pathsim.optim.value module¶

pathsim.optim.value.der(arr, val)[source]¶

Compute the derivative of an array of ‘Value’ objects with respect to ‘val’, fallback to scalar case and non-‘Value’ objects

Parameters:

arr (array[Value]) – array or list of Values
val (Value) – dual number value for AD

Returns:

der – partial derivatives w.r.t. the ‘val’

Return type:

array[float]

pathsim.optim.value.jac(arr, vals)[source]¶

Compute the derivative of an array of ‘Value’ objects with respect to each ‘Value’ object in ‘vals’, fallback to scalars in both cases.

This effectively constructs the jacobian.

Parameters:

arr (array[Value]) – array or list of Values
vals (array[Value]) – array or list of Values

Returns:

jac – partial derivatives w.r.t. all values in ‘vals’, effectively the jacobian

Return type:

array[float]

pathsim.optim.value.var(arr, vals)[source]¶

Compute the variance of an array of ‘Value’ objects with respect to all the ‘Value’ objects in ‘vals’ using the standard deviations of the values and a first order taylor approximation using the partial derivatives.

Parameters:

arr (array[Value]) – array or list of Values
vals (array[Value]) – array or list of Values

Returns:

var – variance from taylor expansion w.r.t. all values in ‘vals’

Return type:

array[float]

pathsim.optim.value.autojac(fnc)[source]¶

Decorator that wraps a function such that it computes its jacobian alongside its evaluaiton.

Parameters:: fnc (callable) – function to wrap and compute jacobian of

pathsim.optim.value.add_funcs(cls)[source]¶: Decorator that adds numpy functions as methods to the Value class while overloading them with their analytical derivatives to propagate the gradient with the chain rule.

class pathsim.optim.value.Value(val=0.0, grad=None, sig=0.0)[source]¶

Bases: object

Dual number ‘Value’ definition for small autograd framework.

The dunder methods of the ‘Value’ class are overloaded to simultaneously compute the partial derivatives with respect to the instance itself and other instances of the value class.

This is realized by a dictionary that handles the reference tracking via the id of the ‘Value’ instances.

Example

Automatic differentiation of numpy functions:

import numpy as np

v = Value(3)

#AD
w = np.sin(v)

#reference
dw_dv = np.cos(v)

print(dw_dv == w(v))
#True

Automatic differentiation of concatenations:

import numpy as np

v = Value(2)

#AD
w = np.exp(np.sin(v) + v**3)

#reference
dw_dv = np.exp(np.sin(v) + v**3) * (np.cos(v) + 3*v**2)

print(dw_dv == w(v))
#True

Automatic differentiation with multiple values:

import numpy as np

u = Value(0.5)
v = Value(2)

#AD
w = u * v + np.sin(u + v)

#references
dw_du = v + np.cos(u + v)
dw_dv = u + np.cos(u + v)

print(dw_du == w(u), dw_dv == w(v))
#True True

Parameters:

val (float, int, complex) – The numerical value.
grad (defaultdict, None) – The gradient dictionary. If None, initializes with self derivative.
sig (float, int, complex) – Standard deviation of value for uncertainty analysis (optional)

_id¶

id for reference tracking in gradient dict

Type:: int

val¶

grad¶

sig¶

property real¶

property imag¶

classmethod numeric(arr)[source]¶

Cast an array with value objects to an array of numeric values.

Numeric entries are just passed through.

Parameters:: arr (array[obj]) – array of mixed value, numeric objects
Returns:: numeric – array of numeric values
Return type:: array[numeric]

classmethod array(arr)[source]¶

Cast an array or list to an array of value objects.

For Value entries, their numeric values are used to create a new Value instance.

Example

Instantiate multiple values from 1d array:

a, b, c = Value.array([1, 2, 3])

Instantiate numpy array filled with values:

import numpy as np

A_num = np.random.rand(3, 5)

A = Value.array(A_num)

Parameters:: arr (array[obj]) – array of mixed value, numeric objects
Returns:: array – array of Value objects
Return type:: array[Value]

static der(arr, val)[source]¶

static jac(arr, vals)[source]¶

static var(arr, vals)[source]¶

absolute¶

Calculate the absolute value element-wise.

np.abs is a shorthand for this function.

Parameters:

x (array_like) – Input array.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

absolute – An ndarray containing the absolute value of each element in x. For complex input, a + ib, the absolute value is \(\sqrt{ a^2 + b^2 }\). This is a scalar if x is a scalar.

Return type:

ndarray

Examples

>>> x = np.array([-1.2, 1.2])
>>> np.absolute(x)
array([ 1.2,  1.2])
>>> np.absolute(1.2 + 1j)
1.5620499351813308

Plot the function over [-10, 10]:

>>> import matplotlib.pyplot as plt

>>> x = np.linspace(start=-10, stop=10, num=101)
>>> plt.plot(x, np.absolute(x))
>>> plt.show()

Plot the function over the complex plane:

>>> xx = x + 1j * x[:, np.newaxis]
>>> plt.imshow(np.abs(xx), extent=[-10, 10, -10, 10], cmap='gray')
>>> plt.show()

The abs function can be used as a shorthand for np.absolute on ndarrays.

>>> x = np.array([-1.2, 1.2])
>>> abs(x)
array([1.2, 1.2])

angle(deg=False)¶

Return the angle of the complex argument.

Parameters:

z (array_like) – A complex number or sequence of complex numbers.
deg (bool, optional) – Return angle in degrees if True, radians if False (default).

Returns:

angle – The counterclockwise angle from the positive real axis on the complex plane in the range (-pi, pi], with dtype as numpy.float64.

Changed in version 1.16.0: This function works on subclasses of ndarray like ma.array.

Return type:

ndarray or scalar

See also

arctan2, absolute

Notes

Although the angle of the complex number 0 is undefined, numpy.angle(0) returns the value 0.

Examples

>>> np.angle([1.0, 1.0j, 1+1j])               # in radians
array([ 0.        ,  1.57079633,  0.78539816]) # may vary
>>> np.angle(1+1j, deg=True)                  # in degrees
45.0

arccos¶

Trigonometric inverse cosine, element-wise.

The inverse of cos so that, if y = cos(x), then x = arccos(y).

Parameters:

x (array_like) – x-coordinate on the unit circle. For real arguments, the domain is [-1, 1].
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

angle – The angle of the ray intersecting the unit circle at the given x-coordinate in radians [0, pi]. This is a scalar if x is a scalar.

Return type:

ndarray

See also

cos, arctan, arcsin, emath.arccos

Notes

arccos is a multivalued function: for each x there are infinitely many numbers z such that cos(z) = x. The convention is to return the angle z whose real part lies in [0, pi].

For real-valued input data types, arccos always returns real output. For each value that cannot be expressed as a real number or infinity, it yields nan and sets the invalid floating point error flag.

For complex-valued input, arccos is a complex analytic function that has branch cuts [-inf, -1] and [1, inf] and is continuous from above on the former and from below on the latter.

The inverse cos is also known as acos or cos^-1.

References

M. Abramowitz and I.A. Stegun, “Handbook of Mathematical Functions”, 10th printing, 1964, pp. 79. https://personal.math.ubc.ca/~cbm/aands/page_79.htm

Examples

We expect the arccos of 1 to be 0, and of -1 to be pi:

>>> np.arccos([1, -1])
array([ 0.        ,  3.14159265])

Plot arccos:

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-1, 1, num=100)
>>> plt.plot(x, np.arccos(x))
>>> plt.axis('tight')
>>> plt.show()

arccosh¶

Inverse hyperbolic cosine, element-wise.

Parameters:

x (array_like) – Input array.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

arccosh – Array of the same shape as x. This is a scalar if x is a scalar.

Return type:

ndarray

Notes

arccosh is a multivalued function: for each x there are infinitely many numbers z such that cosh(z) = x. The convention is to return the z whose imaginary part lies in [-pi, pi] and the real part in [0, inf].

For real-valued input data types, arccosh always returns real output. For each value that cannot be expressed as a real number or infinity, it yields nan and sets the invalid floating point error flag.

For complex-valued input, arccosh is a complex analytical function that has a branch cut [-inf, 1] and is continuous from above on it.

References

Examples

>>> np.arccosh([np.e, 10.0])
array([ 1.65745445,  2.99322285])
>>> np.arccosh(1)
0.0

arcsin¶

Inverse sine, element-wise.

Parameters:

x (array_like) – y-coordinate on the unit circle.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

angle – The inverse sine of each element in x, in radians and in the closed interval [-pi/2, pi/2]. This is a scalar if x is a scalar.

Return type:

ndarray

See also

sin, cos, arccos, tan, arctan, arctan2, emath.arcsin

Notes

arcsin is a multivalued function: for each x there are infinitely many numbers z such that \(sin(z) = x\). The convention is to return the angle z whose real part lies in [-pi/2, pi/2].

For real-valued input data types, arcsin always returns real output. For each value that cannot be expressed as a real number or infinity, it yields nan and sets the invalid floating point error flag.

For complex-valued input, arcsin is a complex analytic function that has, by convention, the branch cuts [-inf, -1] and [1, inf] and is continuous from above on the former and from below on the latter.

The inverse sine is also known as asin or sin^{-1}.

References

Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, 10th printing, New York: Dover, 1964, pp. 79ff. https://personal.math.ubc.ca/~cbm/aands/page_79.htm

Examples

>>> np.arcsin(1)     # pi/2
1.5707963267948966
>>> np.arcsin(-1)    # -pi/2
-1.5707963267948966
>>> np.arcsin(0)
0.0

arcsinh¶

Inverse hyperbolic sine element-wise.

Parameters:

x (array_like) – Input array.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

out – Array of the same shape as x. This is a scalar if x is a scalar.

Return type:

ndarray or scalar

Notes

arcsinh is a multivalued function: for each x there are infinitely many numbers z such that sinh(z) = x. The convention is to return the z whose imaginary part lies in [-pi/2, pi/2].

For real-valued input data types, arcsinh always returns real output. For each value that cannot be expressed as a real number or infinity, it returns nan and sets the invalid floating point error flag.

For complex-valued input, arccos is a complex analytical function that has branch cuts [1j, infj] and [-1j, -infj] and is continuous from the right on the former and from the left on the latter.

The inverse hyperbolic sine is also known as asinh or sinh^-1.

References

Examples

>>> np.arcsinh(np.array([np.e, 10.0]))
array([ 1.72538256,  2.99822295])

arctan¶

Trigonometric inverse tangent, element-wise.

The inverse of tan, so that if y = tan(x) then x = arctan(y).

Parameters:

x (array_like)
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

out – Out has the same shape as x. Its real part is in [-pi/2, pi/2] (arctan(+/-inf) returns +/-pi/2). This is a scalar if x is a scalar.

Return type:

ndarray or scalar

See also

arctan2: The “four quadrant” arctan of the angle formed by (x, y) and the positive x-axis.
angle: Argument of complex values.

Notes

arctan is a multi-valued function: for each x there are infinitely many numbers z such that tan(z) = x. The convention is to return the angle z whose real part lies in [-pi/2, pi/2].

For real-valued input data types, arctan always returns real output. For each value that cannot be expressed as a real number or infinity, it yields nan and sets the invalid floating point error flag.

For complex-valued input, arctan is a complex analytic function that has [1j, infj] and [-1j, -infj] as branch cuts, and is continuous from the left on the former and from the right on the latter.

The inverse tangent is also known as atan or tan^{-1}.

References

Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, 10th printing, New York: Dover, 1964, pp. 79. https://personal.math.ubc.ca/~cbm/aands/page_79.htm

Examples

We expect the arctan of 0 to be 0, and of 1 to be pi/4:

>>> np.arctan([0, 1])
array([ 0.        ,  0.78539816])

>>> np.pi/4
0.78539816339744828

Plot arctan:

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-10, 10)
>>> plt.plot(x, np.arctan(x))
>>> plt.axis('tight')
>>> plt.show()

arctanh¶

Inverse hyperbolic tangent element-wise.

Parameters:

x (array_like) – Input array.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

out – Array of the same shape as x. This is a scalar if x is a scalar.

Return type:

ndarray or scalar

See also

emath.arctanh

Notes

arctanh is a multivalued function: for each x there are infinitely many numbers z such that tanh(z) = x. The convention is to return the z whose imaginary part lies in [-pi/2, pi/2].

For real-valued input data types, arctanh always returns real output. For each value that cannot be expressed as a real number or infinity, it yields nan and sets the invalid floating point error flag.

For complex-valued input, arctanh is a complex analytical function that has branch cuts [-1, -inf] and [1, inf] and is continuous from above on the former and from below on the latter.

The inverse hyperbolic tangent is also known as atanh or tanh^-1.

References

Examples

>>> np.arctanh([0, -0.5])
array([ 0.        , -0.54930614])

cbrt¶

Return the cube-root of an array, element-wise.

Added in version 1.10.0.

Parameters:

x (array_like) – The values whose cube-roots are required.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

y – An array of the same shape as x, containing the cube cube-root of each element in x. If out was provided, y is a reference to it. This is a scalar if x is a scalar.

Return type:

ndarray

Examples

>>> np.cbrt([1,8,27])
array([ 1.,  2.,  3.])

conjugate¶

Return the complex conjugate, element-wise.

The complex conjugate of a complex number is obtained by changing the sign of its imaginary part.

Parameters:

x (array_like) – Input value.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

y – The complex conjugate of x, with same dtype as y. This is a scalar if x is a scalar.

Return type:

ndarray

Notes

conj is an alias for conjugate:

>>> np.conj is np.conjugate
True

Examples

>>> np.conjugate(1+2j)
(1-2j)

>>> x = np.eye(2) + 1j * np.eye(2)
>>> np.conjugate(x)
array([[ 1.-1.j,  0.-0.j],
       [ 0.-0.j,  1.-1.j]])

cos¶

Cosine element-wise.

Parameters:

x (array_like) – Input array in radians.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

y – The corresponding cosine values. This is a scalar if x is a scalar.

Return type:

ndarray

Notes

If out is provided, the function writes the result into it, and returns a reference to out. (See Examples)

References

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York, NY: Dover, 1972.

Examples

>>> np.cos(np.array([0, np.pi/2, np.pi]))
array([  1.00000000e+00,   6.12303177e-17,  -1.00000000e+00])
>>>
>>> # Example of providing the optional output parameter
>>> out1 = np.array([0], dtype='d')
>>> out2 = np.cos([0.1], out1)
>>> out2 is out1
True
>>>
>>> # Example of ValueError due to provision of shape mis-matched `out`
>>> np.cos(np.zeros((3,3)),np.zeros((2,2)))
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: operands could not be broadcast together with shapes (3,3) (2,2)

cosh¶

Hyperbolic cosine, element-wise.

Equivalent to 1/2 * (np.exp(x) + np.exp(-x)) and np.cos(1j*x).

Parameters:

x (array_like) – Input array.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

out – Output array of same shape as x. This is a scalar if x is a scalar.

Return type:

ndarray or scalar

Examples

>>> np.cosh(0)
1.0

The hyperbolic cosine describes the shape of a hanging cable:

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-4, 4, 1000)
>>> plt.plot(x, np.cosh(x))
>>> plt.show()

exp¶

Calculate the exponential of all elements in the input array.

Parameters:

x (array_like) – Input values.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

out – Output array, element-wise exponential of x. This is a scalar if x is a scalar.

Return type:

ndarray or scalar

See also

expm1: Calculate exp(x) - 1 for all elements in the array.
exp2: Calculate 2**x for all elements in the array.

Notes

The irrational number e is also known as Euler’s number. It is approximately 2.718281, and is the base of the natural logarithm, ln (this means that, if \(x = \ln y = \log_e y\), then \(e^x = y\). For real input, exp(x) is always positive.

For complex arguments, x = a + ib, we can write \(e^x = e^a e^{ib}\). The first term, \(e^a\), is already known (it is the real argument, described above). The second term, \(e^{ib}\), is \(\cos b + i \sin b\), a function with magnitude 1 and a periodic phase.

References

Examples

Plot the magnitude and phase of exp(x) in the complex plane:

>>> import matplotlib.pyplot as plt

>>> x = np.linspace(-2*np.pi, 2*np.pi, 100)
>>> xx = x + 1j * x[:, np.newaxis] # a + ib over complex plane
>>> out = np.exp(xx)

>>> plt.subplot(121)
>>> plt.imshow(np.abs(out),
...            extent=[-2*np.pi, 2*np.pi, -2*np.pi, 2*np.pi], cmap='gray')
>>> plt.title('Magnitude of exp(x)')

>>> plt.subplot(122)
>>> plt.imshow(np.angle(out),
...            extent=[-2*np.pi, 2*np.pi, -2*np.pi, 2*np.pi], cmap='hsv')
>>> plt.title('Phase (angle) of exp(x)')
>>> plt.show()

exp2¶

Calculate 2**p for all p in the input array.

Parameters:

x (array_like) – Input values.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

out – Element-wise 2 to the power x. This is a scalar if x is a scalar.

Return type:

ndarray or scalar

See also

power

Notes

Added in version 1.3.0.

Examples

>>> np.exp2([2, 3])
array([ 4.,  8.])

expm1¶

Calculate exp(x) - 1 for all elements in the array.

Parameters:

x (array_like) – Input values.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

out – Element-wise exponential minus one: out = exp(x) - 1. This is a scalar if x is a scalar.

Return type:

ndarray or scalar

See also

log1p: log(1 + x), the inverse of expm1.

Notes

This function provides greater precision than exp(x) - 1 for small values of x.

Examples

The true value of exp(1e-10) - 1 is 1.00000000005e-10 to about 32 significant digits. This example shows the superiority of expm1 in this case.

>>> np.expm1(1e-10)
1.00000000005e-10
>>> np.exp(1e-10) - 1
1.000000082740371e-10

log¶

Natural logarithm, element-wise.

The natural logarithm log is the inverse of the exponential function, so that log(exp(x)) = x. The natural logarithm is logarithm in base e.

Parameters:

x (array_like) – Input value.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

y – The natural logarithm of x, element-wise. This is a scalar if x is a scalar.

Return type:

ndarray

See also

log10, log2, log1p, emath.log

Notes

Logarithm is a multivalued function: for each x there is an infinite number of z such that exp(z) = x. The convention is to return the z whose imaginary part lies in (-pi, pi].

For real-valued input data types, log always returns real output. For each value that cannot be expressed as a real number or infinity, it yields nan and sets the invalid floating point error flag.

For complex-valued input, log is a complex analytical function that has a branch cut [-inf, 0] and is continuous from above on it. log handles the floating-point negative zero as an infinitesimal negative number, conforming to the C99 standard.

In the cases where the input has a negative real part and a very small negative complex part (approaching 0), the result is so close to -pi that it evaluates to exactly -pi.

References

Examples

>>> np.log([1, np.e, np.e**2, 0])
array([  0.,   1.,   2., -Inf])

log10¶

Return the base 10 logarithm of the input array, element-wise.

Parameters:

x (array_like) – Input values.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

y – The logarithm to the base 10 of x, element-wise. NaNs are returned where x is negative. This is a scalar if x is a scalar.

Return type:

ndarray

See also

emath.log10

Notes

Logarithm is a multivalued function: for each x there is an infinite number of z such that 10**z = x. The convention is to return the z whose imaginary part lies in (-pi, pi].

For real-valued input data types, log10 always returns real output. For each value that cannot be expressed as a real number or infinity, it yields nan and sets the invalid floating point error flag.

For complex-valued input, log10 is a complex analytical function that has a branch cut [-inf, 0] and is continuous from above on it. log10 handles the floating-point negative zero as an infinitesimal negative number, conforming to the C99 standard.

In the cases where the input has a negative real part and a very small negative complex part (approaching 0), the result is so close to -pi that it evaluates to exactly -pi.

References

Examples

>>> np.log10([1e-15, -3.])
array([-15.,  nan])

log1p¶

Return the natural logarithm of one plus the input array, element-wise.

Calculates log(1 + x).

Parameters:

x (array_like) – Input values.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

y – Natural logarithm of 1 + x, element-wise. This is a scalar if x is a scalar.

Return type:

ndarray

See also

expm1: exp(x) - 1, the inverse of log1p.

Notes

For real-valued input, log1p is accurate also for x so small that 1 + x == 1 in floating-point accuracy.

Logarithm is a multivalued function: for each x there is an infinite number of z such that exp(z) = 1 + x. The convention is to return the z whose imaginary part lies in [-pi, pi].

For real-valued input data types, log1p always returns real output. For each value that cannot be expressed as a real number or infinity, it yields nan and sets the invalid floating point error flag.

For complex-valued input, log1p is a complex analytical function that has a branch cut [-inf, -1] and is continuous from above on it. log1p handles the floating-point negative zero as an infinitesimal negative number, conforming to the C99 standard.

References

Examples

>>> np.log1p(1e-99)
1e-99
>>> np.log(1 + 1e-99)
0.0

log2¶

Base-2 logarithm of x.

Parameters:

x (array_like) – Input values.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

y – Base-2 logarithm of x. This is a scalar if x is a scalar.

Return type:

ndarray

See also

log, log10, log1p, emath.log2

Notes

Added in version 1.3.0.

Logarithm is a multivalued function: for each x there is an infinite number of z such that 2**z = x. The convention is to return the z whose imaginary part lies in (-pi, pi].

For real-valued input data types, log2 always returns real output. For each value that cannot be expressed as a real number or infinity, it yields nan and sets the invalid floating point error flag.

For complex-valued input, log2 is a complex analytical function that has a branch cut [-inf, 0] and is continuous from above on it. log2 handles the floating-point negative zero as an infinitesimal negative number, conforming to the C99 standard.

In the cases where the input has a negative real part and a very small negative complex part (approaching 0), the result is so close to -pi that it evaluates to exactly -pi.

Examples

>>> x = np.array([0, 1, 2, 2**4])
>>> np.log2(x)
array([-Inf,   0.,   1.,   4.])

>>> xi = np.array([0+1.j, 1, 2+0.j, 4.j])
>>> np.log2(xi)
array([ 0.+2.26618007j,  0.+0.j        ,  1.+0.j        ,  2.+2.26618007j])

power¶

First array elements raised to powers from second array, element-wise.

Raise each base in x1 to the positionally-corresponding power in x2. x1 and x2 must be broadcastable to the same shape.

An integer type raised to a negative integer power will raise a ValueError.

Negative values raised to a non-integral value will return nan. To get complex results, cast the input to complex, or specify the dtype to be complex (see the example below).

Parameters:

x1 (array_like) – The bases.
x2 (array_like) – The exponents. If x1.shape != x2.shape, they must be broadcastable to a common shape (which becomes the shape of the output).
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

y – The bases in x1 raised to the exponents in x2. This is a scalar if both x1 and x2 are scalars.

Return type:

ndarray

See also

float_power: power function that promotes integers to float

Examples

Cube each element in an array.

>>> x1 = np.arange(6)
>>> x1
[0, 1, 2, 3, 4, 5]
>>> np.power(x1, 3)
array([  0,   1,   8,  27,  64, 125])

Raise the bases to different exponents.

>>> x2 = [1.0, 2.0, 3.0, 3.0, 2.0, 1.0]
>>> np.power(x1, x2)
array([  0.,   1.,   8.,  27.,  16.,   5.])

The effect of broadcasting.

>>> x2 = np.array([[1, 2, 3, 3, 2, 1], [1, 2, 3, 3, 2, 1]])
>>> x2
array([[1, 2, 3, 3, 2, 1],
       [1, 2, 3, 3, 2, 1]])
>>> np.power(x1, x2)
array([[ 0,  1,  8, 27, 16,  5],
       [ 0,  1,  8, 27, 16,  5]])

The ** operator can be used as a shorthand for np.power on ndarrays.

>>> x2 = np.array([1, 2, 3, 3, 2, 1])
>>> x1 = np.arange(6)
>>> x1 ** x2
array([ 0,  1,  8, 27, 16,  5])

Negative values raised to a non-integral value will result in nan (and a warning will be generated).

>>> x3 = np.array([-1.0, -4.0])
>>> with np.errstate(invalid='ignore'):
...     p = np.power(x3, 1.5)
...
>>> p
array([nan, nan])

To get complex results, give the argument dtype=complex.

>>> np.power(x3, 1.5, dtype=complex)
array([-1.83697020e-16-1.j, -1.46957616e-15-8.j])

sign¶

Returns an element-wise indication of the sign of a number.

The sign function returns -1 if x < 0, 0 if x==0, 1 if x > 0. nan is returned for nan inputs.

For complex inputs, the sign function returns sign(x.real) + 0j if x.real != 0 else sign(x.imag) + 0j.

complex(nan, 0) is returned for complex nan inputs.

Parameters:

x (array_like) – Input values.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

y – The sign of x. This is a scalar if x is a scalar.

Return type:

ndarray

Notes

There is more than one definition of sign in common use for complex numbers. The definition used here is equivalent to \(x/\sqrt{x*x}\) which is different from a common alternative, \(x/|x|\).

Examples

>>> np.sign([-5., 4.5])
array([-1.,  1.])
>>> np.sign(0)
0
>>> np.sign(5-2j)
(1+0j)

sin¶

Trigonometric sine, element-wise.

Parameters:

x (array_like) – Angle, in radians (\(2 \pi\) rad equals 360 degrees).
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

y – The sine of each element of x. This is a scalar if x is a scalar.

Return type:

array_like

See also

arcsin, sinh, cos

Notes

The sine is one of the fundamental functions of trigonometry (the mathematical study of triangles). Consider a circle of radius 1 centered on the origin. A ray comes in from the \(+x\) axis, makes an angle at the origin (measured counter-clockwise from that axis), and departs from the origin. The \(y\) coordinate of the outgoing ray’s intersection with the unit circle is the sine of that angle. It ranges from -1 for \(x=3\pi / 2\) to +1 for \(\pi / 2.\) The function has zeroes where the angle is a multiple of \(\pi\). Sines of angles between \(\pi\) and \(2\pi\) are negative. The numerous properties of the sine and related functions are included in any standard trigonometry text.

Examples

Print sine of one angle:

>>> np.sin(np.pi/2.)
1.0

Print sines of an array of angles given in degrees:

>>> np.sin(np.array((0., 30., 45., 60., 90.)) * np.pi / 180. )
array([ 0.        ,  0.5       ,  0.70710678,  0.8660254 ,  1.        ])

Plot the sine function:

>>> import matplotlib.pylab as plt
>>> x = np.linspace(-np.pi, np.pi, 201)
>>> plt.plot(x, np.sin(x))
>>> plt.xlabel('Angle [rad]')
>>> plt.ylabel('sin(x)')
>>> plt.axis('tight')
>>> plt.show()

sinh¶

Hyperbolic sine, element-wise.

Equivalent to 1/2 * (np.exp(x) - np.exp(-x)) or -1j * np.sin(1j*x).

Parameters:

x (array_like) – Input array.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

y – The corresponding hyperbolic sine values. This is a scalar if x is a scalar.

Return type:

ndarray

Notes

If out is provided, the function writes the result into it, and returns a reference to out. (See Examples)

References

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York, NY: Dover, 1972, pg. 83.

Examples

>>> np.sinh(0)
0.0
>>> np.sinh(np.pi*1j/2)
1j
>>> np.sinh(np.pi*1j) # (exact value is 0)
1.2246063538223773e-016j
>>> # Discrepancy due to vagaries of floating point arithmetic.

>>> # Example of providing the optional output parameter
>>> out1 = np.array([0], dtype='d')
>>> out2 = np.sinh([0.1], out1)
>>> out2 is out1
True

>>> # Example of ValueError due to provision of shape mis-matched `out`
>>> np.sinh(np.zeros((3,3)),np.zeros((2,2)))
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: operands could not be broadcast together with shapes (3,3) (2,2)

sqrt¶

Return the non-negative square-root of an array, element-wise.

Parameters:

x (array_like) – The values whose square-roots are required.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

y – An array of the same shape as x, containing the positive square-root of each element in x. If any element in x is complex, a complex array is returned (and the square-roots of negative reals are calculated). If all of the elements in x are real, so is y, with negative elements returning nan. If out was provided, y is a reference to it. This is a scalar if x is a scalar.

Return type:

ndarray

See also

emath.sqrt: A version which returns complex numbers when given negative reals. Note that 0.0 and -0.0 are handled differently for complex inputs.

Notes

sqrt has–consistent with common convention–as its branch cut the real “interval” [-inf, 0), and is continuous from above on it. A branch cut is a curve in the complex plane across which a given complex function fails to be continuous.

Examples

>>> np.sqrt([1,4,9])
array([ 1.,  2.,  3.])

>>> np.sqrt([4, -1, -3+4J])
array([ 2.+0.j,  0.+1.j,  1.+2.j])

>>> np.sqrt([4, -1, np.inf])
array([ 2., nan, inf])

square¶

Return the element-wise square of the input.

Parameters:

x (array_like) – Input data.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

out – Element-wise x*x, of the same shape and dtype as x. This is a scalar if x is a scalar.

Return type:

ndarray or scalar

Examples

>>> np.square([-1j, 1])
array([-1.-0.j,  1.+0.j])

tan¶

Compute tangent element-wise.

Equivalent to np.sin(x)/np.cos(x) element-wise.

Parameters:

x (array_like) – Input array.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

y – The corresponding tangent values. This is a scalar if x is a scalar.

Return type:

ndarray

Notes

If out is provided, the function writes the result into it, and returns a reference to out. (See Examples)

References

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York, NY: Dover, 1972.

Examples

>>> from math import pi
>>> np.tan(np.array([-pi,pi/2,pi]))
array([  1.22460635e-16,   1.63317787e+16,  -1.22460635e-16])
>>>
>>> # Example of providing the optional output parameter illustrating
>>> # that what is returned is a reference to said parameter
>>> out1 = np.array([0], dtype='d')
>>> out2 = np.cos([0.1], out1)
>>> out2 is out1
True
>>>
>>> # Example of ValueError due to provision of shape mis-matched `out`
>>> np.cos(np.zeros((3,3)),np.zeros((2,2)))
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: operands could not be broadcast together with shapes (3,3) (2,2)

tanh¶

Compute hyperbolic tangent element-wise.

Equivalent to np.sinh(x)/np.cosh(x) or -1j * np.tan(1j*x).

Parameters:

x (array_like) – Input array.
out (ndarray, None, or tuple of ndarray and None, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.
where (array_like, optional) – This condition is broadcast over the input. At locations where the condition is True, the out array will be set to the ufunc result. Elsewhere, the out array will retain its original value. Note that if an uninitialized out array is created via the default out=None, locations within it where the condition is False will remain uninitialized.
**kwargs – For other keyword-only arguments, see the ufunc docs.

Returns:

y – The corresponding hyperbolic tangent values. This is a scalar if x is a scalar.

Return type:

ndarray

Notes

If out is provided, the function writes the result into it, and returns a reference to out. (See Examples)

References

Examples

>>> np.tanh((0, np.pi*1j, np.pi*1j/2))
array([ 0. +0.00000000e+00j,  0. -1.22460635e-16j,  0. +1.63317787e+16j])

>>> # Example of providing the optional output parameter illustrating
>>> # that what is returned is a reference to said parameter
>>> out1 = np.array([0], dtype='d')
>>> out2 = np.tanh([0.1], out1)
>>> out2 is out1
True

>>> # Example of ValueError due to provision of shape mis-matched `out`
>>> np.tanh(np.zeros((3,3)),np.zeros((2,2)))
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: operands could not be broadcast together with shapes (3,3) (2,2)