manyi/venv/Lib/site-packages/scipy/optimize/tests/test__numdiff.py

from __future__ import division

import math
from itertools import product

import numpy as np
from numpy.testing import assert_allclose, assert_equal, assert_
from pytest import raises as assert_raises

from scipy.sparse import csr_matrix, csc_matrix, lil_matrix

from scipy.optimize._numdiff import (
    _adjust_scheme_to_bounds, approx_derivative, check_derivative,
    group_columns)


def test_group_columns():
    structure = [
        [1, 1, 0, 0, 0, 0],
        [1, 1, 1, 0, 0, 0],
        [0, 1, 1, 1, 0, 0],
        [0, 0, 1, 1, 1, 0],
        [0, 0, 0, 1, 1, 1],
        [0, 0, 0, 0, 1, 1],
        [0, 0, 0, 0, 0, 0]
    ]
    for transform in [np.asarray, csr_matrix, csc_matrix, lil_matrix]:
        A = transform(structure)
        order = np.arange(6)
        groups_true = np.array([0, 1, 2, 0, 1, 2])
        groups = group_columns(A, order)
        assert_equal(groups, groups_true)

        order = [1, 2, 4, 3, 5, 0]
        groups_true = np.array([2, 0, 1, 2, 0, 1])
        groups = group_columns(A, order)
        assert_equal(groups, groups_true)

    # Test repeatability.
    groups_1 = group_columns(A)
    groups_2 = group_columns(A)
    assert_equal(groups_1, groups_2)


class TestAdjustSchemeToBounds(object):
    def test_no_bounds(self):
        x0 = np.zeros(3)
        h = np.ones(3) * 1e-2
        inf_lower = np.empty_like(x0)
        inf_upper = np.empty_like(x0)
        inf_lower.fill(-np.inf)
        inf_upper.fill(np.inf)

        h_adjusted, one_sided = _adjust_scheme_to_bounds(
            x0, h, 1, '1-sided', inf_lower, inf_upper)
        assert_allclose(h_adjusted, h)
        assert_(np.all(one_sided))

        h_adjusted, one_sided = _adjust_scheme_to_bounds(
            x0, h, 2, '1-sided', inf_lower, inf_upper)
        assert_allclose(h_adjusted, h)
        assert_(np.all(one_sided))

        h_adjusted, one_sided = _adjust_scheme_to_bounds(
            x0, h, 1, '2-sided', inf_lower, inf_upper)
        assert_allclose(h_adjusted, h)
        assert_(np.all(~one_sided))

        h_adjusted, one_sided = _adjust_scheme_to_bounds(
            x0, h, 2, '2-sided', inf_lower, inf_upper)
        assert_allclose(h_adjusted, h)
        assert_(np.all(~one_sided))

    def test_with_bound(self):
        x0 = np.array([0.0, 0.85, -0.85])
        lb = -np.ones(3)
        ub = np.ones(3)
        h = np.array([1, 1, -1]) * 1e-1

        h_adjusted, _ = _adjust_scheme_to_bounds(x0, h, 1, '1-sided', lb, ub)
        assert_allclose(h_adjusted, h)

        h_adjusted, _ = _adjust_scheme_to_bounds(x0, h, 2, '1-sided', lb, ub)
        assert_allclose(h_adjusted, np.array([1, -1, 1]) * 1e-1)

        h_adjusted, one_sided = _adjust_scheme_to_bounds(
            x0, h, 1, '2-sided', lb, ub)
        assert_allclose(h_adjusted, np.abs(h))
        assert_(np.all(~one_sided))

        h_adjusted, one_sided = _adjust_scheme_to_bounds(
            x0, h, 2, '2-sided', lb, ub)
        assert_allclose(h_adjusted, np.array([1, -1, 1]) * 1e-1)
        assert_equal(one_sided, np.array([False, True, True]))

    def test_tight_bounds(self):
        lb = np.array([-0.03, -0.03])
        ub = np.array([0.05, 0.05])
        x0 = np.array([0.0, 0.03])
        h = np.array([-0.1, -0.1])

        h_adjusted, _ = _adjust_scheme_to_bounds(x0, h, 1, '1-sided', lb, ub)
        assert_allclose(h_adjusted, np.array([0.05, -0.06]))

        h_adjusted, _ = _adjust_scheme_to_bounds(x0, h, 2, '1-sided', lb, ub)
        assert_allclose(h_adjusted, np.array([0.025, -0.03]))

        h_adjusted, one_sided = _adjust_scheme_to_bounds(
            x0, h, 1, '2-sided', lb, ub)
        assert_allclose(h_adjusted, np.array([0.03, -0.03]))
        assert_equal(one_sided, np.array([False, True]))

        h_adjusted, one_sided = _adjust_scheme_to_bounds(
            x0, h, 2, '2-sided', lb, ub)
        assert_allclose(h_adjusted, np.array([0.015, -0.015]))
        assert_equal(one_sided, np.array([False, True]))


class TestApproxDerivativesDense(object):
    def fun_scalar_scalar(self, x):
        return np.sinh(x)

    def jac_scalar_scalar(self, x):
        return np.cosh(x)

    def fun_scalar_vector(self, x):
        return np.array([x[0]**2, np.tan(x[0]), np.exp(x[0])])

    def jac_scalar_vector(self, x):
        return np.array(
            [2 * x[0], np.cos(x[0]) ** -2, np.exp(x[0])]).reshape(-1, 1)

    def fun_vector_scalar(self, x):
        return np.sin(x[0] * x[1]) * np.log(x[0])

    def wrong_dimensions_fun(self, x):
        return np.array([x**2, np.tan(x), np.exp(x)])

    def jac_vector_scalar(self, x):
        return np.array([
            x[1] * np.cos(x[0] * x[1]) * np.log(x[0]) +
            np.sin(x[0] * x[1]) / x[0],
            x[0] * np.cos(x[0] * x[1]) * np.log(x[0])
        ])

    def fun_vector_vector(self, x):
        return np.array([
            x[0] * np.sin(x[1]),
            x[1] * np.cos(x[0]),
            x[0] ** 3 * x[1] ** -0.5
        ])

    def jac_vector_vector(self, x):
        return np.array([
            [np.sin(x[1]), x[0] * np.cos(x[1])],
            [-x[1] * np.sin(x[0]), np.cos(x[0])],
            [3 * x[0] ** 2 * x[1] ** -0.5, -0.5 * x[0] ** 3 * x[1] ** -1.5]
        ])

    def fun_parametrized(self, x, c0, c1=1.0):
        return np.array([np.exp(c0 * x[0]), np.exp(c1 * x[1])])

    def jac_parametrized(self, x, c0, c1=0.1):
        return np.array([
            [c0 * np.exp(c0 * x[0]), 0],
            [0, c1 * np.exp(c1 * x[1])]
        ])

    def fun_with_nan(self, x):
        return x if np.abs(x) <= 1e-8 else np.nan

    def jac_with_nan(self, x):
        return 1.0 if np.abs(x) <= 1e-8 else np.nan

    def fun_zero_jacobian(self, x):
        return np.array([x[0] * x[1], np.cos(x[0] * x[1])])

    def jac_zero_jacobian(self, x):
        return np.array([
            [x[1], x[0]],
            [-x[1] * np.sin(x[0] * x[1]), -x[0] * np.sin(x[0] * x[1])]
        ])

    def fun_non_numpy(self, x):
        return math.exp(x)

    def jac_non_numpy(self, x):
        return math.exp(x)

    def test_scalar_scalar(self):
        x0 = 1.0
        jac_diff_2 = approx_derivative(self.fun_scalar_scalar, x0,
                                       method='2-point')
        jac_diff_3 = approx_derivative(self.fun_scalar_scalar, x0)
        jac_diff_4 = approx_derivative(self.fun_scalar_scalar, x0,
                                       method='cs')
        jac_true = self.jac_scalar_scalar(x0)
        assert_allclose(jac_diff_2, jac_true, rtol=1e-6)
        assert_allclose(jac_diff_3, jac_true, rtol=1e-9)
        assert_allclose(jac_diff_4, jac_true, rtol=1e-12)

    def test_scalar_vector(self):
        x0 = 0.5
        jac_diff_2 = approx_derivative(self.fun_scalar_vector, x0,
                                       method='2-point')
        jac_diff_3 = approx_derivative(self.fun_scalar_vector, x0)
        jac_diff_4 = approx_derivative(self.fun_scalar_vector, x0,
                                       method='cs')
        jac_true = self.jac_scalar_vector(np.atleast_1d(x0))
        assert_allclose(jac_diff_2, jac_true, rtol=1e-6)
        assert_allclose(jac_diff_3, jac_true, rtol=1e-9)
        assert_allclose(jac_diff_4, jac_true, rtol=1e-12)

    def test_vector_scalar(self):
        x0 = np.array([100.0, -0.5])
        jac_diff_2 = approx_derivative(self.fun_vector_scalar, x0,
                                       method='2-point')
        jac_diff_3 = approx_derivative(self.fun_vector_scalar, x0)
        jac_diff_4 = approx_derivative(self.fun_vector_scalar, x0,
                                       method='cs')
        jac_true = self.jac_vector_scalar(x0)
        assert_allclose(jac_diff_2, jac_true, rtol=1e-6)
        assert_allclose(jac_diff_3, jac_true, rtol=1e-7)
        assert_allclose(jac_diff_4, jac_true, rtol=1e-12)

    def test_vector_vector(self):
        x0 = np.array([-100.0, 0.2])
        jac_diff_2 = approx_derivative(self.fun_vector_vector, x0,
                                       method='2-point')
        jac_diff_3 = approx_derivative(self.fun_vector_vector, x0)
        jac_diff_4 = approx_derivative(self.fun_vector_vector, x0,
                                       method='cs')
        jac_true = self.jac_vector_vector(x0)
        assert_allclose(jac_diff_2, jac_true, rtol=1e-5)
        assert_allclose(jac_diff_3, jac_true, rtol=1e-6)
        assert_allclose(jac_diff_4, jac_true, rtol=1e-12)

    def test_wrong_dimensions(self):
        x0 = 1.0
        assert_raises(RuntimeError, approx_derivative,
                      self.wrong_dimensions_fun, x0)
        f0 = self.wrong_dimensions_fun(np.atleast_1d(x0))
        assert_raises(ValueError, approx_derivative,
                      self.wrong_dimensions_fun, x0, f0=f0)

    def test_custom_rel_step(self):
        x0 = np.array([-0.1, 0.1])
        jac_diff_2 = approx_derivative(self.fun_vector_vector, x0,
                                       method='2-point', rel_step=1e-4)
        jac_diff_3 = approx_derivative(self.fun_vector_vector, x0,
                                       rel_step=1e-4)
        jac_true = self.jac_vector_vector(x0)
        assert_allclose(jac_diff_2, jac_true, rtol=1e-2)
        assert_allclose(jac_diff_3, jac_true, rtol=1e-4)

    def test_options(self):
        x0 = np.array([1.0, 1.0])
        c0 = -1.0
        c1 = 1.0
        lb = 0.0
        ub = 2.0
        f0 = self.fun_parametrized(x0, c0, c1=c1)
        rel_step = np.array([-1e-6, 1e-7])
        jac_true = self.jac_parametrized(x0, c0, c1)
        jac_diff_2 = approx_derivative(
            self.fun_parametrized, x0, method='2-point', rel_step=rel_step,
            f0=f0, args=(c0,), kwargs=dict(c1=c1), bounds=(lb, ub))
        jac_diff_3 = approx_derivative(
            self.fun_parametrized, x0, rel_step=rel_step,
            f0=f0, args=(c0,), kwargs=dict(c1=c1), bounds=(lb, ub))
        assert_allclose(jac_diff_2, jac_true, rtol=1e-6)
        assert_allclose(jac_diff_3, jac_true, rtol=1e-9)

    def test_with_bounds_2_point(self):
        lb = -np.ones(2)
        ub = np.ones(2)

        x0 = np.array([-2.0, 0.2])
        assert_raises(ValueError, approx_derivative,
                      self.fun_vector_vector, x0, bounds=(lb, ub))

        x0 = np.array([-1.0, 1.0])
        jac_diff = approx_derivative(self.fun_vector_vector, x0,
                                     method='2-point', bounds=(lb, ub))
        jac_true = self.jac_vector_vector(x0)
        assert_allclose(jac_diff, jac_true, rtol=1e-6)

    def test_with_bounds_3_point(self):
        lb = np.array([1.0, 1.0])
        ub = np.array([2.0, 2.0])

        x0 = np.array([1.0, 2.0])
        jac_true = self.jac_vector_vector(x0)

        jac_diff = approx_derivative(self.fun_vector_vector, x0)
        assert_allclose(jac_diff, jac_true, rtol=1e-9)

        jac_diff = approx_derivative(self.fun_vector_vector, x0,
                                     bounds=(lb, np.inf))
        assert_allclose(jac_diff, jac_true, rtol=1e-9)

        jac_diff = approx_derivative(self.fun_vector_vector, x0,
                                     bounds=(-np.inf, ub))
        assert_allclose(jac_diff, jac_true, rtol=1e-9)

        jac_diff = approx_derivative(self.fun_vector_vector, x0,
                                     bounds=(lb, ub))
        assert_allclose(jac_diff, jac_true, rtol=1e-9)

    def test_tight_bounds(self):
        x0 = np.array([10.0, 10.0])
        lb = x0 - 3e-9
        ub = x0 + 2e-9
        jac_true = self.jac_vector_vector(x0)
        jac_diff = approx_derivative(
            self.fun_vector_vector, x0, method='2-point', bounds=(lb, ub))
        assert_allclose(jac_diff, jac_true, rtol=1e-6)
        jac_diff = approx_derivative(
            self.fun_vector_vector, x0, method='2-point',
            rel_step=1e-6, bounds=(lb, ub))
        assert_allclose(jac_diff, jac_true, rtol=1e-6)

        jac_diff = approx_derivative(
            self.fun_vector_vector, x0, bounds=(lb, ub))
        assert_allclose(jac_diff, jac_true, rtol=1e-6)
        jac_diff = approx_derivative(
            self.fun_vector_vector, x0, rel_step=1e-6, bounds=(lb, ub))
        assert_allclose(jac_true, jac_diff, rtol=1e-6)

    def test_bound_switches(self):
        lb = -1e-8
        ub = 1e-8
        x0 = 0.0
        jac_true = self.jac_with_nan(x0)
        jac_diff_2 = approx_derivative(
            self.fun_with_nan, x0, method='2-point', rel_step=1e-6,
            bounds=(lb, ub))
        jac_diff_3 = approx_derivative(
            self.fun_with_nan, x0, rel_step=1e-6, bounds=(lb, ub))
        assert_allclose(jac_diff_2, jac_true, rtol=1e-6)
        assert_allclose(jac_diff_3, jac_true, rtol=1e-9)

        x0 = 1e-8
        jac_true = self.jac_with_nan(x0)
        jac_diff_2 = approx_derivative(
            self.fun_with_nan, x0, method='2-point', rel_step=1e-6,
            bounds=(lb, ub))
        jac_diff_3 = approx_derivative(
            self.fun_with_nan, x0, rel_step=1e-6, bounds=(lb, ub))
        assert_allclose(jac_diff_2, jac_true, rtol=1e-6)
        assert_allclose(jac_diff_3, jac_true, rtol=1e-9)

    def test_non_numpy(self):
        x0 = 1.0
        jac_true = self.jac_non_numpy(x0)
        jac_diff_2 = approx_derivative(self.jac_non_numpy, x0,
                                       method='2-point')
        jac_diff_3 = approx_derivative(self.jac_non_numpy, x0)
        assert_allclose(jac_diff_2, jac_true, rtol=1e-6)
        assert_allclose(jac_diff_3, jac_true, rtol=1e-8)

        # math.exp cannot handle complex arguments, hence this raises
        assert_raises(TypeError, approx_derivative, self.jac_non_numpy, x0,
                      **dict(method='cs'))

    def test_check_derivative(self):
        x0 = np.array([-10.0, 10])
        accuracy = check_derivative(self.fun_vector_vector,
                                    self.jac_vector_vector, x0)
        assert_(accuracy < 1e-9)
        accuracy = check_derivative(self.fun_vector_vector,
                                    self.jac_vector_vector, x0)
        assert_(accuracy < 1e-6)

        x0 = np.array([0.0, 0.0])
        accuracy = check_derivative(self.fun_zero_jacobian,
                                    self.jac_zero_jacobian, x0)
        assert_(accuracy == 0)
        accuracy = check_derivative(self.fun_zero_jacobian,
                                    self.jac_zero_jacobian, x0)
        assert_(accuracy == 0)


class TestApproxDerivativeSparse(object):
    # Example from Numerical Optimization 2nd edition, p. 198.
    def setup_method(self):
        np.random.seed(0)
        self.n = 50
        self.lb = -0.1 * (1 + np.arange(self.n))
        self.ub = 0.1 * (1 + np.arange(self.n))
        self.x0 = np.empty(self.n)
        self.x0[::2] = (1 - 1e-7) * self.lb[::2]
        self.x0[1::2] = (1 - 1e-7) * self.ub[1::2]

        self.J_true = self.jac(self.x0)

    def fun(self, x):
        e = x[1:]**3 - x[:-1]**2
        return np.hstack((0, 3 * e)) + np.hstack((2 * e, 0))

    def jac(self, x):
        n = x.size
        J = np.zeros((n, n))
        J[0, 0] = -4 * x[0]
        J[0, 1] = 6 * x[1]**2
        for i in range(1, n - 1):
            J[i, i - 1] = -6 * x[i-1]
            J[i, i] = 9 * x[i]**2 - 4 * x[i]
            J[i, i + 1] = 6 * x[i+1]**2
        J[-1, -1] = 9 * x[-1]**2
        J[-1, -2] = -6 * x[-2]

        return J

    def structure(self, n):
        A = np.zeros((n, n), dtype=int)
        A[0, 0] = 1
        A[0, 1] = 1
        for i in range(1, n - 1):
            A[i, i - 1: i + 2] = 1
        A[-1, -1] = 1
        A[-1, -2] = 1

        return A

    def test_all(self):
        A = self.structure(self.n)
        order = np.arange(self.n)
        groups_1 = group_columns(A, order)
        np.random.shuffle(order)
        groups_2 = group_columns(A, order)

        for method, groups, l, u in product(
                ['2-point', '3-point', 'cs'], [groups_1, groups_2],
                [-np.inf, self.lb], [np.inf, self.ub]):
            J = approx_derivative(self.fun, self.x0, method=method,
                                  bounds=(l, u), sparsity=(A, groups))
            assert_(isinstance(J, csr_matrix))
            assert_allclose(J.toarray(), self.J_true, rtol=1e-6)

            rel_step = 1e-8 * np.ones_like(self.x0)
            rel_step[::2] *= -1
            J = approx_derivative(self.fun, self.x0, method=method,
                                  rel_step=rel_step, sparsity=(A, groups))
            assert_allclose(J.toarray(), self.J_true, rtol=1e-5)

    def test_no_precomputed_groups(self):
        A = self.structure(self.n)
        J = approx_derivative(self.fun, self.x0, sparsity=A)
        assert_allclose(J.toarray(), self.J_true, rtol=1e-6)

    def test_equivalence(self):
        structure = np.ones((self.n, self.n), dtype=int)
        groups = np.arange(self.n)
        for method in ['2-point', '3-point', 'cs']:
            J_dense = approx_derivative(self.fun, self.x0, method=method)
            J_sparse = approx_derivative(
                self.fun, self.x0, sparsity=(structure, groups), method=method)
            assert_equal(J_dense, J_sparse.toarray())

    def test_check_derivative(self):
        def jac(x):
            return csr_matrix(self.jac(x))

        accuracy = check_derivative(self.fun, jac, self.x0,
                                    bounds=(self.lb, self.ub))
        assert_(accuracy < 1e-9)

        accuracy = check_derivative(self.fun, jac, self.x0,
                                    bounds=(self.lb, self.ub))
        assert_(accuracy < 1e-9)


class TestApproxDerivativeLinearOperator(object):

    def fun_scalar_scalar(self, x):
        return np.sinh(x)

    def jac_scalar_scalar(self, x):
        return np.cosh(x)

    def fun_scalar_vector(self, x):
        return np.array([x[0]**2, np.tan(x[0]), np.exp(x[0])])

    def jac_scalar_vector(self, x):
        return np.array(
            [2 * x[0], np.cos(x[0]) ** -2, np.exp(x[0])]).reshape(-1, 1)

    def fun_vector_scalar(self, x):
        return np.sin(x[0] * x[1]) * np.log(x[0])

    def jac_vector_scalar(self, x):
        return np.array([
            x[1] * np.cos(x[0] * x[1]) * np.log(x[0]) +
            np.sin(x[0] * x[1]) / x[0],
            x[0] * np.cos(x[0] * x[1]) * np.log(x[0])
        ])

    def fun_vector_vector(self, x):
        return np.array([
            x[0] * np.sin(x[1]),
            x[1] * np.cos(x[0]),
            x[0] ** 3 * x[1] ** -0.5
        ])

    def jac_vector_vector(self, x):
        return np.array([
            [np.sin(x[1]), x[0] * np.cos(x[1])],
            [-x[1] * np.sin(x[0]), np.cos(x[0])],
            [3 * x[0] ** 2 * x[1] ** -0.5, -0.5 * x[0] ** 3 * x[1] ** -1.5]
        ])

    def test_scalar_scalar(self):
        x0 = 1.0
        jac_diff_2 = approx_derivative(self.fun_scalar_scalar, x0,
                                       method='2-point',
                                       as_linear_operator=True)
        jac_diff_3 = approx_derivative(self.fun_scalar_scalar, x0,
                                       as_linear_operator=True)
        jac_diff_4 = approx_derivative(self.fun_scalar_scalar, x0,
                                       method='cs',
                                       as_linear_operator=True)
        jac_true = self.jac_scalar_scalar(x0)
        np.random.seed(1)
        for i in range(10):
            p = np.random.uniform(-10, 10, size=(1,))
            assert_allclose(jac_diff_2.dot(p), jac_true*p,
                            rtol=1e-5)
            assert_allclose(jac_diff_3.dot(p), jac_true*p,
                            rtol=5e-6)
            assert_allclose(jac_diff_4.dot(p), jac_true*p,
                            rtol=5e-6)

    def test_scalar_vector(self):
        x0 = 0.5
        jac_diff_2 = approx_derivative(self.fun_scalar_vector, x0,
                                       method='2-point',
                                       as_linear_operator=True)
        jac_diff_3 = approx_derivative(self.fun_scalar_vector, x0,
                                       as_linear_operator=True)
        jac_diff_4 = approx_derivative(self.fun_scalar_vector, x0,
                                       method='cs',
                                       as_linear_operator=True)
        jac_true = self.jac_scalar_vector(np.atleast_1d(x0))
        np.random.seed(1)
        for i in range(10):
            p = np.random.uniform(-10, 10, size=(1,))
            assert_allclose(jac_diff_2.dot(p), jac_true.dot(p),
                            rtol=1e-5)
            assert_allclose(jac_diff_3.dot(p), jac_true.dot(p),
                            rtol=5e-6)
            assert_allclose(jac_diff_4.dot(p), jac_true.dot(p),
                            rtol=5e-6)

    def test_vector_scalar(self):
        x0 = np.array([100.0, -0.5])
        jac_diff_2 = approx_derivative(self.fun_vector_scalar, x0,
                                       method='2-point',
                                       as_linear_operator=True)
        jac_diff_3 = approx_derivative(self.fun_vector_scalar, x0,
                                       as_linear_operator=True)
        jac_diff_4 = approx_derivative(self.fun_vector_scalar, x0,
                                       method='cs',
                                       as_linear_operator=True)
        jac_true = self.jac_vector_scalar(x0)
        np.random.seed(1)
        for i in range(10):
            p = np.random.uniform(-10, 10, size=x0.shape)
            assert_allclose(jac_diff_2.dot(p), np.atleast_1d(jac_true.dot(p)),
                            rtol=1e-5)
            assert_allclose(jac_diff_3.dot(p), np.atleast_1d(jac_true.dot(p)),
                            rtol=5e-6)
            assert_allclose(jac_diff_4.dot(p), np.atleast_1d(jac_true.dot(p)),
                            rtol=1e-7)

    def test_vector_vector(self):
        x0 = np.array([-100.0, 0.2])
        jac_diff_2 = approx_derivative(self.fun_vector_vector, x0,
                                       method='2-point',
                                       as_linear_operator=True)
        jac_diff_3 = approx_derivative(self.fun_vector_vector, x0,
                                       as_linear_operator=True)
        jac_diff_4 = approx_derivative(self.fun_vector_vector, x0,
                                       method='cs',
                                       as_linear_operator=True)
        jac_true = self.jac_vector_vector(x0)
        np.random.seed(1)
        for i in range(10):
            p = np.random.uniform(-10, 10, size=x0.shape)
            assert_allclose(jac_diff_2.dot(p), jac_true.dot(p), rtol=1e-5)
            assert_allclose(jac_diff_3.dot(p), jac_true.dot(p), rtol=1e-6)
            assert_allclose(jac_diff_4.dot(p), jac_true.dot(p), rtol=1e-7)

    def test_exception(self):
        x0 = np.array([-100.0, 0.2])
        assert_raises(ValueError, approx_derivative,
                      self.fun_vector_vector, x0,
                      method='2-point', bounds=(1, np.inf))