravel transpose conjugateabsoluteamaxsignisfinitetriu) LinAlgError bandwidth)norm)solveinv)svd)schurrsf2csf) expm_frechet expm_cond)sqrtm)pick_pade_structure pade_UV_calc) _funm_loops)expmcosmsinmtanmcoshmsinhmtanhmlogmfunmsignmrfractional_matrix_powerrr khatri_raodf)ilr,r+FDctj|}t|jdks|jd|jdkrt d|S)a Wraps asarray with the extra requirement that the input be a square matrix. The motivation is that the matfuncs module has real functions that have been lifted to square matrix functions. Parameters ---------- A : array_like A square matrix. Returns ------- out : ndarray An ndarray copy or view or other representation of A. rrz expected square array_like input)npasarraylenshape ValueErrorAs L/granary/edev/quantum/lib/python3.11/site-packages/scipy/linalg/_matfuncs.py_asarray_squarer;$sN$ 1 A 17||qAGAJ!'!*44;<<< Hctj|ritj|rU|0tdztdzdt |jj}tj|j d|r|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) r3 isrealobj iscomplexobjfepseps_array_precisiondtypecharallcloseimagreal)r9Btols r: _maybe_realrN<su4 |A2?1-- ;3h3s7++,>> 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)r9tscipys r:r)r)bs9R A)))) < ) B B1a H HHr<Tc&tj|}ddl}|jj|}t ||}dtz}tjdd5tt||z dtjt|d|j j dz }dddn #1swxYwY|r8t|r||kr!d |}tj|t 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)divideinvalidrrGz1logm result may be inaccurate, approximate err = r2) stacklevel)r3r4rPrQrR_logmrNrEerrstaterrrGrKrwarningswarnRuntimeWarning)r9disprUr/errtolerrestmessages r:r&r&s_n 1 A)))) &,,Q//AAqA #XF Hh 7 7 7UUd1ggai##bja17&K&K&K&PQS&TTUUUUUUUUUUUUUUU  A6V#3#3R&RRG M'>a @ @ @ @&ys(AC  CCc h tj|}|jdkrE|jdkr:tjtj|ggS|jdkrtd|jd|jdkrtdt|jdkrCttj d|j j }tj ||S|jdd d krtj|Stj|j tjs |tj}n4|j tjkr|tj}|jd}tj|j|j }tjd ||f|j }t+d |jd dDD]}||}t-|}t/|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]]) rr2z0The input array must be at least two-dimensionalz-Last 2 dimensions of the array must be squarerr[N)rrc,g|]}t|Sr\)range).0xs r: zexpm..=s888aq888r<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@)"r3r4sizendimarrayexpitemrr6minreyerG empty_like issubdtypeinexactastypefloat64float16float32emptyrranyrr MemoryErrorr RuntimeErroreinsumrl _exp_sinchrtril)r9arGneAAmindawlumsinfoeAwdiag_awsdr-exp_sd_s r:rrsF 1 Av{{qvzzx"&**+,---vzzLMMMwr{agbk!!IJJJ AG}RVAQW---..4}Qe,,,, wrss|vvayy ="* - -! HHRZ  BJ   HHRZ   A !' ) ) )B 1a)17 + + +B88173B3<8889>> sV r]]2ww gbfRWR[[1122BsG  1aaa7 "2&&1 EE=78===>> >B"" 199s{{!#B9=#B#B#BCCC #$:26$:$:$:;;;e 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)Nr@rh)r3diffru)rn 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;r3rCrrKr8s 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;r3rCrrJr8s 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;rNrr r!r8s 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;rNrr8s 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]]) rrr8s 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;rNrr#r$r8s r:r%r%Os8F A q%a%((33 4 44r<c t|}t|\}}t||\}}|j\}}t |t |}||jj}t|d}t||||\}}tt||tt|}t||}ttdt |jj}|dkr|}t#dt%|||z t't)|ddz} t+t-t/t1|dr t2j} |r| d|zkrt7d| |S|| fS) a Evaluate a matrix function specified by a callable. Returns the value of matrix-valued function ``f`` at `A`. The function ``f`` is an extension of the scalar-valued function `func` to matrices. Parameters ---------- A : (N, N) array_like Matrix at which to evaluate the function func : callable Callable object that evaluates a scalar function f. Must be vectorized (eg. using vectorize). disp : bool, optional Print warning if error in the result is estimated large instead of returning estimated error. (Default: True) Returns ------- funm : (N, N) ndarray Value of the matrix function specified by func evaluated at `A` errest : float (if disp == False) 1-norm of the estimated error, ||err||_1 / ||A||_1 Notes ----- This function implements the general algorithm based on Schur decomposition (Algorithm 9.1.1. in [1]_). If the input matrix is known to be diagonalizable, then relying on the eigendecomposition is likely to be faster. For example, if your matrix is Hermitian, you can do >>> from scipy.linalg import eigh >>> def funm_herm(a, func, check_finite=False): ... w, v = eigh(a, check_finite=check_finite) ... ## if you further know that your matrix is positive semidefinite, ... ## you can optionally guard against precision errors by doing ... # w = np.maximum(w, 0) ... w = func(w) ... return (v * w).dot(v.conj().T) References ---------- .. [1] Gene H. Golub, Charles F. van Loan, Matrix Computations 4th ed. Examples -------- >>> import numpy as np >>> from scipy.linalg import funm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> funm(a, lambda x: x*x) array([[ 4., 15.], [ 5., 19.]]) >>> a.dot(a) array([[ 4., 15.], [ 5., 19.]]) )rrr?r@rr)axisrWz0funm result may be inaccurate, approximate err =)r;rrr6rr|rGrHabsrrr r rNrDrErFrwmaxrrrrrrr3infprint) r9funcrcTZrr/mindenrMerrs r:r'r'vs~ A 88DAq 1a==DAq 7DAq TT$q'']]A A 4\\FAq!V,,IAv C1IIy1..//AAqAs  ,QW\: ;C }} aS3v:tDAJJ':'::;; < > 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>) r3rKrGrHrDr rEr r )rnrxcs r: rounded_signzsignm..rounded_signse WQZZ 8=C  Da AACQAXb\\A%+,,,r<r)rcr>r?F) compute_uvrdrz1signm result may be inaccurate, approximate err =)r;r'rDrErFrGrHrr3r identityr6rlrrrrr)r9rcrresultrerdvalsmax_svrS0 prev_errestr-iS0Pps r:r(r(s~B A---!\222NFFTc#g & &'7 8I'J KF   qU # # #D WT]]F F A Qr{171:&& & &BK 3ZZ"gg "s(^ #b"++b. !c"bkk"na(( F??kV33 E   O6V#3#3 Ev N N N 6zr<cRtj|}tj|}|jdkr |jdkstd|jd|jdkstd|jdks |jdkr@|jd|jdz}|jd}tj|||fS|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`. 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]]) r2z(The both arrays should be 2-dimensional.rz6The number of columns for both arrays should be equal.r)r6.N)rh) r3r4rsr7r6rrrynewaxisreshape)rbrrrs r:r*r*$sZ 1 A 1 A FaKKAFaKKCDDD 71: # #,-- - v{{afkk GAJ # GAJ}Qq!f---- #qqq"*aaa  1S"*aaa%:#;;A 99UQWQRR[( ) ))r<)N)T)rsDDDDDDDDDDDDDDDDDDDDDDDDDDDD0///////))))))))22222222""""""========(((((( & & &bhsmmrx}}CC   0    L+I+I+I\FFFFRdddN%%%P%%%P$3$3$3N$6$6$6N$6$6$6N$5$5$5N[[[[|MMMM`?*?*?*?*?*r<