5.2. Numdifftools package details

5.2.1. numdifftools.tests package

5.2.2. numdifftools.core module

numerical differentiation functions:

Derivative, Gradient, Jacobian, and Hessian

Author: Per A. Brodtkorb Created: 01.08.2008 Copyright: (c) pab 2008 Licence: New BSD

class Derivative(fun, step=None, method='central', order=2, n=1, **options)[source]

Bases: _Limit

Calculate n-th derivative with finite difference approximation

Parameters:

funfunction

function of one array fun(x, *args, **kwds)

stepfloat, array-like or StepGenerator object, optional

Defines the spacing used in the approximation. Default is MinStepGenerator(**step_options) if method in in [‘complex’, ‘multicomplex’], otherwise

MaxStepGenerator(**step_options)

The results are extrapolated if the StepGenerator generate more than 3 steps.

method{‘central’, ‘complex’, ‘multicomplex’, ‘forward’, ‘backward’}

defines the method used in the approximation

orderint, optional

defines the order of the error term in the Taylor approximation used. For ‘central’ and ‘complex’ methods, it must be an even number.

nint, optional

Order of the derivative.

richardson_terms: scalar integer, default 2.

number of terms used in the Richardson extrapolation.

full_outputbool, optional

If full_output is False, only the derivative is returned. If full_output is True, then (der, r) is returned der is the derivative, and r is a Results object.

**step_options:

options to pass on to the XXXStepGenerator used.

Methods

__call__(x, *args, **kwds)

Call self as a function.

Returns:

derndarray: array of derivatives

See also

Gradient
Hessian

Notes

Complex methods are usually the most accurate provided the function to differentiate is analytic. The complex-step methods also requires fewer steps than the other methods and can work very close to the support of a function. The complex-step derivative has truncation error O(steps**2) for n=1 and O(steps**4) for n larger, so truncation error can be eliminated by choosing steps to be very small. Especially the first order complex-step derivative avoids the problem of round-off error with small steps because there is no subtraction. However, this method fails if fun(x) does not support complex numbers or involves non-analytic functions such as e.g.: abs, max, min. Central difference methods are almost as accurate and has no restriction on type of function. For this reason the ‘central’ method is the default method, but sometimes one can only allow evaluation in forward or backward direction.

For all methods one should be careful in decreasing the step size too much due to round-off errors.

Higher order approximation methods will generally be more accurate, but may also suffer more from numerical problems. First order methods is usually not recommended.

References

Ridout, M.S. (2009) Statistical applications of the complex-step method: of numerical differentiation. The American Statistician, 63, 66-74
K.-L. Lai, J.L. Crassidis, Y. Cheng, J. Kim (2005), New complex step: derivative approximations with application to second-order kalman filtering, AIAA Guidance, Navigation and Control Conference, San Francisco, California, August 2005, AIAA-2005-5944.
Lyness, J. M., Moler, C. B. (1966). Vandermonde Systems and Numerical: Differentiation. Numerische Mathematik.
Lyness, J. M., Moler, C. B. (1969). Generalized Romberg Methods for: Integrals of Derivatives. Numerische Mathematik.

Examples

>>> import numpy as np
>>> import numdifftools as nd

# 1’st derivative of exp(x), at x == 1

>>> fd = nd.Derivative(np.exp)
>>> np.allclose(fd(1), 2.71828183)
True

>>> d2 = fd([1, 2])
>>> np.allclose(d2, [ 2.71828183,  7.3890561 ])
True

>>> def f(x):
...     return x**3 + x**2

>>> df = nd.Derivative(f)
>>> np.allclose(df(1), 5)
True
>>> ddf = nd.Derivative(f, n=2)
>>> np.allclose(ddf(1), 8)
True

class info(f_value, error_estimate, final_step, index)

Bases: tuple

count(value, /): Return number of occurrences of value.

error_estimate: Alias for field number 1

f_value: Alias for field number 0

final_step: Alias for field number 2

index: Alias for field number 3

property method: Defines the method used in the finite difference approximation.

property method_order: Defines the leading order of the error term in the Richardson extrapolation method.

property n: Order of the derivative.

property order: Defines the order of the error term in the Taylor approximation used.

set_richardson_rule(step_ratio, num_terms=2)[source]: Set Richardson extrapolation options

property step: The step spacing(s) used in the approximation

class Gradient(fun, step=None, method='central', order=2, n=1, **options)[source]

Bases: Jacobian

Calculate Gradient with finite difference approximation

Parameters:

funfunction

function of one array fun(x, *args, **kwds)

stepfloat, array-like or StepGenerator object, optional

Defines the spacing used in the approximation. Default is MinStepGenerator(**step_options) if method in in [‘complex’, ‘multicomplex’], otherwise

MaxStepGenerator(**step_options)

The results are extrapolated if the StepGenerator generate more than 3 steps.

method{‘central’, ‘complex’, ‘multicomplex’, ‘forward’, ‘backward’}

defines the method used in the approximation

orderint, optional

defines the order of the error term in the Taylor approximation used. For ‘central’ and ‘complex’ methods, it must be an even number.

richardson_terms: scalar integer, default 2.

number of terms used in the Richardson extrapolation.

full_outputbool, optional

If full_output is False, only the derivative is returned. If full_output is True, then (der, r) is returned der is the derivative, and r is a Results object.

**step_options:

options to pass on to the XXXStepGenerator used.

Methods

__call__(x, *args, **kwds)

Call self as a function.

Returns:

gradarray: gradient

See also

Derivative, Hessian, Jacobian

Notes

Complex methods are usually the most accurate provided the function to differentiate is analytic. The complex-step methods also requires fewer steps than the other methods and can work very close to the support of a function. The complex-step derivative has truncation error O(steps**2) for n=1 and O(steps**4) for n larger, so truncation error can be eliminated by choosing steps to be very small. Especially the first order complex-step derivative avoids the problem of round-off error with small steps because there is no subtraction. However, this method fails if fun(x) does not support complex numbers or involves non-analytic functions such as e.g.: abs, max, min. Central difference methods are almost as accurate and has no restriction on type of function. For this reason the ‘central’ method is the default method, but sometimes one can only allow evaluation in forward or backward direction.

For all methods one should be careful in decreasing the step size too much due to round-off errors.

Higher order approximation methods will generally be more accurate, but may also suffer more from numerical problems. First order methods is usually not recommended.

If x0 is an n x m array, then fun is assumed to be a function of n * m variables.

References

Ridout, M.S. (2009) Statistical applications of the complex-step method: of numerical differentiation. The American Statistician, 63, 66-74
K.-L. Lai, J.L. Crassidis, Y. Cheng, J. Kim (2005), New complex step: derivative approximations with application to second-order kalman filtering, AIAA Guidance, Navigation and Control Conference, San Francisco, California, August 2005, AIAA-2005-5944.
Lyness, J. M., Moler, C. B. (1966). Vandermonde Systems and Numerical: Differentiation. Numerische Mathematik.
Lyness, J. M., Moler, C. B. (1969). Generalized Romberg Methods for: Integrals of Derivatives. Numerische Mathematik.

Examples

>>> import numpy as np
>>> import numdifftools as nd
>>> fun = lambda x: np.sum(x**2)
>>> dfun = nd.Gradient(fun)
>>> np.allclose(dfun([1,2,3]), [ 2.,  4.,  6.])
True

# At [x,y] = [1,1], compute the numerical gradient # of the function sin(x-y) + y*exp(x)

>>> sin = np.sin; exp = np.exp
>>> x, y = 1, 1
>>> z = lambda xy: sin(xy[0]-xy[1]) + xy[1]*exp(xy[0])
>>> dz = nd.Gradient(z)
>>> dz_dx, dz_dy = dz([x, y])
>>> np.allclose([dz_dx, dz_dy],
...             [ 3.7182818284590686, 1.7182818284590162])
True

# At the global minimizer (1,1) of the Rosenbrock function, # compute the gradient. It should be essentially zero.

>>> rosen = lambda x : (1-x[0])**2 + 105.*(x[1]-x[0]**2)**2
>>> grad_rosen = nd.Gradient(rosen)
>>> df_dx, df_dy = grad_rosen([x, y])
>>> np.allclose([df_dx, df_dy], [0, 0])
True

class info(f_value, error_estimate, final_step, index)

Bases: tuple

count(value, /): Return number of occurrences of value.

error_estimate: Alias for field number 1

f_value: Alias for field number 0

final_step: Alias for field number 2

index: Alias for field number 3

property method: Defines the method used in the finite difference approximation.

property method_order: Defines the leading order of the error term in the Richardson extrapolation method.

property n: Order of the derivative.

property order: Defines the order of the error term in the Taylor approximation used.

set_richardson_rule(step_ratio, num_terms=2): Set Richardson extrapolation options

property step: The step spacing(s) used in the approximation

class Hessdiag(f, step=None, method='central', order=2, **options)[source]

Bases: Derivative

Calculate Hessian diagonal with finite difference approximation

Parameters:

funfunction

function of one array fun(x, *args, **kwds)

stepfloat, array-like or StepGenerator object, optional

Defines the spacing used in the approximation. Default is MinStepGenerator(**step_options) if method in in [‘complex’, ‘multicomplex’], otherwise

MaxStepGenerator(**step_options)

The results are extrapolated if the StepGenerator generate more than 3 steps.

method{‘central’, ‘complex’, ‘multicomplex’, ‘forward’, ‘backward’}

defines the method used in the approximationorder : int, optional defines the order of the error term in the Taylor approximation used. For ‘central’ and ‘complex’ methods, it must be an even number.

richardson_terms: scalar integer, default 2.

number of terms used in the Richardson extrapolation.

full_outputbool, optional

If full_output is False, only the derivative is returned. If full_output is True, then (der, r) is returned der is the derivative, and r is a Results object.

**step_options:

options to pass on to the XXXStepGenerator used.

Methods

__call__(x, *args, **kwds)

Call self as a function.

Returns:

hessdiagarray: hessian diagonal

See also

Derivative, Hessian, Jacobian, Gradient

Notes

Complex methods are usually the most accurate provided the function to differentiate is analytic. The complex-step methods also requires fewer steps than the other methods and can work very close to the support of a function. The complex-step derivative has truncation error O(steps**2) for n=1 and O(steps**4) for n larger, so truncation error can be eliminated by choosing steps to be very small. Especially the first order complex-step derivative avoids the problem of round-off error with small steps because there is no subtraction. However, this method fails if fun(x) does not support complex numbers or involves non-analytic functions such as e.g.: abs, max, min. Central difference methods are almost as accurate and has no restriction on type of function. For this reason the ‘central’ method is the default method, but sometimes one can only allow evaluation in forward or backward direction.

For all methods one should be careful in decreasing the step size too much due to round-off errors.

Higher order approximation methods will generally be more accurate, but may also suffer more from numerical problems. First order methods is usually not recommended.

References

Ridout, M.S. (2009) Statistical applications of the complex-step method: of numerical differentiation. The American Statistician, 63, 66-74
K.-L. Lai, J.L. Crassidis, Y. Cheng, J. Kim (2005), New complex step: derivative approximations with application to second-order kalman filtering, AIAA Guidance, Navigation and Control Conference, San Francisco, California, August 2005, AIAA-2005-5944.
Lyness, J. M., Moler, C. B. (1966). Vandermonde Systems and Numerical: Differentiation. Numerische Mathematik.
Lyness, J. M., Moler, C. B. (1969). Generalized Romberg Methods for: Integrals of Derivatives. Numerische Mathematik.

Examples

>>> import numpy as np
>>> import numdifftools as nd
>>> fun = lambda x : x[0] + x[1]**2 + x[2]**3
>>> Hfun = nd.Hessdiag(fun, full_output=True)
>>> hd, info = Hfun([1,2,3])
>>> np.allclose(hd, [0.,   2.,  18.])
True

>>> bool(np.all(info.error_estimate < 1e-11))
True

class info(f_value, error_estimate, final_step, index)

Bases: tuple

count(value, /): Return number of occurrences of value.

error_estimate: Alias for field number 1

f_value: Alias for field number 0

final_step: Alias for field number 2

index: Alias for field number 3

property method: Defines the method used in the finite difference approximation.

property method_order: Defines the leading order of the error term in the Richardson extrapolation method.

property n: Order of the derivative.

property order: Defines the order of the error term in the Taylor approximation used.

set_richardson_rule(step_ratio, num_terms=2): Set Richardson extrapolation options

property step: The step spacing(s) used in the approximation

class Hessian(f, step=None, method='central', order=None, **options)[source]

Bases: Hessdiag

Calculate Hessian with finite difference approximation

Parameters:

funfunction

function of one array fun(x, *args, **kwds)

stepfloat, array-like or StepGenerator object, optional

Defines the spacing used in the approximation. Default is MinStepGenerator(**step_options) if method in in [‘complex’, ‘multicomplex’], otherwise

MaxStepGenerator(**step_options)

The results are extrapolated if the StepGenerator generate more than 3 steps.

method{‘central’, ‘complex’, ‘multicomplex’, ‘forward’, ‘backward’}

defines the method used in the approximation

richardson_terms: scalar integer, default 2.

number of terms used in the Richardson extrapolation.

full_outputbool, optional

If full_output is False, only the derivative is returned. If full_output is True, then (der, r) is returned der is the derivative, and r is a Results object.

**step_options:

options to pass on to the XXXStepGenerator used.

Methods

__call__(x, *args, **kwds)

Call self as a function.

Returns:

hessndarray: array of partial second derivatives, Hessian

See also

Derivative, Hessian

Notes

Complex methods are usually the most accurate provided the function to differentiate is analytic. The complex-step methods also requires fewer steps than the other methods and can work very close to the support of a function. The complex-step derivative has truncation error O(steps**2) for n=1 and O(steps**4) for n larger, so truncation error can be eliminated by choosing steps to be very small. Especially the first order complex-step derivative avoids the problem of round-off error with small steps because there is no subtraction. However, this method fails if fun(x) does not support complex numbers or involves non-analytic functions such as e.g.: abs, max, min. Central difference methods are almost as accurate and has no restriction on type of function. For this reason the ‘central’ method is the default method, but sometimes one can only allow evaluation in forward or backward direction.

For all methods one should be careful in decreasing the step size too much due to round-off errors.

Computes the Hessian according to method as: ‘forward’ (4.7), ‘central’ (4.9) and ‘complex’ (4.10):

(5.4)\[\quad ((f(x + d_j e_j + d_k e_k) + f(x) - f(x + d_j e_j) - f(x + d_k e_k))) / (d_j d_k)\]

(5.5)\[\quad ((f(x + d_j e_j + d_k e_k) - f(x + d_j e_j - d_k e_k)) - (f(x - d_j e_j + d_k e_k) - f(x - d_j e_j - d_k e_k)) / (4 d_j d_k)\]

(5.6)\[imag(f(x + i d_j e_j + d_k e_k) - f(x + i d_j e_j - d_k e_k)) / (2 d_j d_k)\]

where \(e_j\) is a vector with element \(j\) is one and the rest are zero and \(d_j\) is a scalar spacing \(steps_j\).

References

Ridout, M.S. (2009) Statistical applications of the complex-step method: of numerical differentiation. The American Statistician, 63, 66-74
K.-L. Lai, J.L. Crassidis, Y. Cheng, J. Kim (2005), New complex step: derivative approximations with application to second-order kalman filtering, AIAA Guidance, Navigation and Control Conference, San Francisco, California, August 2005, AIAA-2005-5944.
Lyness, J. M., Moler, C. B. (1966). Vandermonde Systems and Numerical: Differentiation. Numerische Mathematik.
Lyness, J. M., Moler, C. B. (1969). Generalized Romberg Methods for: Integrals of Derivatives. Numerische Mathematik.

Examples

>>> import numpy as np
>>> import numdifftools as nd

# Rosenbrock function, minimized at [1,1]

>>> rosen = lambda x : (1.-x[0])**2 + 105*(x[1]-x[0]**2)**2
>>> Hfun = nd.Hessian(rosen)
>>> h = Hfun([1, 1])
>>> h
array([[ 842., -420.],
       [-420.,  210.]])

# cos(x-y), at (0,0)

>>> cos = np.cos
>>> fun = lambda xy : cos(xy[0]-xy[1])
>>> Hfun2 = nd.Hessian(fun)
>>> h2 = Hfun2([0, 0])
>>> h2
array([[-1.,  1.],
       [ 1., -1.]])

class info(f_value, error_estimate, final_step, index)

Bases: tuple

count(value, /): Return number of occurrences of value.

error_estimate: Alias for field number 1

f_value: Alias for field number 0

final_step: Alias for field number 2

index: Alias for field number 3

property method: Defines the method used in the finite difference approximation.

property method_order: Defines the leading order of the error term in the Richardson extrapolation method.

property n: Order of the derivative.

property order: Defines the order of the error term in the Taylor approximation used.

set_richardson_rule(step_ratio, num_terms=2): Set Richardson extrapolation options

property step: The step spacing(s) used in the approximation

class Jacobian(fun, step=None, method='central', order=2, n=1, **options)[source]

Bases: Derivative

Calculate Jacobian with finite difference approximation

Parameters:

funfunction

function of one array fun(x, *args, **kwds)

stepfloat, array-like or StepGenerator object, optional

Defines the spacing used in the approximation. Default is MinStepGenerator(**step_options) if method in in [‘complex’, ‘multicomplex’], otherwise

MaxStepGenerator(**step_options)

The results are extrapolated if the StepGenerator generate more than 3 steps.

method{‘central’, ‘complex’, ‘multicomplex’, ‘forward’, ‘backward’}

defines the method used in the approximation

orderint, optional

defines the order of the error term in the Taylor approximation used. For ‘central’ and ‘complex’ methods, it must be an even number.

richardson_terms: scalar integer, default 2.

number of terms used in the Richardson extrapolation.

full_outputbool, optional

If full_output is False, only the derivative is returned. If full_output is True, then (der, r) is returned der is the derivative, and r is a Results object.

**step_options:

options to pass on to the XXXStepGenerator used.

Methods

__call__(x, *args, **kwds)

Call self as a function.

Returns:

jacobarray: Jacobian

See also

Derivative, Hessian, Gradient

Notes

Complex methods are usually the most accurate provided the function to differentiate is analytic. The complex-step methods also requires fewer steps than the other methods and can work very close to the support of a function. The complex-step derivative has truncation error O(steps**2) for n=1 and O(steps**4) for n larger, so truncation error can be eliminated by choosing steps to be very small. Especially the first order complex-step derivative avoids the problem of round-off error with small steps because there is no subtraction. However, this method fails if fun(x) does not support complex numbers or involves non-analytic functions such as e.g.: abs, max, min. Central difference methods are almost as accurate and has no restriction on type of function. For this reason the ‘central’ method is the default method, but sometimes one can only allow evaluation in forward or backward direction.

For all methods one should be careful in decreasing the step size too much due to round-off errors.

Higher order approximation methods will generally be more accurate, but may also suffer more from numerical problems. First order methods is usually not recommended.

If fun returns a 1d array, it returns a Jacobian. If a 2d array is returned by fun (e.g., with a value for each observation), it returns a 3d array with the Jacobian of each observation with shape xk x nobs x xk. I.e., the Jacobian of the first observation would be [:, 0, :]

References

Ridout, M.S. (2009) Statistical applications of the complex-step method: of numerical differentiation. The American Statistician, 63, 66-74
K.-L. Lai, J.L. Crassidis, Y. Cheng, J. Kim (2005), New complex step: derivative approximations with application to second-order kalman filtering, AIAA Guidance, Navigation and Control Conference, San Francisco, California, August 2005, AIAA-2005-5944.
Lyness, J. M., Moler, C. B. (1966). Vandermonde Systems and Numerical: Differentiation. Numerische Mathematik.
Lyness, J. M., Moler, C. B. (1969). Generalized Romberg Methods for: Integrals of Derivatives. Numerische Mathematik.

Examples

>>> import numpy as np
>>> import numdifftools as nd

#(nonlinear least squares)

>>> xdata = np.arange(0,1,0.1)
>>> ydata = 1+2*np.exp(0.75*xdata)
>>> fun = lambda c: (c[0]+c[1]*np.exp(c[2]*xdata) - ydata)**2
>>> np.allclose(fun([1, 2, 0.75]).shape,  (10,))
True

>>> jfun = nd.Jacobian(fun)
>>> val = jfun([1, 2, 0.75])
>>> np.allclose(val, np.zeros((10,3)))
True

>>> fun2 = lambda x : x[0]*x[1]*x[2]**2
>>> jfun2 = nd.Jacobian(fun2)
>>> np.allclose(jfun2([1.,2.,3.]), [[18., 9., 12.]])
True

>>> fun3 = lambda x : np.vstack((x[0]*x[1]*x[2]**2, x[0]*x[1]*x[2]))
>>> jfun3 = nd.Jacobian(fun3)

>>> np.allclose(jfun3([1., 2., 3.]), [[[18.], [9.], [12.]], [[6.], [3.], [2.]]])
True
>>> np.allclose(jfun3([4., 5., 6.]), [[[180.], [144.], [240.]], [[30.], [24.], [20.]]])
True
>>> np.allclose(jfun3(np.array([[1.,2.,3.]]).T), [[[18.], [9.], [12.]], [[6.], [3.], [2.]]])
True

class info(f_value, error_estimate, final_step, index)

Bases: tuple

count(value, /): Return number of occurrences of value.

error_estimate: Alias for field number 1

f_value: Alias for field number 0

final_step: Alias for field number 2

index: Alias for field number 3

property method: Defines the method used in the finite difference approximation.

property method_order: Defines the leading order of the error term in the Richardson extrapolation method.

property n: Order of the derivative.

property order: Defines the order of the error term in the Taylor approximation used.

set_richardson_rule(step_ratio, num_terms=2): Set Richardson extrapolation options

property step: The step spacing(s) used in the approximation

class MaxStepGenerator(base_step=2.0, step_ratio=None, num_steps=15, step_nom=None, offset=0, num_extrap=9, use_exact_steps=False, check_num_steps=True, scale=500)[source]

Bases: MinStepGenerator

Generates a sequence of steps

where: steps = step_nom * base_step * step_ratio ** (-i + offset)

for i = 0, 1, …, num_steps-1.

Parameters:

base_stepfloat, array-like, default 2.0: Defines the maximum step, if None, the value is set to EPS**(1/scale)
step_ratioreal scalar, optional, default 2 or 1.6: Ratio between sequential steps generated. Note: Ratio > 1 If None then step_ratio is 2 for n=1 otherwise step_ratio is 1.6
num_stepsscalar integer, optional, default min_num_steps + num_extrap: defines number of steps generated. It should be larger than min_num_steps = (n + order - 1) / fact where fact is 1, 2 or 4 depending on differentiation method used.
step_nomdefault maximum(log(exp(1)+|x|), 1): Nominal step where x is supplied at runtime through the __call__ method.
offsetreal scalar, optional, default 0: offset to the base step
num_extrapscalar integer, default 0: number of points used for extrapolation
check_num_stepsboolean, default True: If True make sure num_steps is larger than the minimum required steps.
use_exact_stepsboolean, default True: If true make sure exact steps are generated
scalereal scalar, default 500: scale used in base step.

property base_step: Base step defines the minimum or maximum step when offset==0.

property min_num_steps: Minimum number of steps required given the differentiation method and order.

property num_steps: Defines number of steps generated

property scale: Scale used in base step.

step_generator_function(x, method='forward', n=1, order=2): Step generator function

property step_nom: Nominal step

property step_ratio: Ratio between sequential steps generated

class MinStepGenerator(base_step=None, step_ratio=None, num_steps=None, step_nom=None, offset=0, num_extrap=0, use_exact_steps=True, check_num_steps=True, scale=None)[source]

Bases: object

Generates a sequence of steps

where: steps = step_nom * base_step * step_ratio ** (i + offset)

for i = num_steps-1,… 1, 0.

Parameters:

base_stepfloat, array-like, optional: Defines the minimum step, if None, the value is set to EPS**(1/scale)
step_ratioreal scalar, optional, default 2: Ratio between sequential steps generated. Note: Ratio > 1 If None then step_ratio is 2 for n=1 otherwise step_ratio is 1.6
num_stepsscalar integer, optional, default min_num_steps + num_extrap: defines number of steps generated. It should be larger than min_num_steps = (n + order - 1) / fact where fact is 1, 2 or 4 depending on differentiation method used.
step_nomdefault maximum(log(exp(1)+|x|), 1): Nominal step where x is supplied at runtime through the __call__ method.
offsetreal scalar, optional, default 0: offset to the base step
num_extrapscalar integer, default 0: number of points used for extrapolation
check_num_stepsboolean, default True: If True make sure num_steps is larger than the minimum required steps.
use_exact_stepsboolean, default True: If true make sure exact steps are generated
scalereal scalar, optional: scale used in base step. If not None it will override the default computed with the default_scale function.

property base_step: Base step defines the minimum or maximum step when offset==0.

property min_num_steps: Minimum number of steps required given the differentiation method and order.

property num_steps: Defines number of steps generated

property scale: Scale used in base step.

step_generator_function(x, method='forward', n=1, order=2)[source]: Step generator function

property step_nom: Nominal step

property step_ratio: Ratio between sequential steps generated

class Richardson(step_ratio=2.0, step=1, order=1, num_terms=2)[source]

Bases: object

Extrapolates a sequence with Richardsons method

Parameters:

step_ratio: real scalar: Ratio between sequential steps, h, generated.
step: scalar integer: Defines the step between exponents in the error polynomial, i.e., step = k_1 - k_0 = k_2 - k_1 = … = k_{i+1} - k_i
order: scalar integer: Leading order of truncation error.
num_terms: scalar integer: Number of terms used in the polynomial fit.

Notes

Suppose f(h) is an approximation of L (exact value) that depends on a positive step size h described with a sequence of the form

L = f(h) + a0 * h^k_0 + a1 * h^k_1+ a2 * h^k_2 + …

where the ai are unknown constants and the k_i are known constants such that h^k_i > h^(k_i+1).

If we evaluate the right hand side for different stepsizes h we can fit a polynomial to that sequence of approximations. This is exactly what this class does. Here k_0 is the leading order step size behavior of truncation error as L = f(h)+O(h^k_0) (f(h) -> L as h -> 0, but f(0) != L) and k_i = order + step * i .

Examples

>>> import numpy as np
>>> import numdifftools as nd
>>> n = 3
>>> Ei = np.zeros((n,1))
>>> h = np.zeros((n,1))
>>> linfun = lambda i : np.linspace(0, np.pi/2., 2**(i+5)+1)
>>> for k in np.arange(n):
...    x = linfun(k)
...    h[k] = x[1]
...    Ei[k] = trapz(np.sin(x),x)
>>> En, err, step = nd.Richardson(step=1, order=1)(Ei, h)
>>> truErr = np.abs(En-1.)
>>> bool(np.all(truErr < err))
True
>>> bool(np.all(np.abs(Ei-1)<1e-3))
True
>>> np.allclose(En, 1)
True

extrapolate(sequence, steps)[source]: Extrapolate sequence

rule(sequence_length=None)[source]: Returns extrapolation rule.

dea3(v_0, v_1, v_2, symmetric=False)[source]

Extrapolates a slowly convergent sequence using Shanks transformations.

Parameters:

v_0, v_1, v_2array-like: Three consecutive values of a convergent sequence to extrapolate.
symmetricbool, optional: If True, returns a sequence with symmetric error estimates. Defaults to False.

Returns:

resultarray-like: The extrapolated value(s).
abserrarray-like: The estimated absolute error(s).

See also

Dea

Notes

DEA3 uses nonlinear Shanks transformations based on three values to extrapolate to a better estimate of the sequence’s limit. It is a vectorized translation of the DQELG function from the QUADPACK Fortran library. for LIMEXP=3, see [Rf7ab399ffe8b-2] and [Rf7ab399ffe8b-3].

References

Examples

# integrate sin(x) from 0 to pi/2

>>> import numpy as np
>>> import numdifftools as nd
>>> Ei= np.zeros(3)
>>> linfun = lambda i : np.linspace(0, np.pi/2., 2**(i+5)+1)
>>> for k in np.arange(3):
...    x = linfun(k)
...    Ei[k] = trapz(np.sin(x),x)
>>> [En, err] = nd.dea3(Ei[0], Ei[1], Ei[2])
>>> truErr = np.abs(En-1.)
>>> bool(np.all(truErr < err))
True
>>> np.allclose(En, 1)
True
>>> bool(np.all(np.abs(Ei-1)<1e-3))
True

directionaldiff(f, x0, vec, **options)[source]

Return directional derivative of a function of n variables

Parameters:

f: function: analytical function to differentiate.
x0: array: vector location at which to differentiate ‘f’. If x0 is an nXm array, then ‘f’ is assumed to be a function of n*m variables.
vec: array: vector defining the line along which to take the derivative. It should be the same size as x0, but need not be a vector of unit length.
**options:: optional arguments to pass on to Derivative.

Returns:

dder: scalar: estimate of the first derivative of ‘f’ in the specified direction.

See also

Derivative
Gradient

Examples

At the global minimizer (1,1) of the Rosenbrock function, compute the directional derivative in the direction [1 2]

>>> import numpy as np
>>> import numdifftools as nd
>>> vec = np.r_[1, 2]
>>> rosen = lambda x: (1-x[0])**2 + 105*(x[1]-x[0]**2)**2
>>> dd, info = nd.directionaldiff(rosen, [1, 1], vec, full_output=True)
>>> np.allclose(dd, 0)
True
>>> bool(np.abs(info.error_estimate)<1e-14)
True

5.2.3. numdifftools.extrapolation module

Created on 28. aug. 2015

@author: pab

class Dea(limexp=50)[source]

Bases: object

Extrapolate a slowly convergent sequence using repeated Shanks transformations.

Notes

DEA attempts to extrapolate nonlinearly by Shanks transformations to a better estimate of the sequence’s limiting value, thus improving the rate of convergence. The epsilon algorithm of P. Wynn, see [1]_, is used to perform the non-linear Shanks transformations. The routine is a translation of the DQELG function found in the QUADPACK fortran library, see [2]_ and [3]_.

List of major variables:

LIMEXP: scalar integer: The maximum number of elements the epsilon table data can contain. The epsilon table is stored in the first (LIMEXP+2) entries of EPSTAB.
EPSTAB: real vector or size (LIMEXP+2+3): The first LIMEXP+2 elements contains the two lower diagonals of the triangular epsilon table. The elements are numbered starting at the right-hand corner of the

triangle.
E0,E1,E2,E3: real scalars: The 4 elements on which the computation of a new element in the epsilon table is based.
NRES: scalar integer: Number of extrapolation results actually generated by the epsilon algorithm in prior calls to the routine.
NEWELM: scalar integer: Number of elements to be computed in the new diagonal of the epsilon table. The condensed epsilon table is computed. Only those elements needed for the computation of the next diagonal are preserved.
RES: real scalar: New element in the new diagonal of the epsilon table.
ERROR: real scalar: An estimate of the absolute error of RES. The routine decides whether RESULT=RES or RESULT=SVALUE by comparing ERROR with abserr from the previous call.
RES3LA: real vector of size 3: Contains at most the last 3 results.

property limexp: Maximum number of elements the epsilon table data.

class EpsAlg[source]

Bases: object

Extrapolate a slowly convergent sequence using Shanks transformation.

Notes

The iterated Shanks transformation is computed using the Wynn epsilon algorithm (see equation 4.3-10a to 4.3-10c given on page 25 in [R9678172c97e0-1]).

References

class Richardson(step_ratio=2.0, step=1, order=1, num_terms=2)[source]

Bases: object

Extrapolates a sequence with Richardsons method

Parameters:

step_ratio: real scalar: Ratio between sequential steps, h, generated.
step: scalar integer: Defines the step between exponents in the error polynomial, i.e., step = k_1 - k_0 = k_2 - k_1 = … = k_{i+1} - k_i
order: scalar integer: Leading order of truncation error.
num_terms: scalar integer: Number of terms used in the polynomial fit.

Notes

Suppose f(h) is an approximation of L (exact value) that depends on a positive step size h described with a sequence of the form

L = f(h) + a0 * h^k_0 + a1 * h^k_1+ a2 * h^k_2 + …

where the ai are unknown constants and the k_i are known constants such that h^k_i > h^(k_i+1).

If we evaluate the right hand side for different stepsizes h we can fit a polynomial to that sequence of approximations. This is exactly what this class does. Here k_0 is the leading order step size behavior of truncation error as L = f(h)+O(h^k_0) (f(h) -> L as h -> 0, but f(0) != L) and k_i = order + step * i .

Examples

>>> import numpy as np
>>> import numdifftools as nd
>>> n = 3
>>> Ei = np.zeros((n,1))
>>> h = np.zeros((n,1))
>>> linfun = lambda i : np.linspace(0, np.pi/2., 2**(i+5)+1)
>>> for k in np.arange(n):
...    x = linfun(k)
...    h[k] = x[1]
...    Ei[k] = trapz(np.sin(x),x)
>>> En, err, step = nd.Richardson(step=1, order=1)(Ei, h)
>>> truErr = np.abs(En-1.)
>>> bool(np.all(truErr < err))
True
>>> bool(np.all(np.abs(Ei-1)<1e-3))
True
>>> np.allclose(En, 1)
True

extrapolate(sequence, steps)[source]: Extrapolate sequence

rule(sequence_length=None)[source]: Returns extrapolation rule.

convolve(sequence, rule, **kwds)[source]: Wrapper around scipy.ndimage.convolve1d that allows complex input.

dea3(v_0, v_1, v_2, symmetric=False)[source]

Extrapolates a slowly convergent sequence using Shanks transformations.

Parameters:

v_0, v_1, v_2array-like: Three consecutive values of a convergent sequence to extrapolate.
symmetricbool, optional: If True, returns a sequence with symmetric error estimates. Defaults to False.

Returns:

resultarray-like: The extrapolated value(s).
abserrarray-like: The estimated absolute error(s).

See also

Dea

Notes

DEA3 uses nonlinear Shanks transformations based on three values to extrapolate to a better estimate of the sequence’s limit. It is a vectorized translation of the DQELG function from the QUADPACK Fortran library. for LIMEXP=3, see [Rc8bfc08f7c28-2] and [Rc8bfc08f7c28-3].

References

Examples

# integrate sin(x) from 0 to pi/2

>>> import numpy as np
>>> import numdifftools as nd
>>> Ei= np.zeros(3)
>>> linfun = lambda i : np.linspace(0, np.pi/2., 2**(i+5)+1)
>>> for k in np.arange(3):
...    x = linfun(k)
...    Ei[k] = trapz(np.sin(x),x)
>>> [En, err] = nd.dea3(Ei[0], Ei[1], Ei[2])
>>> truErr = np.abs(En-1.)
>>> bool(np.all(truErr < err))
True
>>> np.allclose(En, 1)
True
>>> bool(np.all(np.abs(Ei-1)<1e-3))
True

dea_demo()[source]

>>> from numdifftools.extrapolation import dea_demo
>>> dea_demo()
NO. PANELS      TRAP. APPROX          APPROX W/EA           abserr
         0.78539816            0.78539816            0.78539816
         0.94805945            0.94805945            0.97596771
         0.98711580            0.99945672            0.21405856
         0.99678517            0.99996674            0.05190729
         0.99919668            0.99999988            0.00057629
         0.99979919            1.00000000            0.00057665
         0.99994980            1.00000000            0.00003338
         0.99998745            1.00000000            0.00000012
         0.99999686            1.00000000            0.00000000
         0.99999922            1.00000000            0.00000000
         0.99999980            1.00000000            0.00000000
         0.99999995            1.00000000            0.00000000

epsalg_demo()[source]

>>> from numdifftools.extrapolation import epsalg_demo
>>> epsalg_demo()
NO. PANELS      TRAP. APPROX          APPROX W/EA           abserr
         0.78539816            0.78539816            0.21460184
         0.94805945            0.94805945            0.05194055
         0.98711580            0.99945672            0.00054328
         0.99678517            0.99996674            0.00003326
         0.99919668            0.99999988            0.00000012
         0.99979919            1.00000000            0.00000000
         0.99994980            1.00000000            0.00000000
         0.99998745            1.00000000            0.00000000
         0.99999686            1.00000000            0.00000000
         0.99999922            1.00000000            0.00000000

max_abs(a, b)[source]: Returns element-wise maximum of absulute value of array elements

richardson_demo()[source]

>>> from numdifftools.extrapolation import richardson_demo
>>> richardson_demo()
NO. PANELS      TRAP. APPROX          APPROX W/R            abserr
         0.78539816            0.78539816            0.21460184
         0.94805945            1.11072073            0.11072073
         0.98711580            0.99798929            0.00201071
         0.99678517            0.99988201            0.00011799
         0.99919668            0.99999274            0.00000726
         0.99979919            0.99999955            0.00000045
         0.99994980            0.99999997            0.00000003
         0.99998745            1.00000000            0.00000000
         0.99999686            1.00000000            0.00000000
         0.99999922            1.00000000            0.00000000

5.2.4. numdifftools.finite_difference module

Finite difference methods module.

class DifferenceFunctions[source]

Bases: object

Class defining difference functions

Notes

The d

class HessdiagDifferenceFunctions[source]

Bases: object

Class defining Hessdiag difference functions

References

Ridout, M.S. (2009) Statistical applications of the complex-step method: of numerical differentiation. The American Statistician, 63, 66-74

class HessianDifferenceFunctions[source]

Bases: object

Class defining Hessian difference functions

References

Ridout, M.S. (2009) “Statistical applications of the complex-step method of numerical differentiation”, The American Statistician, 63, 66-74

class JacobianDifferenceFunctions[source]

Bases: object

Class defining Jacobian difference functions

static increments(n, h)[source]: Returns Jacobian steps

class LogHessdiagRule(n=1, method='central', order=2)[source]

Bases: LogRule

Log spaced finite difference Hessdiag rule class

Parameters:

n2: Order of the derivative.
method{‘central’, ‘complex’, ‘multicomplex’, ‘forward’, ‘backward’}: defines the method used in the approximation
orderint, optional: defines the order of the error term in the Taylor approximation used. For ‘central’ and ‘complex’ methods, it must be an even number.

Examples

>>> import numpy as np
>>> from numdifftools.finite_difference import LogHessdiagRule as Rule

>>> np.allclose(Rule(method='central', order=2).rule(step_ratio=2.0), 2)
True
>>> np.allclose(Rule(method='central', order=4).rule(step_ratio=2.),
...             [-0.66666667, 10.66666667])
True
>>> np.allclose(Rule(method='central', order=6).rule(step_ratio=2.),
...             [ 4.44444444e-02, -3.55555556e+00,  4.55111111e+01])
True
>>> np.allclose(Rule(method='forward', order=2).rule(step_ratio=2.), [ -4.,  40., -64.])
True
>>> np.allclose(Rule(method='forward', order=4).rule(step_ratio=2.),
...          [-1.90476190e-01,  1.10476190e+01, -1.92000000e+02,
...          1.12152381e+03, -1.56038095e+03])
True
>>> np.allclose(Rule(method='forward', order=6).rule(step_ratio=2.),
...    [-4.09626216e-04,  1.02406554e-01, -8.33015873e+00,  2.76317460e+02,
...     -3.84893968e+03,  2.04024004e+04, -2.74895500e+04])
True
>>> step_ratio=2.0
>>> fd_rule = Rule(method='forward', order=4)
>>> steps = 0.002*(1./step_ratio)**np.arange(6)
>>> x0 = np.array([0., 0.])
>>> f = lambda xy : np.cos(xy[0]-xy[1])
>>> f_x0 = f(x0)
>>> f_del = [f(x0+h) - f_x0  for h in steps] # forward difference
>>> f_del = [fd_rule.diff(f, f_x0, x0, h) for h in steps]  # or alternatively
>>> fder, h, shape = fd_rule.apply(f_del, steps, step_ratio)

>>> np.allclose(fder, [[-1., -1.], [-1., -1.]])
True

property n

class LogHessianRule(n=1, method='central', order=2)[source]

Bases: LogRule

Log spaced finite difference Hessian rule class

Parameters:

n2: Order of the derivative.
method{‘central’, ‘complex’, ‘multicomplex’, ‘forward’, ‘backward’}: defines the method used in the approximation
orderint, optional: defines the order of the error term in the Taylor approximation used. For ‘central’ and ‘complex’ methods, it must be an even number.

apply(sequence, steps, step_ratio=2.0)[source]

Apply finite difference rule along the first axis.

Return derivative estimates of fun at x0 for a sequence of stepsizes h

Parameters:

sequence: finite differences
steps: steps

property n

property order: The order of the error term in the Taylor approximation used

class LogJacobianRule(n=1, method='central', order=2)[source]

Bases: LogRule

Log spaced finite difference Jacobian rule class

Parameters:

n1: Order of the derivative.
method{‘central’, ‘complex’, ‘multicomplex’, ‘forward’, ‘backward’}: defines the method used in the approximation
orderint, optional: defines the order of the error term in the Taylor approximation used. For ‘central’ and ‘complex’ methods, it must be an even number.

Examples

>>> from numdifftools.finite_difference import LogJacobianRule as Rule
>>> np.allclose(Rule(n=1, method='central', order=2).rule(step_ratio=2.0), 1)
True
>>> np.allclose(Rule(n=1, method='central', order=4).rule(step_ratio=2.),
...             [-0.33333333,  2.66666667])
True
>>> np.allclose(Rule(n=1, method='central', order=6).rule(step_ratio=2.),
...             [ 0.02222222, -0.88888889,  5.68888889])
True

>>> np.allclose(Rule(n=1, method='forward', order=2).rule(step_ratio=2.), [-1.,  4.])
True

>>> np.allclose(Rule(n=1, method='forward', order=4).rule(step_ratio=2.),
...             [ -0.04761905,   1.33333333, -10.66666667,  24.38095238])
True
>>> np.allclose(Rule(n=1, method='forward', order=6).rule(step_ratio=2.),
...    [ -1.02406554e-04,   1.26984127e-02,  -5.07936508e-01,
...       8.12698413e+00,  -5.20126984e+01,   1.07381055e+02])
True
>>> step_ratio=2.0
>>> fd_rule = Rule(n=1, method='forward', order=4)
>>> steps = 0.002*(1./step_ratio)**np.arange(6)

>>> x0 = np.atleast_1d(1.)
>>> f = np.exp
>>> f_x0 = f(x0)
>>> f_del = [f(x0+h) - f_x0  for h in steps] # forward difference
>>> f_del = [fd_rule.diff(f, f_x0, x0, h) for h in steps[:, None]]  # or alternatively
>>> fder, h, shape = fd_rule.apply(f_del, steps, step_ratio)
>>> np.allclose(fder, f(x0))
True

class LogRule(n=1, method='central', order=2)[source]

Bases: object

Log spaced finite difference rule class

Parameters:

nint, optional: Order of the derivative.
method{‘central’, ‘complex’, ‘multicomplex’, ‘forward’, ‘backward’}: defines the method used in the approximation
orderint, optional: defines the order of the error term in the Taylor approximation used. For ‘central’ and ‘complex’ methods, it must be an even number.

Examples

>>> from numdifftools.finite_difference import LogRule
>>> np.allclose(LogRule(n=1, method='central', order=2).rule(step_ratio=2.0), 1)
True
>>> np.allclose(LogRule(n=1, method='central', order=4).rule(step_ratio=2.),
...             [-0.33333333,  2.66666667])
True
>>> np.allclose(LogRule(n=1, method='central', order=6).rule(step_ratio=2.),
...             [ 0.02222222, -0.88888889,  5.68888889])
True

>>> np.allclose(LogRule(n=1, method='forward', order=2).rule(step_ratio=2.), [-1.,  4.])
True

>>> np.allclose(LogRule(n=1, method='forward', order=4).rule(step_ratio=2.),
...             [ -0.04761905,   1.33333333, -10.66666667,  24.38095238])
True
>>> np.allclose(LogRule(n=1, method='forward', order=6).rule(step_ratio=2.),
...    [ -1.02406554e-04,   1.26984127e-02,  -5.07936508e-01,
...       8.12698413e+00,  -5.20126984e+01,   1.07381055e+02])
True
>>> step_ratio=2.0
>>> fd_rule = LogRule(n=2, method='forward', order=4)
>>> h = 0.002*(1./step_ratio)**np.arange(6)
>>> x0 = 1.
>>> f = np.exp
>>> f_x0 = f(x0)
>>> f_del = f(x0+h) - f_x0  # forward difference
>>> f_del = fd_rule.diff(f, f_x0, x0, h)  # or alternatively
>>> fder, h, shape = fd_rule.apply(f_del, h, step_ratio)
>>> np.allclose(fder, f(x0))
True

apply(sequence, steps, step_ratio=2.0)[source]

Apply finite difference rule along the first axis.

Return derivative estimates of fun at x0 for a sequence of stepsizes h

Parameters:

sequence: finite differences
steps: steps

property diff: The difference function

property eval_first_condition: True if f(x0) needs to be evaluated given the differentiation method.

property method_order: The leading order of the truncation error of the Richardson extrapolation.

property richardson_step: The step between exponents in the error polynomial of the Richardson extrapolation.

rule(step_ratio=2.0)[source]

Return finite differencing rule.

Parameters:

step_ratioreal scalar, optional, default 2.0: Ratio between sequential steps generated.

Notes

The rule is for a nominal unit step size, and must be scaled later to reflect the local step size.

Member method used: _fd_matrix

Member variables used: n order method

make_exact(h)[source]

Make sure h is an exact representable number

This is important when calculating numerical derivatives and is accomplished by adding 1.0 and then subtracting 1.0.

5.2.5. numdifftools.fornberg module

class Taylor(fun, n=1, r=0.0059, num_extrap=3, step_ratio=1.6, **kwds)[source]

Bases: object

Return Taylor coefficients of complex analytic function using FFT

Parameters:

funcallable: function to differentiate
z0real or complex scalar at which to evaluate the derivatives
nscalar integer, default 1: Number of taylor coefficents to compute. Maximum number is 100.
rreal scalar, default 0.0059: Initial radius at which to evaluate. For well-behaved functions, the computation should be insensitive to the initial radius to within about four orders of magnitude.
num_extrapscalar integer, default 3: number of extrapolation steps used in the calculation
step_ratioreal scalar, default 1.6: Initial grow/shrinking factor for finding the best radius.
max_iterscalar integer, default 30: Maximum number of iterations
min_iterscalar integer, default max_iter // 2: Minimum number of iterations before the solution may be deemed degenerate. A larger number allows the algorithm to correct a bad initial radius.
full_outputbool, optional: If full_output is False, only the coefficents is returned (default). If full_output is True, then (coefs, status) is returned

Returns:

coefsndarray: array of taylor coefficents
status: Optional object into which output information is written:: degenerate: True if the algorithm was unable to bound the error iterations: Number of iterations executed function_count: Number of function calls final_radius: Ending radius of the algorithm failed: True if the maximum number of iterations was reached error_estimate: approximate bounds of the rounding error.

Notes

This module uses the method of Fornberg to compute the Taylor series coefficients of a complex analytic function along with error bounds. The method uses a Fast Fourier Transform to invert function evaluations around a circle into Taylor series coefficients and uses Richardson Extrapolation to improve and bound the estimate. Unlike real-valued finite differences, the method searches for a desirable radius and so is reasonably insensitive to the initial radius-to within a number of orders of magnitude at least. For most cases, the default configuration is likely to succeed.

Restrictions: The method uses the coefficients themselves to control the truncation error, so the error will not be properly bounded for functions like low-order polynomials whose Taylor series coefficients are nearly zero. If the error cannot be bounded, degenerate flag will be set to true, and an answer will still be computed and returned but should be used with caution.

References

[1] Fornberg, B. (1981).: Numerical Differentiation of Analytic Functions. ACM Transactions on Mathematical Software (TOMS), 7(4), 512-526. http://doi.org/10.1145/355972.355979

Examples

Compute the first 6 taylor coefficients 1 / (1 - z) expanded round z0 = 0:

>>> import numdifftools.fornberg as ndf
>>> import numpy as np
>>> c, info = ndf.Taylor(lambda x: 1./(1-x), n=6, full_output=True)(z0=0)
>>> np.allclose(c, np.ones(8))
True
>>> bool(np.all(info.error_estimate < 1e-9))
True
>>> (info.function_count, info.iterations, info.failed) == (136, 17, False)
True

derivative(fun, z0, n=1, **kwds)[source]

Calculate n-th derivative of complex analytic function using FFT

Parameters:

funcallable: function to differentiate
z0real or complex scalar at which to evaluate the derivatives
nscalar integer, default 1: Number of derivatives to compute where 0 represents the value of the function and n represents the nth derivative. Maximum number is 100.
rreal scalar, default 0.0061: Initial radius at which to evaluate. For well-behaved functions, the computation should be insensitive to the initial radius to within about four orders of magnitude.
max_iterscalar integer, default 30: Maximum number of iterations
min_iterscalar integer, default max_iter // 2: Minimum number of iterations before the solution may be deemed degenerate. A larger number allows the algorithm to correct a bad initial radius.
step_ratioreal scalar, default 1.6: Initial grow/shrinking factor for finding the best radius.
num_extrapscalar integer, default 3: number of extrapolation steps used in the calculation
full_outputbool, optional: If full_output is False, only the derivative is returned (default). If full_output is True, then (der, status) is returned der is the derivative, and status is a Results object.

Returns:

derndarray: array of derivatives
status: Optional object into which output information is written. Fields:: degenerate: True if the algorithm was unable to bound the error iterations: Number of iterations executed function_count: Number of function calls final_radius: Ending radius of the algorithm failed: True if the maximum number of iterations was reached error_estimate: approximate bounds of the rounding error.

Notes

This module uses the method of Fornberg to compute the derivatives of a complex analytic function along with error bounds. The method uses a Fast Fourier Transform to invert function evaluations around a circle into Taylor series coefficients, uses Richardson Extrapolation to improve and bound the estimate, then multiplies by a factorial to compute the derivatives. Unlike real-valued finite differences, the method searches for a desirable radius and so is reasonably insensitive to the initial radius-to within a number of orders of magnitude at least. For most cases, the default configuration is likely to succeed.

Restrictions: The method uses the coefficients themselves to control the truncation error, so the error will not be properly bounded for functions like low-order polynomials whose Taylor series coefficients are nearly zero. If the error cannot be bounded, degenerate flag will be set to true, and an answer will still be computed and returned but should be used with caution.

References

[1] Fornberg, B. (1981).: Numerical Differentiation of Analytic Functions. ACM Transactions on Mathematical Software (TOMS), 7(4), 512-526. http://doi.org/10.1145/355972.355979

Examples

To compute the first five derivatives of 1 / (1 - z) at z = 0: Compute the first 6 taylor derivatives of 1 / (1 - z) at z0 = 0:

>>> import numdifftools.fornberg as ndf
>>> import numpy as np
>>> def fun(x):
...    return 1./(1-x)
>>> c, info = ndf.derivative(fun, z0=0, n=6, full_output=True)
>>> np.allclose(c, [1, 1, 2, 6, 24, 120, 720, 5040])
True
>>> bool(np.all(info.error_estimate < 1e-9*c.real))
True
>>> (info.function_count, info.iterations, info.failed) == (136, 17, False)
True

fd_derivative(fx, x, n=1, m=2)[source]

Return the n’th derivative for all points using Finite Difference method.

Parameters:

fxvector: function values which are evaluated on x i.e. fx[i] = f(x[i])
xvector: abscissas on which fx is evaluated. The x values can be arbitrarily spaced but must be distinct and len(x) > n.
nscalar integer: order of derivative.
mscalar integer: defines the stencil size. The stencil size is of 2 * mm + 1 points in the interior, and 2 * mm + 2 points for each of the 2 * mm boundary points where mm = n // 2 + m.
fd_derivative evaluates an approximation for the n’th derivative of the
vector function f(x) using the Fornberg finite difference method.
Restrictions: 0 < n < len(x) and 2*mm+2 <= len(x)

See also

fd_weights

Examples

>>> import numpy as np
>>> import numdifftools.fornberg as ndf
>>> x = np.linspace(-1, 1, 25)
>>> fx = np.exp(x)
>>> df = ndf.fd_derivative(fx, x, n=1)
>>> np.allclose(df, fx)
True

fd_weights(x, x0=0, n=1)[source]

Return finite difference weights for the n’th derivative.

Parameters:

xvector: abscissas used for the evaluation for the derivative at x0.
x0scalar: location where approximations are to be accurate
nscalar integer: order of derivative. Note for n=0 this can be used to evaluate the interpolating polynomial itself.

See also

fd_weights_all

Examples

>>> import numpy as np
>>> import numdifftools.fornberg as ndf
>>> x = np.linspace(-1, 1, 5) * 1e-3
>>> w = ndf.fd_weights(x, x0=0, n=1)
>>> df = np.dot(w, np.exp(x))
>>> np.allclose(df, 1)
True

fd_weights_all(x, x0=0, n=1)[source]

Return finite difference weights for derivatives of all orders up to n.

Parameters:

xvector, length m: x-coordinates for grid points
x0scalar: location where approximations are to be accurate
nscalar integer: highest derivative that we want to find weights for

Returns:

weightsarray, shape n+1 x m: contains coefficients for the j’th derivative in row j (0 <= j <= n)

See also

fd_weights

Notes

The x values can be arbitrarily spaced but must be distinct and len(x) > n.

The Fornberg algorithm is much more stable numerically than regular vandermonde systems for large values of n.

References

B. Fornberg (1998) “Calculation of weights_and_points in finite difference formulas”, SIAM Review 40, pp. 685-691.

http://www.scholarpedia.org/article/Finite_difference_method

richardson(vals, k, c=None)[source]: Richardson extrapolation with parameter estimation

richardson_parameter(vals, k)[source]

taylor(fun, z0=0, n=1, r=0.0059, num_extrap=3, step_ratio=1.6, **kwds)[source]

Return Taylor coefficients of complex analytic function using FFT

Parameters:

funcallable: function to differentiate
z0real or complex scalar at which to evaluate the derivatives
nscalar integer, default 1: Number of taylor coefficents to compute. Maximum number is 100.
rreal scalar, default 0.0059: Initial radius at which to evaluate. For well-behaved functions, the computation should be insensitive to the initial radius to within about four orders of magnitude.
num_extrapscalar integer, default 3: number of extrapolation steps used in the calculation
step_ratioreal scalar, default 1.6: Initial grow/shrinking factor for finding the best radius.
max_iterscalar integer, default 30: Maximum number of iterations
min_iterscalar integer, default max_iter // 2: Minimum number of iterations before the solution may be deemed degenerate. A larger number allows the algorithm to correct a bad initial radius.
full_outputbool, optional: If full_output is False, only the coefficents is returned (default). If full_output is True, then (coefs, status) is returned

Returns:

coefsndarray: array of taylor coefficents
status: Optional object into which output information is written:: degenerate: True if the algorithm was unable to bound the error iterations: Number of iterations executed function_count: Number of function calls final_radius: Ending radius of the algorithm failed: True if the maximum number of iterations was reached error_estimate: approximate bounds of the rounding error.

Notes

This module uses the method of Fornberg to compute the Taylor series coefficents of a complex analytic function along with error bounds. The method uses a Fast Fourier Transform to invert function evaluations around a circle into Taylor series coefficients and uses Richardson Extrapolation to improve and bound the estimate. Unlike real-valued finite differences, the method searches for a desirable radius and so is reasonably insensitive to the initial radius-to within a number of orders of magnitude at least. For most cases, the default configuration is likely to succeed.

Restrictions: The method uses the coefficients themselves to control the truncation error, so the error will not be properly bounded for functions like low-order polynomials whose Taylor series coefficients are nearly zero. If the error cannot be bounded, degenerate flag will be set to true, and an answer will still be computed and returned but should be used with caution.

References

[1] Fornberg, B. (1981).: Numerical Differentiation of Analytic Functions. ACM Transactions on Mathematical Software (TOMS), 7(4), 512-526. http://doi.org/10.1145/355972.355979

Examples

Compute the first 6 taylor coefficients 1 / (1 - z) expanded round z0 = 0:

>>> import numdifftools.fornberg as ndf
>>> import numpy as np
>>> c, info = ndf.taylor(lambda x: 1./(1-x), z0=0, n=6, full_output=True)
>>> np.allclose(c, np.ones(8))
True
>>> bool(np.all(info.error_estimate < 1e-9))
True
>>> (info.function_count, info.iterations, info.failed) == (136, 17, False)
True

5.2.6. numdifftools.limits module

Created on 27. aug. 2015

@author: pab Author: John D’Errico e-mail: woodchips@rochester.rr.com Release: 1.0 Release date: 5/23/2008

class CStepGenerator(base_step=None, step_ratio=4.0, num_steps=None, step_nom=None, offset=0, scale=1.2, **options)[source]

Bases: MinStepGenerator

Generates a sequence of steps

where: steps = base_step * step_nom * (exp(1j*dtheta) * step_ratio) ** (i + offset)

for i = 0, 1, …, num_steps-1

Parameters:

base_stepfloat, array-like, default None: Defines the minimum step, if None, the value is set to EPS**(1/scale)
step_ratioreal scalar, optional, default 4.0: Ratio between sequential steps generated.
num_stepsscalar integer, optional,: defines number of steps generated. If None the value is 2 * int(round(16.0/log(abs(step_ratio)))) + 1
step_nomdefault maximum(log(exp(1)+|x|), 1): Nominal step where x is supplied at runtime through the __call__ method.
offsetreal scalar, optional, default 0: offset to the base step
use_exact_stepsboolean, default True.: If true make sure exact steps are generated.
scalereal scalar, default 1.2: scale used in base step.
path‘radial’ or ‘spiral’: Specifies the type of path to take the limit along. Default ‘radial’.
dtheta: real scalar, default pi/8: If the path is ‘spiral’ it will follow an exponential spiral into the limit, with angular steps at dtheta radians.

property dtheta: Angular steps in radians used for the exponential spiral path.

property num_steps: The number of steps generated

property step_ratio: Ratio between sequential steps generated.

class Limit(fun, step=None, method='above', order=4, full_output=False, **options)[source]

Bases: _Limit

Compute limit of a function at a given point

Parameters:

funcallable: function fun(z, *args, **kwds) to compute the limit for z->z0. The function, fun, is assumed to return a result of the same shape and size as its input, z.
step: float, complex, array-like or StepGenerator object, optional: Defines the spacing used in the approximation. Default is CStepGenerator(base_step=step, **options)
method{‘above’, ‘below’}: defines if the limit is taken from above or below
order: positive scalar integer, optional.: defines the order of approximation used to find the specified limit. The order must be member of [1 2 3 4 5 6 7 8]. 4 is a good compromise.
full_output: bool: If true return additional info.
options:: options to pass on to CStepGenerator

Returns:

limit_fz: array like

estimated limit of f(z) as z –> z0

info:

Only given if full_output is True and contains the following:

error estimate: ndarray: 95 % uncertainty estimate around the limit, such that abs(limit_fz - lim z->z0 f(z)) < error_estimate
final_step: ndarray: final step used in approximation

Notes

Limit computes the limit of a given function at a specified point, z0. When the function is evaluable at the point in question, this is a simple task. But when the function cannot be evaluated at that location due to a singularity, you may need a tool to compute the limit. Limit does this, as well as produce an uncertainty estimate in the final result.

The methods used by Limit are Richardson extrapolation in a combination with Wynn’s epsilon algorithm which also yield an error estimate. The user can specify the method order, as well as the path into z0. z0 may be real or complex. Limit uses a proportionally cascaded series of function evaluations, moving away from your point of evaluation along a path along the real line (or in the complex plane for complex z0 or step.) The step_ratio is the ratio used between sequential steps. The sign of step allows you to specify a limit from above or below. Negative values of step will cause the limit to be taken approaching z0 from below.

A smaller step_ratio means that Limit will take more function evaluations to evaluate the limit, but the result will potentially be less accurate. The step_ratio MUST be a scalar larger than 1. A value in the range [2,100] is recommended. 4 seems a good compromise.

>>> import numpy as np
>>> from numdifftools.limits import Limit
>>> def f(x): return np.sin(x)/x
>>> lim_f0, err = Limit(f, full_output=True)(0)
>>> np.allclose(lim_f0, 1)
True
>>> np.allclose(err.error_estimate, 1.77249444610966e-15)
True

Compute the derivative of cos(x) at x == pi/2. It should be -1. The limit will be taken as a function of the differential parameter, dx.

>>> x0 = np.pi/2;
>>> def g(x): return (np.cos(x0+x)-np.cos(x0))/x
>>> lim_g0, err = Limit(g, full_output=True)(0)
>>> np.allclose(lim_g0, -1)
True
>>> bool(err.error_estimate < 1e-14)
True

Compute the residue at a first order pole at z = 0 The function 1./(1-exp(2*z)) has a pole at z == 0. The residue is given by the limit of z*fun(z) as z –> 0. Here, that residue should be -0.5.

>>> def h(z): return -z/(np.expm1(2*z))
>>> lim_h0, err = Limit(h, full_output=True)(0)
>>> np.allclose(lim_h0, -0.5)
True
>>> bool(err.error_estimate < 1e-14)
True

Compute the residue of function 1./sin(z)**2 at z = 0. This pole is of second order thus the residue is given by the limit of z**2*fun(z) as z –> 0.

>>> def g(z): return z**2/(np.sin(z)**2)
>>> lim_gpi, err = Limit(g, full_output=True)(0)
>>> np.allclose(lim_gpi, 1)
True
>>> bool(err.error_estimate < 1e-14)
True

A more difficult limit is one where there is significant subtractive cancellation at the limit point. In the following example, the cancellation is second order. The true limit should be 0.5.

>>> def k(x): return (x*np.exp(x)-np.expm1(x))/x**2
>>> lim_k0,err = Limit(k, full_output=True)(0)
>>> np.allclose(lim_k0, 0.5)
True
>>> bool(err.error_estimate < 1.0e-8)
True

>>> def h(x): return  (x-np.sin(x))/x**3
>>> lim_h0, err = Limit(h, full_output=True)(0)
>>> np.allclose(lim_h0, 1./6)
True
>>> bool(err.error_estimate < 1e-8)
True

limit(x, *args, **kwds)[source]: Return lim f(z) as z-> x

class Residue(f, step=None, method='above', order=None, pole_order=1, full_output=False, **options)[source]

Bases: Limit

Compute residue of a function at a given point

Parameters:

funcallable: function fun(z, *args, **kwds) to compute the Residue at z=z0. The function, fun, is assumed to return a result of the same shape and size as its input, z.
step: float, complex, array-like or StepGenerator object, optional: Defines the spacing used in the approximation. Default is CStepGenerator(base_step=step, **options)
method{‘above’, ‘below’}: defines if the limit is taken from above or below
order: positive scalar integer, optional.: defines the order of approximation used to find the specified limit. The order must be member of [1 2 3 4 5 6 7 8]. 4 is a good compromise.
pole_orderscalar integer: specifies the order of the pole at z0.
full_output: bool: If true return additional info.
options:: options to pass on to CStepGenerator

Returns:

res_fz: array like

estimated residue, i.e., limit of f(z)*(z-z0)**pole_order as z –> z0 When the residue is estimated as approximately zero,

the wrong order pole may have been specified.

info: namedtuple,

Only given if full_output is True and contains the following:

error estimate: ndarray: 95 % uncertainty estimate around the residue, such that abs(res_fz - lim z->z0 f(z)*(z-z0)**pole_order) < error_estimate Large uncertainties here suggest that the wrong order pole was specified for f(z0).
final_step: ndarray: final step used in approximation

Notes

Residue computes the residue of a given function at a simple first order pole, or at a second order pole.

The methods used by residue are polynomial extrapolants, which also yield an error estimate. The user can specify the method order, as well as the order of the pole.

z0 - scalar point at which to compute the residue. z0 may be: real or complex.

See the document DERIVEST.pdf for more explanation of the algorithms behind the parameters of Residue. In most cases, the user should never need to specify anything other than possibly the PoleOrder.

Examples

A first order pole at z = 0

>>> import numpy as np
>>> from numdifftools.limits import Residue
>>> def f(z): return -1./(np.expm1(2*z))
>>> res_f, info = Residue(f, full_output=True)(0)
>>> np.allclose(res_f, -0.5)
True
>>> bool(info.error_estimate < 1e-14)
True

A second order pole around z = 0 and z = pi >>> def h(z): return 1.0/np.sin(z)**2 >>> res_h, info = Residue(h, full_output=True, pole_order=2)([0, np.pi]) >>> np.allclose(res_h, 1) True >>> bool((info.error_estimate < 1e-10).all()) True

5.2.7. numdifftools.multicomplex module

Created on 22. apr. 2015

@author: pab

5.2.7.1. References

A methodology for robust optimization of low-thrust trajectories in multi-body environments Gregory Lantoine (2010) Phd thesis, Georgia Institute of Technology

Using multicomplex variables for automatic computation of high-order derivatives Gregory Lantoine, Ryan P. Russell , and Thierry Dargent ACM Transactions on Mathematical Software, Vol. 38, No. 3, Article 16, April 2012, 21 pages, http://doi.acm.org/10.1145/2168773.2168774

Bicomplex Numbers and Their Elementary Functions M.E. Luna-Elizarraras, M. Shapiro, D.C. Struppa1, A. Vajiac (2012) CUBO A Mathematical Journal Vol. 14, No 2, (61-80). June 2012.

Computation of higher-order derivatives using the multi-complex step method Adriaen Verheyleweghen, (2014) Project report, NTNU

class Bicomplex(z1, z2)[source]

Bases: object

Creates an instance of a Bicomplex object. zeta = z1 + j*z2, where z1 and z2 are complex numbers.

arccos()[source]

arccosh()[source]

arcsin()[source]

arcsinh()[source]

arctan()[source]

arctanh()[source]

arg_c()[source]

arg_c1p()[source]

static asarray(other)[source]

conjugate()[source]

cos()[source]

cosh()[source]

cot()[source]

coth()[source]

csc()[source]

csch()[source]

dot(other)[source]

exp()[source]

exp2()[source]

expm1()[source]

flat(index)[source]

property imag

property imag1

property imag12

property imag2

log()[source]

log10()[source]

log1p()[source]

log2()[source]

logaddexp(other)[source]

logaddexp2(other)[source]

static mat2bicomp(arr)[source]

mod_c()[source]: Complex modulus

norm()[source]

property real

sec()[source]

sech()[source]

property shape

sin()[source]

sinh()[source]

property size

sqrt()[source]

tan()[source]

tanh()[source]

z1

z2

c_abs(z)[source]

c_atan2(x, y)[source]

c_max(x, y)[source]

c_min(x, y)[source]

5.2.8. numdifftools.nd_algopy module

5.2.8.1. Numdifftools.nd_algopy

This module provide an easy to use interface to derivatives calculated with AlgoPy. Algopy stands for Algorithmic Differentiation in Python.

The purpose of AlgoPy is the evaluation of higher-order derivatives in the forward and reverse mode of Algorithmic Differentiation (AD) of functions that are implemented as Python programs. Particular focus are functions that contain numerical linear algebra functions as they often appear in statistically motivated functions. The intended use of AlgoPy is for easy prototyping at reasonable execution speeds. More precisely, for a typical program a directional derivative takes order 10 times as much time as time as the function evaluation. This is approximately also true for the gradient.

5.2.8.2. Algoritmic differentiation

Algorithmic differentiation (AD) is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.

Algorithmic differentiation is not:

Symbolic differentiation, nor Numerical differentiation (the method of finite differences). These classical methods run into problems: symbolic differentiation leads to inefficient code (unless carefully done) and faces the difficulty of converting a computer program into a single expression, while numerical differentiation can introduce round-off errors in the discretization process and cancellation. Both classical methods have problems with calculating higher derivatives, where the complexity and errors increase. Finally, both classical methods are slow at computing the partial derivatives of a function with respect to many inputs, as is needed for gradient-based optimization algorithms. Algoritmic differentiation solves all of these problems.

5.2.8.2.1. References

Sebastian F. Walter and Lutz Lehmann 2013, “Algorithmic differentiation in Python with AlgoPy”, in Journal of Computational Science, vol 4, no 5, pp 334 - 344, http://www.sciencedirect.com/science/article/pii/S1877750311001013

https://en.wikipedia.org/wiki/Automatic_differentiation

https://pythonhosted.org/algopy/index.html

class Derivative(fun, n=1, method='forward', full_output=False)[source]

Bases: _Derivative

Calculate n-th derivative with Algorithmic Differentiation method

Parameters:

fun: function: function of one array fun(x, *args, **kwds)
n: int, optional: Order of the derivative.
method: string, optional {‘forward’, ‘reverse’}: defines method used in the approximation

Methods

__call__: callable with the following parameters:

x: array_like value at which function derivative is evaluated args: tuple Arguments for function fun. kwds: dict Keyword arguments for function fun.

Returns:

der: ndarray: array of derivatives

See also

Gradient
Hessdiag
Hessian
Jacobian

Notes

Algorithmic differentiation is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.

References

Sebastian F. Walter and Lutz Lehmann 2013, “Algorithmic differentiation in Python with AlgoPy”, in Journal of Computational Science, vol 4, no 5, pp 334 - 344, http://www.sciencedirect.com/science/article/pii/S1877750311001013

https://en.wikipedia.org/wiki/Automatic_differentiation

Examples

# 1’st and 2’nd derivative of exp(x), at x == 1

>>> import numpy as np
>>> import numdifftools.nd_algopy as nda
>>> fd = nda.Derivative(np.exp)              # 1'st derivative
>>> np.allclose(fd(1), 2.718281828459045)
True
>>> fd5 = nda.Derivative(np.exp, n=5)         # 5'th derivative
>>> np.allclose(fd5(1), 2.718281828459045)
True

# 1’st derivative of x^3+x^4, at x = [0,1]

>>> fun = lambda x: x**3 + x**4
>>> fd3 = nda.Derivative(fun)
>>> np.allclose(fd3([0,1]), [ 0.,  7.])
True

class Gradient(fun, n=1, method='forward', full_output=False)[source]

Bases: _Derivative

Calculate Gradient with Algorithmic Differentiation method

Parameters:

fun: function: function of one array fun(x, *args, **kwds)
method: string, optional {‘forward’, ‘reverse’}: defines method used in the approximation

Methods

__call__: callable with the following parameters:

x: array_like value at which function derivative is evaluated args: tuple Arguments for function fun. kwds: dict Keyword arguments for function fun.

Returns:

grad: array: gradient

See also

Derivative
Jacobian
Hessdiag
Hessian

Notes

Algorithmic differentiation is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.

References

Sebastian F. Walter and Lutz Lehmann 2013, “Algorithmic differentiation in Python with AlgoPy”, in Journal of Computational Science, vol 4, no 5, pp 334 - 344, http://www.sciencedirect.com/science/article/pii/S1877750311001013

https://en.wikipedia.org/wiki/Automatic_differentiation

Examples

>>> import numpy as np
>>> import numdifftools.nd_algopy as nda
>>> fun = lambda x: np.sum(x**2)
>>> df = nda.Gradient(fun, method='reverse')
>>> np.allclose(df([1,2,3]), [ 2.,  4.,  6.])
True

#At [x,y] = [1,1], compute the numerical gradient #of the function sin(x-y) + y*exp(x)

>>> sin = np.sin; exp = np.exp
>>> z = lambda xy: sin(xy[0]-xy[1]) + xy[1]*exp(xy[0])
>>> dz = nda.Gradient(z)
>>> grad2 = dz([1, 1])
>>> np.allclose(grad2, [ 3.71828183,  1.71828183])
True

#At the global minimizer (1,1) of the Rosenbrock function, #compute the gradient. It should be essentially zero.

>>> rosen = lambda x : (1-x[0])**2 + 105.*(x[1]-x[0]**2)**2
>>> rd = nda.Gradient(rosen)
>>> grad3 = rd([1,1])
>>> np.allclose(grad3, [ 0.,  0.])
True

class Hessdiag(f, method='forward', full_output=False)[source]

Bases: Hessian

Calculate Hessian diagonal with Algorithmic Differentiation method

Parameters:

fun: function: function of one array fun(x, *args, **kwds)
method: string, optional {‘forward’, ‘reverse’}: defines method used in the approximation

Methods

__call__: callable with the following parameters:

x: array_like value at which function derivative is evaluated args: tuple Arguments for function fun. kwds: dict Keyword arguments for function fun.

Returns:

hessdiagndarray: Hessian diagonal array of partial second order derivatives.

See also

Derivative
Gradient
Jacobian
Hessian

Notes

Algorithmic differentiation is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.

References

Sebastian F. Walter and Lutz Lehmann 2013, “Algorithmic differentiation in Python with AlgoPy”, in Journal of Computational Science, vol 4, no 5, pp 334 - 344, http://www.sciencedirect.com/science/article/pii/S1877750311001013

https://en.wikipedia.org/wiki/Automatic_differentiation

Examples

>>> import numpy as np
>>> import numdifftools.nd_algopy as nda

# Rosenbrock function, minimized at [1,1]

>>> rosen = lambda x : (1.-x[0])**2 + 105*(x[1]-x[0]**2)**2
>>> Hfun = nda.Hessdiag(rosen)
>>> h = Hfun([1, 1]) #  h =[ 842, 210]
>>> np.allclose(h, [ 842.,  210.])
True

# cos(x-y), at (0,0)

>>> cos = np.cos
>>> fun = lambda xy : cos(xy[0]-xy[1])
>>> Hfun2 = nda.Hessdiag(fun)
>>> h2 = Hfun2([0, 0]) # h2 = [-1, -1]
>>> np.allclose(h2, [-1., -1.])
True

>>> Hfun3 = nda.Hessdiag(fun, method='reverse')
>>> h3 = Hfun3([0, 0]) # h2 = [-1, -1];
>>> np.allclose(h3, [-1., -1.])
True

class Hessian(f, method='forward', full_output=False)[source]

Bases: _Derivative

Calculate Hessian with Algorithmic Differentiation method

Parameters:

fun: function: function of one array fun(x, *args, **kwds)
method: string, optional {‘forward’, ‘reverse’}: defines method used in the approximation

Methods

__call__: callable with the following parameters:

x: array_like value at which function derivative is evaluated args: tuple Arguments for function fun. kwds: dict Keyword arguments for function fun.

Returns:

hessndarray: array of partial second derivatives, Hessian

See also

Derivative
Gradient
Jacobian
Hessdiag

Notes

Algorithmic differentiation is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.

References

Sebastian F. Walter and Lutz Lehmann 2013, “Algorithmic differentiation in Python with AlgoPy”, in Journal of Computational Science, vol 4, no 5, pp 334 - 344, http://www.sciencedirect.com/science/article/pii/S1877750311001013

https://en.wikipedia.org/wiki/Automatic_differentiation

Examples

>>> import numpy as np
>>> import numdifftools.nd_algopy as nda

# Rosenbrock function, minimized at [1,1]

>>> rosen = lambda x : (1.-x[0])**2 + 105*(x[1]-x[0]**2)**2
>>> Hf = nda.Hessian(rosen)
>>> h = Hf([1, 1]) #  h =[ 842 -420; -420, 210];
>>> np.allclose(h, [[ 842., -420.],
...                 [-420.,  210.]])
True

# cos(x-y), at (0,0)

>>> cos = np.cos
>>> fun = lambda xy : cos(xy[0]-xy[1])
>>> Hfun2 = nda.Hessian(fun)
>>> h2 = Hfun2([0, 0]) # h2 = [-1 1; 1 -1]
>>> np.allclose(h2, [[-1.,  1.],
...                  [ 1., -1.]])
True

>>> Hfun3 = nda.Hessian(fun, method='reverse')
>>> h3 = Hfun3([0, 0]) # h2 = [-1, 1; 1, -1];
>>> np.allclose(h3, [[-1.,  1.],
...                  [ 1., -1.]])
True

class Jacobian(fun, n=1, method='forward', full_output=False)[source]

Bases: Gradient

Calculate Jacobian with Algorithmic Differentiation method

Parameters:

fun: function: function of one array fun(x, *args, **kwds)
method: string, optional {‘forward’, ‘reverse’}: defines method used in the approximation

Methods

__call__: callable with the following parameters:

x: array_like value at which function derivative is evaluated args: tuple Arguments for function fun. kwds: dict Keyword arguments for function fun.

Returns:

jacob: array: Jacobian

See also

Derivative
Gradient
Hessdiag
Hessian

Notes

Algorithmic differentiation is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.

References

Sebastian F. Walter and Lutz Lehmann 2013, “Algorithmic differentiation in Python with AlgoPy”, in Journal of Computational Science, vol 4, no 5, pp 334 - 344, http://www.sciencedirect.com/science/article/pii/S1877750311001013

https://en.wikipedia.org/wiki/Automatic_differentiation

Examples

>>> import numpy as np
>>> import numdifftools.nd_algopy as nda

#(nonlinear least squares)

>>> xdata = np.arange(0,1,0.1)
>>> ydata = 1+2*np.exp(0.75*xdata)
>>> fun = lambda c: (c[0]+c[1]*np.exp(c[2]*xdata) - ydata)**2

Jfun = nda.Jacobian(fun) # Todo: This does not work Jfun([1,2,0.75]).T # should be numerically zero array([[ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],

[ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.], [ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.]])

>>> Jfun2 = nda.Jacobian(fun, method='reverse')
>>> res = Jfun2([1,2,0.75]).T # should be numerically zero
>>> np.allclose(res,
...                [[ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
...                 [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
...                 [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.]])
True

>>> f2 = lambda x : x[0]*x[1]*x[2]**2
>>> Jfun2 = nda.Jacobian(f2)
>>> np.allclose(Jfun2([1., 2., 3.]), [[ 18., 9., 12.]])
True

>>> Jfun21 = nda.Jacobian(f2, method='reverse')
>>> np.allclose(Jfun21([1., 2., 3.]), [[ 18., 9., 12.]])
True

>>> def fun3(x):
...     n = int(np.prod(np.shape(x[0])))
...     out = nda.algopy.zeros((2, n), dtype=x)
...     out[0] = x[0]*x[1]*x[2]**2
...     out[1] = x[0]*x[1]*x[2]
...     return out
>>> Jfun3 = nda.Jacobian(fun3)

>>> np.allclose(Jfun3([1., 2., 3.]), [[[18., 9., 12.]], [[6., 3., 2.]]])
True
>>> np.allclose(Jfun3([4., 5., 6.]), [[[180., 144., 240.]],
...                                   [[30., 24., 20.]]])
True
>>> np.allclose(Jfun3(np.array([[1.,2.,3.], [4., 5., 6.]]).T),
...             [[[18.,    0.,    9.,    0.,   12.,    0.],
...               [0.,  180.,    0.,  144.,    0.,  240.]],
...              [[6.,    0.,    3.,    0.,    2.,    0.],
...               [0.,   30.,    0.,   24.,    0.,   20.]]])
True

class Taylor(fun, n=1)[source]

Bases: object

Return Taylor coefficients of function using algorithmic differentiation

Parameters:

fun: callable: function to differentiate
z0: real or complex array: at which to evaluate the derivatives
n: scalar integer, default 1: Number of taylor coefficents to compute.

Returns:

coefs: ndarray: array of taylor coefficents

Examples

Compute the first 6 taylor coefficients 1 + 2*z + 3*z**2 expanded round z0 = 0:

>>> import numpy as np
>>> import numdifftools.nd_algopy as nda
>>> c = nda.Taylor(lambda x: 1+2*x+3*x**2, n=6)(z0=0)
>>> np.allclose(c, [1, 2, 3, 0, 0, 0])
True

directionaldiff(f, x0, vec, **options)[source]

Return directional derivative of a function of n variables

Parameters:

fun: callable: analytical function to differentiate.
x0: array: vector location at which to differentiate fun. If x0 is an nxm array, then fun is assumed to be a function of n*m variables.
vec: array: vector defining the line along which to take the derivative. It should be the same size as x0, but need not be a vector of unit length.
**options:: optional arguments to pass on to Derivative.

Returns:

dder: scalar: estimate of the first derivative of fun in the specified direction.

See also

Derivative
Gradient

Examples

At the global minimizer (1,1) of the Rosenbrock function, compute the directional derivative in the direction [1 2]

>>> import numpy as np
>>> import numdifftools.nd_algopy as nda
>>> vec = np.r_[1, 2]
>>> rosen = lambda x: (1-x[0])**2 + 105*(x[1]-x[0]**2)**2
>>> dd = nda.directionaldiff(rosen, [1, 1], vec)
>>> np.allclose(dd, 0)
True

5.2.9. numdifftools.nd_scipy module

class Gradient(fun, step=None, method='central', order=2, bounds=(-inf, inf), sparsity=None)[source]

Bases: Jacobian

Calculate Gradient with finite difference approximation

Parameters:

funfunction: function of one array fun(x, *args, **kwds)
stepfloat, optional: Stepsize, if None, optimal stepsize is used, i.e., x * _EPS for method==`complex` x * _EPS**(1/2) for method==`forward` x * _EPS**(1/3) for method==`central`.
method{‘central’, ‘complex’, ‘forward’}: defines the method used in the approximation.

See also

Hessian, Jacobian

Examples

>>> import numpy as np
>>> import numdifftools.nd_scipy as nd
>>> fun = lambda x: np.sum(x**2)
>>> dfun = nd.Gradient(fun)
>>> np.allclose(dfun([1,2,3]), [ 2.,  4.,  6.])
True

# At [x,y] = [1,1], compute the numerical gradient # of the function sin(x-y) + y*exp(x)

>>> sin = np.sin; exp = np.exp
>>> z = lambda xy: sin(xy[0]-xy[1]) + xy[1]*exp(xy[0])
>>> dz = nd.Gradient(z)
>>> grad2 = dz([1, 1])
>>> np.allclose(grad2, [ 3.71828183,  1.71828183])
True

# At the global minimizer (1,1) of the Rosenbrock function, # compute the gradient. It should be essentially zero.

>>> rosen = lambda x : (1-x[0])**2 + 105.*(x[1]-x[0]**2)**2
>>> rd = nd.Gradient(rosen)
>>> grad3 = rd([1,1])
>>> np.allclose(grad3,[0, 0], atol=1e-7)
True

class Jacobian(fun, step=None, method='central', order=2, bounds=(-inf, inf), sparsity=None)[source]

Bases: _Common

Calculate Jacobian with finite difference approximation

Parameters:

funfunction: function of one array fun(x, *args, **kwds)
stepfloat, optional: Stepsize, if None, optimal stepsize is used, i.e., x * _EPS for method==`complex` x * _EPS**(1/2) for method==`forward` x * _EPS**(1/3) for method==`central`.
method{‘central’, ‘complex’, ‘forward’}: defines the method used in the approximation.

Examples

>>> import numpy as np
>>> import numdifftools.nd_scipy as nd

#(nonlinear least squares)

>>> xdata = np.arange(0,1,0.1)
>>> ydata = 1+2*np.exp(0.75*xdata)
>>> fun = lambda c: (c[0]+c[1]*np.exp(c[2]*xdata) - ydata)**2
>>> np.allclose(fun([1, 2, 0.75]).shape, (10,))
True
>>> dfun = nd.Jacobian(fun)
>>> np.allclose(dfun([1, 2, 0.75]), np.zeros((10,3)))
True

>>> fun2 = lambda x : x[0]*x[1]*x[2]**2
>>> dfun2 = nd.Jacobian(fun2)
>>> np.allclose(dfun2([1.,2.,3.]), [[18., 9., 12.]])
True

>>> fun3 = lambda x : np.vstack((x[0]*x[1]*x[2]**2, x[0]*x[1]*x[2]))

TODO: The following does not work: der3 = nd.Jacobian(fun3)([1., 2., 3.]) np.allclose(der3, … [[18., 9., 12.], [6., 3., 2.]]) True np.allclose(nd.Jacobian(fun3)([4., 5., 6.]), … [[180., 144., 240.], [30., 24., 20.]]) True

np.allclose(nd.Jacobian(fun3)(np.array([[1.,2.,3.], [4., 5., 6.]]).T), … [[[ 18., 180.], … [ 9., 144.], … [ 12., 240.]], … [[ 6., 30.], … [ 3., 24.], … [ 2., 20.]]]) True

5.2.10. numdifftools.nd_statsmodels module

5.2.10.1. Numdifftools.nd_statsmodels

This module provides an easy to use interface to derivatives calculated with statsmodels.numdiff.

class Gradient(fun, step=None, method='central', order=None)[source]

Bases: Jacobian

Calculate Gradient with finite difference approximation

Parameters:

funfunction: function of one array fun(x, *args, **kwds)
stepfloat, optional: Stepsize, if None, optimal stepsize is used, i.e., x * _EPS for method==`complex` x * _EPS**(1/2) for method==`forward` x * _EPS**(1/3) for method==`central`.
method{‘central’, ‘complex’, ‘forward’, ‘backward’}: defines the method used in the approximation.

See also

Hessian, Jacobian

Examples

>>> import numpy as np
>>> import numdifftools.nd_statsmodels as nd
>>> fun = lambda x: np.sum(x**2)
>>> dfun = nd.Gradient(fun)
>>> np.allclose(dfun([1,2,3]), [ 2.,  4.,  6.])
True

# At [x,y] = [1,1], compute the numerical gradient # of the function sin(x-y) + y*exp(x)

>>> sin = np.sin; exp = np.exp
>>> z = lambda xy: sin(xy[0]-xy[1]) + xy[1]*exp(xy[0])
>>> dz = nd.Gradient(z)
>>> grad2 = dz([1, 1])
>>> np.allclose(grad2, [ 3.71828183,  1.71828183])
True

# At the global minimizer (1,1) of the Rosenbrock function, # compute the gradient. It should be essentially zero.

>>> rosen = lambda x : (1-x[0])**2 + 105.*(x[1]-x[0]**2)**2
>>> rd = nd.Gradient(rosen)
>>> grad3 = rd([1,1])
>>> np.allclose(grad3,[0, 0])
True

class Hessian(fun, step=None, method='central', order=None)[source]

Bases: _Common

Calculate Hessian with finite difference approximation

Parameters:

funfunction: function of one array fun(x, *args, **kwds)
stepfloat, optional: Stepsize, if None, optimal stepsize is used, i.e., x * _EPS**(1/3) for method==`forward`, complex or central2 x * _EPS**(1/4) for method==`central`.
method{‘central’, ‘complex’, ‘forward’, ‘backward’}: defines the method used in the approximation.

See also

Jacobian, Gradient

Examples

>>> import numpy as np
>>> import numdifftools.nd_statsmodels as nd

# Rosenbrock function, minimized at [1,1]

>>> rosen = lambda x : (1.-x[0])**2 + 105*(x[1]-x[0]**2)**2
>>> Hfun = nd.Hessian(rosen)
>>> h = Hfun([1, 1])
>>> np.allclose(h, [[ 842., -420.], [-420.,  210.]])
True

# cos(x-y), at (0,0)

>>> cos = np.cos
>>> fun = lambda xy : cos(xy[0]-xy[1])
>>> Hfun2 = nd.Hessian(fun)
>>> h2 = Hfun2([0, 0])
>>> np.allclose(h2, [[-1.,  1.], [ 1., -1.]])
True

property n

class Jacobian(fun, step=None, method='central', order=None)[source]

Bases: _Common

Calculate Jacobian with finite difference approximation

Parameters:

funfunction: function of one array fun(x, *args, **kwds)
stepfloat, optional: Stepsize, if None, optimal stepsize is used, i.e., x * _EPS for method==`complex` x * _EPS**(1/2) for method==`forward` x * _EPS**(1/3) for method==`central`.
method{‘central’, ‘complex’, ‘forward’, ‘backward’}: defines the method used in the approximation.

Examples

>>> import numpy as np
>>> import numdifftools.nd_statsmodels as nd

#(nonlinear least squares)

>>> xdata = np.arange(0,1,0.1)
>>> ydata = 1+2*np.exp(0.75*xdata)
>>> fun = lambda c: (c[0]+c[1]*np.exp(c[2]*xdata) - ydata)**2
>>> np.allclose(fun([1, 2, 0.75]).shape, (10,))
True
>>> dfun = nd.Jacobian(fun)
>>> np.allclose(dfun([1, 2, 0.75]), np.zeros((10,3)))
True

>>> fun2 = lambda x : x[0]*x[1]*x[2]**2
>>> dfun2 = nd.Jacobian(fun2)
>>> np.allclose(dfun2([1.,2.,3.]), [[18., 9., 12.]])
True

>>> fun3 = lambda x : np.vstack((x[0]*x[1]*x[2]**2, x[0]*x[1]*x[2]))
>>> np.allclose(nd.Jacobian(fun3)([1., 2., 3.]), [[[18.], [9.], [12.]], [[6.], [3.], [2.]]])
True
>>> np.allclose(nd.Jacobian(fun3)([4., 5., 6.]),
...            [[[180.], [144.], [240.]], [[30.], [24.], [20.]]])
True

>>> np.allclose(nd.Jacobian(fun3)(np.array([[1.,2.,3.], [4., 5., 6.]]).T),
...            [[[  18.,  180.],
...              [   9.,  144.],
...              [  12.,  240.]],
...             [[   6.,   30.],
...              [   3.,   24.],
...              [   2.,   20.]]])
True

approx_fprime(x, f, epsilon=None, args=(), kwargs=None, centered=True)[source]

Gradient of function, or Jacobian if function fun returns 1d array

Parameters:

xarray: parameters at which the derivative is evaluated
funfunction: fun(*((x,)+args), **kwargs) returning either one value or 1d array
epsilonfloat, optional: Stepsize, if None, optimal stepsize is used. This is _EPS**(1/2)*x for centered == False and _EPS**(1/3)*x for centered == True.
argstuple: Tuple of additional arguments for function fun.
kwargsdict: Dictionary of additional keyword arguments for function fun.
centeredbool: Whether central difference should be returned. If not, does forward differencing.

Returns:

gradarray: gradient or Jacobian

Notes

If fun returns a 1d array, it returns a Jacobian. If a 2d array is returned by fun (e.g., with a value for each observation), it returns a 3d array with the Jacobian of each observation with shape xk x nobs x xk. I.e., the Jacobian of the first observation would be [:, 0, :]

approx_fprime_cs(x, f, epsilon=None, args=(), kwargs=None)[source]

Calculate gradient or Jacobian with complex step derivative approximation

Parameters:

xarray: parameters at which the derivative is evaluated
ffunction: f(*((x,)+args), **kwargs) returning either one value or 1d array
epsilonfloat, optional: Stepsize, if None, optimal stepsize is used. Optimal step-size is EPS*x. See note.
argstuple: Tuple of additional arguments for function f.
kwargsdict: Dictionary of additional keyword arguments for function f.

Returns:

partialsndarray: array of partial derivatives, Gradient or Jacobian

Notes

The complex-step derivative has truncation error O(epsilon**2), so truncation error can be eliminated by choosing epsilon to be very small. The complex-step derivative avoids the problem of round-off error with small epsilon because there is no subtraction.

5.2.11. numdifftools.step_generators module

Step generators module

class BasicMaxStepGenerator(base_step, step_ratio, num_steps, offset=0)[source]

Bases: object

Generates a sequence of steps of decreasing magnitude

where: steps = base_step * step_ratio ** (-i + offset)

for i=0, 1,.., num_steps-1.

Parameters:

base_stepfloat, array-like.: Defines the start step, i.e., maximum step
step_ratioreal scalar.: Ratio between sequential steps generated. Note: Ratio > 1
num_stepsscalar integer.: defines number of steps generated.
offsetreal scalar, optional, default 0: offset to the base step

Examples

>>> from numdifftools.step_generators import BasicMaxStepGenerator
>>> step_gen = BasicMaxStepGenerator(base_step=2.0, step_ratio=2,
...                                  num_steps=4)
>>> [s for s in step_gen()]
[2.0, 1.0, 0.5, 0.25]

class BasicMinStepGenerator(base_step, step_ratio, num_steps, offset=0)[source]

Bases: BasicMaxStepGenerator

Generates a sequence of steps of decreasing magnitude

where: steps = base_step * step_ratio ** (i + offset), i=num_steps-1,… 1, 0.

Parameters:

base_stepfloat, array-like.: Defines the end step, i.e., minimum step
step_ratioreal scalar.: Ratio between sequential steps generated. Note: Ratio > 1
num_stepsscalar integer.: defines number of steps generated.
offsetreal scalar, optional, default 0: offset to the base step

Examples

>>> from numdifftools.step_generators import BasicMinStepGenerator
>>> step_gen = BasicMinStepGenerator(base_step=0.25, step_ratio=2,
...                                  num_steps=4)
>>> [s for s in step_gen()]
[2.0, 1.0, 0.5, 0.25]

class MaxStepGenerator(base_step=2.0, step_ratio=None, num_steps=15, step_nom=None, offset=0, num_extrap=9, use_exact_steps=False, check_num_steps=True, scale=500)[source]

Bases: MinStepGenerator

Generates a sequence of steps

where: steps = step_nom * base_step * step_ratio ** (-i + offset)

for i = 0, 1, …, num_steps-1.

Parameters:

base_stepfloat, array-like, default 2.0: Defines the maximum step, if None, the value is set to EPS**(1/scale)
step_ratioreal scalar, optional, default 2 or 1.6: Ratio between sequential steps generated. Note: Ratio > 1 If None then step_ratio is 2 for n=1 otherwise step_ratio is 1.6
num_stepsscalar integer, optional, default min_num_steps + num_extrap: defines number of steps generated. It should be larger than min_num_steps = (n + order - 1) / fact where fact is 1, 2 or 4 depending on differentiation method used.
step_nomdefault maximum(log(exp(1)+|x|), 1): Nominal step where x is supplied at runtime through the __call__ method.
offsetreal scalar, optional, default 0: offset to the base step
num_extrapscalar integer, default 0: number of points used for extrapolation
check_num_stepsboolean, default True: If True make sure num_steps is larger than the minimum required steps.
use_exact_stepsboolean, default True: If true make sure exact steps are generated
scalereal scalar, default 500: scale used in base step.

class MinStepGenerator(base_step=None, step_ratio=None, num_steps=None, step_nom=None, offset=0, num_extrap=0, use_exact_steps=True, check_num_steps=True, scale=None)[source]

Bases: object

Generates a sequence of steps

where: steps = step_nom * base_step * step_ratio ** (i + offset)

for i = num_steps-1,… 1, 0.

Parameters:

base_stepfloat, array-like, optional: Defines the minimum step, if None, the value is set to EPS**(1/scale)
step_ratioreal scalar, optional, default 2: Ratio between sequential steps generated. Note: Ratio > 1 If None then step_ratio is 2 for n=1 otherwise step_ratio is 1.6
num_stepsscalar integer, optional, default min_num_steps + num_extrap: defines number of steps generated. It should be larger than min_num_steps = (n + order - 1) / fact where fact is 1, 2 or 4 depending on differentiation method used.
step_nomdefault maximum(log(exp(1)+|x|), 1): Nominal step where x is supplied at runtime through the __call__ method.
offsetreal scalar, optional, default 0: offset to the base step
num_extrapscalar integer, default 0: number of points used for extrapolation
check_num_stepsboolean, default True: If True make sure num_steps is larger than the minimum required steps.
use_exact_stepsboolean, default True: If true make sure exact steps are generated
scalereal scalar, optional: scale used in base step. If not None it will override the default computed with the default_scale function.

property base_step: Base step defines the minimum or maximum step when offset==0.

property min_num_steps: Minimum number of steps required given the differentiation method and order.

property num_steps: Defines number of steps generated

property scale: Scale used in base step.

step_generator_function(x, method='forward', n=1, order=2)[source]: Step generator function

property step_nom: Nominal step

property step_ratio: Ratio between sequential steps generated

default_scale(method='forward', n=1, order=2)[source]: Returns good scale for MinStepGenerator

get_base_step(scale)[source]: Return base_step = EPS ** (1. / scale)

get_nominal_step(x=None)[source]: Return nominal step

make_exact(h)[source]

Make sure h is an exact representable number

This is important when calculating numerical derivatives and is accomplished by adding 1.0 and then subtracting 1.0.