(ph+cSr/SQrSSKrSSKrSSKrSSKJrJr SSKJ r SSK J r J r J r JrJrJrJrJrJrJrJrJrJrJrJrJrJrJrJrJrJrJ r J!r!J"r"J#r#J$r$J%r%J&r&J'r'J(r(J)r)J*r*J+r+J,r,J-r-J.r.J/r/J0r0J1r1J2r2J3r3J4r4 SSK5J6r6 SS K J7r7 SS K8J9r9J:r: SS K;Jr> "S S\5r?"SS\5r@"SS\5rA"SS\5rB"SS\5rC\R"\7RSS9rE\rF\ "S5"SS\G55rHSrI\I"5rJCISrKSrLSrMSrNS rOS!rPS"rQS#rRS$rS\\\\\\\\0rT\\\\\\\\0rU\4S%jrV\4S&jrWS'rXS(rYS)rZS*r[S+r\S,r]S-r^S.r_SWS/jr`\E"\`5SWS0j5raS1rb\E"\b5S25rcSWS3jrd\E"\d5SXS4j5reS5rf\E"\f5S65rgS7rh\E"\h5S85ri\E"\f5S95rjSWS:jrk\E"\k5SYS;j5rl\E"\f5S<5rmSWS=jrn\E"\n5SZS>j5roS?rp\E"\f5S@5rq\E"\n5SZSAj5rrS[SBjrs\E"\s5S\SCj5rtSWSDjru\E"\u5SWSEj5rvS]SFjrw\E"\w5S^SGj5rxS]SHjry\E"\y5S_SIj5rz\E"\f5SJ5r{\E"\f5SK5r|SWSLjr}\E"\}5S`SMj5r~SNrS[SOjr\E"\5SaSPj5rSSQ.SRjr\E"\5SSQ.SSj5rSWSTjrSbSUjrSWSVjrg)caxLite version of scipy.linalg. Notes ----- This module is a lite version of the linalg.py module in SciPy which contains high-level Python interface to the LAPACK library. The lite version only accesses the following LAPACK functions: dgesv, zgesv, dgeev, zgeev, dgesdd, zgesdd, dgelsd, zgelsd, dsyevd, zheevd, dgetrf, zgetrf, dpotrf, zpotrf, dgeqrf, zgeqrf, zungqr, dorgqr. ) matrix_powersolve tensorsolve tensorinvinvcholeskyeigvalseigvalshpinvslogdetdetsvdeigeighlstsqnormqrcond matrix_rank LinAlgError multi_dotN) NamedTupleAny) set_module)*arrayasarrayzerosempty empty_likeintcsingledoublecsinglecdoubleinexactcomplexfloatingnewaxisallInfdotaddmultiplysqrtsumisfinitefinfoerrstate geterrobjmoveaxisaminamaxprodabs atleast_2dintp asanyarrayobject_matmulswapaxesdivide count_nonzeroisnansignargsortsort reciprocal)normalize_axis_index) overrides)triueye) _umath_linalg)NDArrayc6\rSrSr%\\\S'\\\S'Srg) EigResult' eigenvalues eigenvectorsN__name__ __module__ __qualname____firstlineno__rKr__annotations____static_attributes__rQF/var/www/html/venv/lib/python3.13/site-packages/numpy/linalg/linalg.pyrMrM'#,rYrMc6\rSrSr%\\\S'\\\S'Srg) EighResult+rOrPrQNrRrQrYrZr]r]+r[rYr]c6\rSrSr%\\\S'\\\S'Srg)QRResult/QRrQNrRrQrYrZr`r`/ss|Os|OrYr`c6\rSrSr%\\\S'\\\S'Srg) SlogdetResult3rB logabsdetrQNrRrQrYrZrere3s #,s|rYrecF\rSrSr%\\\S'\\\S'\\\S'Srg) SVDResult7USVhrQNrRrQrYrZriri7ss|Os|O rYriz numpy.linalg)modulec\rSrSrSrSrg)rCa Generic Python-exception-derived object raised by linalg functions. General purpose exception class, derived from Python's ValueError class, programmatically raised in linalg functions when a Linear Algebra-related condition would prevent further correct execution of the function. Parameters ---------- None Examples -------- >>> from numpy import linalg as LA >>> LA.inv(np.zeros((2,2))) Traceback (most recent call last): File "", line 1, in File "...linalg.py", line 350, in inv return wrap(solve(a, identity(a.shape[0], dtype=a.dtype))) File "...linalg.py", line 249, in solve raise LinAlgError('Singular matrix') numpy.linalg.LinAlgError: Singular matrix rQN)rSrTrUrV__doc__rXrQrYrZrrCsrYrc[5nUSn[SSSSS9 [5SnSSS5 UWS/$!,(df  N=f)Nrcallignore)invalidoverr?under)r3r2)errobjbufsizeinvalid_call_errmasks rZ_determine_error_statesr|asO [FQiG &x! 3({1~ 3 )4 00  3 3s 7 Ac[S5e)NzSingular matrixrerrflags rZ_raise_linalgerror_singularros ' ((rYc[S5e)NzMatrix is not positive definiter~rs rZ_raise_linalgerror_nonposdefrrs 7 88rYc[S5e)NzEigenvalues did not converger~rs rZ-_raise_linalgerror_eigenvalues_nonconvergencerus 4 55rYc[S5e)NzSVD did not converger~rs rZ%_raise_linalgerror_svd_nonconvergencerxs , --rYc[S5e)Nz,SVD did not converge in Linear Least Squaresr~rs rZ_raise_linalgerror_lstsqr{s D EErYc[S5e)Nz:Incorrect argument found while performing QR factorizationr~rs rZ_raise_linalgerror_qrr~s ) **rYc,[[5nXS'U$)Nr)list_linalg_error_extobj)callbackextobjs rZget_linalg_error_extobjrs & 'F1I MrYcL[U5n[USUR5nX4$)N__array_prepare__)rgetattr__array_wrap__)anewwraps rZ _makearrayrs( !*C 1)3+=+= >D 9rYc"[U[5$N) issubclassr')ts rZ isComplexTypers a ))rYc,[RX5$r)_real_types_mapgetrdefaults rZ _realTypers   q **rYc,[RX5$r)_complex_types_maprrs rZ _complexTypers  ! !! --rYcr[nSnUHnURRn[U[5(aV[ U5(aSn[ USS9nU[La[nM[Uc$[SURR<S35eM[nM U(a[Un[U4$[U4$)NFT)rz array type z is unsupported in linalg) r"dtypetyperr&rrr# TypeErrornamerr%)arrays result_type is_complexrtype_rts rZ _commonTypersKJ   eW % %U##! 5$/BV|$ !()) !K(5  ##{""rYc /nUHanURRS;a3UR[X"RR S5S95 MPURU5 Mc [ U5S:XaUS$U$)N)=|rrrxr)r byteorderappendr newbyteorderlen)rretarrs rZ_to_native_byte_orderrsi C 99  j 0 JJws))*@*@*EF G JJsO   3x1}1v  rYcfUH+nURS:wdM[SUR-5e g)Nrz9%d-dimensional array given. Array must be two-dimensionalndimrrrs rZ _assert_2drs5  66Q;&()/0 0rYcfUH+nURS:dM[SUR-5e g)NrzB%d-dimensional array given. Array must be at least two-dimensionalrrs rZ_assert_stacked_2drs5  66A:/1289 9rYcXUH$nURSSup#X#:wdM[S5e g)Nz-Last 2 dimensions of the array must be square)shaper)rrmns rZ_assert_stacked_squarers. wwrs| 6MN NrYchUH,n[U5R5(aM#[S5e g)Nz#Array must not contain infs or NaNs)r0r)rrs rZ_assert_finiters) {  CD DrYcdURS:H=(a [URSS5S:H$)Nrr)sizer7r)rs rZ _is_empty_2drs) 88q= 6T#))BC.1Q66rYc[USS5$)z Transpose each matrix in a stack of matrices. Unlike np.transpose, this only swaps the last two axes, rather than all of them Parameters ---------- a : (...,M,N) array_like Returns ------- aT : (...,N,M) ndarray r)r>rs rZ transposers Ar2 rYcX4$rrQ)rbaxess rZ_tensorsolve_dispatcherr 6MrYc[U5up[U5nURnUbQ[[ SU55nUH%nUR U5 UR XF5 M' URU5nURXAR- *SnSnUHnX-nM URUS-:wa [S5eURX5nUR5nU"[X55n XylU $)az Solve the tensor equation ``a x = b`` for x. It is assumed that all indices of `x` are summed over in the product, together with the rightmost indices of `a`, as is done in, for example, ``tensordot(a, x, axes=x.ndim)``. Parameters ---------- a : array_like Coefficient tensor, of shape ``b.shape + Q``. `Q`, a tuple, equals the shape of that sub-tensor of `a` consisting of the appropriate number of its rightmost indices, and must be such that ``prod(Q) == prod(b.shape)`` (in which sense `a` is said to be 'square'). b : array_like Right-hand tensor, which can be of any shape. axes : tuple of ints, optional Axes in `a` to reorder to the right, before inversion. If None (default), no reordering is done. Returns ------- x : ndarray, shape Q Raises ------ LinAlgError If `a` is singular or not 'square' (in the above sense). See Also -------- numpy.tensordot, tensorinv, numpy.einsum Examples -------- >>> a = np.eye(2*3*4) >>> a.shape = (2*3, 4, 2, 3, 4) >>> b = np.random.randn(2*3, 4) >>> x = np.linalg.tensorsolve(a, b) >>> x.shape (2, 3, 4) >>> np.allclose(np.tensordot(a, x, axes=3), b) True NrrxrzfInput arrays must satisfy the requirement prod(a.shape[b.ndim:]) == prod(a.shape[:b.ndim]))rrrrrangeremoveinsertrrrrreshaperavelr) rrrranallaxeskoldshaper7ress rZrrs`mGA A B uQ|$A NN1  NN2 ! KK wwFF|}%H D   vv >  $A  A uQ{ CI JrYcX4$rrQ)rrs rZ_solve_dispatcherrErrYcx[U5up[U5 [U5 [U5up[X5upEURURS- :Xa[ R nO[ Rn[U5(aSOSn[[5nU"XXxS9n U"U RUSS95$)a Solve a linear matrix equation, or system of linear scalar equations. Computes the "exact" solution, `x`, of the well-determined, i.e., full rank, linear matrix equation `ax = b`. Parameters ---------- a : (..., M, M) array_like Coefficient matrix. b : {(..., M,), (..., M, K)}, array_like Ordinate or "dependent variable" values. Returns ------- x : {(..., M,), (..., M, K)} ndarray Solution to the system a x = b. Returned shape is identical to `b`. Raises ------ LinAlgError If `a` is singular or not square. See Also -------- scipy.linalg.solve : Similar function in SciPy. Notes ----- .. versionadded:: 1.8.0 Broadcasting rules apply, see the `numpy.linalg` documentation for details. The solutions are computed using LAPACK routine ``_gesv``. `a` must be square and of full-rank, i.e., all rows (or, equivalently, columns) must be linearly independent; if either is not true, use `lstsq` for the least-squares best "solution" of the system/equation. References ---------- .. [1] G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pg. 22. Examples -------- Solve the system of equations ``x0 + 2 * x1 = 1`` and ``3 * x0 + 5 * x1 = 2``: >>> a = np.array([[1, 2], [3, 5]]) >>> b = np.array([1, 2]) >>> x = np.linalg.solve(a, b) >>> x array([-1., 1.]) Check that the solution is correct: >>> np.allclose(np.dot(a, x), b) True rxDD->Ddd->d signaturerFcopy) rrrrrrJsolve1rrrrastype) rr_rrresult_tgufuncrrrs rZrrIsB a=DAq1mGAa#KA vv!%%$$(++I $%@ AFqy8A . //rYcU4$rrQ)rinds rZ_tensorinv_dispatcherr 4KrYc[U5nURnSnUS:aX!SUSU-nX!SHnX5-nM O [S5eURUS5n[ U5nUR"U6$)a Compute the 'inverse' of an N-dimensional array. The result is an inverse for `a` relative to the tensordot operation ``tensordot(a, b, ind)``, i. e., up to floating-point accuracy, ``tensordot(tensorinv(a), a, ind)`` is the "identity" tensor for the tensordot operation. Parameters ---------- a : array_like Tensor to 'invert'. Its shape must be 'square', i. e., ``prod(a.shape[:ind]) == prod(a.shape[ind:])``. ind : int, optional Number of first indices that are involved in the inverse sum. Must be a positive integer, default is 2. Returns ------- b : ndarray `a`'s tensordot inverse, shape ``a.shape[ind:] + a.shape[:ind]``. Raises ------ LinAlgError If `a` is singular or not 'square' (in the above sense). See Also -------- numpy.tensordot, tensorsolve Examples -------- >>> a = np.eye(4*6) >>> a.shape = (4, 6, 8, 3) >>> ainv = np.linalg.tensorinv(a, ind=2) >>> ainv.shape (8, 3, 4, 6) >>> b = np.random.randn(4, 6) >>> np.allclose(np.tensordot(ainv, b), np.linalg.tensorsolve(a, b)) True >>> a = np.eye(4*6) >>> a.shape = (24, 8, 3) >>> ainv = np.linalg.tensorinv(a, ind=1) >>> ainv.shape (8, 3, 24) >>> b = np.random.randn(24) >>> np.allclose(np.tensordot(ainv, b, 1), np.linalg.tensorsolve(a, b)) True rxrNzInvalid ind argument.r)rr ValueErrorrr)rrrr7invshaperias rZrrsl  AwwH D QwD>HTcN2$A ID 011 $A QB ::x  rYcU4$rrQrs rZ_unary_dispatcherrrrYc[U5up[U5 [U5 [U5up#[ U5(aSOSn[ [ 5n[R"XUS9nU"URUSS95$)a Compute the (multiplicative) inverse of a matrix. Given a square matrix `a`, return the matrix `ainv` satisfying ``dot(a, ainv) = dot(ainv, a) = eye(a.shape[0])``. Parameters ---------- a : (..., M, M) array_like Matrix to be inverted. Returns ------- ainv : (..., M, M) ndarray or matrix (Multiplicative) inverse of the matrix `a`. Raises ------ LinAlgError If `a` is not square or inversion fails. See Also -------- scipy.linalg.inv : Similar function in SciPy. Notes ----- .. versionadded:: 1.8.0 Broadcasting rules apply, see the `numpy.linalg` documentation for details. Examples -------- >>> from numpy.linalg import inv >>> a = np.array([[1., 2.], [3., 4.]]) >>> ainv = inv(a) >>> np.allclose(np.dot(a, ainv), np.eye(2)) True >>> np.allclose(np.dot(ainv, a), np.eye(2)) True If a is a matrix object, then the return value is a matrix as well: >>> ainv = inv(np.matrix(a)) >>> ainv matrix([[-2. , 1. ], [ 1.5, -0.5]]) Inverses of several matrices can be computed at once: >>> a = np.array([[[1., 2.], [3., 4.]], [[1, 3], [3, 5]]]) >>> inv(a) array([[[-2. , 1. ], [ 1.5 , -0.5 ]], [[-1.25, 0.75], [ 0.75, -0.25]]]) D->Dd->drFr) rrrrrrrrJrr)rrrrrrainvs rZrrsq|mGAq1a.KA'**I $%@ AF   QF CD  H5 1 22rYcU4$rrQ)rrs rZ_matrix_power_dispatcherr5rrYc[U5n[U5 [U5 [R"U5nUR [:wa[nO"URS:Xa[nO [S5eUS:Xa1[U5n[URSUR S9US'U$US:a[U5n[!U5nUS :XaU$US:XaU"X5$US :XaU"U"X5U5$S=pEUS:a7UcUOU"XD5n[#US5upU(a UcUOU"XT5nUS:aM7U$![ an[ S5UeSnAff=f) a Raise a square matrix to the (integer) power `n`. For positive integers `n`, the power is computed by repeated matrix squarings and matrix multiplications. If ``n == 0``, the identity matrix of the same shape as M is returned. If ``n < 0``, the inverse is computed and then raised to the ``abs(n)``. .. note:: Stacks of object matrices are not currently supported. Parameters ---------- a : (..., M, M) array_like Matrix to be "powered". n : int The exponent can be any integer or long integer, positive, negative, or zero. Returns ------- a**n : (..., M, M) ndarray or matrix object The return value is the same shape and type as `M`; if the exponent is positive or zero then the type of the elements is the same as those of `M`. If the exponent is negative the elements are floating-point. Raises ------ LinAlgError For matrices that are not square or that (for negative powers) cannot be inverted numerically. Examples -------- >>> from numpy.linalg import matrix_power >>> i = np.array([[0, 1], [-1, 0]]) # matrix equiv. of the imaginary unit >>> matrix_power(i, 3) # should = -i array([[ 0, -1], [ 1, 0]]) >>> matrix_power(i, 0) array([[1, 0], [0, 1]]) >>> matrix_power(i, -3) # should = 1/(-i) = i, but w/ f.p. elements array([[ 0., 1.], [-1., 0.]]) Somewhat more sophisticated example >>> q = np.zeros((4, 4)) >>> q[0:2, 0:2] = -i >>> q[2:4, 2:4] = i >>> q # one of the three quaternion units not equal to 1 array([[ 0., -1., 0., 0.], [ 1., 0., 0., 0.], [ 0., 0., 0., 1.], [ 0., 0., -1., 0.]]) >>> matrix_power(q, 2) # = -np.eye(4) array([[-1., 0., 0., 0.], [ 0., -1., 0., 0.], [ 0., 0., -1., 0.], [ 0., 0., 0., -1.]]) zexponent must be an integerNrz6matrix_power not supported for stacks of object arraysrrr.rx)r;rroperatorindexrrobjectr=rr+NotImplementedErrorr rIrrr8divmod)rrefmatmulzresultbits rZrr9sZB 1 Aq1> NN1   ww& 1! DF F Av qMQWWR[0# Q F F Av aq} awq}a(( A a%A 1  .Qgf.@F a% MU >56A=>sD66 E E  Ec[[5n[Rn[ U5up[ U5 [ U5 [U5upE[U5(aSOSnU"XUS9nU"URUSS95$)a Cholesky decomposition. Return the Cholesky decomposition, `L * L.H`, of the square matrix `a`, where `L` is lower-triangular and .H is the conjugate transpose operator (which is the ordinary transpose if `a` is real-valued). `a` must be Hermitian (symmetric if real-valued) and positive-definite. No checking is performed to verify whether `a` is Hermitian or not. In addition, only the lower-triangular and diagonal elements of `a` are used. Only `L` is actually returned. Parameters ---------- a : (..., M, M) array_like Hermitian (symmetric if all elements are real), positive-definite input matrix. Returns ------- L : (..., M, M) array_like Lower-triangular Cholesky factor of `a`. Returns a matrix object if `a` is a matrix object. Raises ------ LinAlgError If the decomposition fails, for example, if `a` is not positive-definite. See Also -------- scipy.linalg.cholesky : Similar function in SciPy. scipy.linalg.cholesky_banded : Cholesky decompose a banded Hermitian positive-definite matrix. scipy.linalg.cho_factor : Cholesky decomposition of a matrix, to use in `scipy.linalg.cho_solve`. Notes ----- .. versionadded:: 1.8.0 Broadcasting rules apply, see the `numpy.linalg` documentation for details. The Cholesky decomposition is often used as a fast way of solving .. math:: A \mathbf{x} = \mathbf{b} (when `A` is both Hermitian/symmetric and positive-definite). First, we solve for :math:`\mathbf{y}` in .. math:: L \mathbf{y} = \mathbf{b}, and then for :math:`\mathbf{x}` in .. math:: L.H \mathbf{x} = \mathbf{y}. Examples -------- >>> A = np.array([[1,-2j],[2j,5]]) >>> A array([[ 1.+0.j, -0.-2.j], [ 0.+2.j, 5.+0.j]]) >>> L = np.linalg.cholesky(A) >>> L array([[1.+0.j, 0.+0.j], [0.+2.j, 1.+0.j]]) >>> np.dot(L, L.T.conj()) # verify that L * L.H = A array([[1.+0.j, 0.-2.j], [0.+2.j, 5.+0.j]]) >>> A = [[1,-2j],[2j,5]] # what happens if A is only array_like? >>> np.linalg.cholesky(A) # an ndarray object is returned array([[1.+0.j, 0.+0.j], [0.+2.j, 1.+0.j]]) >>> # But a matrix object is returned if A is a matrix object >>> np.linalg.cholesky(np.matrix(A)) matrix([[ 1.+0.j, 0.+0.j], [ 0.+2.j, 1.+0.j]]) rrrFr) rrrJ cholesky_lorrrrrr)rrrrrrrrs rZrrsvh%%A BF  & &FmGAq1a.KA'**Iqf5A . //rYcU4$rrQ)rmodes rZ_qr_dispatcherrrrYcUS;ahUS;a.SRS5n[R"U[SS9 SnO4US;aS n[R"U[SS9 S nO[ S US 35e[ U5up[ U5 URS SupE[U5upgURUSS9n[U5n[XE5nXE::a[Rn O[Rn [U5(aSOSn [!["5n U "X U S9n US:Xa-[%USSU2SS245n U RUSS9n U"U 5$US:Xa5['U5nURUSS9nU RUSS9n U"U5U 4$US :XaURUSS9nU"U5$US:XaXE:aUn[R(n OUn[R*n [U5(aSOSn [!["5n U "X XS9n[%USSU2SS245n URUSS9nU RUSS9n [-U"U5U"U 55$)a Compute the qr factorization of a matrix. Factor the matrix `a` as *qr*, where `q` is orthonormal and `r` is upper-triangular. Parameters ---------- a : array_like, shape (..., M, N) An array-like object with the dimensionality of at least 2. mode : {'reduced', 'complete', 'r', 'raw'}, optional If K = min(M, N), then * 'reduced' : returns Q, R with dimensions (..., M, K), (..., K, N) (default) * 'complete' : returns Q, R with dimensions (..., M, M), (..., M, N) * 'r' : returns R only with dimensions (..., K, N) * 'raw' : returns h, tau with dimensions (..., N, M), (..., K,) The options 'reduced', 'complete, and 'raw' are new in numpy 1.8, see the notes for more information. The default is 'reduced', and to maintain backward compatibility with earlier versions of numpy both it and the old default 'full' can be omitted. Note that array h returned in 'raw' mode is transposed for calling Fortran. The 'economic' mode is deprecated. The modes 'full' and 'economic' may be passed using only the first letter for backwards compatibility, but all others must be spelled out. See the Notes for more explanation. Returns ------- When mode is 'reduced' or 'complete', the result will be a namedtuple with the attributes `Q` and `R`. Q : ndarray of float or complex, optional A matrix with orthonormal columns. When mode = 'complete' the result is an orthogonal/unitary matrix depending on whether or not a is real/complex. The determinant may be either +/- 1 in that case. In case the number of dimensions in the input array is greater than 2 then a stack of the matrices with above properties is returned. R : ndarray of float or complex, optional The upper-triangular matrix or a stack of upper-triangular matrices if the number of dimensions in the input array is greater than 2. (h, tau) : ndarrays of np.double or np.cdouble, optional The array h contains the Householder reflectors that generate q along with r. The tau array contains scaling factors for the reflectors. In the deprecated 'economic' mode only h is returned. Raises ------ LinAlgError If factoring fails. See Also -------- scipy.linalg.qr : Similar function in SciPy. scipy.linalg.rq : Compute RQ decomposition of a matrix. Notes ----- This is an interface to the LAPACK routines ``dgeqrf``, ``zgeqrf``, ``dorgqr``, and ``zungqr``. For more information on the qr factorization, see for example: https://en.wikipedia.org/wiki/QR_factorization Subclasses of `ndarray` are preserved except for the 'raw' mode. So if `a` is of type `matrix`, all the return values will be matrices too. New 'reduced', 'complete', and 'raw' options for mode were added in NumPy 1.8.0 and the old option 'full' was made an alias of 'reduced'. In addition the options 'full' and 'economic' were deprecated. Because 'full' was the previous default and 'reduced' is the new default, backward compatibility can be maintained by letting `mode` default. The 'raw' option was added so that LAPACK routines that can multiply arrays by q using the Householder reflectors can be used. Note that in this case the returned arrays are of type np.double or np.cdouble and the h array is transposed to be FORTRAN compatible. No routines using the 'raw' return are currently exposed by numpy, but some are available in lapack_lite and just await the necessary work. Examples -------- >>> a = np.random.randn(9, 6) >>> Q, R = np.linalg.qr(a) >>> np.allclose(a, np.dot(Q, R)) # a does equal QR True >>> R2 = np.linalg.qr(a, mode='r') >>> np.allclose(R, R2) # mode='r' returns the same R as mode='full' True >>> a = np.random.normal(size=(3, 2, 2)) # Stack of 2 x 2 matrices as input >>> Q, R = np.linalg.qr(a) >>> Q.shape (3, 2, 2) >>> R.shape (3, 2, 2) >>> np.allclose(a, np.matmul(Q, R)) True Example illustrating a common use of `qr`: solving of least squares problems What are the least-squares-best `m` and `y0` in ``y = y0 + mx`` for the following data: {(0,1), (1,0), (1,2), (2,1)}. (Graph the points and you'll see that it should be y0 = 0, m = 1.) The answer is provided by solving the over-determined matrix equation ``Ax = b``, where:: A = array([[0, 1], [1, 1], [1, 1], [2, 1]]) x = array([[y0], [m]]) b = array([[1], [0], [2], [1]]) If A = QR such that Q is orthonormal (which is always possible via Gram-Schmidt), then ``x = inv(R) * (Q.T) * b``. (In numpy practice, however, we simply use `lstsq`.) >>> A = np.array([[0, 1], [1, 1], [1, 1], [2, 1]]) >>> A array([[0, 1], [1, 1], [1, 1], [2, 1]]) >>> b = np.array([1, 2, 2, 3]) >>> Q, R = np.linalg.qr(A) >>> p = np.dot(Q.T, b) >>> np.dot(np.linalg.inv(R), p) array([ 1., 1.]) )reducedcompleterraw)ffull)z7The 'full' option is deprecated in favor of 'reduced'. z,For backward compatibility let mode default.r stacklevelr)r economicz$The 'economic' option is deprecated.rzUnrecognized mode ''rNTrrrrr.Frrrr)joinwarningswarnDeprecationWarningrrrrrrrminrJ qr_r_raw_m qr_r_raw_nrrrrHr qr_complete qr_reducedr`)rrmsgrrrrrmnrrrtaurqmcs rZrrs[H 66 = ''DEC MM#1a @D & &8C MM#1a @D24&:; ;mGAq 7723>> from numpy import linalg as LA >>> x = np.random.random() >>> Q = np.array([[np.cos(x), -np.sin(x)], [np.sin(x), np.cos(x)]]) >>> LA.norm(Q[0, :]), LA.norm(Q[1, :]), np.dot(Q[0, :],Q[1, :]) (1.0, 1.0, 0.0) Now multiply a diagonal matrix by ``Q`` on one side and by ``Q.T`` on the other: >>> D = np.diag((-1,1)) >>> LA.eigvals(D) array([-1., 1.]) >>> A = np.dot(Q, D) >>> A = np.dot(A, Q.T) >>> LA.eigvals(A) array([ 1., -1.]) # random rzd->DrrFr)rrrrrrrrrJrr)imagrealrrr)rrrrrrws rZrrsNmGAq11a.KA $57F'**IaVDA    qvv{  A *H#H-H 88H58 ))rYcU4$rrQ)rUPLOs rZ_eigvalsh_dispatcherr4>rrYcxUR5nUS;a [S5e[[5nUS:Xa[R nO[R n[U5up[U5 [U5 [U5upV[U5(aSOSnU"XUS9nUR[U5SS9$) a Compute the eigenvalues of a complex Hermitian or real symmetric matrix. Main difference from eigh: the eigenvectors are not computed. Parameters ---------- a : (..., M, M) array_like A complex- or real-valued matrix whose eigenvalues are to be computed. UPLO : {'L', 'U'}, optional Specifies whether the calculation is done with the lower triangular part of `a` ('L', default) or the upper triangular part ('U'). Irrespective of this value only the real parts of the diagonal will be considered in the computation to preserve the notion of a Hermitian matrix. It therefore follows that the imaginary part of the diagonal will always be treated as zero. Returns ------- w : (..., M,) ndarray The eigenvalues in ascending order, each repeated according to its multiplicity. Raises ------ LinAlgError If the eigenvalue computation does not converge. See Also -------- eigh : eigenvalues and eigenvectors of real symmetric or complex Hermitian (conjugate symmetric) arrays. eigvals : eigenvalues of general real or complex arrays. eig : eigenvalues and right eigenvectors of general real or complex arrays. scipy.linalg.eigvalsh : Similar function in SciPy. Notes ----- .. versionadded:: 1.8.0 Broadcasting rules apply, see the `numpy.linalg` documentation for details. The eigenvalues are computed using LAPACK routines ``_syevd``, ``_heevd``. Examples -------- >>> from numpy import linalg as LA >>> a = np.array([[1, -2j], [2j, 5]]) >>> LA.eigvalsh(a) array([ 0.17157288, 5.82842712]) # may vary >>> # demonstrate the treatment of the imaginary part of the diagonal >>> a = np.array([[5+2j, 9-2j], [0+2j, 2-1j]]) >>> a array([[5.+2.j, 9.-2.j], [0.+2.j, 2.-1.j]]) >>> # with UPLO='L' this is numerically equivalent to using LA.eigvals() >>> # with: >>> b = np.array([[5.+0.j, 0.-2.j], [0.+2.j, 2.-0.j]]) >>> b array([[5.+0.j, 0.-2.j], [0.+2.j, 2.+0.j]]) >>> wa = LA.eigvalsh(a) >>> wb = LA.eigvals(b) >>> wa; wb array([1., 6.]) array([6.+0.j, 1.+0.j]) Lrk UPLO argument must be 'L' or 'U'r7D->drrFr)upperrrrrJ eigvalsh_lo eigvalsh_uprrrrrrr) rr3rrrrrrr1s rZr r BsV ::rYc[U5up[U5 [U5 [U5 [ U5up#[ [ 5n[U5(aSOSn[R"XUS9upg[U5(dA[URS:H5(a$URnURn[U5nO [U5nURUSS9n[!URUSS9U"U55$)a Compute the eigenvalues and right eigenvectors of a square array. Parameters ---------- a : (..., M, M) array Matrices for which the eigenvalues and right eigenvectors will be computed Returns ------- A namedtuple with the following attributes: eigenvalues : (..., M) array The eigenvalues, each repeated according to its multiplicity. The eigenvalues are not necessarily ordered. The resulting array will be of complex type, unless the imaginary part is zero in which case it will be cast to a real type. When `a` is real the resulting eigenvalues will be real (0 imaginary part) or occur in conjugate pairs eigenvectors : (..., M, M) array The normalized (unit "length") eigenvectors, such that the column ``eigenvectors[:,i]`` is the eigenvector corresponding to the eigenvalue ``eigenvalues[i]``. Raises ------ LinAlgError If the eigenvalue computation does not converge. See Also -------- eigvals : eigenvalues of a non-symmetric array. eigh : eigenvalues and eigenvectors of a real symmetric or complex Hermitian (conjugate symmetric) array. eigvalsh : eigenvalues of a real symmetric or complex Hermitian (conjugate symmetric) array. scipy.linalg.eig : Similar function in SciPy that also solves the generalized eigenvalue problem. scipy.linalg.schur : Best choice for unitary and other non-Hermitian normal matrices. Notes ----- .. versionadded:: 1.8.0 Broadcasting rules apply, see the `numpy.linalg` documentation for details. This is implemented using the ``_geev`` LAPACK routines which compute the eigenvalues and eigenvectors of general square arrays. The number `w` is an eigenvalue of `a` if there exists a vector `v` such that ``a @ v = w * v``. Thus, the arrays `a`, `eigenvalues`, and `eigenvectors` satisfy the equations ``a @ eigenvectors[:,i] = eigenvalues[i] * eigenvalues[:,i]`` for :math:`i \in \{0,...,M-1\}`. The array `eigenvectors` may not be of maximum rank, that is, some of the columns may be linearly dependent, although round-off error may obscure that fact. If the eigenvalues are all different, then theoretically the eigenvectors are linearly independent and `a` can be diagonalized by a similarity transformation using `eigenvectors`, i.e, ``inv(eigenvectors) @ a @ eigenvectors`` is diagonal. For non-Hermitian normal matrices the SciPy function `scipy.linalg.schur` is preferred because the matrix `eigenvectors` is guaranteed to be unitary, which is not the case when using `eig`. The Schur factorization produces an upper triangular matrix rather than a diagonal matrix, but for normal matrices only the diagonal of the upper triangular matrix is needed, the rest is roundoff error. Finally, it is emphasized that `eigenvectors` consists of the *right* (as in right-hand side) eigenvectors of `a`. A vector `y` satisfying ``y.T @ a = z * y.T`` for some number `z` is called a *left* eigenvector of `a`, and, in general, the left and right eigenvectors of a matrix are not necessarily the (perhaps conjugate) transposes of each other. References ---------- G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, Various pp. Examples -------- >>> from numpy import linalg as LA (Almost) trivial example with real eigenvalues and eigenvectors. >>> eigenvalues, eigenvectors = LA.eig(np.diag((1, 2, 3))) >>> eigenvalues array([1., 2., 3.]) >>> eigenvectors array([[1., 0., 0.], [0., 1., 0.], [0., 0., 1.]]) Real matrix possessing complex eigenvalues and eigenvectors; note that the eigenvalues are complex conjugates of each other. >>> eigenvalues, eigenvectors = LA.eig(np.array([[1, -1], [1, 1]])) >>> eigenvalues array([1.+1.j, 1.-1.j]) >>> eigenvectors array([[0.70710678+0.j , 0.70710678-0.j ], [0. -0.70710678j, 0. +0.70710678j]]) Complex-valued matrix with real eigenvalues (but complex-valued eigenvectors); note that ``a.conj().T == a``, i.e., `a` is Hermitian. >>> a = np.array([[1, 1j], [-1j, 1]]) >>> eigenvalues, eigenvectors = LA.eig(a) >>> eigenvalues array([2.+0.j, 0.+0.j]) >>> eigenvectors array([[ 0. +0.70710678j, 0.70710678+0.j ], # may vary [ 0.70710678+0.j , -0. +0.70710678j]]) Be careful about round-off error! >>> a = np.array([[1 + 1e-9, 0], [0, 1 - 1e-9]]) >>> # Theor. eigenvalues are 1 +/- 1e-9 >>> eigenvalues, eigenvectors = LA.eig(a) >>> eigenvalues array([1., 1.]) >>> eigenvectors array([[1., 0.], [0., 1.]]) zD->DDzd->DDrgFr)rrrrrrrrrJrr)r/r0rrrrM)rrrrrrr1vts rZrrsJmGAq11a.KA $57F(++I   aV DEA   AFFcM 2 2 FF WWX&) 8% (B QXXhUX3T"X >>rYcUR5nUS;a [S5e[U5up[U5 [ U5 [ U5up4[ [5nUS:Xa[RnO[Rn[U5(aSOSnU"XUS9upUR[U5SS9nU RUSS9n [X"U 55$) a" Return the eigenvalues and eigenvectors of a complex Hermitian (conjugate symmetric) or a real symmetric matrix. Returns two objects, a 1-D array containing the eigenvalues of `a`, and a 2-D square array or matrix (depending on the input type) of the corresponding eigenvectors (in columns). Parameters ---------- a : (..., M, M) array Hermitian or real symmetric matrices whose eigenvalues and eigenvectors are to be computed. UPLO : {'L', 'U'}, optional Specifies whether the calculation is done with the lower triangular part of `a` ('L', default) or the upper triangular part ('U'). Irrespective of this value only the real parts of the diagonal will be considered in the computation to preserve the notion of a Hermitian matrix. It therefore follows that the imaginary part of the diagonal will always be treated as zero. Returns ------- A namedtuple with the following attributes: eigenvalues : (..., M) ndarray The eigenvalues in ascending order, each repeated according to its multiplicity. eigenvectors : {(..., M, M) ndarray, (..., M, M) matrix} The column ``eigenvectors[:, i]`` is the normalized eigenvector corresponding to the eigenvalue ``eigenvalues[i]``. Will return a matrix object if `a` is a matrix object. Raises ------ LinAlgError If the eigenvalue computation does not converge. See Also -------- eigvalsh : eigenvalues of real symmetric or complex Hermitian (conjugate symmetric) arrays. eig : eigenvalues and right eigenvectors for non-symmetric arrays. eigvals : eigenvalues of non-symmetric arrays. scipy.linalg.eigh : Similar function in SciPy (but also solves the generalized eigenvalue problem). Notes ----- .. versionadded:: 1.8.0 Broadcasting rules apply, see the `numpy.linalg` documentation for details. The eigenvalues/eigenvectors are computed using LAPACK routines ``_syevd``, ``_heevd``. The eigenvalues of real symmetric or complex Hermitian matrices are always real. [1]_ The array `eigenvalues` of (column) eigenvectors is unitary and `a`, `eigenvalues`, and `eigenvectors` satisfy the equations ``dot(a, eigenvectors[:, i]) = eigenvalues[i] * eigenvectors[:, i]``. References ---------- .. [1] G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pg. 222. Examples -------- >>> from numpy import linalg as LA >>> a = np.array([[1, -2j], [2j, 5]]) >>> a array([[ 1.+0.j, -0.-2.j], [ 0.+2.j, 5.+0.j]]) >>> eigenvalues, eigenvectors = LA.eigh(a) >>> eigenvalues array([0.17157288, 5.82842712]) >>> eigenvectors array([[-0.92387953+0.j , -0.38268343+0.j ], # may vary [ 0. +0.38268343j, 0. -0.92387953j]]) >>> np.dot(a, eigenvectors[:, 0]) - eigenvalues[0] * eigenvectors[:, 0] # verify 1st eigenval/vec pair array([5.55111512e-17+0.0000000e+00j, 0.00000000e+00+1.2490009e-16j]) >>> np.dot(a, eigenvectors[:, 1]) - eigenvalues[1] * eigenvectors[:, 1] # verify 2nd eigenval/vec pair array([0.+0.j, 0.+0.j]) >>> A = np.matrix(a) # what happens if input is a matrix object >>> A matrix([[ 1.+0.j, -0.-2.j], [ 0.+2.j, 5.+0.j]]) >>> eigenvalues, eigenvectors = LA.eigh(A) >>> eigenvalues array([0.17157288, 5.82842712]) >>> eigenvectors matrix([[-0.92387953+0.j , -0.38268343+0.j ], # may vary [ 0. +0.38268343j, 0. -0.92387953j]]) >>> # demonstrate the treatment of the imaginary part of the diagonal >>> a = np.array([[5+2j, 9-2j], [0+2j, 2-1j]]) >>> a array([[5.+2.j, 9.-2.j], [0.+2.j, 2.-1.j]]) >>> # with UPLO='L' this is numerically equivalent to using LA.eig() with: >>> b = np.array([[5.+0.j, 0.-2.j], [0.+2.j, 2.-0.j]]) >>> b array([[5.+0.j, 0.-2.j], [0.+2.j, 2.+0.j]]) >>> wa, va = LA.eigh(a) >>> wb, vb = LA.eig(b) >>> wa; wb array([1., 6.]) array([6.+0.j, 1.+0.j]) >>> va; vb array([[-0.4472136 +0.j , -0.89442719+0.j ], # may vary [ 0. +0.89442719j, 0. -0.4472136j ]]) array([[ 0.89442719+0.j , -0. +0.4472136j], [-0. +0.4472136j, 0.89442719+0.j ]]) r6r8r7zD->dDd->ddrFr)r:rrrrrrrrJeigh_loeigh_uprrrr]) rr3rrrrrrr1rAs rZrrDst ::= 2``. full_matrices : bool, optional If True (default), `u` and `vh` have the shapes ``(..., M, M)`` and ``(..., N, N)``, respectively. Otherwise, the shapes are ``(..., M, K)`` and ``(..., K, N)``, respectively, where ``K = min(M, N)``. compute_uv : bool, optional Whether or not to compute `u` and `vh` in addition to `s`. True by default. hermitian : bool, optional If True, `a` is assumed to be Hermitian (symmetric if real-valued), enabling a more efficient method for finding singular values. Defaults to False. .. versionadded:: 1.17.0 Returns ------- When `compute_uv` is True, the result is a namedtuple with the following attribute names: U : { (..., M, M), (..., M, K) } array Unitary array(s). The first ``a.ndim - 2`` dimensions have the same size as those of the input `a`. The size of the last two dimensions depends on the value of `full_matrices`. Only returned when `compute_uv` is True. S : (..., K) array Vector(s) with the singular values, within each vector sorted in descending order. The first ``a.ndim - 2`` dimensions have the same size as those of the input `a`. Vh : { (..., N, N), (..., K, N) } array Unitary array(s). The first ``a.ndim - 2`` dimensions have the same size as those of the input `a`. The size of the last two dimensions depends on the value of `full_matrices`. Only returned when `compute_uv` is True. Raises ------ LinAlgError If SVD computation does not converge. See Also -------- scipy.linalg.svd : Similar function in SciPy. scipy.linalg.svdvals : Compute singular values of a matrix. Notes ----- .. versionchanged:: 1.8.0 Broadcasting rules apply, see the `numpy.linalg` documentation for details. The decomposition is performed using LAPACK routine ``_gesdd``. SVD is usually described for the factorization of a 2D matrix :math:`A`. The higher-dimensional case will be discussed below. In the 2D case, SVD is written as :math:`A = U S V^H`, where :math:`A = a`, :math:`U= u`, :math:`S= \mathtt{np.diag}(s)` and :math:`V^H = vh`. The 1D array `s` contains the singular values of `a` and `u` and `vh` are unitary. The rows of `vh` are the eigenvectors of :math:`A^H A` and the columns of `u` are the eigenvectors of :math:`A A^H`. In both cases the corresponding (possibly non-zero) eigenvalues are given by ``s**2``. If `a` has more than two dimensions, then broadcasting rules apply, as explained in :ref:`routines.linalg-broadcasting`. This means that SVD is working in "stacked" mode: it iterates over all indices of the first ``a.ndim - 2`` dimensions and for each combination SVD is applied to the last two indices. The matrix `a` can be reconstructed from the decomposition with either ``(u * s[..., None, :]) @ vh`` or ``u @ (s[..., None] * vh)``. (The ``@`` operator can be replaced by the function ``np.matmul`` for python versions below 3.5.) If `a` is a ``matrix`` object (as opposed to an ``ndarray``), then so are all the return values. Examples -------- >>> a = np.random.randn(9, 6) + 1j*np.random.randn(9, 6) >>> b = np.random.randn(2, 7, 8, 3) + 1j*np.random.randn(2, 7, 8, 3) Reconstruction based on full SVD, 2D case: >>> U, S, Vh = np.linalg.svd(a, full_matrices=True) >>> U.shape, S.shape, Vh.shape ((9, 9), (6,), (6, 6)) >>> np.allclose(a, np.dot(U[:, :6] * S, Vh)) True >>> smat = np.zeros((9, 6), dtype=complex) >>> smat[:6, :6] = np.diag(S) >>> np.allclose(a, np.dot(U, np.dot(smat, Vh))) True Reconstruction based on reduced SVD, 2D case: >>> U, S, Vh = np.linalg.svd(a, full_matrices=False) >>> U.shape, S.shape, Vh.shape ((9, 6), (6,), (6, 6)) >>> np.allclose(a, np.dot(U * S, Vh)) True >>> smat = np.diag(S) >>> np.allclose(a, np.dot(U, np.dot(smat, Vh))) True Reconstruction based on full SVD, 4D case: >>> U, S, Vh = np.linalg.svd(b, full_matrices=True) >>> U.shape, S.shape, Vh.shape ((2, 7, 8, 8), (2, 7, 3), (2, 7, 3, 3)) >>> np.allclose(b, np.matmul(U[..., :3] * S[..., None, :], Vh)) True >>> np.allclose(b, np.matmul(U[..., :3], S[..., None] * Vh)) True Reconstruction based on reduced SVD, 4D case: >>> U, S, Vh = np.linalg.svd(b, full_matrices=False) >>> U.shape, S.shape, Vh.shape ((2, 7, 8, 3), (2, 7, 3), (2, 7, 3, 3)) >>> np.allclose(b, np.matmul(U * S[..., None, :], Vh)) True >>> np.allclose(b, np.matmul(U, S[..., None] * Vh)) True rN.raxisrzD->DdDzd->dddrFrr9r)numpyrrrBr8rCtake_along_axisr conjugaterir rDrrrrrrJsvd_m_fsvd_n_fsvd_m_ssvd_n_srrrsvd_msvd_n)rrGrHrI_nxrsusgnsidxrArrrrrrrvhs rZr r s/XmGA 7DAq'CAA1:c4R4i(D%%cb%9C##A"#5A##ACqL'9#CA13a<001;;=BT!Wab2 2 AAA73"9% %qa.KA $%J KF 7723>> from numpy import linalg as LA >>> a = np.array([[1, 0, -1], [0, 1, 0], [1, 0, 1]]) >>> a array([[ 1, 0, -1], [ 0, 1, 0], [ 1, 0, 1]]) >>> LA.cond(a) 1.4142135623730951 >>> LA.cond(a, 'fro') 3.1622776601683795 >>> LA.cond(a, np.inf) 2.0 >>> LA.cond(a, -np.inf) 1.0 >>> LA.cond(a, 1) 2.0 >>> LA.cond(a, -1) 1.0 >>> LA.cond(a, 2) 1.4142135623730951 >>> LA.cond(a, -2) 0.70710678118654746 # may vary >>> min(LA.svd(a, compute_uv=False))*min(LA.svd(LA.inv(a), compute_uv=False)) 0.70710678118654746 # may vary z#cond is not defined on empty arraysNrrFrHrt)r)).r).rrrrrrrLrrrQ)rrrr r2rrrrrJrrrrAanyrr*) r^r_rXrrrrinvxnan_masks rZrrsx`  AA?@@yAFa2g e $ ( #BwgJ6*fI' * $ # 1q!!!n +A..FF ( # $$Q>> from numpy.linalg import matrix_rank >>> matrix_rank(np.eye(4)) # Full rank matrix 4 >>> I=np.eye(4); I[-1,-1] = 0. # rank deficient matrix >>> matrix_rank(I) 3 >>> matrix_rank(np.ones((4,))) # 1 dimension - rank 1 unless all 0 1 >>> matrix_rank(np.zeros((4,))) 0 rrF)rHrINrTrMkeepdimsr.rL) rrintr)r maxrr1repsr(r@)rirjrIrls rZrr!s|  Avvzs1a4y=!! A%95A {eede+c!''"#,.??%.BTBTTcl3<( r **rYcU4$rrQ)rrcondrIs rZ_pinv_dispatcherrtrrYc [U5up[U5n[U5(a>URSSupE[ URSSXT4-UR S9nU"U5$UR 5n[USUS9upxn US[4[USSS 9-n X:n [S XUS 9nS X)'[[U 5[US[4[U555nU"U5$) a Compute the (Moore-Penrose) pseudo-inverse of a matrix. Calculate the generalized inverse of a matrix using its singular-value decomposition (SVD) and including all *large* singular values. .. versionchanged:: 1.14 Can now operate on stacks of matrices Parameters ---------- a : (..., M, N) array_like Matrix or stack of matrices to be pseudo-inverted. rcond : (...) array_like of float Cutoff for small singular values. Singular values less than or equal to ``rcond * largest_singular_value`` are set to zero. Broadcasts against the stack of matrices. hermitian : bool, optional If True, `a` is assumed to be Hermitian (symmetric if real-valued), enabling a more efficient method for finding singular values. Defaults to False. .. versionadded:: 1.17.0 Returns ------- B : (..., N, M) ndarray The pseudo-inverse of `a`. If `a` is a `matrix` instance, then so is `B`. Raises ------ LinAlgError If the SVD computation does not converge. See Also -------- scipy.linalg.pinv : Similar function in SciPy. scipy.linalg.pinvh : Compute the (Moore-Penrose) pseudo-inverse of a Hermitian matrix. Notes ----- The pseudo-inverse of a matrix A, denoted :math:`A^+`, is defined as: "the matrix that 'solves' [the least-squares problem] :math:`Ax = b`," i.e., if :math:`\bar{x}` is said solution, then :math:`A^+` is that matrix such that :math:`\bar{x} = A^+b`. It can be shown that if :math:`Q_1 \Sigma Q_2^T = A` is the singular value decomposition of A, then :math:`A^+ = Q_2 \Sigma^+ Q_1^T`, where :math:`Q_{1,2}` are orthogonal matrices, :math:`\Sigma` is a diagonal matrix consisting of A's so-called singular values, (followed, typically, by zeros), and then :math:`\Sigma^+` is simply the diagonal matrix consisting of the reciprocals of A's singular values (again, followed by zeros). [1]_ References ---------- .. [1] G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pp. 139-142. Examples -------- The following example checks that ``a * a+ * a == a`` and ``a+ * a * a+ == a+``: >>> a = np.random.randn(9, 6) >>> B = np.linalg.pinv(a) >>> np.allclose(a, np.dot(a, np.dot(B, a))) True >>> np.allclose(B, np.dot(B, np.dot(a, B))) True rNrF)rGrI.rTrmrx)whereoutr)rrrrrrrPr r(r6r?r=rr-) rrsrIrrrrrYrXrAcutofflarges rZr r s^mGA ENEAwwrs|AGGCRLA6)9Cy A1EY?HA"3< 4T#B BF JEq!a(AAfI 23<)A, G HC 9rYc [U5n[U5 [U5 [U5up[ U5n[ U5(aSOSn[ R"XS9upVUR"USS9nURUSS9n[XV5$)a Compute the sign and (natural) logarithm of the determinant of an array. If an array has a very small or very large determinant, then a call to `det` may overflow or underflow. This routine is more robust against such issues, because it computes the logarithm of the determinant rather than the determinant itself. Parameters ---------- a : (..., M, M) array_like Input array, has to be a square 2-D array. Returns ------- A namedtuple with the following attributes: sign : (...) array_like A number representing the sign of the determinant. For a real matrix, this is 1, 0, or -1. For a complex matrix, this is a complex number with absolute value 1 (i.e., it is on the unit circle), or else 0. logabsdet : (...) array_like The natural log of the absolute value of the determinant. If the determinant is zero, then `sign` will be 0 and `logabsdet` will be -Inf. In all cases, the determinant is equal to ``sign * np.exp(logabsdet)``. See Also -------- det Notes ----- .. versionadded:: 1.8.0 Broadcasting rules apply, see the `numpy.linalg` documentation for details. .. versionadded:: 1.6.0 The determinant is computed via LU factorization using the LAPACK routine ``z/dgetrf``. Examples -------- The determinant of a 2-D array ``[[a, b], [c, d]]`` is ``ad - bc``: >>> a = np.array([[1, 2], [3, 4]]) >>> (sign, logabsdet) = np.linalg.slogdet(a) >>> (sign, logabsdet) (-1, 0.69314718055994529) # may vary >>> sign * np.exp(logabsdet) -2.0 Computing log-determinants for a stack of matrices: >>> a = np.array([ [[1, 2], [3, 4]], [[1, 2], [2, 1]], [[1, 3], [3, 1]] ]) >>> a.shape (3, 2, 2) >>> sign, logabsdet = np.linalg.slogdet(a) >>> (sign, logabsdet) (array([-1., -1., -1.]), array([ 0.69314718, 1.09861229, 2.07944154])) >>> sign * np.exp(logabsdet) array([-2., -3., -8.]) This routine succeeds where ordinary `det` does not: >>> np.linalg.det(np.eye(500) * 0.1) 0.0 >>> np.linalg.slogdet(np.eye(500) * 0.1) (1, -1151.2925464970228) zD->DdrCrcFr) rrrrrrrJr rre)rrrreal_trrBlogdets rZr r sZ  Aq1a.KA x F(++I ((@LD ;;xe ,D ]]6] .F  &&rYc[U5n[U5 [U5 [U5up[ U5(aSOSn[ R "XS9nURUSS9nU$)a Compute the determinant of an array. Parameters ---------- a : (..., M, M) array_like Input array to compute determinants for. Returns ------- det : (...) array_like Determinant of `a`. See Also -------- slogdet : Another way to represent the determinant, more suitable for large matrices where underflow/overflow may occur. scipy.linalg.det : Similar function in SciPy. Notes ----- .. versionadded:: 1.8.0 Broadcasting rules apply, see the `numpy.linalg` documentation for details. The determinant is computed via LU factorization using the LAPACK routine ``z/dgetrf``. Examples -------- The determinant of a 2-D array [[a, b], [c, d]] is ad - bc: >>> a = np.array([[1, 2], [3, 4]]) >>> np.linalg.det(a) -2.0 # may vary Computing determinants for a stack of matrices: >>> a = np.array([ [[1, 2], [3, 4]], [[1, 2], [2, 1]], [[1, 3], [3, 1]] ]) >>> a.shape (3, 2, 2) >>> np.linalg.det(a) array([-2., -3., -8.]) rrrcFr)rrrrrrJr r)rrrrrs rZr r Nsab  Aq1a.KA'**I!1A &A HrYcX4$rrQ)rrrss rZ_lstsq_dispatcherrrrYc[U5up[U5upURS:HnU(a USS2[4n[X5 URSSupgURSSupXh:wa [ S5e[ X5up[U 5n US:Xa[R"S[SS9 S nUc![U 5R[Xv5-nXg::a[Rn O[R n [#U 5(aS OS n[%[&5nU S :Xa([)URSSXiS-4-UR*S 9nU "XX.US9unnnnUS :XaS US'U S :XaUSSU 24nUSSU 24nU(aUR-S S9nUU:wdXg::a [//U 5nUR1U SS9nUR1U SS9nUR1U SS9nU"U5U"U5UU4$)a Return the least-squares solution to a linear matrix equation. Computes the vector `x` that approximately solves the equation ``a @ x = b``. The equation may be under-, well-, or over-determined (i.e., the number of linearly independent rows of `a` can be less than, equal to, or greater than its number of linearly independent columns). If `a` is square and of full rank, then `x` (but for round-off error) is the "exact" solution of the equation. Else, `x` minimizes the Euclidean 2-norm :math:`||b - ax||`. If there are multiple minimizing solutions, the one with the smallest 2-norm :math:`||x||` is returned. Parameters ---------- a : (M, N) array_like "Coefficient" matrix. b : {(M,), (M, K)} array_like Ordinate or "dependent variable" values. If `b` is two-dimensional, the least-squares solution is calculated for each of the `K` columns of `b`. rcond : float, optional Cut-off ratio for small singular values of `a`. For the purposes of rank determination, singular values are treated as zero if they are smaller than `rcond` times the largest singular value of `a`. .. versionchanged:: 1.14.0 If not set, a FutureWarning is given. The previous default of ``-1`` will use the machine precision as `rcond` parameter, the new default will use the machine precision times `max(M, N)`. To silence the warning and use the new default, use ``rcond=None``, to keep using the old behavior, use ``rcond=-1``. Returns ------- x : {(N,), (N, K)} ndarray Least-squares solution. If `b` is two-dimensional, the solutions are in the `K` columns of `x`. residuals : {(1,), (K,), (0,)} ndarray Sums of squared residuals: Squared Euclidean 2-norm for each column in ``b - a @ x``. If the rank of `a` is < N or M <= N, this is an empty array. If `b` is 1-dimensional, this is a (1,) shape array. Otherwise the shape is (K,). rank : int Rank of matrix `a`. s : (min(M, N),) ndarray Singular values of `a`. Raises ------ LinAlgError If computation does not converge. See Also -------- scipy.linalg.lstsq : Similar function in SciPy. Notes ----- If `b` is a matrix, then all array results are returned as matrices. Examples -------- Fit a line, ``y = mx + c``, through some noisy data-points: >>> x = np.array([0, 1, 2, 3]) >>> y = np.array([-1, 0.2, 0.9, 2.1]) By examining the coefficients, we see that the line should have a gradient of roughly 1 and cut the y-axis at, more or less, -1. We can rewrite the line equation as ``y = Ap``, where ``A = [[x 1]]`` and ``p = [[m], [c]]``. Now use `lstsq` to solve for `p`: >>> A = np.vstack([x, np.ones(len(x))]).T >>> A array([[ 0., 1.], [ 1., 1.], [ 2., 1.], [ 3., 1.]]) >>> m, c = np.linalg.lstsq(A, y, rcond=None)[0] >>> m, c (1.0 -0.95) # may vary Plot the data along with the fitted line: >>> import matplotlib.pyplot as plt >>> _ = plt.plot(x, y, 'o', label='Original data', markersize=10) >>> _ = plt.plot(x, m*x + c, 'r', label='Fitted line') >>> _ = plt.legend() >>> plt.show() rxNrzIncompatible dimensionsr"a `rcond` parameter will change to the default of machine precision times ``max(M, N)`` where M and N are the input matrix dimensions. To use the future default and silence this warning we advise to pass `rcond=None`, to keep using the old, explicitly pass `rcond=-1`.rrrz DDd->Ddidz ddd->ddidrrr.rLFrT)rrr(rrrrrr!r" FutureWarningr1rqrprJlstsq_mlstsq_nrrrrrsqueezerr)rrrsrris_1drrm2n_rhsrr result_real_trrrr^residsrankrXs rZrrsB a=DAmGA FFaKE ajMq 7723F z !''#2,!QY/qww ?ePAvtQAv# z c6E6kNVeV $ II2I   qyAFr=) U+A ]]=u] 5F %A 7DL$ ))rYc@[XU4S5nU"[USS9SS9nU$)aCompute a function of the singular values of the 2-D matrices in `x`. This is a private utility function used by `numpy.linalg.norm()`. Parameters ---------- x : ndarray row_axis, col_axis : int The axes of `x` that hold the 2-D matrices. op : callable This should be either numpy.amin or `numpy.amax` or `numpy.sum`. Returns ------- result : float or ndarray If `x` is 2-D, the return values is a float. Otherwise, it is an array with ``x.ndim - 2`` dimensions. The return values are either the minimum or maximum or sum of the singular values of the matrices, depending on whether `op` is `numpy.amin` or `numpy.amax` or `numpy.sum`. rdFrbrrL)r4r )r^row_axiscol_axisopyrs rZ_multi_svd_normr/ s-. x((3A A%(r 2F MrYcU4$rrQ)r^ordrMrns rZ_norm_dispatcherrK rrYc [U5n[URR[[ 45(dUR [5nUcURnUbUS;aUS:Xd US:XaUS:XaURSS9n[URR5(a<URnURnURU5URU5-nOURU5n[U5nU(aURUS/-5nU$URn Uc[![#U 55nO$[%U[ 5(d['U5nU4n[-U5S:XGaHU[.:Xa[1U5R3X#S9$U[.*:Xa[1U5R5X#S9$US :Xa5US :gR URR5R7X#S9$US:Xa[8R:"[1U5X#S9$UbUS:Xa;UR=5U-Rn [[8R:"XUS95$[%U[>5(a[AS US 35e[1U5n X-n [8R:"XUS9nU[CXRS 9-nU$[-U5S:XGaUup[EX5n [EX5nX:Xa [AS 5eUS:Xa[GX U[H5nGOvUS:Xa[GX U[J5nGO]US:Xa7X:aUS-n[8R:"[1U5U S9R3US9nGO U[.:Xa6X:aU S-n [8R:"[1U5US9R3U S9nOUS:Xa6X:aUS-n[8R:"[1U5U S9R5US9nOU[.*:Xa6X:aU S-n [8R:"[1U5US9R5U S9nOcUS;a:[[8R:"UR=5U-RUS95nO#US:Xa[GX U[65nO [AS5eU(a4[MURN5nSXS 'SXS'URU5nU$[AS5e![(an [+S5U eSn A ff=f)a Matrix or vector norm. This function is able to return one of eight different matrix norms, or one of an infinite number of vector norms (described below), depending on the value of the ``ord`` parameter. Parameters ---------- x : array_like Input array. If `axis` is None, `x` must be 1-D or 2-D, unless `ord` is None. If both `axis` and `ord` are None, the 2-norm of ``x.ravel`` will be returned. ord : {non-zero int, inf, -inf, 'fro', 'nuc'}, optional Order of the norm (see table under ``Notes``). inf means numpy's `inf` object. The default is None. axis : {None, int, 2-tuple of ints}, optional. If `axis` is an integer, it specifies the axis of `x` along which to compute the vector norms. If `axis` is a 2-tuple, it specifies the axes that hold 2-D matrices, and the matrix norms of these matrices are computed. If `axis` is None then either a vector norm (when `x` is 1-D) or a matrix norm (when `x` is 2-D) is returned. The default is None. .. versionadded:: 1.8.0 keepdims : bool, optional If this is set to True, the axes which are normed over are left in the result as dimensions with size one. With this option the result will broadcast correctly against the original `x`. .. versionadded:: 1.10.0 Returns ------- n : float or ndarray Norm of the matrix or vector(s). See Also -------- scipy.linalg.norm : Similar function in SciPy. Notes ----- For values of ``ord < 1``, the result is, strictly speaking, not a mathematical 'norm', but it may still be useful for various numerical purposes. The following norms can be calculated: ===== ============================ ========================== ord norm for matrices norm for vectors ===== ============================ ========================== None Frobenius norm 2-norm 'fro' Frobenius norm -- 'nuc' nuclear norm -- inf max(sum(abs(x), axis=1)) max(abs(x)) -inf min(sum(abs(x), axis=1)) min(abs(x)) 0 -- sum(x != 0) 1 max(sum(abs(x), axis=0)) as below -1 min(sum(abs(x), axis=0)) as below 2 2-norm (largest sing. value) as below -2 smallest singular value as below other -- sum(abs(x)**ord)**(1./ord) ===== ============================ ========================== The Frobenius norm is given by [1]_: :math:`||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}` The nuclear norm is the sum of the singular values. Both the Frobenius and nuclear norm orders are only defined for matrices and raise a ValueError when ``x.ndim != 2``. References ---------- .. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15 Examples -------- >>> from numpy import linalg as LA >>> a = np.arange(9) - 4 >>> a array([-4, -3, -2, ..., 2, 3, 4]) >>> b = a.reshape((3, 3)) >>> b array([[-4, -3, -2], [-1, 0, 1], [ 2, 3, 4]]) >>> LA.norm(a) 7.745966692414834 >>> LA.norm(b) 7.745966692414834 >>> LA.norm(b, 'fro') 7.745966692414834 >>> LA.norm(a, np.inf) 4.0 >>> LA.norm(b, np.inf) 9.0 >>> LA.norm(a, -np.inf) 0.0 >>> LA.norm(b, -np.inf) 2.0 >>> LA.norm(a, 1) 20.0 >>> LA.norm(b, 1) 7.0 >>> LA.norm(a, -1) -4.6566128774142013e-010 >>> LA.norm(b, -1) 6.0 >>> LA.norm(a, 2) 7.745966692414834 >>> LA.norm(b, 2) 7.3484692283495345 >>> LA.norm(a, -2) 0.0 >>> LA.norm(b, -2) 1.8570331885190563e-016 # may vary >>> LA.norm(a, 3) 5.8480354764257312 # may vary >>> LA.norm(a, -3) 0.0 Using the `axis` argument to compute vector norms: >>> c = np.array([[ 1, 2, 3], ... [-1, 1, 4]]) >>> LA.norm(c, axis=0) array([ 1.41421356, 2.23606798, 5. ]) >>> LA.norm(c, axis=1) array([ 3.74165739, 4.24264069]) >>> LA.norm(c, ord=1, axis=1) array([ 6., 6.]) Using the `axis` argument to compute matrix norms: >>> m = np.arange(8).reshape(2,2,2) >>> LA.norm(m, axis=(1,2)) array([ 3.74165739, 11.22497216]) >>> LA.norm(m[0, :, :]), LA.norm(m[1, :, :]) (3.7416573867739413, 11.224972160321824) N)rfrorrxK)orderz6'axis' must be None, an integer or a tuple of integersrmrzInvalid norm order 'z ' for vectorsrzDuplicate axes given.rrLr)Nrrnucz Invalid norm order for matrices.z&Improper number of dimensions to norm.)(rrrrr&r<rfloatrrrr0r/r+r.rtupler isinstancero Exceptionrrr*r8rpr$r/r,reduceconjstrrrErFrr6r5rr)r^rrMrnrx_realx_imagsqnormrndr rXabsxrr ret_shapes rZrrO sMn  A aggllWg$6 7 7 HHUO |vv [ L TQY AX$!)c"AQWW\\**F+fjj.@@qv,Ckk$s(+J B |U2Y e $ $ ]t9Dw 4yA~ #:q6::4:; ; SD[q6::4:; ; AXF??166<<044$4R R AX::c!f4C C [C1HA##A 1(CD DS ! !3C5 FG Gq6D LD**Tx@C Js))4 4CJ Ta!'5'5  45 5 !8"1$?C BY!!x>C AX"A **SV(377X7FC CZ"A **SV(377X7FC BY"A **SV(377X7FC SD["A **SV(377X7FC & &szz1668a<"5"5DABC E\!!x=C?@ @ QWW I!"I1g !"I1g ++i(C ABBI ]TU[\ \ ]s S S/ S**S/rwc#,# UShvN Uv gN 7frrQ)rrws rZ_multidot_dispatcherrN s Is  c`[U5nUS:a [S5eUS:Xa[USUSUS9$UVs/sHn[U5PM nnUSRUSRpTUSRS:Xa[ US5US'USRS:Xa[ US5R US'[U6 US:Xa[USUSUSUS9nO[U5n[XSUS- US9nUS:Xa US:XaUS$US:XdUS:XaUR5$U$s snf) ai Compute the dot product of two or more arrays in a single function call, while automatically selecting the fastest evaluation order. `multi_dot` chains `numpy.dot` and uses optimal parenthesization of the matrices [1]_ [2]_. Depending on the shapes of the matrices, this can speed up the multiplication a lot. If the first argument is 1-D it is treated as a row vector. If the last argument is 1-D it is treated as a column vector. The other arguments must be 2-D. Think of `multi_dot` as:: def multi_dot(arrays): return functools.reduce(np.dot, arrays) Parameters ---------- arrays : sequence of array_like If the first argument is 1-D it is treated as row vector. If the last argument is 1-D it is treated as column vector. The other arguments must be 2-D. out : ndarray, optional Output argument. This must have the exact kind that would be returned if it was not used. In particular, it must have the right type, must be C-contiguous, and its dtype must be the dtype that would be returned for `dot(a, b)`. This is a performance feature. Therefore, if these conditions are not met, an exception is raised, instead of attempting to be flexible. .. versionadded:: 1.19.0 Returns ------- output : ndarray Returns the dot product of the supplied arrays. See Also -------- numpy.dot : dot multiplication with two arguments. References ---------- .. [1] Cormen, "Introduction to Algorithms", Chapter 15.2, p. 370-378 .. [2] https://en.wikipedia.org/wiki/Matrix_chain_multiplication Examples -------- `multi_dot` allows you to write:: >>> from numpy.linalg import multi_dot >>> # Prepare some data >>> A = np.random.random((10000, 100)) >>> B = np.random.random((100, 1000)) >>> C = np.random.random((1000, 5)) >>> D = np.random.random((5, 333)) >>> # the actual dot multiplication >>> _ = multi_dot([A, B, C, D]) instead of:: >>> _ = np.dot(np.dot(np.dot(A, B), C), D) >>> # or >>> _ = A.dot(B).dot(C).dot(D) Notes ----- The cost for a matrix multiplication can be calculated with the following function:: def cost(A, B): return A.shape[0] * A.shape[1] * B.shape[1] Assume we have three matrices :math:`A_{10x100}, B_{100x5}, C_{5x50}`. The costs for the two different parenthesizations are as follows:: cost((AB)C) = 10*100*5 + 10*5*50 = 5000 + 2500 = 7500 cost(A(BC)) = 10*100*50 + 100*5*50 = 50000 + 25000 = 75000 rzExpecting at least two arrays.rrxrrr)rr) rrr+r;rr9r>r_multi_dot_three_multi_dot_matrix_chain_order _multi_dotr)rrwrr ndim_first ndim_lastrrs rZrrS sCl F A1u9:: a6!9fQiS11%+ ,VjmVF ,#1INNF2JOO ay~~vay)q  bz!r +--r  Av!&)VAYq sK-f5F1a!e=Q9>d| qIN||~ 3-sD+cURupEURupgXF-XW--nXW-XF--n X:a[[X5X#S9$[U[X5US9$)z Find the best order for three arrays and do the multiplication. For three arguments `_multi_dot_three` is approximately 15 times faster than `_multi_dot_matrix_chain_order` r)rr+) riBCrwa0a1b0b1c0c1cost1cost2s rZrr s_wwHBwwHD I #E I #E }3q9a))1c!iS))rYc[U5nUVs/sHo3RSPM snUSRS/-n[X"4[S9n[ X"4[ S9n[ SU5Hqn[ X'- 5H]nX-n [XXU 4'[ X5H=n XXU 4XZS-U 4-XHXJS--XIS---n XX4:dM3XX4'XX4'M? M_ Ms U(aXe4$U$s snf)a Return a np.array that encodes the optimal order of mutiplications. The optimal order array is then used by `_multi_dot()` to do the multiplication. Also return the cost matrix if `return_costs` is `True` The implementation CLOSELY follows Cormen, "Introduction to Algorithms", Chapter 15.2, p. 370-378. Note that Cormen uses 1-based indices. cost[i, j] = min([ cost[prefix] + cost[suffix] + cost_mult(prefix, suffix) for k in range(i, j)]) rrrxr)rrrr#rr:rr*) r return_costsrrr_rrXlijrr,s rZrr s " F A$$VV$r (8(8(;'<r?r@rArBrCrDrEnumpy.core.multiarrayrFrGnumpy.lib.twodim_baserHrI numpy.linalgrJ numpy._typingrKrMr]r`reripartialarray_function_dispatch fortran_intrrr|rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr4r r?rrrJr r`rrkrrtr r r rrrrrrrrrrrQrYrZrsO  ' "7 +&! zJ  $++ %%n>  N*:1/0)96.F*  *FFVV% ww)  +$.#2 0 9 O E 7 &01I2IX*+Q0,Q0h./@!0@!J*+E3,E3P12p3pl*+[0,[0@(G&)G&X*+X*,X*v-.[5/[5z*+W?,W?t-.M#/M#d)C*CL)*s +s l01e+2e+T)*^+^H*+U',U'p*+7 ,7 x*+\*,\*~8)*yC+yC|)- -.!u/up**&)R rY