image-match/python/py/Lib/site-packages/scipy/linalg/tests/test_special_matrices.py

import pytest
import numpy as np
from numpy import arange, array, eye, copy, sqrt
from numpy.testing import (assert_equal, assert_array_equal,
                           assert_array_almost_equal, assert_allclose)
from pytest import raises as assert_raises

from scipy.fft import fft
from scipy.special import comb
from scipy.linalg import (toeplitz, hankel, circulant, hadamard, leslie, dft,
                          companion, block_diag,
                          helmert, hilbert, invhilbert, pascal, invpascal,
                          fiedler, fiedler_companion, eigvals,
                          convolution_matrix)
from numpy.linalg import cond
from scipy._lib._array_api import (make_xp_test_case, xp_assert_equal, xp_size,
                                   xp_default_dtype)


class TestToeplitz:

    def test_basic(self):
        y = toeplitz([1, 2, 3])
        assert_array_equal(y, [[1, 2, 3], [2, 1, 2], [3, 2, 1]])
        y = toeplitz([1, 2, 3], [1, 4, 5])
        assert_array_equal(y, [[1, 4, 5], [2, 1, 4], [3, 2, 1]])

    def test_complex_01(self):
        data = (1.0 + arange(3.0)) * (1.0 + 1.0j)
        x = copy(data)
        t = toeplitz(x)
        # Calling toeplitz should not change x.
        assert_array_equal(x, data)
        # According to the docstring, x should be the first column of t.
        col0 = t[:, 0]
        assert_array_equal(col0, data)
        assert_array_equal(t[0, 1:], data[1:].conj())

    def test_scalar_00(self):
        """Scalar arguments still produce a 2D array."""
        t = toeplitz(10)
        assert_array_equal(t, [[10]])
        t = toeplitz(10, 20)
        assert_array_equal(t, [[10]])

    def test_scalar_01(self):
        c = array([1, 2, 3])
        t = toeplitz(c, 1)
        assert_array_equal(t, [[1], [2], [3]])

    def test_scalar_02(self):
        c = array([1, 2, 3])
        t = toeplitz(c, array(1))
        assert_array_equal(t, [[1], [2], [3]])

    def test_scalar_03(self):
        c = array([1, 2, 3])
        t = toeplitz(c, array([1]))
        assert_array_equal(t, [[1], [2], [3]])

    def test_scalar_04(self):
        r = array([10, 2, 3])
        t = toeplitz(1, r)
        assert_array_equal(t, [[1, 2, 3]])


class TestHankel:
    def test_basic(self):
        y = hankel([1, 2, 3])
        assert_array_equal(y, [[1, 2, 3], [2, 3, 0], [3, 0, 0]])
        y = hankel([1, 2, 3], [3, 4, 5])
        assert_array_equal(y, [[1, 2, 3], [2, 3, 4], [3, 4, 5]])


class TestCirculant:
    def test_basic(self):
        y = circulant([1, 2, 3])
        assert_array_equal(y, [[1, 3, 2], [2, 1, 3], [3, 2, 1]])


class TestHadamard:

    def test_basic(self):

        y = hadamard(1)
        assert_array_equal(y, [[1]])

        y = hadamard(2, dtype=float)
        assert_array_equal(y, [[1.0, 1.0], [1.0, -1.0]])

        y = hadamard(4)
        assert_array_equal(y, [[1, 1, 1, 1],
                               [1, -1, 1, -1],
                               [1, 1, -1, -1],
                               [1, -1, -1, 1]])

        assert_raises(ValueError, hadamard, 0)
        assert_raises(ValueError, hadamard, 5)


class TestLeslie:

    def test_bad_shapes(self):
        assert_raises(ValueError, leslie, [[1, 1], [2, 2]], [3, 4, 5])
        assert_raises(ValueError, leslie, [1, 2], [1, 2])
        assert_raises(ValueError, leslie, [1], [])

    def test_basic(self):
        a = leslie([1, 2, 3], [0.25, 0.5])
        expected = array([[1.0, 2.0, 3.0],
                          [0.25, 0.0, 0.0],
                          [0.0, 0.5, 0.0]])
        assert_array_equal(a, expected)


class TestCompanion:

    def test_bad_shapes(self):
        assert_raises(ValueError, companion, [0, 4, 5])
        assert_raises(ValueError, companion, [1])
        assert_raises(ValueError, companion, [])

    def test_basic(self):
        c = companion([1, 2, 3])
        expected = array([
            [-2.0, -3.0],
            [1.0, 0.0]])
        assert_array_equal(c, expected)

        c = companion([2.0, 5.0, -10.0])
        expected = array([
            [-2.5, 5.0],
            [1.0, 0.0]])
        assert_array_equal(c, expected)

        c = companion([(1.0, 2.0, 3.0),
                       (4.0, 5.0, 6.0)])
        expected = array([
            ([-2.00, -3.00],
             [+1.00, +0.00]),
            ([-1.25, -1.50],
             [+1.00, +0.00])
        ])
        assert_array_equal(c, expected)


@make_xp_test_case(block_diag)
class TestBlockDiag:
    def test_basic(self, xp):
        dtype = xp.asarray(1).dtype
        x = block_diag(xp.eye(2, dtype=dtype), xp.asarray([[1, 2], [3, 4], [5, 6]]),
                       xp.asarray([[1, 2, 3]]))
        xp_assert_equal(x, xp.asarray([[1, 0, 0, 0, 0, 0, 0],
                                       [0, 1, 0, 0, 0, 0, 0],
                                       [0, 0, 1, 2, 0, 0, 0],
                                       [0, 0, 3, 4, 0, 0, 0],
                                       [0, 0, 5, 6, 0, 0, 0],
                                       [0, 0, 0, 0, 1, 2, 3]]))

    def test_dtype(self, xp):
        x = block_diag(xp.asarray([[1.5]]))
        assert x.dtype == xp_default_dtype(xp)

        x = block_diag(xp.asarray([[True]]))
        assert x.dtype == xp.bool

    def test_mixed_dtypes(self, xp):
        actual = block_diag(xp.asarray([[1.]]), xp.asarray([[1j]]))
        desired = xp.asarray([[1, 0], [0, 1j]])
        xp_assert_equal(actual, desired)

    def test_scalar_and_1d_args(self, xp):
        a = block_diag(xp.asarray(1))
        assert a.shape == (1, 1)
        xp_assert_equal(a, xp.asarray([[1]]))

        a = block_diag(xp.asarray([2, 3]), xp.asarray(4))
        xp_assert_equal(a, xp.asarray([[2, 3, 0], [0, 0, 4]]))

    def test_no_args(self):
        a = block_diag()
        assert a.ndim == 2
        assert a.nbytes == 0

    def test_empty_matrix_arg(self, xp):
        # regression test for gh-4596: check the shape of the result
        # for empty matrix inputs. Empty matrices are no longer ignored
        # (gh-4908) it is viewed as a shape (1, 0) matrix.
        dtype = xp.asarray(1).dtype
        a = block_diag(xp.asarray([[1, 0], [0, 1]]),
                       xp.asarray([], dtype=dtype),
                       xp.asarray([[2, 3], [4, 5], [6, 7]]))
        xp_assert_equal(a, xp.asarray([[1, 0, 0, 0],
                                       [0, 1, 0, 0],
                                       [0, 0, 0, 0],
                                       [0, 0, 2, 3],
                                       [0, 0, 4, 5],
                                       [0, 0, 6, 7]]))

    @pytest.mark.skip_xp_backends("dask.array", reason="dask/dask#11800")
    def test_zerosized_matrix_arg(self, xp):
        # test for gh-4908: check the shape of the result for
        # zero-sized matrix inputs, i.e. matrices with shape (0,n) or (n,0).
        # note that [[]] takes shape (1,0)
        dtype = xp.asarray(1).dtype
        a = block_diag(xp.asarray([[1, 0], [0, 1]]),
                       xp.asarray([[]], dtype=dtype),
                       xp.asarray([[2, 3], [4, 5], [6, 7]]),
                       xp.zeros([0, 2], dtype=dtype))
        xp_assert_equal(a, xp.asarray([[1, 0, 0, 0, 0, 0],
                                       [0, 1, 0, 0, 0, 0],
                                       [0, 0, 0, 0, 0, 0],
                                       [0, 0, 2, 3, 0, 0],
                                       [0, 0, 4, 5, 0, 0],
                                       [0, 0, 6, 7, 0, 0]]))


class TestHelmert:

    def test_orthogonality(self):
        for n in range(1, 7):
            H = helmert(n, full=True)
            Id = np.eye(n)
            assert_allclose(H.dot(H.T), Id, atol=1e-12)
            assert_allclose(H.T.dot(H), Id, atol=1e-12)

    def test_subspace(self):
        for n in range(2, 7):
            H_full = helmert(n, full=True)
            H_partial = helmert(n)
            for U in H_full[1:, :].T, H_partial.T:
                C = np.eye(n) - np.full((n, n), 1 / n)
                assert_allclose(U.dot(U.T), C)
                assert_allclose(U.T.dot(U), np.eye(n-1), atol=1e-12)


class TestHilbert:

    def test_basic(self):
        h3 = array([[1.0, 1/2., 1/3.],
                    [1/2., 1/3., 1/4.],
                    [1/3., 1/4., 1/5.]])
        assert_array_almost_equal(hilbert(3), h3)

        assert_array_equal(hilbert(1), [[1.0]])

        h0 = hilbert(0)
        assert_equal(h0.shape, (0, 0))


class TestInvHilbert:

    def test_basic(self):
        invh1 = array([[1]])
        assert_array_equal(invhilbert(1, exact=True), invh1)
        assert_array_equal(invhilbert(1), invh1)

        invh2 = array([[4, -6],
                       [-6, 12]])
        assert_array_equal(invhilbert(2, exact=True), invh2)
        assert_array_almost_equal(invhilbert(2), invh2)

        invh3 = array([[9, -36, 30],
                       [-36, 192, -180],
                       [30, -180, 180]])
        assert_array_equal(invhilbert(3, exact=True), invh3)
        assert_array_almost_equal(invhilbert(3), invh3)

        invh4 = array([[16, -120, 240, -140],
                       [-120, 1200, -2700, 1680],
                       [240, -2700, 6480, -4200],
                       [-140, 1680, -4200, 2800]])
        assert_array_equal(invhilbert(4, exact=True), invh4)
        assert_array_almost_equal(invhilbert(4), invh4)

        invh5 = array([[25, -300, 1050, -1400, 630],
                       [-300, 4800, -18900, 26880, -12600],
                       [1050, -18900, 79380, -117600, 56700],
                       [-1400, 26880, -117600, 179200, -88200],
                       [630, -12600, 56700, -88200, 44100]])
        assert_array_equal(invhilbert(5, exact=True), invh5)
        assert_array_almost_equal(invhilbert(5), invh5)

        invh17 = array([
            [289, -41616, 1976760, -46124400, 629598060, -5540462928,
             33374693352, -143034400080, 446982500250, -1033026222800,
             1774926873720, -2258997839280, 2099709530100, -1384423866000,
             613101997800, -163493866080, 19835652870],
            [-41616, 7990272, -426980160, 10627061760, -151103534400,
             1367702848512, -8410422724704, 36616806420480, -115857864064800,
             270465047424000, -468580694662080, 600545887119360,
             -561522320049600, 372133135180800, -165537539406000,
             44316454993920, -5395297580640],
            [1976760, -426980160, 24337869120, -630981792000, 9228108708000,
             -85267724461920, 532660105897920, -2348052711713280,
             7504429831470000, -17664748409880000, 30818191841236800,
             -39732544853164800, 37341234283298400, -24857330514030000,
             11100752642520000, -2982128117299200, 364182586693200],
            [-46124400, 10627061760, -630981792000, 16826181120000,
             -251209625940000, 2358021022156800, -14914482965141760,
             66409571644416000, -214015221119700000, 507295338950400000,
             -890303319857952000, 1153715376477081600, -1089119333262870000,
             727848632044800000, -326170262829600000, 87894302404608000,
             -10763618673376800],
            [629598060, -151103534400, 9228108708000,
             -251209625940000, 3810012660090000, -36210360321495360,
             231343968720664800, -1038687206500944000, 3370739732635275000,
             -8037460526495400000, 14178080368737885600, -18454939322943942000,
             17489975175339030000, -11728977435138600000, 5272370630081100000,
             -1424711708039692800, 174908803442373000],
            [-5540462928, 1367702848512, -85267724461920, 2358021022156800,
             -36210360321495360, 347619459086355456, -2239409617216035264,
             10124803292907663360, -33052510749726468000,
             79217210949138662400, -140362995650505067440,
             183420385176741672960, -174433352415381259200,
             117339159519533952000, -52892422160973595200,
             14328529177999196160, -1763080738699119840],
            [33374693352, -8410422724704, 532660105897920,
             -14914482965141760, 231343968720664800, -2239409617216035264,
             14527452132196331328, -66072377044391477760,
             216799987176909536400, -521925895055522958000,
             928414062734059661760, -1217424500995626443520,
             1161358898976091015200, -783401860847777371200,
             354015418167362952000, -96120549902411274240,
             11851820521255194480],
            [-143034400080, 36616806420480, -2348052711713280,
             66409571644416000, -1038687206500944000, 10124803292907663360,
             -66072377044391477760, 302045152202932469760,
             -995510145200094810000, 2405996923185123840000,
             -4294704507885446054400, 5649058909023744614400,
             -5403874060541811254400, 3654352703663101440000,
             -1655137020003255360000, 450325202737117593600,
             -55630994283442749600],
            [446982500250, -115857864064800, 7504429831470000,
             -214015221119700000, 3370739732635275000, -33052510749726468000,
             216799987176909536400, -995510145200094810000,
             3293967392206196062500, -7988661659013106500000,
             14303908928401362270000, -18866974090684772052000,
             18093328327706957325000, -12263364009096700500000,
             5565847995255512250000, -1517208935002984080000,
             187754605706619279900],
            [-1033026222800, 270465047424000, -17664748409880000,
             507295338950400000, -8037460526495400000, 79217210949138662400,
             -521925895055522958000, 2405996923185123840000,
             -7988661659013106500000, 19434404971634224000000,
             -34894474126569249192000, 46141453390504792320000,
             -44349976506971935800000, 30121928988527376000000,
             -13697025107665828500000, 3740200989399948902400,
             -463591619028689580000],
            [1774926873720, -468580694662080,
             30818191841236800, -890303319857952000, 14178080368737885600,
             -140362995650505067440, 928414062734059661760,
             -4294704507885446054400, 14303908928401362270000,
             -34894474126569249192000, 62810053427824648545600,
             -83243376594051600326400, 80177044485212743068000,
             -54558343880470209780000, 24851882355348879230400,
             -6797096028813368678400, 843736746632215035600],
            [-2258997839280, 600545887119360, -39732544853164800,
             1153715376477081600, -18454939322943942000, 183420385176741672960,
             -1217424500995626443520, 5649058909023744614400,
             -18866974090684772052000, 46141453390504792320000,
             -83243376594051600326400, 110552468520163390156800,
             -106681852579497947388000, 72720410752415168870400,
             -33177973900974346080000, 9087761081682520473600,
             -1129631016152221783200],
            [2099709530100, -561522320049600, 37341234283298400,
             -1089119333262870000, 17489975175339030000,
             -174433352415381259200, 1161358898976091015200,
             -5403874060541811254400, 18093328327706957325000,
             -44349976506971935800000, 80177044485212743068000,
             -106681852579497947388000, 103125790826848015808400,
             -70409051543137015800000, 32171029219823375700000,
             -8824053728865840192000, 1098252376814660067000],
            [-1384423866000, 372133135180800,
             -24857330514030000, 727848632044800000, -11728977435138600000,
             117339159519533952000, -783401860847777371200,
             3654352703663101440000, -12263364009096700500000,
             30121928988527376000000, -54558343880470209780000,
             72720410752415168870400, -70409051543137015800000,
             48142941226076592000000, -22027500987368499000000,
             6049545098753157120000, -753830033789944188000],
            [613101997800, -165537539406000,
             11100752642520000, -326170262829600000, 5272370630081100000,
             -52892422160973595200, 354015418167362952000,
             -1655137020003255360000, 5565847995255512250000,
             -13697025107665828500000, 24851882355348879230400,
             -33177973900974346080000, 32171029219823375700000,
             -22027500987368499000000, 10091416708498869000000,
             -2774765838662800128000, 346146444087219270000],
            [-163493866080, 44316454993920, -2982128117299200,
             87894302404608000, -1424711708039692800,
             14328529177999196160, -96120549902411274240,
             450325202737117593600, -1517208935002984080000,
             3740200989399948902400, -6797096028813368678400,
             9087761081682520473600, -8824053728865840192000,
             6049545098753157120000, -2774765838662800128000,
             763806510427609497600, -95382575704033754400],
            [19835652870, -5395297580640, 364182586693200, -10763618673376800,
             174908803442373000, -1763080738699119840, 11851820521255194480,
             -55630994283442749600, 187754605706619279900,
             -463591619028689580000, 843736746632215035600,
             -1129631016152221783200, 1098252376814660067000,
             -753830033789944188000, 346146444087219270000,
             -95382575704033754400, 11922821963004219300]
        ])
        assert_array_equal(invhilbert(17, exact=True), invh17)
        assert_allclose(invhilbert(17), invh17.astype(float), rtol=1e-12)

    def test_inverse(self):
        for n in range(1, 10):
            a = hilbert(n)
            b = invhilbert(n)
            # The Hilbert matrix is increasingly badly conditioned,
            # so take that into account in the test
            c = cond(a)
            assert_allclose(a.dot(b), eye(n), atol=1e-15*c, rtol=1e-15*c)


class TestPascal:

    cases = [
        (1, array([[1]]), array([[1]])),
        (2, array([[1, 1],
                   [1, 2]]),
            array([[1, 0],
                   [1, 1]])),
        (3, array([[1, 1, 1],
                   [1, 2, 3],
                   [1, 3, 6]]),
            array([[1, 0, 0],
                   [1, 1, 0],
                   [1, 2, 1]])),
        (4, array([[1, 1, 1, 1],
                   [1, 2, 3, 4],
                   [1, 3, 6, 10],
                   [1, 4, 10, 20]]),
            array([[1, 0, 0, 0],
                   [1, 1, 0, 0],
                   [1, 2, 1, 0],
                   [1, 3, 3, 1]])),
    ]

    def check_case(self, n, sym, low):
        assert_array_equal(pascal(n), sym)
        assert_array_equal(pascal(n, kind='lower'), low)
        assert_array_equal(pascal(n, kind='upper'), low.T)
        assert_array_almost_equal(pascal(n, exact=False), sym)
        assert_array_almost_equal(pascal(n, exact=False, kind='lower'), low)
        assert_array_almost_equal(pascal(n, exact=False, kind='upper'), low.T)

    def test_cases(self):
        for n, sym, low in self.cases:
            self.check_case(n, sym, low)

    def test_big(self):
        p = pascal(50)
        assert p[-1, -1] == comb(98, 49, exact=True)

    def test_threshold(self):
        # Regression test.  An early version of `pascal` returned an
        # array of type np.uint64 for n=35, but that data type is too small
        # to hold p[-1, -1].  The second assert_equal below would fail
        # because p[-1, -1] overflowed.
        p = pascal(34)
        assert_equal(2*p.item(-1, -2), p.item(-1, -1), err_msg="n = 34")
        p = pascal(35)
        assert_equal(2.*p.item(-1, -2), 1.*p.item(-1, -1), err_msg="n = 35")


def test_invpascal():

    def check_invpascal(n, kind, exact):
        ip = invpascal(n, kind=kind, exact=exact)
        p = pascal(n, kind=kind, exact=exact)
        # Matrix-multiply ip and p, and check that we get the identity matrix.
        # We can't use the simple expression e = ip.dot(p), because when
        # n < 35 and exact is True, p.dtype is np.uint64 and ip.dtype is
        # np.int64. The product of those dtypes is np.float64, which loses
        # precision when n is greater than 18.  Instead we'll cast both to
        # object arrays, and then multiply.
        e = ip.astype(object).dot(p.astype(object))
        assert_array_equal(e, eye(n), err_msg=f"n={n}  kind={kind!r} exact={exact!r}")

    kinds = ['symmetric', 'lower', 'upper']

    ns = [1, 2, 5, 18]
    for n in ns:
        for kind in kinds:
            for exact in [True, False]:
                check_invpascal(n, kind, exact)

    ns = [19, 34, 35, 50]
    for n in ns:
        for kind in kinds:
            check_invpascal(n, kind, True)


def test_dft():
    m = dft(2)
    expected = array([[1.0, 1.0], [1.0, -1.0]])
    assert_array_almost_equal(m, expected)
    m = dft(2, scale='n')
    assert_array_almost_equal(m, expected/2.0)
    m = dft(2, scale='sqrtn')
    assert_array_almost_equal(m, expected/sqrt(2.0))

    x = array([0, 1, 2, 3, 4, 5, 0, 1])
    m = dft(8)
    mx = m.dot(x)
    fx = fft(x)
    assert_array_almost_equal(mx, fx)


@make_xp_test_case(fiedler)
def test_fiedler(xp):
    f = fiedler(xp.asarray([]))
    assert xp_size(f) == 0

    f = fiedler(xp.asarray([123.]))
    xp_assert_equal(f, xp.asarray([[0.]]))

    f = fiedler(xp.arange(1, 7))
    des = xp.asarray([[0, 1, 2, 3, 4, 5],
                      [1, 0, 1, 2, 3, 4],
                      [2, 1, 0, 1, 2, 3],
                      [3, 2, 1, 0, 1, 2],
                      [4, 3, 2, 1, 0, 1],
                      [5, 4, 3, 2, 1, 0]])
    xp_assert_equal(f, des)


def test_fiedler_companion():
    fc = fiedler_companion([])
    assert_equal(fc.size, 0)
    fc = fiedler_companion([1.])
    assert_equal(fc.size, 0)
    fc = fiedler_companion([1., 2.])
    assert_array_equal(fc, np.array([[-2.]]))
    fc = fiedler_companion([1e-12, 2., 3.])
    assert_array_almost_equal(fc, companion([1e-12, 2., 3.]))
    with assert_raises(ValueError):
        fiedler_companion([0, 1, 2])
    fc = fiedler_companion([1., -16., 86., -176., 105.])
    assert_array_almost_equal(eigvals(fc),
                              np.array([7., 5., 3., 1.]))


class TestConvolutionMatrix:
    """
    Test convolution_matrix vs. numpy.convolve for various parameters.
    """

    def create_vector(self, n, cpx):
        """Make a complex or real test vector of length n."""
        x = np.linspace(-2.5, 2.2, n)
        if cpx:
            x = x + 1j*np.linspace(-1.5, 3.1, n)
        return x

    def test_bad_n(self):
        # n must be a positive integer
        with pytest.raises(ValueError, match='n must be a positive integer'):
            convolution_matrix([1, 2, 3], 0)

    def test_empty_first_arg(self):
        # first arg must have at least one value
        with pytest.raises(ValueError, match=r'len\(a\)'):
            convolution_matrix([], 4)

    def test_bad_mode(self):
        # mode must be in ('full', 'valid', 'same')
        with pytest.raises(ValueError, match='mode.*must be one of'):
            convolution_matrix((1, 1), 4, mode='invalid argument')

    @pytest.mark.parametrize('cpx', [False, True])
    @pytest.mark.parametrize('na', [1, 2, 9])
    @pytest.mark.parametrize('nv', [1, 2, 9])
    @pytest.mark.parametrize('mode', [None, 'full', 'valid', 'same'])
    def test_against_numpy_convolve(self, cpx, na, nv, mode):
        a = self.create_vector(na, cpx)
        v = self.create_vector(nv, cpx)
        if mode is None:
            y1 = np.convolve(v, a)
            A = convolution_matrix(a, nv)
        else:
            y1 = np.convolve(v, a, mode)
            A = convolution_matrix(a, nv, mode)
        y2 = A @ v
        assert_array_almost_equal(y1, y2)


@pytest.mark.fail_slow(5)  # `leslie` has an import in the function
@pytest.mark.parametrize('f, args', [(circulant, ()),
                                     (companion, ()),
                                     (convolution_matrix, (5, 'same')),
                                     (fiedler, ()),
                                     (fiedler_companion, ()),
                                     (hankel, (np.arange(9),)),
                                     (leslie, (np.arange(9),)),
                                     (toeplitz, (np.arange(9),)),
                                     ])
def test_batch(f, args):
    rng = np.random.default_rng(283592436523456)
    batch_shape = (2, 3)
    m = 10
    A = rng.random(batch_shape + (m,))

    if f in {hankel}:
        message = "Beginning in SciPy 1.19, multidimensional input will be..."
        with pytest.warns(FutureWarning, match=message):
            f(A, *args)
        return

    res = f(A, *args)
    ref = np.asarray([f(a, *args) for a in A.reshape(-1, m)])
    ref = ref.reshape(A.shape[:-1] + ref.shape[-2:])
    assert_allclose(res, ref)