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# # Author: Travis Oliphant, March 2002 #
'tanhm','logm','funm','signm','sqrtm', 'expm_frechet', 'expm_cond', 'fractional_matrix_power']
transpose, conjugate, absolute, amax, sign, isfinite, single)
# Local imports
############################################################################### # Utility functions.
""" Wraps asarray with the extra requirement that the input be a square matrix.
The motivation is that the matfuncs module has real functions that have been lifted to square matrix functions.
Parameters ---------- A : array_like A square matrix.
Returns ------- out : ndarray An ndarray copy or view or other representation of A.
""" A = np.asarray(A) if len(A.shape) != 2 or A.shape[0] != A.shape[1]: raise ValueError('expected square array_like input') return A
""" Return either B or the real part of B, depending on properties of A and B.
The motivation is that B has been computed as a complicated function of A, and B may be perturbed by negligible imaginary components. If A is real and B is complex with small imaginary components, then return a real copy of B. The assumption in that case would be that the imaginary components of B are numerical artifacts.
Parameters ---------- A : ndarray Input array whose type is to be checked as real vs. complex. B : ndarray Array to be returned, possibly without its imaginary part. tol : float Absolute tolerance.
Returns ------- out : real or complex array Either the input array B or only the real part of the input array B.
""" # Note that booleans and integers compare as real. if np.isrealobj(A) and np.iscomplexobj(B): if tol is None: tol = {0:feps*1e3, 1:eps*1e6}[_array_precision[B.dtype.char]] if np.allclose(B.imag, 0.0, atol=tol): B = B.real return B
############################################################################### # Matrix functions.
""" Compute the fractional power of a matrix.
Proceeds according to the discussion in section (6) of [1]_.
Parameters ---------- A : (N, N) array_like Matrix whose fractional power to evaluate. t : float Fractional power.
Returns ------- X : (N, N) array_like The fractional power of the matrix.
References ---------- .. [1] Nicholas J. Higham and Lijing lin (2011) "A Schur-Pade Algorithm for Fractional Powers of a Matrix." SIAM Journal on Matrix Analysis and Applications, 32 (3). pp. 1056-1078. ISSN 0895-4798
Examples -------- >>> from scipy.linalg import fractional_matrix_power >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> b = fractional_matrix_power(a, 0.5) >>> b array([[ 0.75592895, 1.13389342], [ 0.37796447, 1.88982237]]) >>> np.dot(b, b) # Verify square root array([[ 1., 3.], [ 1., 4.]])
""" # This fixes some issue with imports; # this function calls onenormest which is in scipy.sparse. A = _asarray_square(A) import scipy.linalg._matfuncs_inv_ssq return scipy.linalg._matfuncs_inv_ssq._fractional_matrix_power(A, t)
""" Compute matrix logarithm.
The matrix logarithm is the inverse of expm: expm(logm(`A`)) == `A`
Parameters ---------- A : (N, N) array_like Matrix whose logarithm to evaluate disp : bool, optional Print warning if error in the result is estimated large instead of returning estimated error. (Default: True)
Returns ------- logm : (N, N) ndarray Matrix logarithm of `A` errest : float (if disp == False)
1-norm of the estimated error, ||err||_1 / ||A||_1
References ---------- .. [1] Awad H. Al-Mohy and Nicholas J. Higham (2012) "Improved Inverse Scaling and Squaring Algorithms for the Matrix Logarithm." SIAM Journal on Scientific Computing, 34 (4). C152-C169. ISSN 1095-7197
.. [2] Nicholas J. Higham (2008) "Functions of Matrices: Theory and Computation" ISBN 978-0-898716-46-7
.. [3] Nicholas J. Higham and Lijing lin (2011) "A Schur-Pade Algorithm for Fractional Powers of a Matrix." SIAM Journal on Matrix Analysis and Applications, 32 (3). pp. 1056-1078. ISSN 0895-4798
Examples -------- >>> from scipy.linalg import logm, expm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> b = logm(a) >>> b array([[-1.02571087, 2.05142174], [ 0.68380725, 1.02571087]]) >>> expm(b) # Verify expm(logm(a)) returns a array([[ 1., 3.], [ 1., 4.]])
""" A = _asarray_square(A) # Avoid circular import ... this is OK, right? import scipy.linalg._matfuncs_inv_ssq F = scipy.linalg._matfuncs_inv_ssq._logm(A) F = _maybe_real(A, F) errtol = 1000*eps #TODO use a better error approximation errest = norm(expm(F)-A,1) / norm(A,1) if disp: if not isfinite(errest) or errest >= errtol: print("logm result may be inaccurate, approximate err =", errest) return F else: return F, errest
""" Compute the matrix exponential using Pade approximation.
Parameters ---------- A : (N, N) array_like or sparse matrix Matrix to be exponentiated.
Returns ------- expm : (N, N) ndarray Matrix exponential of `A`.
References ---------- .. [1] Awad H. Al-Mohy and Nicholas J. Higham (2009) "A New Scaling and Squaring Algorithm for the Matrix Exponential." SIAM Journal on Matrix Analysis and Applications. 31 (3). pp. 970-989. ISSN 1095-7162
Examples -------- >>> from scipy.linalg import expm, sinm, cosm
Matrix version of the formula exp(0) = 1:
>>> expm(np.zeros((2,2))) array([[ 1., 0.], [ 0., 1.]])
Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta)) applied to a matrix:
>>> a = np.array([[1.0, 2.0], [-1.0, 3.0]]) >>> expm(1j*a) array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) >>> cosm(a) + 1j*sinm(a) array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])
""" # Input checking and conversion is provided by sparse.linalg.expm(). import scipy.sparse.linalg return scipy.sparse.linalg.expm(A)
""" Compute the matrix cosine.
This routine uses expm to compute the matrix exponentials.
Parameters ---------- A : (N, N) array_like Input array
Returns ------- cosm : (N, N) ndarray Matrix cosine of A
Examples -------- >>> from scipy.linalg import expm, sinm, cosm
Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta)) applied to a matrix:
>>> a = np.array([[1.0, 2.0], [-1.0, 3.0]]) >>> expm(1j*a) array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) >>> cosm(a) + 1j*sinm(a) array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])
""" A = _asarray_square(A) if np.iscomplexobj(A): return 0.5*(expm(1j*A) + expm(-1j*A)) else: return expm(1j*A).real
""" Compute the matrix sine.
This routine uses expm to compute the matrix exponentials.
Parameters ---------- A : (N, N) array_like Input array.
Returns ------- sinm : (N, N) ndarray Matrix sine of `A`
Examples -------- >>> from scipy.linalg import expm, sinm, cosm
Euler's identity (exp(i*theta) = cos(theta) + i*sin(theta)) applied to a matrix:
>>> a = np.array([[1.0, 2.0], [-1.0, 3.0]]) >>> expm(1j*a) array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]]) >>> cosm(a) + 1j*sinm(a) array([[ 0.42645930+1.89217551j, -2.13721484-0.97811252j], [ 1.06860742+0.48905626j, -1.71075555+0.91406299j]])
""" A = _asarray_square(A) if np.iscomplexobj(A): return -0.5j*(expm(1j*A) - expm(-1j*A)) else: return expm(1j*A).imag
""" Compute the matrix tangent.
This routine uses expm to compute the matrix exponentials.
Parameters ---------- A : (N, N) array_like Input array.
Returns ------- tanm : (N, N) ndarray Matrix tangent of `A`
Examples -------- >>> from scipy.linalg import tanm, sinm, cosm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> t = tanm(a) >>> t array([[ -2.00876993, -8.41880636], [ -2.80626879, -10.42757629]])
Verify tanm(a) = sinm(a).dot(inv(cosm(a)))
>>> s = sinm(a) >>> c = cosm(a) >>> s.dot(np.linalg.inv(c)) array([[ -2.00876993, -8.41880636], [ -2.80626879, -10.42757629]])
""" A = _asarray_square(A) return _maybe_real(A, solve(cosm(A), sinm(A)))
""" Compute the hyperbolic matrix cosine.
This routine uses expm to compute the matrix exponentials.
Parameters ---------- A : (N, N) array_like Input array.
Returns ------- coshm : (N, N) ndarray Hyperbolic matrix cosine of `A`
Examples -------- >>> from scipy.linalg import tanhm, sinhm, coshm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> c = coshm(a) >>> c array([[ 11.24592233, 38.76236492], [ 12.92078831, 50.00828725]])
Verify tanhm(a) = sinhm(a).dot(inv(coshm(a)))
>>> t = tanhm(a) >>> s = sinhm(a) >>> t - s.dot(np.linalg.inv(c)) array([[ 2.72004641e-15, 4.55191440e-15], [ 0.00000000e+00, -5.55111512e-16]])
""" A = _asarray_square(A) return _maybe_real(A, 0.5 * (expm(A) + expm(-A)))
""" Compute the hyperbolic matrix sine.
This routine uses expm to compute the matrix exponentials.
Parameters ---------- A : (N, N) array_like Input array.
Returns ------- sinhm : (N, N) ndarray Hyperbolic matrix sine of `A`
Examples -------- >>> from scipy.linalg import tanhm, sinhm, coshm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> s = sinhm(a) >>> s array([[ 10.57300653, 39.28826594], [ 13.09608865, 49.86127247]])
Verify tanhm(a) = sinhm(a).dot(inv(coshm(a)))
>>> t = tanhm(a) >>> c = coshm(a) >>> t - s.dot(np.linalg.inv(c)) array([[ 2.72004641e-15, 4.55191440e-15], [ 0.00000000e+00, -5.55111512e-16]])
""" A = _asarray_square(A) return _maybe_real(A, 0.5 * (expm(A) - expm(-A)))
""" Compute the hyperbolic matrix tangent.
This routine uses expm to compute the matrix exponentials.
Parameters ---------- A : (N, N) array_like Input array
Returns ------- tanhm : (N, N) ndarray Hyperbolic matrix tangent of `A`
Examples -------- >>> from scipy.linalg import tanhm, sinhm, coshm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> t = tanhm(a) >>> t array([[ 0.3428582 , 0.51987926], [ 0.17329309, 0.86273746]])
Verify tanhm(a) = sinhm(a).dot(inv(coshm(a)))
>>> s = sinhm(a) >>> c = coshm(a) >>> t - s.dot(np.linalg.inv(c)) array([[ 2.72004641e-15, 4.55191440e-15], [ 0.00000000e+00, -5.55111512e-16]])
""" A = _asarray_square(A) return _maybe_real(A, solve(coshm(A), sinhm(A)))
""" Evaluate a matrix function specified by a callable.
Returns the value of matrix-valued function ``f`` at `A`. The function ``f`` is an extension of the scalar-valued function `func` to matrices.
Parameters ---------- A : (N, N) array_like Matrix at which to evaluate the function func : callable Callable object that evaluates a scalar function f. Must be vectorized (eg. using vectorize). disp : bool, optional Print warning if error in the result is estimated large instead of returning estimated error. (Default: True)
Returns ------- funm : (N, N) ndarray Value of the matrix function specified by func evaluated at `A` errest : float (if disp == False)
1-norm of the estimated error, ||err||_1 / ||A||_1
Examples -------- >>> from scipy.linalg import funm >>> a = np.array([[1.0, 3.0], [1.0, 4.0]]) >>> funm(a, lambda x: x*x) array([[ 4., 15.], [ 5., 19.]]) >>> a.dot(a) array([[ 4., 15.], [ 5., 19.]])
Notes ----- This function implements the general algorithm based on Schur decomposition (Algorithm 9.1.1. in [1]_).
If the input matrix is known to be diagonalizable, then relying on the eigendecomposition is likely to be faster. For example, if your matrix is Hermitian, you can do
>>> from scipy.linalg import eigh >>> def funm_herm(a, func, check_finite=False): ... w, v = eigh(a, check_finite=check_finite) ... ## if you further know that your matrix is positive semidefinite, ... ## you can optionally guard against precision errors by doing ... # w = np.maximum(w, 0) ... w = func(w) ... return (v * w).dot(v.conj().T)
References ---------- .. [1] Gene H. Golub, Charles F. van Loan, Matrix Computations 4th ed.
""" A = _asarray_square(A) # Perform Shur decomposition (lapack ?gees) T, Z = schur(A) T, Z = rsf2csf(T,Z) n,n = T.shape F = diag(func(diag(T))) # apply function to diagonal elements F = F.astype(T.dtype.char) # e.g. when F is real but T is complex
minden = abs(T[0,0])
# implement Algorithm 11.1.1 from Golub and Van Loan # "matrix Computations." for p in range(1,n): for i in range(1,n-p+1): j = i + p s = T[i-1,j-1] * (F[j-1,j-1] - F[i-1,i-1]) ksl = slice(i,j-1) val = dot(T[i-1,ksl],F[ksl,j-1]) - dot(F[i-1,ksl],T[ksl,j-1]) s = s + val den = T[j-1,j-1] - T[i-1,i-1] if den != 0.0: s = s / den F[i-1,j-1] = s minden = min(minden,abs(den))
F = dot(dot(Z, F), transpose(conjugate(Z))) F = _maybe_real(A, F)
tol = {0:feps, 1:eps}[_array_precision[F.dtype.char]] if minden == 0.0: minden = tol err = min(1, max(tol,(tol/minden)*norm(triu(T,1),1))) if product(ravel(logical_not(isfinite(F))),axis=0): err = Inf if disp: if err > 1000*tol: print("funm result may be inaccurate, approximate err =", err) return F else: return F, err
""" Matrix sign function.
Extension of the scalar sign(x) to matrices.
Parameters ---------- A : (N, N) array_like Matrix at which to evaluate the sign function disp : bool, optional Print warning if error in the result is estimated large instead of returning estimated error. (Default: True)
Returns ------- signm : (N, N) ndarray Value of the sign function at `A` errest : float (if disp == False)
1-norm of the estimated error, ||err||_1 / ||A||_1
Examples -------- >>> from scipy.linalg import signm, eigvals >>> a = [[1,2,3], [1,2,1], [1,1,1]] >>> eigvals(a) array([ 4.12488542+0.j, -0.76155718+0.j, 0.63667176+0.j]) >>> eigvals(signm(a)) array([-1.+0.j, 1.+0.j, 1.+0.j])
""" A = _asarray_square(A)
def rounded_sign(x): rx = np.real(x) if rx.dtype.char == 'f': c = 1e3*feps*amax(x) else: c = 1e3*eps*amax(x) return sign((absolute(rx) > c) * rx) result, errest = funm(A, rounded_sign, disp=0) errtol = {0:1e3*feps, 1:1e3*eps}[_array_precision[result.dtype.char]] if errest < errtol: return result
# Handle signm of defective matrices:
# See "E.D.Denman and J.Leyva-Ramos, Appl.Math.Comp., # 8:237-250,1981" for how to improve the following (currently a # rather naive) iteration process:
# a = result # sometimes iteration converges faster but where??
# Shifting to avoid zero eigenvalues. How to ensure that shifting does # not change the spectrum too much? vals = svd(A, compute_uv=0) max_sv = np.amax(vals) # min_nonzero_sv = vals[(vals>max_sv*errtol).tolist().count(1)-1] # c = 0.5/min_nonzero_sv c = 0.5/max_sv S0 = A + c*np.identity(A.shape[0]) prev_errest = errest for i in range(100): iS0 = inv(S0) S0 = 0.5*(S0 + iS0) Pp = 0.5*(dot(S0,S0)+S0) errest = norm(dot(Pp,Pp)-Pp,1) if errest < errtol or prev_errest == errest: break prev_errest = errest if disp: if not isfinite(errest) or errest >= errtol: print("signm result may be inaccurate, approximate err =", errest) return S0 else: return S0, errest |