|
"""Lite 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. """
'cholesky', 'eigvals', 'eigvalsh', 'pinv', 'slogdet', 'det', 'svd', 'eig', 'eigh', 'lstsq', 'norm', 'qr', 'cond', 'matrix_rank', 'LinAlgError', 'multi_dot']
array, asarray, zeros, empty, empty_like, intc, single, double, csingle, cdouble, inexact, complexfloating, newaxis, all, Inf, dot, add, multiply, sqrt, fastCopyAndTranspose, sum, isfinite, finfo, errstate, geterrobj, moveaxis, amin, amax, product, abs, atleast_2d, intp, asanyarray, object_, matmul, swapaxes, divide, count_nonzero, isnan )
overrides.array_function_dispatch, module='numpy.linalg')
# For Python2/3 compatibility
""" Generic Python-exception-derived object raised by linalg functions.
General purpose exception class, derived from Python's exception.Exception 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 "<stdin>", line 1, in <module> 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
"""
divide='ignore', under='ignore'):
# Dealing with errors in _umath_linalg
def _raise_linalgerror_singular(err, flag): raise LinAlgError("Singular matrix")
def _raise_linalgerror_nonposdef(err, flag): raise LinAlgError("Matrix is not positive definite")
def _raise_linalgerror_eigenvalues_nonconvergence(err, flag): raise LinAlgError("Eigenvalues did not converge")
def _raise_linalgerror_svd_nonconvergence(err, flag): raise LinAlgError("SVD did not converge")
def _raise_linalgerror_lstsq(err, flag): raise LinAlgError("SVD did not converge in Linear Least Squares")
double : double, csingle : single, cdouble : double}
double : cdouble, csingle : csingle, cdouble : cdouble}
return _complex_types_map.get(t, default)
"""Cast the type t to either double or cdouble.""" return double
# in lite version, use higher precision (always double or cdouble) is_complex = True # unsupported inexact scalar raise TypeError("array type %s is unsupported in linalg" % (a.dtype.name,)) else: rt = double t = cdouble result_type = _complex_types_map[result_type] else:
# _fastCopyAndTranpose assumes the input is 2D (as all the calls in here are).
ret = [] for arr in arrays: if arr.dtype.byteorder not in ('=', '|'): ret.append(asarray(arr, dtype=arr.dtype.newbyteorder('='))) else: ret.append(arr) if len(ret) == 1: return ret[0] else: return ret
cast_arrays = () for a in arrays: if a.dtype.type is type: cast_arrays = cast_arrays + (_fastCT(a),) else: cast_arrays = cast_arrays + (_fastCT(a.astype(type)),) if len(cast_arrays) == 1: return cast_arrays[0] else: return cast_arrays
for a in arrays: if a.ndim != 2: raise LinAlgError('%d-dimensional array given. Array must be ' 'two-dimensional' % a.ndim)
raise LinAlgError('%d-dimensional array given. Array must be ' 'at least two-dimensional' % a.ndim)
raise LinAlgError('Last 2 dimensions of the array must be square')
for a in arrays: if not (isfinite(a).all()): raise LinAlgError("Array must not contain infs or NaNs")
# check size first for efficiency return arr.size == 0 and product(arr.shape[-2:]) == 0
for a in arrays: if _isEmpty2d(a): raise LinAlgError("Arrays cannot be empty")
""" 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 """ return swapaxes(a, -1, -2)
# Linear equations
return (a, b)
""" 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=b.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
""" a, wrap = _makearray(a) b = asarray(b) an = a.ndim
if axes is not None: allaxes = list(range(0, an)) for k in axes: allaxes.remove(k) allaxes.insert(an, k) a = a.transpose(allaxes)
oldshape = a.shape[-(an-b.ndim):] prod = 1 for k in oldshape: prod *= k
a = a.reshape(-1, prod) b = b.ravel() res = wrap(solve(a, b)) res.shape = oldshape return res
return (a, b)
def solve(a, b): """ 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.
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 ``3 * x0 + x1 = 9`` and ``x0 + 2 * x1 = 8``:
>>> a = np.array([[3,1], [1,2]]) >>> b = np.array([9,8]) >>> x = np.linalg.solve(a, b) >>> x array([ 2., 3.])
Check that the solution is correct:
>>> np.allclose(np.dot(a, x), b) True
"""
# We use the b = (..., M,) logic, only if the number of extra dimensions # match exactly else: gufunc = _umath_linalg.solve
extobj = get_linalg_error_extobj(_raise_linalgerror_singular)
return (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
""" a = asarray(a) oldshape = a.shape prod = 1 if ind > 0: invshape = oldshape[ind:] + oldshape[:ind] for k in oldshape[ind:]: prod *= k else: raise ValueError("Invalid ind argument.") a = a.reshape(prod, -1) ia = inv(a) return ia.reshape(*invshape)
# Matrix inversion
return (a,)
def inv(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.
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]], [[-5. , 2. ], [ 3. , -1. ]]])
"""
extobj = get_linalg_error_extobj(_raise_linalgerror_singular)
return (a,)
def matrix_power(a, n): """ 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.]])
""" a = asanyarray(a) _assertRankAtLeast2(a) _assertNdSquareness(a)
try: n = operator.index(n) except TypeError: raise TypeError("exponent must be an integer")
# Fall back on dot for object arrays. Object arrays are not supported by # the current implementation of matmul using einsum if a.dtype != object: fmatmul = matmul elif a.ndim == 2: fmatmul = dot else: raise NotImplementedError( "matrix_power not supported for stacks of object arrays")
if n == 0: a = empty_like(a) a[...] = eye(a.shape[-2], dtype=a.dtype) return a
elif n < 0: a = inv(a) n = abs(n)
# short-cuts. if n == 1: return a
elif n == 2: return fmatmul(a, a)
elif n == 3: return fmatmul(fmatmul(a, a), a)
# Use binary decomposition to reduce the number of matrix multiplications. # Here, we iterate over the bits of n, from LSB to MSB, raise `a` to # increasing powers of 2, and multiply into the result as needed. z = result = None while n > 0: z = a if z is None else fmatmul(z, z) n, bit = divmod(n, 2) if bit: result = z if result is None else fmatmul(result, z)
return result
# Cholesky decomposition
def cholesky(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. 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 Upper or 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.
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 >>> LA.cholesky(np.matrix(A)) matrix([[ 1.+0.j, 0.+0.j], [ 0.+2.j, 1.+0.j]])
""" extobj = get_linalg_error_extobj(_raise_linalgerror_nonposdef) gufunc = _umath_linalg.cholesky_lo a, wrap = _makearray(a) _assertRankAtLeast2(a) _assertNdSquareness(a) t, result_t = _commonType(a) signature = 'D->D' if isComplexType(t) else 'd->d' r = gufunc(a, signature=signature, extobj=extobj) return wrap(r.astype(result_t, copy=False))
# QR decompostion
return (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) Matrix to be factored. mode : {'reduced', 'complete', 'r', 'raw', 'full', 'economic'}, 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,) * 'full' : alias of 'reduced', deprecated * 'economic' : returns h from 'raw', deprecated.
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 ------- 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. r : ndarray of float or complex, optional The upper-triangular matrix. (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.
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') >>> r3 = np.linalg.qr(a, mode='economic') >>> np.allclose(r, r2) # mode='r' returns the same r as mode='full' True >>> # But only triu parts are guaranteed equal when mode='economic' >>> np.allclose(r, np.triu(r3[:6,:6], k=0)) 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, 0, 2, 1]) >>> q, r = LA.qr(A) >>> p = np.dot(q.T, b) >>> np.dot(LA.inv(r), p) array([ 1.1e-16, 1.0e+00])
""" if mode not in ('reduced', 'complete', 'r', 'raw'): if mode in ('f', 'full'): # 2013-04-01, 1.8 msg = "".join(( "The 'full' option is deprecated in favor of 'reduced'.\n", "For backward compatibility let mode default.")) warnings.warn(msg, DeprecationWarning, stacklevel=2) mode = 'reduced' elif mode in ('e', 'economic'): # 2013-04-01, 1.8 msg = "The 'economic' option is deprecated." warnings.warn(msg, DeprecationWarning, stacklevel=2) mode = 'economic' else: raise ValueError("Unrecognized mode '%s'" % mode)
a, wrap = _makearray(a) _assertRank2(a) m, n = a.shape t, result_t = _commonType(a) a = _fastCopyAndTranspose(t, a) a = _to_native_byte_order(a) mn = min(m, n) tau = zeros((mn,), t)
if isComplexType(t): lapack_routine = lapack_lite.zgeqrf routine_name = 'zgeqrf' else: lapack_routine = lapack_lite.dgeqrf routine_name = 'dgeqrf'
# calculate optimal size of work data 'work' lwork = 1 work = zeros((lwork,), t) results = lapack_routine(m, n, a, max(1, m), tau, work, -1, 0) if results['info'] != 0: raise LinAlgError('%s returns %d' % (routine_name, results['info']))
# do qr decomposition lwork = max(1, n, int(abs(work[0]))) work = zeros((lwork,), t) results = lapack_routine(m, n, a, max(1, m), tau, work, lwork, 0) if results['info'] != 0: raise LinAlgError('%s returns %d' % (routine_name, results['info']))
# handle modes that don't return q if mode == 'r': r = _fastCopyAndTranspose(result_t, a[:, :mn]) return wrap(triu(r))
if mode == 'raw': return a, tau
if mode == 'economic': if t != result_t : a = a.astype(result_t, copy=False) return wrap(a.T)
# generate q from a if mode == 'complete' and m > n: mc = m q = empty((m, m), t) else: mc = mn q = empty((n, m), t) q[:n] = a
if isComplexType(t): lapack_routine = lapack_lite.zungqr routine_name = 'zungqr' else: lapack_routine = lapack_lite.dorgqr routine_name = 'dorgqr'
# determine optimal lwork lwork = 1 work = zeros((lwork,), t) results = lapack_routine(m, mc, mn, q, max(1, m), tau, work, -1, 0) if results['info'] != 0: raise LinAlgError('%s returns %d' % (routine_name, results['info']))
# compute q lwork = max(1, n, int(abs(work[0]))) work = zeros((lwork,), t) results = lapack_routine(m, mc, mn, q, max(1, m), tau, work, lwork, 0) if results['info'] != 0: raise LinAlgError('%s returns %d' % (routine_name, results['info']))
q = _fastCopyAndTranspose(result_t, q[:mc]) r = _fastCopyAndTranspose(result_t, a[:, :mc])
return wrap(q), wrap(triu(r))
# Eigenvalues
def eigvals(a): """ Compute the eigenvalues of a general matrix.
Main difference between `eigvals` and `eig`: the eigenvectors aren't returned.
Parameters ---------- a : (..., M, M) array_like A complex- or real-valued matrix whose eigenvalues will be computed.
Returns ------- w : (..., M,) ndarray The eigenvalues, each repeated according to its multiplicity. They are not necessarily ordered, nor are they necessarily real for real matrices.
Raises ------ LinAlgError If the eigenvalue computation does not converge.
See Also -------- eig : eigenvalues and right eigenvectors of general arrays eigvalsh : eigenvalues of real symmetric or complex Hermitian (conjugate symmetric) arrays. eigh : eigenvalues and eigenvectors of real symmetric or complex Hermitian (conjugate symmetric) arrays.
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.
Examples -------- Illustration, using the fact that the eigenvalues of a diagonal matrix are its diagonal elements, that multiplying a matrix on the left by an orthogonal matrix, `Q`, and on the right by `Q.T` (the transpose of `Q`), preserves the eigenvalues of the "middle" matrix. In other words, if `Q` is orthogonal, then ``Q * A * Q.T`` has the same eigenvalues as ``A``:
>>> 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.])
""" a, wrap = _makearray(a) _assertRankAtLeast2(a) _assertNdSquareness(a) _assertFinite(a) t, result_t = _commonType(a)
extobj = get_linalg_error_extobj( _raise_linalgerror_eigenvalues_nonconvergence) signature = 'D->D' if isComplexType(t) else 'd->D' w = _umath_linalg.eigvals(a, signature=signature, extobj=extobj)
if not isComplexType(t): if all(w.imag == 0): w = w.real result_t = _realType(result_t) else: result_t = _complexType(result_t)
return w.astype(result_t, copy=False)
return (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.
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])
>>> # 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])
""" UPLO = UPLO.upper() if UPLO not in ('L', 'U'): raise ValueError("UPLO argument must be 'L' or 'U'")
extobj = get_linalg_error_extobj( _raise_linalgerror_eigenvalues_nonconvergence) if UPLO == 'L': gufunc = _umath_linalg.eigvalsh_lo else: gufunc = _umath_linalg.eigvalsh_up
a, wrap = _makearray(a) _assertRankAtLeast2(a) _assertNdSquareness(a) t, result_t = _commonType(a) signature = 'D->d' if isComplexType(t) else 'd->d' w = gufunc(a, signature=signature, extobj=extobj) return w.astype(_realType(result_t), copy=False)
t, result_t = _commonType(a) a = _fastCT(a.astype(t)) return a, t, result_t
# Eigenvectors
def eig(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 ------- w : (..., 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
v : (..., M, M) array The normalized (unit "length") eigenvectors, such that the column ``v[:,i]`` is the eigenvector corresponding to the eigenvalue ``w[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.
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 ``dot(a,v) = w * v``. Thus, the arrays `a`, `w`, and `v` satisfy the equations ``dot(a[:,:], v[:,i]) = w[i] * v[:,i]`` for :math:`i \\in \\{0,...,M-1\\}`.
The array `v` of 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. Likewise, the (complex-valued) matrix of eigenvectors `v` is unitary if the matrix `a` is normal, i.e., if ``dot(a, a.H) = dot(a.H, a)``, where `a.H` denotes the conjugate transpose of `a`.
Finally, it is emphasized that `v` consists of the *right* (as in right-hand side) eigenvectors of `a`. A vector `y` satisfying ``dot(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 e-values and e-vectors.
>>> w, v = LA.eig(np.diag((1, 2, 3))) >>> w; v array([ 1., 2., 3.]) array([[ 1., 0., 0.], [ 0., 1., 0.], [ 0., 0., 1.]])
Real matrix possessing complex e-values and e-vectors; note that the e-values are complex conjugates of each other.
>>> w, v = LA.eig(np.array([[1, -1], [1, 1]])) >>> w; v array([ 1. + 1.j, 1. - 1.j]) array([[ 0.70710678+0.j , 0.70710678+0.j ], [ 0.00000000-0.70710678j, 0.00000000+0.70710678j]])
Complex-valued matrix with real e-values (but complex-valued e-vectors); note that a.conj().T = a, i.e., a is Hermitian.
>>> a = np.array([[1, 1j], [-1j, 1]]) >>> w, v = LA.eig(a) >>> w; v array([ 2.00000000e+00+0.j, 5.98651912e-36+0.j]) # i.e., {2, 0} array([[ 0.00000000+0.70710678j, 0.70710678+0.j ], [ 0.70710678+0.j , 0.00000000+0.70710678j]])
Be careful about round-off error!
>>> a = np.array([[1 + 1e-9, 0], [0, 1 - 1e-9]]) >>> # Theor. e-values are 1 +/- 1e-9 >>> w, v = LA.eig(a) >>> w; v array([ 1., 1.]) array([[ 1., 0.], [ 0., 1.]])
""" a, wrap = _makearray(a) _assertRankAtLeast2(a) _assertNdSquareness(a) _assertFinite(a) t, result_t = _commonType(a)
extobj = get_linalg_error_extobj( _raise_linalgerror_eigenvalues_nonconvergence) signature = 'D->DD' if isComplexType(t) else 'd->DD' w, vt = _umath_linalg.eig(a, signature=signature, extobj=extobj)
if not isComplexType(t) and all(w.imag == 0.0): w = w.real vt = vt.real result_t = _realType(result_t) else: result_t = _complexType(result_t)
vt = vt.astype(result_t, copy=False) return w.astype(result_t, copy=False), wrap(vt)
""" 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 ------- w : (..., M) ndarray The eigenvalues in ascending order, each repeated according to its multiplicity. v : {(..., M, M) ndarray, (..., M, M) matrix} The column ``v[:, i]`` is the normalized eigenvector corresponding to the eigenvalue ``w[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.
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 `v` of (column) eigenvectors is unitary and `a`, `w`, and `v` satisfy the equations ``dot(a, v[:, i]) = w[i] * v[:, 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]]) >>> w, v = LA.eigh(a) >>> w; v array([ 0.17157288, 5.82842712]) array([[-0.92387953+0.j , -0.38268343+0.j ], [ 0.00000000+0.38268343j, 0.00000000-0.92387953j]])
>>> np.dot(a, v[:, 0]) - w[0] * v[:, 0] # verify 1st e-val/vec pair array([2.77555756e-17 + 0.j, 0. + 1.38777878e-16j]) >>> np.dot(a, v[:, 1]) - w[1] * v[:, 1] # verify 2nd e-val/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]]) >>> w, v = LA.eigh(A) >>> w; v array([ 0.17157288, 5.82842712]) matrix([[-0.92387953+0.j , -0.38268343+0.j ], [ 0.00000000+0.38268343j, 0.00000000-0.92387953j]])
>>> # 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([[-0.44721360-0.j , -0.89442719+0.j ], [ 0.00000000+0.89442719j, 0.00000000-0.4472136j ]]) array([[ 0.89442719+0.j , 0.00000000-0.4472136j], [ 0.00000000-0.4472136j, 0.89442719+0.j ]]) """ raise ValueError("UPLO argument must be 'L' or 'U'")
extobj = get_linalg_error_extobj( _raise_linalgerror_eigenvalues_nonconvergence) else: gufunc = _umath_linalg.eigh_up
# Singular value decomposition
return (a,)
""" Singular Value Decomposition.
When `a` is a 2D array, it is factorized as ``u @ np.diag(s) @ vh = (u * s) @ vh``, where `u` and `vh` are 2D unitary arrays and `s` is a 1D array of `a`'s singular values. When `a` is higher-dimensional, SVD is applied in stacked mode as explained below.
Parameters ---------- a : (..., M, N) array_like A real or complex array with ``a.ndim >= 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.
Returns ------- 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.
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
""" a, wrap = _makearray(a) _assertRankAtLeast2(a) t, result_t = _commonType(a)
extobj = get_linalg_error_extobj(_raise_linalgerror_svd_nonconvergence)
m, n = a.shape[-2:] if compute_uv: if full_matrices: if m < n: gufunc = _umath_linalg.svd_m_f else: gufunc = _umath_linalg.svd_n_f else: if m < n: gufunc = _umath_linalg.svd_m_s else: gufunc = _umath_linalg.svd_n_s
signature = 'D->DdD' if isComplexType(t) else 'd->ddd' u, s, vh = gufunc(a, signature=signature, extobj=extobj) u = u.astype(result_t, copy=False) s = s.astype(_realType(result_t), copy=False) vh = vh.astype(result_t, copy=False) return wrap(u), s, wrap(vh) else: if m < n: gufunc = _umath_linalg.svd_m else: gufunc = _umath_linalg.svd_n
signature = 'D->d' if isComplexType(t) else 'd->d' s = gufunc(a, signature=signature, extobj=extobj) s = s.astype(_realType(result_t), copy=False) return s
return (x,)
""" Compute the condition number of a matrix.
This function is capable of returning the condition number using one of seven different norms, depending on the value of `p` (see Parameters below).
Parameters ---------- x : (..., M, N) array_like The matrix whose condition number is sought. p : {None, 1, -1, 2, -2, inf, -inf, 'fro'}, optional Order of the norm:
===== ============================ p norm for matrices ===== ============================ None 2-norm, computed directly using the ``SVD`` 'fro' Frobenius norm inf max(sum(abs(x), axis=1)) -inf min(sum(abs(x), axis=1)) 1 max(sum(abs(x), axis=0)) -1 min(sum(abs(x), axis=0)) 2 2-norm (largest sing. value) -2 smallest singular value ===== ============================
inf means the numpy.inf object, and the Frobenius norm is the root-of-sum-of-squares norm.
Returns ------- c : {float, inf} The condition number of the matrix. May be infinite.
See Also -------- numpy.linalg.norm
Notes ----- The condition number of `x` is defined as the norm of `x` times the norm of the inverse of `x` [1]_; the norm can be the usual L2-norm (root-of-sum-of-squares) or one of a number of other matrix norms.
References ---------- .. [1] G. Strang, *Linear Algebra and Its Applications*, Orlando, FL, Academic Press, Inc., 1980, pg. 285.
Examples -------- >>> 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 >>> min(LA.svd(a, compute_uv=0))*min(LA.svd(LA.inv(a), compute_uv=0)) 0.70710678118654746
""" x = asarray(x) # in case we have a matrix _assertNoEmpty2d(x) if p is None or p == 2 or p == -2: s = svd(x, compute_uv=False) with errstate(all='ignore'): if p == -2: r = s[..., -1] / s[..., 0] else: r = s[..., 0] / s[..., -1] else: # Call inv(x) ignoring errors. The result array will # contain nans in the entries where inversion failed. _assertRankAtLeast2(x) _assertNdSquareness(x) t, result_t = _commonType(x) signature = 'D->D' if isComplexType(t) else 'd->d' with errstate(all='ignore'): invx = _umath_linalg.inv(x, signature=signature) r = norm(x, p, axis=(-2, -1)) * norm(invx, p, axis=(-2, -1)) r = r.astype(result_t, copy=False)
# Convert nans to infs unless the original array had nan entries r = asarray(r) nan_mask = isnan(r) if nan_mask.any(): nan_mask &= ~isnan(x).any(axis=(-2, -1)) if r.ndim > 0: r[nan_mask] = Inf elif nan_mask: r[()] = Inf
# Convention is to return scalars instead of 0d arrays if r.ndim == 0: r = r[()]
return r
return (M,)
""" Return matrix rank of array using SVD method
Rank of the array is the number of singular values of the array that are greater than `tol`.
.. versionchanged:: 1.14 Can now operate on stacks of matrices
Parameters ---------- M : {(M,), (..., M, N)} array_like input vector or stack of matrices tol : (...) array_like, float, optional threshold below which SVD values are considered zero. If `tol` is None, and ``S`` is an array with singular values for `M`, and ``eps`` is the epsilon value for datatype of ``S``, then `tol` is set to ``S.max() * max(M.shape) * eps``.
.. versionchanged:: 1.14 Broadcasted against the stack of matrices hermitian : bool, optional If True, `M` is assumed to be Hermitian (symmetric if real-valued), enabling a more efficient method for finding singular values. Defaults to False.
.. versionadded:: 1.14
Notes ----- The default threshold to detect rank deficiency is a test on the magnitude of the singular values of `M`. By default, we identify singular values less than ``S.max() * max(M.shape) * eps`` as indicating rank deficiency (with the symbols defined above). This is the algorithm MATLAB uses [1]. It also appears in *Numerical recipes* in the discussion of SVD solutions for linear least squares [2].
This default threshold is designed to detect rank deficiency accounting for the numerical errors of the SVD computation. Imagine that there is a column in `M` that is an exact (in floating point) linear combination of other columns in `M`. Computing the SVD on `M` will not produce a singular value exactly equal to 0 in general: any difference of the smallest SVD value from 0 will be caused by numerical imprecision in the calculation of the SVD. Our threshold for small SVD values takes this numerical imprecision into account, and the default threshold will detect such numerical rank deficiency. The threshold may declare a matrix `M` rank deficient even if the linear combination of some columns of `M` is not exactly equal to another column of `M` but only numerically very close to another column of `M`.
We chose our default threshold because it is in wide use. Other thresholds are possible. For example, elsewhere in the 2007 edition of *Numerical recipes* there is an alternative threshold of ``S.max() * np.finfo(M.dtype).eps / 2. * np.sqrt(m + n + 1.)``. The authors describe this threshold as being based on "expected roundoff error" (p 71).
The thresholds above deal with floating point roundoff error in the calculation of the SVD. However, you may have more information about the sources of error in `M` that would make you consider other tolerance values to detect *effective* rank deficiency. The most useful measure of the tolerance depends on the operations you intend to use on your matrix. For example, if your data come from uncertain measurements with uncertainties greater than floating point epsilon, choosing a tolerance near that uncertainty may be preferable. The tolerance may be absolute if the uncertainties are absolute rather than relative.
References ---------- .. [1] MATLAB reference documention, "Rank" https://www.mathworks.com/help/techdoc/ref/rank.html .. [2] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, "Numerical Recipes (3rd edition)", Cambridge University Press, 2007, page 795.
Examples -------- >>> 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 """ M = asarray(M) if M.ndim < 2: return int(not all(M==0)) if hermitian: S = abs(eigvalsh(M)) else: S = svd(M, compute_uv=False) if tol is None: tol = S.max(axis=-1, keepdims=True) * max(M.shape[-2:]) * finfo(S.dtype).eps else: tol = asarray(tol)[..., newaxis] return count_nonzero(S > tol, axis=-1)
# Generalized inverse
return (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 smaller (in modulus) than `rcond` * largest_singular_value (again, in modulus) are set to zero. Broadcasts against the stack of matrices
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.
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
""" a, wrap = _makearray(a) rcond = asarray(rcond) if _isEmpty2d(a): m, n = a.shape[-2:] res = empty(a.shape[:-2] + (n, m), dtype=a.dtype) return wrap(res) a = a.conjugate() u, s, vt = svd(a, full_matrices=False)
# discard small singular values cutoff = rcond[..., newaxis] * amax(s, axis=-1, keepdims=True) large = s > cutoff s = divide(1, s, where=large, out=s) s[~large] = 0
res = matmul(transpose(vt), multiply(s[..., newaxis], transpose(u))) return wrap(res)
# Determinant
def slogdet(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 ------- 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. logdet : (...) array_like The natural log of the absolute value of the determinant.
If the determinant is zero, then `sign` will be 0 and `logdet` will be -Inf. In all cases, the determinant is equal to ``sign * np.exp(logdet)``.
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, logdet) = np.linalg.slogdet(a) >>> (sign, logdet) (-1, 0.69314718055994529) >>> sign * np.exp(logdet) -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, logdet = np.linalg.slogdet(a) >>> (sign, logdet) (array([-1., -1., -1.]), array([ 0.69314718, 1.09861229, 2.07944154])) >>> sign * np.exp(logdet) 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)
""" a = asarray(a) _assertRankAtLeast2(a) _assertNdSquareness(a) t, result_t = _commonType(a) real_t = _realType(result_t) signature = 'D->Dd' if isComplexType(t) else 'd->dd' sign, logdet = _umath_linalg.slogdet(a, signature=signature) sign = sign.astype(result_t, copy=False) logdet = logdet.astype(real_t, copy=False) return sign, logdet
def det(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.
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
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.])
"""
# Linear Least Squares
return (a, b)
""" Return the least-squares solution to a linear matrix equation.
Solves the equation `a x = b` by computing a vector `x` that minimizes the Euclidean 2-norm `|| b - a x ||^2`. 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.
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 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.
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] >>> print(m, c) 1.0 -0.95
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()
""" a, _ = _makearray(a) b, wrap = _makearray(b) is_1d = b.ndim == 1 if is_1d: b = b[:, newaxis] _assertRank2(a, b) m, n = a.shape[-2:] m2, n_rhs = b.shape[-2:] if m != m2: raise LinAlgError('Incompatible dimensions')
t, result_t = _commonType(a, b) # FIXME: real_t is unused real_t = _linalgRealType(t) result_real_t = _realType(result_t)
# Determine default rcond value if rcond == "warn": # 2017-08-19, 1.14.0 warnings.warn("`rcond` parameter will change to the default of " "machine precision times ``max(M, N)`` where M and N " "are the input matrix dimensions.\n" "To use the future default and silence this warning " "we advise to pass `rcond=None`, to keep using the old, " "explicitly pass `rcond=-1`.", FutureWarning, stacklevel=2) rcond = -1 if rcond is None: rcond = finfo(t).eps * max(n, m)
if m <= n: gufunc = _umath_linalg.lstsq_m else: gufunc = _umath_linalg.lstsq_n
signature = 'DDd->Ddid' if isComplexType(t) else 'ddd->ddid' extobj = get_linalg_error_extobj(_raise_linalgerror_lstsq) if n_rhs == 0: # lapack can't handle n_rhs = 0 - so allocate the array one larger in that axis b = zeros(b.shape[:-2] + (m, n_rhs + 1), dtype=b.dtype) x, resids, rank, s = gufunc(a, b, rcond, signature=signature, extobj=extobj) if m == 0: x[...] = 0 if n_rhs == 0: # remove the item we added x = x[..., :n_rhs] resids = resids[..., :n_rhs]
# remove the axis we added if is_1d: x = x.squeeze(axis=-1) # we probably should squeeze resids too, but we can't # without breaking compatibility.
# as documented if rank != n or m <= n: resids = array([], result_real_t)
# coerce output arrays s = s.astype(result_real_t, copy=False) resids = resids.astype(result_real_t, copy=False) x = x.astype(result_t, copy=True) # Copying lets the memory in r_parts be freed return wrap(x), wrap(resids), rank, s
"""Compute 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`.
""" y = moveaxis(x, (row_axis, col_axis), (-2, -1)) result = op(svd(y, compute_uv=0), axis=-1) return result
return (x,)
""" 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. ord : {non-zero int, inf, -inf, 'fro', 'nuc'}, optional Order of the norm (see table under ``Notes``). inf means numpy's `inf` object. axis : {int, 2-tuple of ints, None}, 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.
.. 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).
Notes ----- For values of ``ord <= 0``, 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.
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, -1, 0, 1, 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) nan >>> LA.norm(b, -2) 1.8570331885190563e-016 >>> LA.norm(a, 3) 5.8480354764257312 >>> LA.norm(a, -3) nan
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)
"""
x = x.astype(float)
# Immediately handle some default, simple, fast, and common cases. (ord in ('f', 'fro') and ndim == 2) or (ord == 2 and ndim == 1)):
sqnorm = dot(x.real, x.real) + dot(x.imag, x.imag) else: ret = ret.reshape(ndim*[1])
# Normalize the `axis` argument to a tuple. nd = x.ndim if axis is None: axis = tuple(range(nd)) elif not isinstance(axis, tuple): try: axis = int(axis) except Exception: raise TypeError("'axis' must be None, an integer or a tuple of integers") axis = (axis,)
if len(axis) == 1: if ord == Inf: return abs(x).max(axis=axis, keepdims=keepdims) elif ord == -Inf: return abs(x).min(axis=axis, keepdims=keepdims) elif ord == 0: # Zero norm return (x != 0).astype(x.real.dtype).sum(axis=axis, keepdims=keepdims) elif ord == 1: # special case for speedup return add.reduce(abs(x), axis=axis, keepdims=keepdims) elif ord is None or ord == 2: # special case for speedup s = (x.conj() * x).real return sqrt(add.reduce(s, axis=axis, keepdims=keepdims)) else: try: ord + 1 except TypeError: raise ValueError("Invalid norm order for vectors.") absx = abs(x) absx **= ord ret = add.reduce(absx, axis=axis, keepdims=keepdims) ret **= (1 / ord) return ret elif len(axis) == 2: row_axis, col_axis = axis row_axis = normalize_axis_index(row_axis, nd) col_axis = normalize_axis_index(col_axis, nd) if row_axis == col_axis: raise ValueError('Duplicate axes given.') if ord == 2: ret = _multi_svd_norm(x, row_axis, col_axis, amax) elif ord == -2: ret = _multi_svd_norm(x, row_axis, col_axis, amin) elif ord == 1: if col_axis > row_axis: col_axis -= 1 ret = add.reduce(abs(x), axis=row_axis).max(axis=col_axis) elif ord == Inf: if row_axis > col_axis: row_axis -= 1 ret = add.reduce(abs(x), axis=col_axis).max(axis=row_axis) elif ord == -1: if col_axis > row_axis: col_axis -= 1 ret = add.reduce(abs(x), axis=row_axis).min(axis=col_axis) elif ord == -Inf: if row_axis > col_axis: row_axis -= 1 ret = add.reduce(abs(x), axis=col_axis).min(axis=row_axis) elif ord in [None, 'fro', 'f']: ret = sqrt(add.reduce((x.conj() * x).real, axis=axis)) elif ord == 'nuc': ret = _multi_svd_norm(x, row_axis, col_axis, sum) else: raise ValueError("Invalid norm order for matrices.") if keepdims: ret_shape = list(x.shape) ret_shape[axis[0]] = 1 ret_shape[axis[1]] = 1 ret = ret.reshape(ret_shape) return ret else: raise ValueError("Improper number of dimensions to norm.")
# multi_dot
return arrays
def multi_dot(arrays): """ 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.
Returns ------- output : ndarray Returns the dot product of the supplied arrays.
See Also -------- 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]
Let's 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
""" n = len(arrays) # optimization only makes sense for len(arrays) > 2 if n < 2: raise ValueError("Expecting at least two arrays.") elif n == 2: return dot(arrays[0], arrays[1])
arrays = [asanyarray(a) for a in arrays]
# save original ndim to reshape the result array into the proper form later ndim_first, ndim_last = arrays[0].ndim, arrays[-1].ndim # Explicitly convert vectors to 2D arrays to keep the logic of the internal # _multi_dot_* functions as simple as possible. if arrays[0].ndim == 1: arrays[0] = atleast_2d(arrays[0]) if arrays[-1].ndim == 1: arrays[-1] = atleast_2d(arrays[-1]).T _assertRank2(*arrays)
# _multi_dot_three is much faster than _multi_dot_matrix_chain_order if n == 3: result = _multi_dot_three(arrays[0], arrays[1], arrays[2]) else: order = _multi_dot_matrix_chain_order(arrays) result = _multi_dot(arrays, order, 0, n - 1)
# return proper shape if ndim_first == 1 and ndim_last == 1: return result[0, 0] # scalar elif ndim_first == 1 or ndim_last == 1: return result.ravel() # 1-D else: return result
""" 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`
""" a0, a1b0 = A.shape b1c0, c1 = C.shape # cost1 = cost((AB)C) = a0*a1b0*b1c0 + a0*b1c0*c1 cost1 = a0 * b1c0 * (a1b0 + c1) # cost2 = cost(A(BC)) = a1b0*b1c0*c1 + a0*a1b0*c1 cost2 = a1b0 * c1 * (a0 + b1c0)
if cost1 < cost2: return dot(dot(A, B), C) else: return dot(A, dot(B, C))
""" 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)])
""" n = len(arrays) # p stores the dimensions of the matrices # Example for p: A_{10x100}, B_{100x5}, C_{5x50} --> p = [10, 100, 5, 50] p = [a.shape[0] for a in arrays] + [arrays[-1].shape[1]] # m is a matrix of costs of the subproblems # m[i,j]: min number of scalar multiplications needed to compute A_{i..j} m = zeros((n, n), dtype=double) # s is the actual ordering # s[i, j] is the value of k at which we split the product A_i..A_j s = empty((n, n), dtype=intp)
for l in range(1, n): for i in range(n - l): j = i + l m[i, j] = Inf for k in range(i, j): q = m[i, k] + m[k+1, j] + p[i]*p[k+1]*p[j+1] if q < m[i, j]: m[i, j] = q s[i, j] = k # Note that Cormen uses 1-based index
return (s, m) if return_costs else s
"""Actually do the multiplication with the given order.""" if i == j: return arrays[i] else: return dot(_multi_dot(arrays, order, i, order[i, j]), _multi_dot(arrays, order, order[i, j] + 1, j)) |