up follow livre

venv/lib/python3.13/site-packages/scipy/odr/__init__.py

@@ -0,0 +1,131 @@
+"""
+=================================================
+Orthogonal distance regression (:mod:`scipy.odr`)
+=================================================
+
+.. currentmodule:: scipy.odr
+
+Package Content
+===============
+
+.. autosummary::
+   :toctree: generated/
+
+   Data          -- The data to fit.
+   RealData      -- Data with weights as actual std. dev.s and/or covariances.
+   Model         -- Stores information about the function to be fit.
+   ODR           -- Gathers all info & manages the main fitting routine.
+   Output        -- Result from the fit.
+   odr           -- Low-level function for ODR.
+
+   OdrWarning    -- Warning about potential problems when running ODR.
+   OdrError      -- Error exception.
+   OdrStop       -- Stop exception.
+
+   polynomial    -- Factory function for a general polynomial model.
+   exponential   -- Exponential model
+   multilinear   -- Arbitrary-dimensional linear model
+   unilinear     -- Univariate linear model
+   quadratic     -- Quadratic model
+
+Usage information
+=================
+
+Introduction
+------------
+
+Why Orthogonal Distance Regression (ODR)? Sometimes one has
+measurement errors in the explanatory (a.k.a., "independent")
+variable(s), not just the response (a.k.a., "dependent") variable(s).
+Ordinary Least Squares (OLS) fitting procedures treat the data for
+explanatory variables as fixed, i.e., not subject to error of any kind.
+Furthermore, OLS procedures require that the response variables be an
+explicit function of the explanatory variables; sometimes making the
+equation explicit is impractical and/or introduces errors.  ODR can
+handle both of these cases with ease, and can even reduce to the OLS
+case if that is sufficient for the problem.
+
+ODRPACK is a FORTRAN-77 library for performing ODR with possibly
+non-linear fitting functions. It uses a modified trust-region
+Levenberg-Marquardt-type algorithm [1]_ to estimate the function
+parameters.  The fitting functions are provided by Python functions
+operating on NumPy arrays. The required derivatives may be provided
+by Python functions as well, or may be estimated numerically. ODRPACK
+can do explicit or implicit ODR fits, or it can do OLS. Input and
+output variables may be multidimensional. Weights can be provided to
+account for different variances of the observations, and even
+covariances between dimensions of the variables.
+
+The `scipy.odr` package offers an object-oriented interface to
+ODRPACK, in addition to the low-level `odr` function.
+
+Additional background information about ODRPACK can be found in the
+`ODRPACK User's Guide
+<https://docs.scipy.org/doc/external/odrpack_guide.pdf>`_, reading
+which is recommended.
+
+Basic usage
+-----------
+
+1. Define the function you want to fit against.::
+
+       def f(B, x):
+           '''Linear function y = m*x + b'''
+           # B is a vector of the parameters.
+           # x is an array of the current x values.
+           # x is in the same format as the x passed to Data or RealData.
+           #
+           # Return an array in the same format as y passed to Data or RealData.
+           return B[0]*x + B[1]
+
+2. Create a Model.::
+
+       linear = Model(f)
+
+3. Create a Data or RealData instance.::
+
+       mydata = Data(x, y, wd=1./power(sx,2), we=1./power(sy,2))
+
+   or, when the actual covariances are known::
+
+       mydata = RealData(x, y, sx=sx, sy=sy)
+
+4. Instantiate ODR with your data, model and initial parameter estimate.::
+
+       myodr = ODR(mydata, linear, beta0=[1., 2.])
+
+5. Run the fit.::
+
+       myoutput = myodr.run()
+
+6. Examine output.::
+
+       myoutput.pprint()
+
+
+References
+----------
+.. [1] P. T. Boggs and J. E. Rogers, "Orthogonal Distance Regression,"
+   in "Statistical analysis of measurement error models and
+   applications: proceedings of the AMS-IMS-SIAM joint summer research
+   conference held June 10-16, 1989," Contemporary Mathematics,
+   vol. 112, pg. 186, 1990.
+
+"""
+# version: 0.7
+# author: Robert Kern <robert.kern@gmail.com>
+# date: 2006-09-21
+
+from ._odrpack import *
+from ._models import *
+from . import _add_newdocs
+
+# Deprecated namespaces, to be removed in v2.0.0
+from . import models, odrpack
+
+__all__ = [s for s in dir()
+           if not (s.startswith('_') or s in ('odr_stop', 'odr_error'))]
+
+from scipy._lib._testutils import PytestTester
+test = PytestTester(__name__)
+del PytestTester

venv/lib/python3.13/site-packages/scipy/odr/_add_newdocs.py

@@ -0,0 +1,34 @@
+from numpy.lib import add_newdoc
+
+add_newdoc('scipy.odr', 'odr',
+    """
+    odr(fcn, beta0, y, x, we=None, wd=None, fjacb=None, fjacd=None, extra_args=None,
+        ifixx=None, ifixb=None, job=0, iprint=0, errfile=None, rptfile=None, ndigit=0,
+        taufac=0.0, sstol=-1.0, partol=-1.0, maxit=-1, stpb=None, stpd=None, sclb=None,
+        scld=None, work=None, iwork=None, full_output=0)
+
+    Low-level function for ODR.
+
+    See Also
+    --------
+    ODR : The ODR class gathers all information and coordinates the running of the
+          main fitting routine.
+    Model : The Model class stores information about the function you wish to fit.
+    Data : The data to fit.
+    RealData : Data with weights as actual std. dev.s and/or covariances.
+
+    Notes
+    -----
+    This is a function performing the same operation as the `ODR`,
+    `Model`, and `Data` classes together. The parameters of this
+    function are explained in the class documentation.
+
+    """)
+
+add_newdoc('scipy.odr.__odrpack', '_set_exceptions',
+    """
+    _set_exceptions(odr_error, odr_stop)
+
+    Internal function: set exception classes.
+
+    """)

venv/lib/python3.13/site-packages/scipy/odr/_models.py

@@ -0,0 +1,315 @@
+""" Collection of Model instances for use with the odrpack fitting package.
+"""
+import numpy as np
+from scipy.odr._odrpack import Model
+
+__all__ = ['Model', 'exponential', 'multilinear', 'unilinear', 'quadratic',
+           'polynomial']
+
+
+def _lin_fcn(B, x):
+    a, b = B[0], B[1:]
+    b.shape = (b.shape[0], 1)
+
+    return a + (x*b).sum(axis=0)
+
+
+def _lin_fjb(B, x):
+    a = np.ones(x.shape[-1], float)
+    res = np.concatenate((a, x.ravel()))
+    res.shape = (B.shape[-1], x.shape[-1])
+    return res
+
+
+def _lin_fjd(B, x):
+    b = B[1:]
+    b = np.repeat(b, (x.shape[-1],)*b.shape[-1], axis=0)
+    b.shape = x.shape
+    return b
+
+
+def _lin_est(data):
+    # Eh. The answer is analytical, so just return all ones.
+    # Don't return zeros since that will interfere with
+    # ODRPACK's auto-scaling procedures.
+
+    if len(data.x.shape) == 2:
+        m = data.x.shape[0]
+    else:
+        m = 1
+
+    return np.ones((m + 1,), float)
+
+
+def _poly_fcn(B, x, powers):
+    a, b = B[0], B[1:]
+    b.shape = (b.shape[0], 1)
+
+    return a + np.sum(b * np.power(x, powers), axis=0)
+
+
+def _poly_fjacb(B, x, powers):
+    res = np.concatenate((np.ones(x.shape[-1], float),
+                          np.power(x, powers).flat))
+    res.shape = (B.shape[-1], x.shape[-1])
+    return res
+
+
+def _poly_fjacd(B, x, powers):
+    b = B[1:]
+    b.shape = (b.shape[0], 1)
+
+    b = b * powers
+
+    return np.sum(b * np.power(x, powers-1), axis=0)
+
+
+def _exp_fcn(B, x):
+    return B[0] + np.exp(B[1] * x)
+
+
+def _exp_fjd(B, x):
+    return B[1] * np.exp(B[1] * x)
+
+
+def _exp_fjb(B, x):
+    res = np.concatenate((np.ones(x.shape[-1], float), x * np.exp(B[1] * x)))
+    res.shape = (2, x.shape[-1])
+    return res
+
+
+def _exp_est(data):
+    # Eh.
+    return np.array([1., 1.])
+
+
+class _MultilinearModel(Model):
+    r"""
+    Arbitrary-dimensional linear model
+
+    This model is defined by :math:`y=\beta_0 + \sum_{i=1}^m \beta_i x_i`
+
+    Examples
+    --------
+    We can calculate orthogonal distance regression with an arbitrary
+    dimensional linear model:
+
+    >>> from scipy import odr
+    >>> import numpy as np
+    >>> x = np.linspace(0.0, 5.0)
+    >>> y = 10.0 + 5.0 * x
+    >>> data = odr.Data(x, y)
+    >>> odr_obj = odr.ODR(data, odr.multilinear)
+    >>> output = odr_obj.run()
+    >>> print(output.beta)
+    [10.  5.]
+
+    """
+
+    def __init__(self):
+        super().__init__(
+            _lin_fcn, fjacb=_lin_fjb, fjacd=_lin_fjd, estimate=_lin_est,
+            meta={'name': 'Arbitrary-dimensional Linear',
+                  'equ': 'y = B_0 + Sum[i=1..m, B_i * x_i]',
+                  'TeXequ': r'$y=\beta_0 + \sum_{i=1}^m \beta_i x_i$'})
+
+
+multilinear = _MultilinearModel()
+
+
+def polynomial(order):
+    """
+    Factory function for a general polynomial model.
+
+    Parameters
+    ----------
+    order : int or sequence
+        If an integer, it becomes the order of the polynomial to fit. If
+        a sequence of numbers, then these are the explicit powers in the
+        polynomial.
+        A constant term (power 0) is always included, so don't include 0.
+        Thus, polynomial(n) is equivalent to polynomial(range(1, n+1)).
+
+    Returns
+    -------
+    polynomial : Model instance
+        Model instance.
+
+    Examples
+    --------
+    We can fit an input data using orthogonal distance regression (ODR) with
+    a polynomial model:
+
+    >>> import numpy as np
+    >>> import matplotlib.pyplot as plt
+    >>> from scipy import odr
+    >>> x = np.linspace(0.0, 5.0)
+    >>> y = np.sin(x)
+    >>> poly_model = odr.polynomial(3)  # using third order polynomial model
+    >>> data = odr.Data(x, y)
+    >>> odr_obj = odr.ODR(data, poly_model)
+    >>> output = odr_obj.run()  # running ODR fitting
+    >>> poly = np.poly1d(output.beta[::-1])
+    >>> poly_y = poly(x)
+    >>> plt.plot(x, y, label="input data")
+    >>> plt.plot(x, poly_y, label="polynomial ODR")
+    >>> plt.legend()
+    >>> plt.show()
+
+    """
+
+    powers = np.asarray(order)
+    if powers.shape == ():
+        # Scalar.
+        powers = np.arange(1, powers + 1)
+
+    powers.shape = (len(powers), 1)
+    len_beta = len(powers) + 1
+
+    def _poly_est(data, len_beta=len_beta):
+        # Eh. Ignore data and return all ones.
+        return np.ones((len_beta,), float)
+
+    return Model(_poly_fcn, fjacd=_poly_fjacd, fjacb=_poly_fjacb,
+                 estimate=_poly_est, extra_args=(powers,),
+                 meta={'name': 'Sorta-general Polynomial',
+                 'equ': 'y = B_0 + Sum[i=1..%s, B_i * (x**i)]' % (len_beta-1),
+                 'TeXequ': r'$y=\beta_0 + \sum_{i=1}^{%s} \beta_i x^i$' %
+                        (len_beta-1)})
+
+
+class _ExponentialModel(Model):
+    r"""
+    Exponential model
+
+    This model is defined by :math:`y=\beta_0 + e^{\beta_1 x}`
+
+    Examples
+    --------
+    We can calculate orthogonal distance regression with an exponential model:
+
+    >>> from scipy import odr
+    >>> import numpy as np
+    >>> x = np.linspace(0.0, 5.0)
+    >>> y = -10.0 + np.exp(0.5*x)
+    >>> data = odr.Data(x, y)
+    >>> odr_obj = odr.ODR(data, odr.exponential)
+    >>> output = odr_obj.run()
+    >>> print(output.beta)
+    [-10.    0.5]
+
+    """
+
+    def __init__(self):
+        super().__init__(_exp_fcn, fjacd=_exp_fjd, fjacb=_exp_fjb,
+                         estimate=_exp_est,
+                         meta={'name': 'Exponential',
+                               'equ': 'y= B_0 + exp(B_1 * x)',
+                               'TeXequ': r'$y=\beta_0 + e^{\beta_1 x}$'})
+
+
+exponential = _ExponentialModel()
+
+
+def _unilin(B, x):
+    return x*B[0] + B[1]
+
+
+def _unilin_fjd(B, x):
+    return np.ones(x.shape, float) * B[0]
+
+
+def _unilin_fjb(B, x):
+    _ret = np.concatenate((x, np.ones(x.shape, float)))
+    _ret.shape = (2,) + x.shape
+
+    return _ret
+
+
+def _unilin_est(data):
+    return (1., 1.)
+
+
+def _quadratic(B, x):
+    return x*(x*B[0] + B[1]) + B[2]
+
+
+def _quad_fjd(B, x):
+    return 2*x*B[0] + B[1]
+
+
+def _quad_fjb(B, x):
+    _ret = np.concatenate((x*x, x, np.ones(x.shape, float)))
+    _ret.shape = (3,) + x.shape
+
+    return _ret
+
+
+def _quad_est(data):
+    return (1.,1.,1.)
+
+
+class _UnilinearModel(Model):
+    r"""
+    Univariate linear model
+
+    This model is defined by :math:`y = \beta_0 x + \beta_1`
+
+    Examples
+    --------
+    We can calculate orthogonal distance regression with an unilinear model:
+
+    >>> from scipy import odr
+    >>> import numpy as np
+    >>> x = np.linspace(0.0, 5.0)
+    >>> y = 1.0 * x + 2.0
+    >>> data = odr.Data(x, y)
+    >>> odr_obj = odr.ODR(data, odr.unilinear)
+    >>> output = odr_obj.run()
+    >>> print(output.beta)
+    [1. 2.]
+
+    """
+
+    def __init__(self):
+        super().__init__(_unilin, fjacd=_unilin_fjd, fjacb=_unilin_fjb,
+                         estimate=_unilin_est,
+                         meta={'name': 'Univariate Linear',
+                               'equ': 'y = B_0 * x + B_1',
+                               'TeXequ': '$y = \\beta_0 x + \\beta_1$'})
+
+
+unilinear = _UnilinearModel()
+
+
+class _QuadraticModel(Model):
+    r"""
+    Quadratic model
+
+    This model is defined by :math:`y = \beta_0 x^2 + \beta_1 x + \beta_2`
+
+    Examples
+    --------
+    We can calculate orthogonal distance regression with a quadratic model:
+
+    >>> from scipy import odr
+    >>> import numpy as np
+    >>> x = np.linspace(0.0, 5.0)
+    >>> y = 1.0 * x ** 2 + 2.0 * x + 3.0
+    >>> data = odr.Data(x, y)
+    >>> odr_obj = odr.ODR(data, odr.quadratic)
+    >>> output = odr_obj.run()
+    >>> print(output.beta)
+    [1. 2. 3.]
+
+    """
+
+    def __init__(self):
+        super().__init__(
+            _quadratic, fjacd=_quad_fjd, fjacb=_quad_fjb, estimate=_quad_est,
+            meta={'name': 'Quadratic',
+                  'equ': 'y = B_0*x**2 + B_1*x + B_2',
+                  'TeXequ': '$y = \\beta_0 x^2 + \\beta_1 x + \\beta_2'})
+
+
+quadratic = _QuadraticModel()

venv/lib/python3.13/site-packages/scipy/odr/models.py

@@ -0,0 +1,20 @@
+# This file is not meant for public use and will be removed in SciPy v2.0.0.
+# Use the `scipy.odr` namespace for importing the functions
+# included below.
+
+from scipy._lib.deprecation import _sub_module_deprecation
+
+__all__ = [  # noqa: F822
+    'Model', 'exponential', 'multilinear', 'unilinear',
+    'quadratic', 'polynomial'
+]
+
+
+def __dir__():
+    return __all__
+
+
+def __getattr__(name):
+    return _sub_module_deprecation(sub_package="odr", module="models",
+                                   private_modules=["_models"], all=__all__,
+                                   attribute=name)

venv/lib/python3.13/site-packages/scipy/odr/odrpack.py

@@ -0,0 +1,21 @@
+# This file is not meant for public use and will be removed in SciPy v2.0.0.
+# Use the `scipy.odr` namespace for importing the functions
+# included below.
+
+from scipy._lib.deprecation import _sub_module_deprecation
+
+__all__ = [  # noqa: F822
+    'odr', 'OdrWarning', 'OdrError', 'OdrStop',
+    'Data', 'RealData', 'Model', 'Output', 'ODR',
+    'odr_error', 'odr_stop'
+]
+
+
+def __dir__():
+    return __all__
+
+
+def __getattr__(name):
+    return _sub_module_deprecation(sub_package="odr", module="odrpack",
+                                   private_modules=["_odrpack"], all=__all__,
+                                   attribute=name)

venv/lib/python3.13/site-packages/scipy/odr/tests/test_odr.py

@@ -0,0 +1,607 @@
+import pickle
+import tempfile
+import shutil
+import os
+
+import numpy as np
+from numpy import pi
+from numpy.testing import (assert_array_almost_equal,
+                           assert_equal, assert_warns,
+                           assert_allclose)
+import pytest
+from pytest import raises as assert_raises
+
+from scipy.odr import (Data, Model, ODR, RealData, OdrStop, OdrWarning,
+                       multilinear, exponential, unilinear, quadratic,
+                       polynomial)
+
+
+class TestODR:
+
+    # Bad Data for 'x'
+
+    def test_bad_data(self):
+        assert_raises(ValueError, Data, 2, 1)
+        assert_raises(ValueError, RealData, 2, 1)
+
+    # Empty Data for 'x'
+    def empty_data_func(self, B, x):
+        return B[0]*x + B[1]
+
+    @pytest.mark.thread_unsafe
+    def test_empty_data(self):
+        beta0 = [0.02, 0.0]
+        linear = Model(self.empty_data_func)
+
+        empty_dat = Data([], [])
+        assert_warns(OdrWarning, ODR,
+                     empty_dat, linear, beta0=beta0)
+
+        empty_dat = RealData([], [])
+        assert_warns(OdrWarning, ODR,
+                     empty_dat, linear, beta0=beta0)
+
+    # Explicit Example
+
+    def explicit_fcn(self, B, x):
+        ret = B[0] + B[1] * np.power(np.exp(B[2]*x) - 1.0, 2)
+        return ret
+
+    def explicit_fjd(self, B, x):
+        eBx = np.exp(B[2]*x)
+        ret = B[1] * 2.0 * (eBx-1.0) * B[2] * eBx
+        return ret
+
+    def explicit_fjb(self, B, x):
+        eBx = np.exp(B[2]*x)
+        res = np.vstack([np.ones(x.shape[-1]),
+                         np.power(eBx-1.0, 2),
+                         B[1]*2.0*(eBx-1.0)*eBx*x])
+        return res
+
+    def test_explicit(self):
+        explicit_mod = Model(
+            self.explicit_fcn,
+            fjacb=self.explicit_fjb,
+            fjacd=self.explicit_fjd,
+            meta=dict(name='Sample Explicit Model',
+                      ref='ODRPACK UG, pg. 39'),
+        )
+        explicit_dat = Data([0.,0.,5.,7.,7.5,10.,16.,26.,30.,34.,34.5,100.],
+                        [1265.,1263.6,1258.,1254.,1253.,1249.8,1237.,1218.,1220.6,
+                         1213.8,1215.5,1212.])
+        explicit_odr = ODR(explicit_dat, explicit_mod, beta0=[1500.0, -50.0, -0.1],
+                       ifixx=[0,0,1,1,1,1,1,1,1,1,1,0])
+        explicit_odr.set_job(deriv=2)
+        explicit_odr.set_iprint(init=0, iter=0, final=0)
+
+        out = explicit_odr.run()
+        assert_array_almost_equal(
+            out.beta,
+            np.array([1.2646548050648876e+03, -5.4018409956678255e+01,
+                -8.7849712165253724e-02]),
+        )
+        assert_array_almost_equal(
+            out.sd_beta,
+            np.array([1.0349270280543437, 1.583997785262061, 0.0063321988657267]),
+        )
+        assert_array_almost_equal(
+            out.cov_beta,
+            np.array([[4.4949592379003039e-01, -3.7421976890364739e-01,
+                 -8.0978217468468912e-04],
+               [-3.7421976890364739e-01, 1.0529686462751804e+00,
+                 -1.9453521827942002e-03],
+               [-8.0978217468468912e-04, -1.9453521827942002e-03,
+                  1.6827336938454476e-05]]),
+        )
+
+    # Implicit Example
+
+    def implicit_fcn(self, B, x):
+        return (B[2]*np.power(x[0]-B[0], 2) +
+                2.0*B[3]*(x[0]-B[0])*(x[1]-B[1]) +
+                B[4]*np.power(x[1]-B[1], 2) - 1.0)
+
+    def test_implicit(self):
+        implicit_mod = Model(
+            self.implicit_fcn,
+            implicit=1,
+            meta=dict(name='Sample Implicit Model',
+                      ref='ODRPACK UG, pg. 49'),
+        )
+        implicit_dat = Data([
+            [0.5,1.2,1.6,1.86,2.12,2.36,2.44,2.36,2.06,1.74,1.34,0.9,-0.28,
+             -0.78,-1.36,-1.9,-2.5,-2.88,-3.18,-3.44],
+            [-0.12,-0.6,-1.,-1.4,-2.54,-3.36,-4.,-4.75,-5.25,-5.64,-5.97,-6.32,
+             -6.44,-6.44,-6.41,-6.25,-5.88,-5.5,-5.24,-4.86]],
+            1,
+        )
+        implicit_odr = ODR(implicit_dat, implicit_mod,
+            beta0=[-1.0, -3.0, 0.09, 0.02, 0.08])
+
+        out = implicit_odr.run()
+        assert_array_almost_equal(
+            out.beta,
+            np.array([-0.9993809167281279, -2.9310484652026476, 0.0875730502693354,
+                0.0162299708984738, 0.0797537982976416]),
+        )
+        assert_array_almost_equal(
+            out.sd_beta,
+            np.array([0.1113840353364371, 0.1097673310686467, 0.0041060738314314,
+                0.0027500347539902, 0.0034962501532468]),
+        )
+        assert_allclose(
+            out.cov_beta,
+            np.array([[2.1089274602333052e+00, -1.9437686411979040e+00,
+                  7.0263550868344446e-02, -4.7175267373474862e-02,
+                  5.2515575927380355e-02],
+               [-1.9437686411979040e+00, 2.0481509222414456e+00,
+                 -6.1600515853057307e-02, 4.6268827806232933e-02,
+                 -5.8822307501391467e-02],
+               [7.0263550868344446e-02, -6.1600515853057307e-02,
+                  2.8659542561579308e-03, -1.4628662260014491e-03,
+                  1.4528860663055824e-03],
+               [-4.7175267373474862e-02, 4.6268827806232933e-02,
+                 -1.4628662260014491e-03, 1.2855592885514335e-03,
+                 -1.2692942951415293e-03],
+               [5.2515575927380355e-02, -5.8822307501391467e-02,
+                  1.4528860663055824e-03, -1.2692942951415293e-03,
+                  2.0778813389755596e-03]]),
+            rtol=1e-6, atol=2e-6,
+        )
+
+    # Multi-variable Example
+
+    def multi_fcn(self, B, x):
+        if (x < 0.0).any():
+            raise OdrStop
+        theta = pi*B[3]/2.
+        ctheta = np.cos(theta)
+        stheta = np.sin(theta)
+        omega = np.power(2.*pi*x*np.exp(-B[2]), B[3])
+        phi = np.arctan2((omega*stheta), (1.0 + omega*ctheta))
+        r = (B[0] - B[1]) * np.power(np.sqrt(np.power(1.0 + omega*ctheta, 2) +
+             np.power(omega*stheta, 2)), -B[4])
+        ret = np.vstack([B[1] + r*np.cos(B[4]*phi),
+                         r*np.sin(B[4]*phi)])
+        return ret
+
+    def test_multi(self):
+        multi_mod = Model(
+            self.multi_fcn,
+            meta=dict(name='Sample Multi-Response Model',
+                      ref='ODRPACK UG, pg. 56'),
+        )
+
+        multi_x = np.array([30.0, 50.0, 70.0, 100.0, 150.0, 200.0, 300.0, 500.0,
+            700.0, 1000.0, 1500.0, 2000.0, 3000.0, 5000.0, 7000.0, 10000.0,
+            15000.0, 20000.0, 30000.0, 50000.0, 70000.0, 100000.0, 150000.0])
+        multi_y = np.array([
+            [4.22, 4.167, 4.132, 4.038, 4.019, 3.956, 3.884, 3.784, 3.713,
+             3.633, 3.54, 3.433, 3.358, 3.258, 3.193, 3.128, 3.059, 2.984,
+             2.934, 2.876, 2.838, 2.798, 2.759],
+            [0.136, 0.167, 0.188, 0.212, 0.236, 0.257, 0.276, 0.297, 0.309,
+             0.311, 0.314, 0.311, 0.305, 0.289, 0.277, 0.255, 0.24, 0.218,
+             0.202, 0.182, 0.168, 0.153, 0.139],
+        ])
+        n = len(multi_x)
+        multi_we = np.zeros((2, 2, n), dtype=float)
+        multi_ifixx = np.ones(n, dtype=int)
+        multi_delta = np.zeros(n, dtype=float)
+
+        multi_we[0,0,:] = 559.6
+        multi_we[1,0,:] = multi_we[0,1,:] = -1634.0
+        multi_we[1,1,:] = 8397.0
+
+        for i in range(n):
+            if multi_x[i] < 100.0:
+                multi_ifixx[i] = 0
+            elif multi_x[i] <= 150.0:
+                pass  # defaults are fine
+            elif multi_x[i] <= 1000.0:
+                multi_delta[i] = 25.0
+            elif multi_x[i] <= 10000.0:
+                multi_delta[i] = 560.0
+            elif multi_x[i] <= 100000.0:
+                multi_delta[i] = 9500.0
+            else:
+                multi_delta[i] = 144000.0
+            if multi_x[i] == 100.0 or multi_x[i] == 150.0:
+                multi_we[:,:,i] = 0.0
+
+        multi_dat = Data(multi_x, multi_y, wd=1e-4/np.power(multi_x, 2),
+            we=multi_we)
+        multi_odr = ODR(multi_dat, multi_mod, beta0=[4.,2.,7.,.4,.5],
+            delta0=multi_delta, ifixx=multi_ifixx)
+        multi_odr.set_job(deriv=1, del_init=1)
+
+        out = multi_odr.run()
+        assert_array_almost_equal(
+            out.beta,
+            np.array([4.3799880305938963, 2.4333057577497703, 8.0028845899503978,
+                0.5101147161764654, 0.5173902330489161]),
+        )
+        assert_array_almost_equal(
+            out.sd_beta,
+            np.array([0.0130625231081944, 0.0130499785273277, 0.1167085962217757,
+                0.0132642749596149, 0.0288529201353984]),
+        )
+        assert_array_almost_equal(
+            out.cov_beta,
+            np.array([[0.0064918418231375, 0.0036159705923791, 0.0438637051470406,
+                -0.0058700836512467, 0.011281212888768],
+               [0.0036159705923791, 0.0064793789429006, 0.0517610978353126,
+                -0.0051181304940204, 0.0130726943624117],
+               [0.0438637051470406, 0.0517610978353126, 0.5182263323095322,
+                -0.0563083340093696, 0.1269490939468611],
+               [-0.0058700836512467, -0.0051181304940204, -0.0563083340093696,
+                 0.0066939246261263, -0.0140184391377962],
+               [0.011281212888768, 0.0130726943624117, 0.1269490939468611,
+                -0.0140184391377962, 0.0316733013820852]]),
+        )
+
+    # Pearson's Data
+    # K. Pearson, Philosophical Magazine, 2, 559 (1901)
+
+    def pearson_fcn(self, B, x):
+        return B[0] + B[1]*x
+
+    def test_pearson(self):
+        p_x = np.array([0.,.9,1.8,2.6,3.3,4.4,5.2,6.1,6.5,7.4])
+        p_y = np.array([5.9,5.4,4.4,4.6,3.5,3.7,2.8,2.8,2.4,1.5])
+        p_sx = np.array([.03,.03,.04,.035,.07,.11,.13,.22,.74,1.])
+        p_sy = np.array([1.,.74,.5,.35,.22,.22,.12,.12,.1,.04])
+
+        p_dat = RealData(p_x, p_y, sx=p_sx, sy=p_sy)
+
+        # Reverse the data to test invariance of results
+        pr_dat = RealData(p_y, p_x, sx=p_sy, sy=p_sx)
+
+        p_mod = Model(self.pearson_fcn, meta=dict(name='Uni-linear Fit'))
+
+        p_odr = ODR(p_dat, p_mod, beta0=[1.,1.])
+        pr_odr = ODR(pr_dat, p_mod, beta0=[1.,1.])
+
+        out = p_odr.run()
+        assert_array_almost_equal(
+            out.beta,
+            np.array([5.4767400299231674, -0.4796082367610305]),
+        )
+        assert_array_almost_equal(
+            out.sd_beta,
+            np.array([0.3590121690702467, 0.0706291186037444]),
+        )
+        assert_array_almost_equal(
+            out.cov_beta,
+            np.array([[0.0854275622946333, -0.0161807025443155],
+               [-0.0161807025443155, 0.003306337993922]]),
+        )
+
+        rout = pr_odr.run()
+        assert_array_almost_equal(
+            rout.beta,
+            np.array([11.4192022410781231, -2.0850374506165474]),
+        )
+        assert_array_almost_equal(
+            rout.sd_beta,
+            np.array([0.9820231665657161, 0.3070515616198911]),
+        )
+        assert_array_almost_equal(
+            rout.cov_beta,
+            np.array([[0.6391799462548782, -0.1955657291119177],
+               [-0.1955657291119177, 0.0624888159223392]]),
+        )
+
+    # Lorentz Peak
+    # The data is taken from one of the undergraduate physics labs I performed.
+
+    def lorentz(self, beta, x):
+        return (beta[0]*beta[1]*beta[2] / np.sqrt(np.power(x*x -
+            beta[2]*beta[2], 2.0) + np.power(beta[1]*x, 2.0)))
+
+    def test_lorentz(self):
+        l_sy = np.array([.29]*18)
+        l_sx = np.array([.000972971,.000948268,.000707632,.000706679,
+            .000706074, .000703918,.000698955,.000456856,
+            .000455207,.000662717,.000654619,.000652694,
+            .000000859202,.00106589,.00106378,.00125483, .00140818,.00241839])
+
+        l_dat = RealData(
+            [3.9094, 3.85945, 3.84976, 3.84716, 3.84551, 3.83964, 3.82608,
+             3.78847, 3.78163, 3.72558, 3.70274, 3.6973, 3.67373, 3.65982,
+             3.6562, 3.62498, 3.55525, 3.41886],
+            [652, 910.5, 984, 1000, 1007.5, 1053, 1160.5, 1409.5, 1430, 1122,
+             957.5, 920, 777.5, 709.5, 698, 578.5, 418.5, 275.5],
+            sx=l_sx,
+            sy=l_sy,
+        )
+        l_mod = Model(self.lorentz, meta=dict(name='Lorentz Peak'))
+        l_odr = ODR(l_dat, l_mod, beta0=(1000., .1, 3.8))
+
+        out = l_odr.run()
+        assert_array_almost_equal(
+            out.beta,
+            np.array([1.4306780846149925e+03, 1.3390509034538309e-01,
+                 3.7798193600109009e+00]),
+        )
+        assert_array_almost_equal(
+            out.sd_beta,
+            np.array([7.3621186811330963e-01, 3.5068899941471650e-04,
+                 2.4451209281408992e-04]),
+        )
+        assert_array_almost_equal(
+            out.cov_beta,
+            np.array([[2.4714409064597873e-01, -6.9067261911110836e-05,
+                 -3.1236953270424990e-05],
+               [-6.9067261911110836e-05, 5.6077531517333009e-08,
+                  3.6133261832722601e-08],
+               [-3.1236953270424990e-05, 3.6133261832722601e-08,
+                  2.7261220025171730e-08]]),
+        )
+
+    def test_ticket_1253(self):
+        def linear(c, x):
+            return c[0]*x+c[1]
+
+        c = [2.0, 3.0]
+        x = np.linspace(0, 10)
+        y = linear(c, x)
+
+        model = Model(linear)
+        data = Data(x, y, wd=1.0, we=1.0)
+        job = ODR(data, model, beta0=[1.0, 1.0])
+        result = job.run()
+        assert_equal(result.info, 2)
+
+    # Verify fix for gh-9140
+
+    def test_ifixx(self):
+        x1 = [-2.01, -0.99, -0.001, 1.02, 1.98]
+        x2 = [3.98, 1.01, 0.001, 0.998, 4.01]
+        fix = np.vstack((np.zeros_like(x1, dtype=int), np.ones_like(x2, dtype=int)))
+        data = Data(np.vstack((x1, x2)), y=1, fix=fix)
+        model = Model(lambda beta, x: x[1, :] - beta[0] * x[0, :]**2., implicit=True)
+
+        odr1 = ODR(data, model, beta0=np.array([1.]))
+        sol1 = odr1.run()
+        odr2 = ODR(data, model, beta0=np.array([1.]), ifixx=fix)
+        sol2 = odr2.run()
+        assert_equal(sol1.beta, sol2.beta)
+
+    # verify bugfix for #11800 in #11802
+    def test_ticket_11800(self):
+        # parameters
+        beta_true = np.array([1.0, 2.3, 1.1, -1.0, 1.3, 0.5])
+        nr_measurements = 10
+
+        std_dev_x = 0.01
+        x_error = np.array([[0.00063445, 0.00515731, 0.00162719, 0.01022866,
+            -0.01624845, 0.00482652, 0.00275988, -0.00714734, -0.00929201, -0.00687301],
+            [-0.00831623, -0.00821211, -0.00203459, 0.00938266, -0.00701829,
+            0.0032169, 0.00259194, -0.00581017, -0.0030283, 0.01014164]])
+
+        std_dev_y = 0.05
+        y_error = np.array([[0.05275304, 0.04519563, -0.07524086, 0.03575642,
+            0.04745194, 0.03806645, 0.07061601, -0.00753604, -0.02592543, -0.02394929],
+            [0.03632366, 0.06642266, 0.08373122, 0.03988822, -0.0092536,
+            -0.03750469, -0.03198903, 0.01642066, 0.01293648, -0.05627085]])
+
+        beta_solution = np.array([
+            2.62920235756665876536e+00, -1.26608484996299608838e+02,
+            1.29703572775403074502e+02, -1.88560985401185465804e+00,
+            7.83834160771274923718e+01, -7.64124076838087091801e+01])
+
+        # model's function and Jacobians
+        def func(beta, x):
+            y0 = beta[0] + beta[1] * x[0, :] + beta[2] * x[1, :]
+            y1 = beta[3] + beta[4] * x[0, :] + beta[5] * x[1, :]
+
+            return np.vstack((y0, y1))
+
+        def df_dbeta_odr(beta, x):
+            nr_meas = np.shape(x)[1]
+            zeros = np.zeros(nr_meas)
+            ones = np.ones(nr_meas)
+
+            dy0 = np.array([ones, x[0, :], x[1, :], zeros, zeros, zeros])
+            dy1 = np.array([zeros, zeros, zeros, ones, x[0, :], x[1, :]])
+
+            return np.stack((dy0, dy1))
+
+        def df_dx_odr(beta, x):
+            nr_meas = np.shape(x)[1]
+            ones = np.ones(nr_meas)
+
+            dy0 = np.array([beta[1] * ones, beta[2] * ones])
+            dy1 = np.array([beta[4] * ones, beta[5] * ones])
+            return np.stack((dy0, dy1))
+
+        # do measurements with errors in independent and dependent variables
+        x0_true = np.linspace(1, 10, nr_measurements)
+        x1_true = np.linspace(1, 10, nr_measurements)
+        x_true = np.array([x0_true, x1_true])
+
+        y_true = func(beta_true, x_true)
+
+        x_meas = x_true + x_error
+        y_meas = y_true + y_error
+
+        # estimate model's parameters
+        model_f = Model(func, fjacb=df_dbeta_odr, fjacd=df_dx_odr)
+
+        data = RealData(x_meas, y_meas, sx=std_dev_x, sy=std_dev_y)
+
+        odr_obj = ODR(data, model_f, beta0=0.9 * beta_true, maxit=100)
+        #odr_obj.set_iprint(init=2, iter=0, iter_step=1, final=1)
+        odr_obj.set_job(deriv=3)
+
+        odr_out = odr_obj.run()
+
+        # check results
+        assert_equal(odr_out.info, 1)
+        assert_array_almost_equal(odr_out.beta, beta_solution)
+
+    def test_multilinear_model(self):
+        x = np.linspace(0.0, 5.0)
+        y = 10.0 + 5.0 * x
+        data = Data(x, y)
+        odr_obj = ODR(data, multilinear)
+        output = odr_obj.run()
+        assert_array_almost_equal(output.beta, [10.0, 5.0])
+
+    def test_exponential_model(self):
+        x = np.linspace(0.0, 5.0)
+        y = -10.0 + np.exp(0.5*x)
+        data = Data(x, y)
+        odr_obj = ODR(data, exponential)
+        output = odr_obj.run()
+        assert_array_almost_equal(output.beta, [-10.0, 0.5])
+
+    def test_polynomial_model(self):
+        x = np.linspace(0.0, 5.0)
+        y = 1.0 + 2.0 * x + 3.0 * x ** 2 + 4.0 * x ** 3
+        poly_model = polynomial(3)
+        data = Data(x, y)
+        odr_obj = ODR(data, poly_model)
+        output = odr_obj.run()
+        assert_array_almost_equal(output.beta, [1.0, 2.0, 3.0, 4.0])
+
+    def test_unilinear_model(self):
+        x = np.linspace(0.0, 5.0)
+        y = 1.0 * x + 2.0
+        data = Data(x, y)
+        odr_obj = ODR(data, unilinear)
+        output = odr_obj.run()
+        assert_array_almost_equal(output.beta, [1.0, 2.0])
+
+    def test_quadratic_model(self):
+        x = np.linspace(0.0, 5.0)
+        y = 1.0 * x ** 2 + 2.0 * x + 3.0
+        data = Data(x, y)
+        odr_obj = ODR(data, quadratic)
+        output = odr_obj.run()
+        assert_array_almost_equal(output.beta, [1.0, 2.0, 3.0])
+
+    def test_work_ind(self):
+
+        def func(par, x):
+            b0, b1 = par
+            return b0 + b1 * x
+
+        # generate some data
+        n_data = 4
+        x = np.arange(n_data)
+        y = np.where(x % 2, x + 0.1, x - 0.1)
+        x_err = np.full(n_data, 0.1)
+        y_err = np.full(n_data, 0.1)
+
+        # do the fitting
+        linear_model = Model(func)
+        real_data = RealData(x, y, sx=x_err, sy=y_err)
+        odr_obj = ODR(real_data, linear_model, beta0=[0.4, 0.4])
+        odr_obj.set_job(fit_type=0)
+        out = odr_obj.run()
+
+        sd_ind = out.work_ind['sd']
+        assert_array_almost_equal(out.sd_beta,
+                                  out.work[sd_ind:sd_ind + len(out.sd_beta)])
+
+    @pytest.mark.skipif(True, reason="Fortran I/O prone to crashing so better "
+                                     "not to run this test, see gh-13127")
+    def test_output_file_overwrite(self):
+        """
+        Verify fix for gh-1892
+        """
+        def func(b, x):
+            return b[0] + b[1] * x
+
+        p = Model(func)
+        data = Data(np.arange(10), 12 * np.arange(10))
+        tmp_dir = tempfile.mkdtemp()
+        error_file_path = os.path.join(tmp_dir, "error.dat")
+        report_file_path = os.path.join(tmp_dir, "report.dat")
+        try:
+            ODR(data, p, beta0=[0.1, 13], errfile=error_file_path,
+                rptfile=report_file_path).run()
+            ODR(data, p, beta0=[0.1, 13], errfile=error_file_path,
+                rptfile=report_file_path, overwrite=True).run()
+        finally:
+            # remove output files for clean up
+            shutil.rmtree(tmp_dir)
+
+    def test_odr_model_default_meta(self):
+        def func(b, x):
+            return b[0] + b[1] * x
+
+        p = Model(func)
+        p.set_meta(name='Sample Model Meta', ref='ODRPACK')
+        assert_equal(p.meta, {'name': 'Sample Model Meta', 'ref': 'ODRPACK'})
+
+    def test_work_array_del_init(self):
+        """
+        Verify fix for gh-18739 where del_init=1 fails.
+        """
+        def func(b, x):
+            return b[0] + b[1] * x
+
+        # generate some data
+        n_data = 4
+        x = np.arange(n_data)
+        y = np.where(x % 2, x + 0.1, x - 0.1)
+        x_err = np.full(n_data, 0.1)
+        y_err = np.full(n_data, 0.1)
+
+        linear_model = Model(func)
+        # Try various shapes of the `we` array from various `sy` and `covy`
+        rd0 = RealData(x, y, sx=x_err, sy=y_err)
+        rd1 = RealData(x, y, sx=x_err, sy=0.1)
+        rd2 = RealData(x, y, sx=x_err, sy=[0.1])
+        rd3 = RealData(x, y, sx=x_err, sy=np.full((1, n_data), 0.1))
+        rd4 = RealData(x, y, sx=x_err, covy=[[0.01]])
+        rd5 = RealData(x, y, sx=x_err, covy=np.full((1, 1, n_data), 0.01))
+        for rd in [rd0, rd1, rd2, rd3, rd4, rd5]:
+            odr_obj = ODR(rd, linear_model, beta0=[0.4, 0.4],
+                          delta0=np.full(n_data, -0.1))
+            odr_obj.set_job(fit_type=0, del_init=1)
+            # Just make sure that it runs without raising an exception.
+            odr_obj.run()
+
+    def test_pickling_data(self):
+        x = np.linspace(0.0, 5.0)
+        y = 1.0 * x + 2.0
+        data = Data(x, y)
+
+        obj_pickle = pickle.dumps(data)
+        del data
+        pickle.loads(obj_pickle)
+
+    def test_pickling_real_data(self):
+        x = np.linspace(0.0, 5.0)
+        y = 1.0 * x + 2.0
+        data = RealData(x, y)
+
+        obj_pickle = pickle.dumps(data)
+        del data
+        pickle.loads(obj_pickle)
+
+    def test_pickling_model(self):
+        obj_pickle = pickle.dumps(unilinear)
+        pickle.loads(obj_pickle)
+
+    def test_pickling_odr(self):
+        x = np.linspace(0.0, 5.0)
+        y = 1.0 * x + 2.0
+        odr_obj = ODR(Data(x, y), unilinear)
+
+        obj_pickle = pickle.dumps(odr_obj)
+        del odr_obj
+        pickle.loads(obj_pickle)
+
+    def test_pickling_output(self):
+        x = np.linspace(0.0, 5.0)
+        y = 1.0 * x + 2.0
+        output = ODR(Data(x, y), unilinear).run
+
+        obj_pickle = pickle.dumps(output)
+        del output
+        pickle.loads(obj_pickle)