σ Λ½χXc@`s{ddlmZmZmZdddddddd d d d d ddddgZddlmZmZmZm Z m Z m Z m Z m Z mZmZmZmZmZmZmZmZddlZddlZddlmZddlmZmZddlmZddlm Z ddl!m"Z"ddl#m$Z$m%Z%ddl&m'Z'm(Z(ddl)m*Z*ej+e,ƒj-Z-ej+eƒj-Z.idd6dd6dd6dd 6dd!6dd"6Z/d#„Z0dd$„Z2d%„Z3e4d&„Z5dd'„Z6ej7d(dƒd)„ƒZ8ej7d(dƒd*d+„ƒZ9d,„Z:d-„Z;d.„Z<d/„Z=d0„Z>d1„Z?e4d2„Z@e4d3„ZAdS(4i(tdivisiontprint_functiontabsolute_importtexpmtexpm2texpm3tcosmtsinmttanmtcoshmtsinhmttanhmtlogmtfunmtsignmtsqrtmt expm_frechett expm_condtfractional_matrix_power(tInftdottdiagtexptproductt logical_nottcasttravelt transposet conjugatetabsolutetamaxtsigntisfinitetsqrttsingleNi(tnorm(tsolvetinv(ttriu(teig(tsvd(tschurtrsf2csf(RR(RtitltftdtFtDcC`sQtj|ƒ}t|jƒdks>|jd|jdkrMtdƒ‚n|S(sƒ Wraps asarray with the extra requirement that the input be a square matrix. The motivation is that the matfuncs module has real functions that have been lifted to square matrix functions. Parameters ---------- A : array_like A square matrix. Returns ------- out : ndarray An ndarray copy or view or other representation of A. iiis expected square array_like input(tnptasarraytlentshapet ValueError(tA((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pyt_asarray_square$s/cC`s…tj|ƒrtj|ƒr|dkrWitdd6tdd6t|jj}ntj |j dd|ƒr|j }qn|S(s( 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. g@@ig€„.AigtatolN( R1t isrealobjt iscomplexobjtNonetfepstepst_array_precisiontdtypetchartallclosetimagtreal(R6tBttol((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pyt _maybe_real<s  -cC`s.t|ƒ}ddl}|jjj||ƒS(sΧ 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.]]) iN(R7tscipy.linalg._matfuncs_inv_ssqtlinalgt_matfuncs_inv_ssqt_fractional_matrix_power(R6tttscipy((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pyRbs(  cC`s­t|ƒ}ddl}|jjj|ƒ}t||ƒ}dt}tt|ƒ|dƒt|dƒ}|rŸt |ƒ s‹||kr›t d|ƒn|S||fSdS(sϋ 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.]]) iNiθis0logm result may be inaccurate, approximate err =( R7RGRHRIt_logmRFR=R#RR tprint(R6tdispRLR/terrtolterrest((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pyR s6   &cC`sD|dk r%d}tj|tƒnddl}|jjj|ƒS(sœ 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]]) s2argument q=... in scipy.linalg.expm is deprecated.iN(R;twarningstwarntDeprecationWarningtscipy.sparse.linalgtsparseRHR(R6tqtmsgRL((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pyRΥs +  tnew_namecC`s«t|ƒ}|jj}|dkr<|jdƒ}d}nt|ƒ\}}t|ƒ}tt|tt|ƒƒƒ|ƒ}|dkrš|j j|ƒS|j|ƒSdS(sϋ Compute the matrix exponential using eigenvalue decomposition. Parameters ---------- A : (N, N) array_like Matrix to be exponentiated Returns ------- expm2 : (N, N) ndarray Matrix exponential of `A` R-R/R.R0N(R-R/R.R0(R-R.( R7R?R@tastypeR'R%RRRRC(R6RKtstvrtvritr((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pyR s     $ icC`sΏt|ƒ}|jd}|jj}|dkrI|jdƒ}d}ntj|d|ƒ}tj|d|ƒ}t|}x;td|ƒD]*}|j |ƒ||ƒ|(||7}qW|S( s- Compute the matrix exponential using Taylor series. Parameters ---------- A : (N, N) array_like Matrix to be exponentiated q : int Order of the Taylor series used is `q-1` Returns ------- expm3 : (N, N) ndarray Matrix exponential of `A` iR-R/R.R0R?i(R-R/R.R0( R7R4R?R@RZR1tidentityRtrangeR(R6RWtnRKteAttrmtcastfunctk((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pyR(s      cC`sPt|ƒ}tj|ƒr;dtd|ƒtd|ƒStd|ƒjSdS(s 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]]) gΰ?yπ?yπΏN(R7R1R:RRC(R6((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pyRIs  cC`sPt|ƒ}tj|ƒr;dtd|ƒtd|ƒStd|ƒjSdS(s 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]]) yΰΏyπ?yπΏN(R7R1R:RRB(R6((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pyRps  cC`s.t|ƒ}t|tt|ƒt|ƒƒƒS(sΎ 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]]) (R7RFR$RR(R6((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pyR—s" cC`s.t|ƒ}t|dt|ƒt| ƒƒS(sπ 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]]) gΰ?(R7RFR(R6((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pyR ½s" cC`s.t|ƒ}t|dt|ƒt| ƒƒS(sμ 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]]) gΰ?(R7RFR(R6((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pyR γs" cC`s.t|ƒ}t|tt|ƒt|ƒƒƒS(sν 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]]) (R7RFR$R R (R6((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pyR s" c C`sοt|ƒ}t|ƒ\}}t||ƒ\}}|j\}}t|t|ƒƒƒ}|j|jjƒ}t|dƒ}xlt d|ƒD][}xRt d||dƒD]9} | |} || d| df|| d| df|| d| df} t | | dƒ} t || d| f|| | dfƒt || d| f|| | dfƒ} | | } || d| df|| d| df}|dkrΉ| |} n| || d| df>> 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. iigtaxisiθs0funm result may be inaccurate, approximate err =N(ii(R7R)R*R4RRZR?R@tabsR`tsliceRtminRRRFR<R=R>tmaxR#R&RRRR RRN(R6tfuncROtTtZRaR/tmindentpR+tjR[tksltvaltdenREterr((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pyR /s@>  DT .  $"  2$ cC`s€t|ƒ}d„}t||ddƒ\}}idtd6dtd6t|jj}||krj|St|ddƒ}tj |ƒ}d|}||tj |j dƒ} |} xƒt dƒD]u} t | ƒ} d| | } dt| | ƒ| } tt| | ƒ| dƒ}||ks1| |kr5Pn|} qΖW|rrt|ƒ s^||krntd |ƒn| S| |fSd S( s' 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]) cS`sftj|ƒ}|jjdkr8dtt|ƒ}ndtt|ƒ}tt|ƒ|k|ƒS(NR-g@@( R1RCR?R@R<RR=RR(txtrxtc((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pyt rounded_signΊs ROig@@it compute_uvgΰ?ids1signm result may be inaccurate, approximate err =N(R7R R<R=R>R?R@R(R1RR_R4R`R%RR#R RN(R6RORxtresultRQRPtvalstmax_svRwtS0t prev_errestR+tiS0tPp((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pyR—s0!  *     (Bt __future__RRRt__all__tnumpyRRRRRRRRRRRRRR R!R"R1RRtmiscR#tbasicR$R%tspecial_matricesR&tdecompR't decomp_svdR(t decomp_schurR)R*t _expm_frechetRRt_matfuncs_sqrtmRtfinfotfloatR=R<R>R7R;RFRtTrueR Rt deprecateRRRRRR R R R R(((s4/tmp/pip-build-7oUkmx/scipy/scipy/linalg/matfuncs.pytsBj  0  & - F 4 ' ' & & & & h