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Return evenly spaced numbers over a specified interval.
Returns `num` evenly spaced samples, calculated over the
interval [`start`, `stop`].
The endpoint of the interval can optionally be excluded.
Parameters
----------
start : scalar
The starting value of the sequence.
stop : scalar
The end value of the sequence, unless `endpoint` is set to False.
In that case, the sequence consists of all but the last of ``num + 1``
evenly spaced samples, so that `stop` is excluded. Note that the step
size changes when `endpoint` is False.
num : int, optional
Number of samples to generate. Default is 50. Must be non-negative.
endpoint : bool, optional
If True, `stop` is the last sample. Otherwise, it is not included.
Default is True.
retstep : bool, optional
If True, return (`samples`, `step`), where `step` is the spacing
between samples.
dtype : dtype, optional
The type of the output array. If `dtype` is not given, infer the data
type from the other input arguments.
.. versionadded:: 1.9.0
Returns
-------
samples : ndarray
There are `num` equally spaced samples in the closed interval
``[start, stop]`` or the half-open interval ``[start, stop)``
(depending on whether `endpoint` is True or False).
step : float, optional
Only returned if `retstep` is True
Size of spacing between samples.
See Also
--------
arange : Similar to `linspace`, but uses a step size (instead of the
number of samples).
logspace : Samples uniformly distributed in log space.
Examples
--------
>>> np.linspace(2.0, 3.0, num=5)
array([ 2. , 2.25, 2.5 , 2.75, 3. ])
>>> np.linspace(2.0, 3.0, num=5, endpoint=False)
array([ 2. , 2.2, 2.4, 2.6, 2.8])
>>> np.linspace(2.0, 3.0, num=5, retstep=True)
(array([ 2. , 2.25, 2.5 , 2.75, 3. ]), 0.25)
Graphical illustration:
>>> import matplotlib.pyplot as plt
>>> N = 8
>>> y = np.zeros(N)
>>> x1 = np.linspace(0, 10, N, endpoint=True)
>>> x2 = np.linspace(0, 10, N, endpoint=False)
>>> plt.plot(x1, y, 'o')
[<matplotlib.lines.Line2D object at 0x...>]
>>> plt.plot(x2, y + 0.5, 'o')
[<matplotlib.lines.Line2D object at 0x...>]
>>> plt.ylim([-0.5, 1])
(-0.5, 1)
>>> plt.show()
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Return numbers spaced evenly on a log scale.
In linear space, the sequence starts at ``base ** start``
(`base` to the power of `start`) and ends with ``base ** stop``
(see `endpoint` below).
Parameters
----------
start : float
``base ** start`` is the starting value of the sequence.
stop : float
``base ** stop`` is the final value of the sequence, unless `endpoint`
is False. In that case, ``num + 1`` values are spaced over the
interval in log-space, of which all but the last (a sequence of
length `num`) are returned.
num : integer, optional
Number of samples to generate. Default is 50.
endpoint : boolean, optional
If true, `stop` is the last sample. Otherwise, it is not included.
Default is True.
base : float, optional
The base of the log space. The step size between the elements in
``ln(samples) / ln(base)`` (or ``log_base(samples)``) is uniform.
Default is 10.0.
dtype : dtype
The type of the output array. If `dtype` is not given, infer the data
type from the other input arguments.
Returns
-------
samples : ndarray
`num` samples, equally spaced on a log scale.
See Also
--------
arange : Similar to linspace, with the step size specified instead of the
number of samples. Note that, when used with a float endpoint, the
endpoint may or may not be included.
linspace : Similar to logspace, but with the samples uniformly distributed
in linear space, instead of log space.
geomspace : Similar to logspace, but with endpoints specified directly.
Notes
-----
Logspace is equivalent to the code
>>> y = np.linspace(start, stop, num=num, endpoint=endpoint)
... # doctest: +SKIP
>>> power(base, y).astype(dtype)
... # doctest: +SKIP
Examples
--------
>>> np.logspace(2.0, 3.0, num=4)
array([ 100. , 215.443469 , 464.15888336, 1000. ])
>>> np.logspace(2.0, 3.0, num=4, endpoint=False)
array([ 100. , 177.827941 , 316.22776602, 562.34132519])
>>> np.logspace(2.0, 3.0, num=4, base=2.0)
array([ 4. , 5.0396842 , 6.34960421, 8. ])
Graphical illustration:
>>> import matplotlib.pyplot as plt
>>> N = 10
>>> x1 = np.logspace(0.1, 1, N, endpoint=True)
>>> x2 = np.logspace(0.1, 1, N, endpoint=False)
>>> y = np.zeros(N)
>>> plt.plot(x1, y, 'o')
[<matplotlib.lines.Line2D object at 0x...>]
>>> plt.plot(x2, y + 0.5, 'o')
[<matplotlib.lines.Line2D object at 0x...>]
>>> plt.ylim([-0.5, 1])
(-0.5, 1)
>>> plt.show()
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Return numbers spaced evenly on a log scale (a geometric progression).
This is similar to `logspace`, but with endpoints specified directly.
Each output sample is a constant multiple of the previous.
Parameters
----------
start : scalar
The starting value of the sequence.
stop : scalar
The final value of the sequence, unless `endpoint` is False.
In that case, ``num + 1`` values are spaced over the
interval in log-space, of which all but the last (a sequence of
length `num`) are returned.
num : integer, optional
Number of samples to generate. Default is 50.
endpoint : boolean, optional
If true, `stop` is the last sample. Otherwise, it is not included.
Default is True.
dtype : dtype
The type of the output array. If `dtype` is not given, infer the data
type from the other input arguments.
Returns
-------
samples : ndarray
`num` samples, equally spaced on a log scale.
See Also
--------
logspace : Similar to geomspace, but with endpoints specified using log
and base.
linspace : Similar to geomspace, but with arithmetic instead of geometric
progression.
arange : Similar to linspace, with the step size specified instead of the
number of samples.
Notes
-----
If the inputs or dtype are complex, the output will follow a logarithmic
spiral in the complex plane. (There are an infinite number of spirals
passing through two points; the output will follow the shortest such path.)
Examples
--------
>>> np.geomspace(1, 1000, num=4)
array([ 1., 10., 100., 1000.])
>>> np.geomspace(1, 1000, num=3, endpoint=False)
array([ 1., 10., 100.])
>>> np.geomspace(1, 1000, num=4, endpoint=False)
array([ 1. , 5.62341325, 31.6227766 , 177.827941 ])
>>> np.geomspace(1, 256, num=9)
array([ 1., 2., 4., 8., 16., 32., 64., 128., 256.])
Note that the above may not produce exact integers:
>>> np.geomspace(1, 256, num=9, dtype=int)
array([ 1, 2, 4, 7, 16, 32, 63, 127, 256])
>>> np.around(np.geomspace(1, 256, num=9)).astype(int)
array([ 1, 2, 4, 8, 16, 32, 64, 128, 256])
Negative, decreasing, and complex inputs are allowed:
>>> np.geomspace(1000, 1, num=4)
array([ 1000., 100., 10., 1.])
>>> np.geomspace(-1000, -1, num=4)
array([-1000., -100., -10., -1.])
>>> np.geomspace(1j, 1000j, num=4) # Straight line
array([ 0. +1.j, 0. +10.j, 0. +100.j, 0.+1000.j])
>>> np.geomspace(-1+0j, 1+0j, num=5) # Circle
array([-1.00000000+0.j , -0.70710678+0.70710678j,
0.00000000+1.j , 0.70710678+0.70710678j,
1.00000000+0.j ])
Graphical illustration of ``endpoint`` parameter:
>>> import matplotlib.pyplot as plt
>>> N = 10
>>> y = np.zeros(N)
>>> plt.semilogx(np.geomspace(1, 1000, N, endpoint=True), y + 1, 'o')
>>> plt.semilogx(np.geomspace(1, 1000, N, endpoint=False), y + 2, 'o')
>>> plt.axis([0.5, 2000, 0, 3])
>>> plt.grid(True, color='0.7', linestyle='-', which='both', axis='both')
>>> plt.show()
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