K iH&ddlmZGddeZy))xrangec@eZdZdZd dZdZdZdZd dZdZ dZ y ) MatrixCalculusMethodsc fd}d}d} ||jkrn|dz }||z }ttdjj |d}|}j }xj |dzz c_ |d|zz }|j}j|} j|} j|} jd} td|dzD]E} | j|| z dzd|z| z dz| zz z} || z} | | z}| |z } | d| z|zz } Gj| | }t|D]} ||z}  |_|dzS#|_wxYw) a, Exponential of a matrix using Pade approximants. See G. H. Golub, C. F. van Loan 'Matrix Computations', third Ed., page 572 TODO: - find a good estimate for q - reduce the number of matrix multiplications to improve performance cjddd|zz zj|dzzjd|zdzd|zdzzz S)Nr)mpf factorial)pctxs ^/mnt/ssd/data/python-lab/Trading/venv/lib/python3.12/site-packages/mpmath/matrices/calculus.pyeps_padez1MatrixCalculusMethods._exp_pade..eps_pades_771:!A#& a !#$%(]]1Q3%7%:acAg%FH Hr infrr) epsintmaxmagmnormprecdpsrowseyer range lu_solve_mat)r arqextraqjextrarnadennumxckcxfs` r _exp_padezMatrixCalculusMethods._exp_pades H {SWW$ FA V  Aswwsyy5123 4xx 519 !Q$AB''"+C''"+C A A1ac] $SWWQUQY'!A#'A+):;;aCqSr Qw|#  $   c*A1X aC CHs CHs CE++ E4c |dk(rZ|j} |j|}|xjd|jzz c_|j|}||_|S|j|}|j}t t d|j |j|d}|t d|dzzz } |xjdd|zzz c_|j}|d|zz }|}|dz|z}d} ||d|j| z zz}|j|d|krn ||z }| dz } ;t|D]} ||z}  ||_|dz}|S#||_wxYw#||_wxYw)a Computes the matrix exponential of a square matrix `A`, which is defined by the power series .. math :: \exp(A) = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \ldots With method='taylor', the matrix exponential is computed using the Taylor series. With method='pade', Pade approximants are used instead. **Examples** Basic examples:: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> expm(zeros(3)) [1.0 0.0 0.0] [0.0 1.0 0.0] [0.0 0.0 1.0] >>> expm(eye(3)) [2.71828182845905 0.0 0.0] [ 0.0 2.71828182845905 0.0] [ 0.0 0.0 2.71828182845905] >>> expm([[1,1,0],[1,0,1],[0,1,0]]) [ 3.86814500615414 2.26812870852145 0.841130841230196] [ 2.26812870852145 2.44114713886289 1.42699786729125] [0.841130841230196 1.42699786729125 1.6000162976327] >>> expm([[1,1,0],[1,0,1],[0,1,0]], method='pade') [ 3.86814500615414 2.26812870852145 0.841130841230196] [ 2.26812870852145 2.44114713886289 1.42699786729125] [0.841130841230196 1.42699786729125 1.6000162976327] >>> expm([[1+j, 0], [1+j,1]]) [(1.46869393991589 + 2.28735528717884j) 0.0] [ (1.03776739863568 + 3.536943175722j) (2.71828182845905 + 0.0j)] Matrices with large entries are allowed:: >>> expm(matrix([[1,2],[2,3]])**25) [5.65024064048415e+2050488462815550 9.14228140091932e+2050488462815550] [9.14228140091932e+2050488462815550 1.47925220414035e+2050488462815551] The identity `\exp(A+B) = \exp(A) \exp(B)` does not hold for noncommuting matrices:: >>> A = hilbert(3) >>> B = A + eye(3) >>> chop(mnorm(A*B - B*A)) 0.0 >>> chop(mnorm(expm(A+B) - expm(A)*expm(B))) 0.0 >>> B = A + ones(3) >>> mnorm(A*B - B*A) 1.8 >>> mnorm(expm(A+B) - expm(A)*expm(B)) 42.0927851137247 paderr r? ) rmatrixrr-rrrrrr r) r Amethodrresr#tolTYr*s rexpmzMatrixCalculusMethods.expm5sz V 88D JJqMAaffH$mmA&J JJqMxx Aswwsyy5123 4 ST3Y   HHQqS H77(C!Q$AA1qAAQ!CGGAJ,''99Q&,QQ  AY aC CH Q1 ,CHsAE"B E." E+. E7c,d|j||jz|j||j zzz}t|j|jjt s|j|j }|S)a Gives the cosine of a square matrix `A`, defined in analogy with the matrix exponential. Examples:: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> X = eye(3) >>> cosm(X) [0.54030230586814 0.0 0.0] [ 0.0 0.54030230586814 0.0] [ 0.0 0.0 0.54030230586814] >>> X = hilbert(3) >>> cosm(X) [ 0.424403834569555 -0.316643413047167 -0.221474945949293] [-0.316643413047167 0.820646708837824 -0.127183694770039] [-0.221474945949293 -0.127183694770039 0.909236687217541] >>> X = matrix([[1+j,-2],[0,-j]]) >>> cosm(X) [(0.833730025131149 - 0.988897705762865j) (1.07485840848393 - 0.17192140544213j)] [ 0.0 (1.54308063481524 + 0.0j)] r0r:r#sumapplyimabsrer r4Bs rcosmzMatrixCalculusMethods.cosmsl0 388AceeG$sxxCEE6 ';; <177366?((-.Arc,d|j||jz|j||j zz z}t|j|jjt s|j|j }|S)a Gives the sine of a square matrix `A`, defined in analogy with the matrix exponential. Examples:: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> X = eye(3) >>> sinm(X) [0.841470984807897 0.0 0.0] [ 0.0 0.841470984807897 0.0] [ 0.0 0.0 0.841470984807897] >>> X = hilbert(3) >>> sinm(X) [0.711608512150994 0.339783913247439 0.220742837314741] [0.339783913247439 0.244113865695532 0.187231271174372] [0.220742837314741 0.187231271174372 0.155816730769635] >>> X = matrix([[1+j,-2],[0,-j]]) >>> sinm(X) [(1.29845758141598 + 0.634963914784736j) (-1.96751511930922 + 0.314700021761367j)] [ 0.0 (0.0 - 1.1752011936438j)] yr<rBs rsinmzMatrixCalculusMethods.sinmsl0sxx#%%(388AvJ+?? @177366?((-.Arcn|jdz}|j||z||j|z S)Ng333333?)r#sqrtmsqrt)r r4 _may_rotateus r _sqrtm_rotz MatrixCalculusMethods._sqrtm_rots3 EE3Jyy1k*SXXa[88rcZ|j|}|dz|k(r|S|j}|rd|j|}t|j |d|j zkr)|j |dkr|j||dz S |xjdz c_|j dz}|}|dzx}}d} |} d||j|zzd||j|zz}}|j|| z d} |j|d} | | |zkrnL|r)| dkDr$| | d zks|j||dz ||_S| dz } | |jkDr |j||_|dz}|S#t$r|r|j||dz }Y0wxYw#||_wxYw) a Computes a square root of the square matrix `A`, i.e. returns a matrix `B = A^{1/2}` such that `B^2 = A`. The square root of a matrix, if it exists, is not unique. **Examples** Square roots of some simple matrices:: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> sqrtm([[1,0], [0,1]]) [1.0 0.0] [0.0 1.0] >>> sqrtm([[0,0], [0,0]]) [0.0 0.0] [0.0 0.0] >>> sqrtm([[2,0],[0,1]]) [1.4142135623731 0.0] [ 0.0 1.0] >>> sqrtm([[1,1],[1,0]]) [ (0.920442065259926 - 0.21728689675164j) (0.568864481005783 + 0.351577584254143j)] [(0.568864481005783 + 0.351577584254143j) (0.351577584254143 - 0.568864481005783j)] >>> sqrtm([[1,0],[0,1]]) [1.0 0.0] [0.0 1.0] >>> sqrtm([[-1,0],[0,1]]) [(0.0 - 1.0j) 0.0] [ 0.0 (1.0 + 0.0j)] >>> sqrtm([[j,0],[0,j]]) [(0.707106781186547 + 0.707106781186547j) 0.0] [ 0.0 (0.707106781186547 + 0.707106781186547j)] A square root of a rotation matrix, giving the corresponding half-angle rotation matrix:: >>> t1 = 0.75 >>> t2 = t1 * 0.5 >>> A1 = matrix([[cos(t1), -sin(t1)], [sin(t1), cos(t1)]]) >>> A2 = matrix([[cos(t2), -sin(t2)], [sin(t2), cos(t2)]]) >>> sqrtm(A1) [0.930507621912314 -0.366272529086048] [0.366272529086048 0.930507621912314] >>> A2 [0.930507621912314 -0.366272529086048] [0.366272529086048 0.930507621912314] The identity `(A^2)^{1/2} = A` does not necessarily hold:: >>> A = matrix([[4,1,4],[7,8,9],[10,2,11]]) >>> sqrtm(A**2) [ 4.0 1.0 4.0] [ 7.0 8.0 9.0] [10.0 2.0 11.0] >>> sqrtm(A)**2 [ 4.0 1.0 4.0] [ 7.0 8.0 9.0] [10.0 2.0 11.0] >>> A = matrix([[-4,1,4],[7,-8,9],[10,2,11]]) >>> sqrtm(A**2) [ 7.43715112194995 -0.324127569985474 1.8481718827526] [-0.251549715716942 9.32699765900402 2.48221180985147] [ 4.11609388833616 0.775751877098258 13.017955697342] >>> chop(sqrtm(A)**2) [-4.0 1.0 4.0] [ 7.0 -8.0 9.0] [10.0 2.0 11.0] For some matrices, a square root does not exist:: >>> sqrtm([[0,1], [0,0]]) Traceback (most recent call last): ... ZeroDivisionError: matrix is numerically singular Two examples from the documentation for Matlab's ``sqrtm``:: >>> mp.dps = 15; mp.pretty = True >>> sqrtm([[7,10],[15,22]]) [1.56669890360128 1.74077655955698] [2.61116483933547 4.17786374293675] >>> >>> X = matrix(\ ... [[5,-4,1,0,0], ... [-4,6,-4,1,0], ... [1,-4,6,-4,1], ... [0,1,-4,6,-4], ... [0,0,1,-4,5]]) >>> Y = matrix(\ ... [[2,-1,-0,-0,-0], ... [-1,2,-1,0,-0], ... [0,-1,2,-1,0], ... [-0,0,-1,2,-1], ... [-0,-0,-0,-1,2]]) >>> mnorm(sqrtm(X) - Y) 4.53155328326114e-19 r2r r1r0rgMbP?) r3rdetr@r?rrArLinverseZeroDivisionErrorr NoConvergence) r r4rJrdr7r9ZIr*Yprevmag1mag2s rrHzMatrixCalculusMethods.sqrtmsF JJqM Q3!8Hxx  A366!9~377 *svvay1}~~aQ77  HHNH''C-CAqDLAA#++a. 013#++a.8H3IqAyy5%0yyE*48#1q5u 1D>>![]; CH Qsxx<+++%(CH Q%)"NN1k!m<  CHs72F!.E:0AF! !F!: FF!FF!! F*c|j|}|j} |xjdz c_|jdz}|dz}|}d} |j|}|dz }|j ||z ddkrn0||z x}}|dz} d} | dzr | || z z } n| || z z} ||z}|j |d|krn!| dz } | |jkDr |j R ||_| d|zz} | S#||_wxYw)a Computes a logarithm of the square matrix `A`, i.e. returns a matrix `B = \log(A)` such that `\exp(B) = A`. The logarithm of a matrix, if it exists, is not unique. **Examples** Logarithms of some simple matrices:: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> X = eye(3) >>> logm(X) [0.0 0.0 0.0] [0.0 0.0 0.0] [0.0 0.0 0.0] >>> logm(2*X) [0.693147180559945 0.0 0.0] [ 0.0 0.693147180559945 0.0] [ 0.0 0.0 0.693147180559945] >>> logm(expm(X)) [1.0 0.0 0.0] [0.0 1.0 0.0] [0.0 0.0 1.0] A logarithm of a complex matrix:: >>> X = matrix([[2+j, 1, 3], [1-j, 1-2*j, 1], [-4, -5, j]]) >>> B = logm(X) >>> nprint(B) [ (0.808757 + 0.107759j) (2.20752 + 0.202762j) (1.07376 - 0.773874j)] [ (0.905709 - 0.107795j) (0.0287395 - 0.824993j) (0.111619 + 0.514272j)] [(-0.930151 + 0.399512j) (-2.06266 - 0.674397j) (0.791552 + 0.519839j)] >>> chop(expm(B)) [(2.0 + 1.0j) 1.0 3.0] [(1.0 - 1.0j) (1.0 - 2.0j) 1.0] [ -4.0 -5.0 (0.0 + 1.0j)] A matrix `X` close to the identity matrix, for which `\log(\exp(X)) = \exp(\log(X)) = X` holds:: >>> X = eye(3) + hilbert(3)/4 >>> X [ 1.25 0.125 0.0833333333333333] [ 0.125 1.08333333333333 0.0625] [0.0833333333333333 0.0625 1.05] >>> logm(expm(X)) [ 1.25 0.125 0.0833333333333333] [ 0.125 1.08333333333333 0.0625] [0.0833333333333333 0.0625 1.05] >>> expm(logm(X)) [ 1.25 0.125 0.0833333333333333] [ 0.125 1.08333333333333 0.0625] [0.0833333333333333 0.0625 1.05] A logarithm of a rotation matrix, giving back the angle of the rotation:: >>> t = 3.7 >>> A = matrix([[cos(t),sin(t)],[-sin(t),cos(t)]]) >>> chop(logm(A)) [ 0.0 -2.58318530717959] [2.58318530717959 0.0] >>> (2*pi-t) 2.58318530717959 For some matrices, a logarithm does not exist:: >>> logm([[1,0], [0,0]]) Traceback (most recent call last): ... ZeroDivisionError: matrix is numerically singular Logarithm of a matrix with large entries:: >>> logm(hilbert(3) * 10**20).apply(re) [ 45.5597513593433 1.27721006042799 0.317662687717978] [ 1.27721006042799 42.5222778973542 2.24003708791604] [0.317662687717978 2.24003708791604 42.395212822267] r1rOr2r rg?r)r3rrrHrrT) r r4rr7rWrCnr8XLr*s rlogmzMatrixCalculusMethods.logm^s5d JJqMxx  HHNH''C-C1AAAIIaLQ99QqS%(50  aCKA!AAq5QJAQJAQ99Q&,Qsxx<+++ CH QT CHs B?C00 C9c|j|}|j|}|j} |xjdz c_|j|r|t |z}nZ|j|dzr#t |dz}|j ||z}n#|j ||j|z}||_|dz}|S#||_wxYw)aF Computes `A^r = \exp(A \log r)` for a matrix `A` and complex number `r`. **Examples** Powers and inverse powers of a matrix:: >>> from mpmath import * >>> mp.dps = 15; mp.pretty = True >>> A = matrix([[4,1,4],[7,8,9],[10,2,11]]) >>> powm(A, 2) [ 63.0 20.0 69.0] [174.0 89.0 199.0] [164.0 48.0 179.0] >>> chop(powm(powm(A, 4), 1/4.)) [ 4.0 1.0 4.0] [ 7.0 8.0 9.0] [10.0 2.0 11.0] >>> powm(extraprec(20)(powm)(A, -4), -1/4.) [ 4.0 1.0 4.0] [ 7.0 8.0 9.0] [10.0 2.0 11.0] >>> chop(powm(powm(A, 1+0.5j), 1/(1+0.5j))) [ 4.0 1.0 4.0] [ 7.0 8.0 9.0] [10.0 2.0 11.0] >>> powm(extraprec(5)(powm)(A, -1.5), -1/(1.5)) [ 4.0 1.0 4.0] [ 7.0 8.0 9.0] [10.0 2.0 11.0] A Fibonacci-generating matrix:: >>> powm([[1,1],[1,0]], 10) [89.0 55.0] [55.0 34.0] >>> fib(10) 55.0 >>> powm([[1,1],[1,0]], 6.5) [(16.5166626964253 - 0.0121089837381789j) (10.2078589271083 + 0.0195927472575932j)] [(10.2078589271083 + 0.0195927472575932j) (6.30880376931698 - 0.0317017309957721j)] >>> (phi**6.5 - (1-phi)**6.5)/sqrt(5) (10.2078589271083 - 0.0195927472575932j) >>> powm([[1,1],[1,0]], 6.2) [ (14.3076953002666 - 0.008222855781077j) (8.81733464837593 + 0.0133048601383712j)] [(8.81733464837593 + 0.0133048601383712j) (5.49036065189071 - 0.0215277159194482j)] >>> (phi**6.2 - (1-phi)**6.2)/sqrt(5) (8.81733464837593 - 0.0133048601383712j) r1rr )r3convertrisintrrHr:r_)r r4rrvys rpowmzMatrixCalculusMethods.powmsh JJqM KKNxx  HHNHyy|QK1Q3!HIIaLA%HHQsxx{]+CH QCHs BC CN)taylor)r) __name__ __module__ __qualname__r-r:rDrFrLrHr_rfrrrrs2,\\|::9 IVpdCrrN) libmp.backendrobjectrrkrrrns"NFNr