def test_integer_matrix(self):
Q = np.array([
[-3, 1, 1, 1],
[1, -3, 1, 1],
[1, 1, -3, 1],
[1, 1, 1, -3]])
assert_allclose(expm(Q), expm(1.0 * Q))
def test_triangularity_perturbation(self):
# Experiment (1) of
# Awad H. Al-Mohy and Nicholas J. Higham (2012)
# Improved Inverse Scaling and Squaring Algorithms
# for the Matrix Logarithm.
A = np.array([
[3.2346e-1, 3e4, 3e4, 3e4],
[0, 3.0089e-1, 3e4, 3e4],
[0, 0, 3.221e-1, 3e4],
[0, 0, 0, 3.0744e-1]],
dtype=float)
A_logm = np.array([
[-1.12867982029050462e+00, 9.61418377142025565e+04,
-4.52485573953179264e+09, 2.92496941103871812e+14],
[0.00000000000000000e+00, -1.20101052953082288e+00,
9.63469687211303099e+04, -4.68104828911105442e+09],
[0.00000000000000000e+00, 0.00000000000000000e+00,
-1.13289322264498393e+00, 9.53249183094775653e+04],
[0.00000000000000000e+00, 0.00000000000000000e+00,
0.00000000000000000e+00, -1.17947533272554850e+00]],
dtype=float)
assert_allclose(expm(A_logm), A, rtol=1e-4)
# Perturb the upper triangular matrix by tiny amounts,
# so that it becomes technically not upper triangular.
random.seed(1234)
tiny = 1e-17
A_logm_perturbed = A_logm.copy()
A_logm_perturbed[1, 0] = tiny
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "Ill-conditioned.*")
A_expm_logm_perturbed = expm(A_logm_perturbed)
rtol = 1e-4
atol = 100 * tiny
assert_(not np.allclose(A_expm_logm_perturbed, A, rtol=rtol, atol=atol))
def test_integer_matrix_2(self):
# Check for integer overflows
Q = np.array([[-500, 500, 0, 0],
[0, -550, 360, 190],
[0, 630, -630, 0],
[0, 0, 0, 0]], dtype=np.int16)
assert_allclose(expm(Q), expm(1.0 * Q))
Q = csc_matrix(Q)
assert_allclose(expm(Q).A, expm(1.0 * Q).A)
def test_burkardt_11(self):
# This is Ward's example #2.
# It is a symmetric matrix.
A = np.array([
[29.87942128909879, 0.7815750847907159, -2.289519314033932],
[0.7815750847907159, 25.72656945571064, 8.680737820540137],
[-2.289519314033932, 8.680737820540137, 34.39400925519054],
], dtype=float)
assert_allclose(scipy.linalg.eigvalsh(A), (20, 30, 40))
desired = np.array([
[
5.496313853692378E+15,
-1.823188097200898E+16,
-3.047577080858001E+16],
[
-1.823188097200899E+16,
6.060522870222108E+16,
1.012918429302482E+17],
[
-3.047577080858001E+16,
1.012918429302482E+17,
1.692944112408493E+17],
], dtype=float)
actual = expm(A)
assert_allclose(actual, desired)
def test_burkardt_13(self):
# This is Ward's example #4.
# This is a version of the Forsythe matrix.
# The eigenvector problem is badly conditioned.
# Ward's algorithm has difficulty esimating the accuracy
# of its results for this problem.
#
# Check the construction of one instance of this family of matrices.
A4_actual = _burkardt_13_power(4, 1)
A4_desired = [[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1],
[1e-4, 0, 0, 0]]
assert_allclose(A4_actual, A4_desired)
# Check the expm for a few instances.
for n in (2, 3, 4, 10):
# Approximate expm using Taylor series.
# This works well for this matrix family
# because each matrix in the summation,
# even before dividing by the factorial,
# is entrywise positive with max entry 10**(-floor(p/n)*n).
k = max(1, int(np.ceil(16/n)))
desired = np.zeros((n, n), dtype=float)
for p in range(n*k):
Ap = _burkardt_13_power(n, p)
assert_equal(np.min(Ap), 0)
assert_allclose(np.max(Ap), np.power(10, -np.floor(p/n)*n))
desired += Ap / factorial(p)
actual = expm(_burkardt_13_power(n, 1))
assert_allclose(actual, desired)
def test_padecases_dtype_float(self):
for dtype in [np.float32, np.float64]:
for scale in [1e-2, 1e-1, 5e-1, 1, 10]:
A = scale * eye(3, dtype=dtype)
observed = expm(A)
expected = exp(scale) * eye(3, dtype=dtype)
assert_array_almost_equal_nulp(observed, expected, nulp=100)
def test_padecases_dtype_sparse_complex(self):
# float32 and complex64 lead to errors in spsolve/UMFpack
dtype = np.complex128
for scale in [1e-2, 1e-1, 5e-1, 1, 10]:
a = scale * speye(3, 3, dtype=dtype, format='csc')
e = exp(scale) * eye(3, dtype=dtype)
assert_array_almost_equal_nulp(expm(a).toarray(), e, nulp=100)
def test_overscaling_example(self):
# See the blog post
# http://blogs.mathworks.com/cleve/2012/07/23/a-balancing-act-for-the-matrix-exponential/
a = 2e10
b = 4e8/6.
c = 200/3.
d = 3
e = 1e-8
A = np.array([[0,e,0],[-(a+b), -d, a], [c, 0, -c]])
# This answer is wrong, and it is caused by overscaling.
wrong_solution = np.array([
[1.7465684381715e+17, -923050477.783131, -1.73117355055901e+17],
[-3.07408665108297e+25, 1.62463553675545e+17, 3.04699053651329e+25],
[1.09189154376804e+17, -577057840.468934, -1.08226721572342e+17]])
# This is the correct answer.
correct_solution = np.array([
[0.446849468283175, 1.54044157383952e-09, 0.462811453558774],
[-5743067.77947947, -0.0152830038686819, -4526542.71278401],
[0.447722977849494, 1.54270484519591e-09, 0.463480648837651]])
# Higham 2005 expm would give the wrong answer.
# Assert that the Higham 2009 expm gives the correct answer.
assert_allclose(expm(A), correct_solution)
def test_padecases_dtype_sparse_complex(self):
# float32 and complex64 lead to errors in spsolve/UMFpack
dtype = np.complex128
with warnings.catch_warnings():
warnings.simplefilter("ignore", category=SparseEfficiencyWarning)
for scale in [1e-2, 1e-1, 5e-1, 1, 10]:
a = scale * speye(3, 3, dtype=dtype, format='csc')
e = exp(scale) * eye(3, dtype=dtype)
assert_array_almost_equal_nulp(expm(a).toarray(), e, nulp=100)
def test_padecases_dtype_sparse_complex(self):
# float32 and complex64 lead to errors in spsolve/UMFpack
dtype = np.complex128
for scale in [1e-2, 1e-1, 5e-1, 1, 10]:
a = scale * speye(3, 3, dtype=dtype, format='csc')
e = exp(scale) * eye(3, dtype=dtype)
with suppress_warnings() as sup:
sup.filter(SparseEfficiencyWarning,
"Changing the sparsity structure of a csc_matrix is expensive.")
assert_array_almost_equal_nulp(expm(a).toarray(), e, nulp=100)
def test_logm_consistency(self):
random.seed(1234)
for dtype in [np.float64, np.complex128]:
for n in range(1, 10):
for scale in [1e-4, 1e-3, 1e-2, 1e-1, 1, 1e1, 1e2]:
# make logm(A) be of a given scale
A = (eye(n) + random.rand(n, n) * scale).astype(dtype)
if np.iscomplexobj(A):
A = A + 1j * random.rand(n, n) * scale
assert_array_almost_equal(expm(logm(A)), A)
def test_burkardt_2(self):
# This matrix is symmetric.
# The calculation of the matrix exponential is straightforward.
A = np.array([
[1, 3],
[3, 2],
], dtype=float)
desired = np.array([
[39.322809708033859, 46.166301438885753],
[46.166301438885768, 54.711576854329110],
], dtype=float)
actual = expm(A)
assert_allclose(actual, desired)
def test_bidiagonal_sparse(self):
A = csc_matrix([
[1, 3, 0],
[0, 1, 5],
[0, 0, 2]], dtype=float)
e1 = math.exp(1)
e2 = math.exp(2)
expected = np.array([
[e1, 3*e1, 15*(e2 - 2*e1)],
[0, e1, 5*(e2 - e1)],
[0, 0, e2]], dtype=float)
observed = expm(A).toarray()
assert_array_almost_equal(observed, expected)
def test_burkardt_4(self):
# This example is due to Moler and Van Loan.
# The example will cause problems for the series summation approach,
# as well as for diagonal Pade approximations.
A = np.array([
[-49, 24],
[-64, 31],
], dtype=float)
U = np.array([[3, 1], [4, 2]], dtype=float)
V = np.array([[1, -1/2], [-2, 3/2]], dtype=float)
w = np.array([-17, -1], dtype=float)
desired = np.dot(U * np.exp(w), V)
actual = expm(A)
assert_allclose(actual, desired)
def test_burkardt_14(self):
# This is Moler's example.
# This badly scaled matrix caused problems for MATLAB's expm().
A = np.array([
[0, 1e-8, 0],
[-(2e10 + 4e8/6.), -3, 2e10],
[200./3., 0, -200./3.],
], dtype=float)
desired = np.array([
[0.446849468283175, 1.54044157383952e-09, 0.462811453558774],
[-5743067.77947947, -0.0152830038686819, -4526542.71278401],
[0.447722977849494, 1.54270484519591e-09, 0.463480648837651],
], dtype=float)
actual = expm(A)
assert_allclose(actual, desired)
def test_burkardt_6(self):
# This example is due to Moler and Van Loan.
# This matrix does not have a complete set of eigenvectors.
# That means the eigenvector approach will fail.
exp1 = np.exp(1)
A = np.array([
[1, 1],
[0, 1],
], dtype=float)
desired = np.array([
[exp1, exp1],
[0, exp1],
], dtype=float)
actual = expm(A)
assert_allclose(actual, desired)
def test_burkardt_8(self):
# This matrix was an example in Wikipedia.
exp4 = np.exp(4)
exp16 = np.exp(16)
A = np.array([
[21, 17, 6],
[-5, -1, -6],
[4, 4, 16],
], dtype=float)
desired = np.array([
[13*exp16 - exp4, 13*exp16 - 5*exp4, 2*exp16 - 2*exp4],
[-9*exp16 + exp4, -9*exp16 + 5*exp4, -2*exp16 + 2*exp4],
[16*exp16, 16*exp16, 4*exp16],
], dtype=float) * 0.25
actual = expm(A)
assert_allclose(actual, desired)
def test_burkardt_10(self):
# This is Ward's example #1.
# It is defective and nonderogatory.
A = np.array([
[4, 2, 0],
[1, 4, 1],
[1, 1, 4],
], dtype=float)
assert_allclose(sorted(scipy.linalg.eigvals(A)), (3, 3, 6))
desired = np.array([
[147.8666224463699, 183.7651386463682, 71.79703239999647],
[127.7810855231823, 183.7651386463682, 91.88256932318415],
[127.7810855231824, 163.6796017231806, 111.9681062463718],
], dtype=float)
actual = expm(A)
assert_allclose(actual, desired)
def test_burkardt_12(self):
# This is Ward's example #3.
# Ward's algorithm has difficulty estimating the accuracy
# of its results.
A = np.array([
[-131, 19, 18],
[-390, 56, 54],
[-387, 57, 52],
], dtype=float)
assert_allclose(sorted(scipy.linalg.eigvals(A)), (-20, -2, -1))
desired = np.array([
[-1.509644158793135, 0.3678794391096522, 0.1353352811751005],
[-5.632570799891469, 1.471517758499875, 0.4060058435250609],
[-4.934938326088363, 1.103638317328798, 0.5413411267617766],
], dtype=float)
actual = expm(A)
assert_allclose(actual, desired)
def test_burkardt_9(self):
# This matrix is due to the NAG Library.
# It is an example for function F01ECF.
A = np.array([
[1, 2, 2, 2],
[3, 1, 1, 2],
[3, 2, 1, 2],
[3, 3, 3, 1],
], dtype=float)
desired = np.array([
[740.7038, 610.8500, 542.2743, 549.1753],
[731.2510, 603.5524, 535.0884, 542.2743],
[823.7630, 679.4257, 603.5524, 610.8500],
[998.4355, 823.7630, 731.2510, 740.7038],
], dtype=float)
actual = expm(A)
assert_allclose(actual, desired)
def test_burkardt_7(self):
# This example is due to Moler and Van Loan.
# This matrix is very close to example 5.
# Mathematically, it has a complete set of eigenvectors.
# Numerically, however, the calculation will be suspect.
exp1 = np.exp(1)
eps = np.spacing(1)
A = np.array([
[1 + eps, 1],
[0, 1 - eps],
], dtype=float)
desired = np.array([
[exp1, exp1],
[0, exp1],
], dtype=float)
actual = expm(A)
assert_allclose(actual, desired)
def test_burkardt_5(self):
# This example is due to Moler and Van Loan.
# This matrix is strictly upper triangular
# All powers of A are zero beyond some (low) limit.
# This example will cause problems for Pade approximations.
A = np.array([
[0, 6, 0, 0],
[0, 0, 6, 0],
[0, 0, 0, 6],
[0, 0, 0, 0],
], dtype=float)
desired = np.array([
[1, 6, 18, 36],
[0, 1, 6, 18],
[0, 0, 1, 6],
[0, 0, 0, 1],
], dtype=float)
actual = expm(A)
assert_allclose(actual, desired)
def test_pascal(self):
# Test pascal triangle.
# Nilpotent exponential, used to trigger a failure (gh-8029)
for scale in [1.0, 1e-3, 1e-6]:
for n in range(120):
A = np.diag(np.arange(1, n + 1), -1) * scale
B = expm(A)
sc = scale**np.arange(n, -1, -1)
if np.any(sc < 1e-300):
continue
got = B
expected = binom(np.arange(n + 1)[:,None],
np.arange(n + 1)[None,:]) * sc[None,:] / sc[:,None]
err = abs(expected - got).max()
atol = 1e-13 * abs(expected).max()
assert_allclose(got, expected, atol=atol)
def test_burkardt_3(self):
# This example is due to Laub.
# This matrix is ill-suited for the Taylor series approach.
# As powers of A are computed, the entries blow up too quickly.
exp1 = np.exp(1)
exp39 = np.exp(39)
A = np.array([
[0, 1],
[-39, -40],
], dtype=float)
desired = np.array([
[
39/(38*exp1) - 1/(38*exp39),
-np.expm1(-38) / (38*exp1)],
[
39*np.expm1(-38) / (38*exp1),
-1/(38*exp1) + 39/(38*exp39)],
], dtype=float)
actual = expm(A)
assert_allclose(actual, desired)
def test_burkardt_1(self):
# This matrix is diagonal.
# The calculation of the matrix exponential is simple.
#
# This is the first of a series of matrix exponential tests
# collected by John Burkardt from the following sources.
#
# Alan Laub,
# Review of "Linear System Theory" by Joao Hespanha,
# SIAM Review,
# Volume 52, Number 4, December 2010, pages 779--781.
#
# Cleve Moler and Charles Van Loan,
# Nineteen Dubious Ways to Compute the Exponential of a Matrix,
# Twenty-Five Years Later,
# SIAM Review,
# Volume 45, Number 1, March 2003, pages 3--49.
#
# Cleve Moler,
# Cleve's Corner: A Balancing Act for the Matrix Exponential,
# 23 July 2012.
#
# Robert Ward,
# Numerical computation of the matrix exponential
# with accuracy estimate,
# SIAM Journal on Numerical Analysis,
# Volume 14, Number 4, September 1977, pages 600--610.
exp1 = np.exp(1)
exp2 = np.exp(2)
A = np.array([
[1, 0],
[0, 2],
], dtype=float)
desired = np.array([
[exp1, 0],
[0, exp2],
], dtype=float)
actual = expm(A)
assert_allclose(actual, desired)
def test_integer_matrix(self):
Q = np.array([[-3, 1, 1, 1], [1, -3, 1, 1], [1, 1, -3, 1],
[1, 1, 1, -3]])
assert_allclose(expm(Q), expm(1.0 * Q))
def test_padecases_dtype_complex(self):
for dtype in [np.complex64, np.complex128]:
for scale in [1e-2, 1e-1, 5e-1, 1, 10]:
a = scale * eye(3, dtype=dtype)
e = exp(scale) * eye(3, dtype=dtype)
assert_array_almost_equal_nulp(expm(a), e, nulp=100)
def test_zero_ndarray(self):
a = array([[0., 0], [0, 0]])
assert_array_almost_equal(expm(a), [[1, 0], [0, 1]])
def test_zero_sparse(self):
a = csc_matrix([[0., 0], [0, 0]])
assert_array_almost_equal(expm(a).toarray(), [[1, 0], [0, 1]])
def test_zero_sparse(self):
a = csc_matrix([[0.,0],[0,0]])
assert_array_almost_equal(expm(a).toarray(),[[1,0],[0,1]])
def test_zero(self):
a = array([[0.,0],[0,0]])
assert_array_almost_equal(expm(a),[[1,0],[0,1]])
def test_misc_types(self):
A = expm(np.array([[1]]))
assert_allclose(expm(((1,),)), A)
assert_allclose(expm([[1]]), A)
assert_allclose(expm(matrix([[1]])), A)
assert_allclose(expm(np.array([[1]])), A)
assert_allclose(expm(csc_matrix([[1]])).A, A)
B = expm(np.array([[1j]]))
assert_allclose(expm(((1j,),)), B)
assert_allclose(expm([[1j]]), B)
assert_allclose(expm(matrix([[1j]])), B)
assert_allclose(expm(csc_matrix([[1j]])).A, B)
def test_zero_matrix(self):
a = np.matrix([[0., 0], [0, 0]])
assert_array_almost_equal(expm(a), [[1, 0], [0, 1]])
def test_misc_types(self):
A = expm(np.array([[1]]))
yield assert_allclose, expm(((1, ), )), A
yield assert_allclose, expm([[1]]), A
yield assert_allclose, expm(np.matrix([[1]])), A
yield assert_allclose, expm(np.array([[1]])), A
yield assert_allclose, expm(csc_matrix([[1]])), A
B = expm(np.array([[1j]]))
yield assert_allclose, expm(((1j, ), )), B
yield assert_allclose, expm([[1j]]), B
yield assert_allclose, expm(np.matrix([[1j]])), B
yield assert_allclose, expm(csc_matrix([[1j]])), B
def test_misc_types(self):
A = expm(np.array([[1]]))
assert_allclose(expm(((1,),)), A)
assert_allclose(expm([[1]]), A)
assert_allclose(expm(np.matrix([[1]])), A)
assert_allclose(expm(np.array([[1]])), A)
assert_allclose(expm(csc_matrix([[1]])).A, A)
B = expm(np.array([[1j]]))
assert_allclose(expm(((1j,),)), B)
assert_allclose(expm([[1j]]), B)
assert_allclose(expm(np.matrix([[1j]])), B)
assert_allclose(expm(csc_matrix([[1j]])).A, B)
def test_zero_matrix(self):
a = matrix([[0.,0],[0,0]])
assert_array_almost_equal(expm(a),[[1,0],[0,1]])