SciPy Notes and Guide

1.1 What is SciPy?

SciPy is both (1) a way to handle large arrays of numerical data in Python (a capability it gets from Numpy) and (2) a way to apply scientific, statistical, and mathematical operations to those arrays of data. When combined with a package such as h5py or PyTables, if is also capable of storing and retrieving large arrays of data in an efficient way. Since much of it's calculations are done in C extension modules, SciPy can be quite fast.

1.2 What is Numpy?

Numpy is an array and vector library. SciPy, and other Python numberical packages as well, is built on top of Numpy.

1.3 Resources and Help

You can find help and documentation here -- SciPy.org -- http://www.scipy.org/

This document mostly discusses the use of Numpy and SciPy. However, if you are processing numerical data, you may also want to look at these:

• Pandas -- http://pandas.pydata.org/

• Numexpr -- https://github.com/pydata/numexpr

• Numba -- http://numba.pydata.org/

• Sympy -- http://sympy.org/en/index.html

• Matplotlib -- http://matplotlib.org/

• IPython -- An advanced Python interactive shell -- http://ipython.org/

• h5py -- HDF5 for Python -- http://www.h5py.org/

• PyTables -- http://pytable.sourceforge.net/

• SciPy -- http://www.scipy.org/

• Numpy -- http://www.numpy.org/

Overviews and collections -- The following Web sites contain summaries of and links to many of the Python (numerical) data analysis packages:

• PyData -- http://pydata.org/downloads/

• Python for data -- Packages at Github -- https://github.com/pydata

• Python for Data Analysis: The Landscape of Tutorials -- http://datacommunitydc.org/blog/2013/07/python-for-data-analysis-the-landscape-of-tutorials/

And there is a great deal of information that may be of help here: http://www.scipy.org/topical-software.html

1.4 Installation, configuration, etc

For instructions on installing Numpy and SciPy, look at the SciPy Web site.

4.1 The array interface and array protocol

The array interface is a specification for a developer who wishes to implement a replacement for the implementation of arrays, e.g. those used in scipy.

The array protocol is the way in which, for example, a scipy user uses arrays. It includes such things as:

• The ability to select elements in an array, for example, with a1[3] or a1[3,4].

• The ability to select slices of an array, for example, with a1[1:3].

• The ability to convert arrays without copying (see Converting arrays, below).

• The iterator protocol -- The ability to iterate over items in an array.

• Can respond to request for its length, for example len(my_array).

You should be aware of the difference between (1) a1[3,4] and (2) a1[3][4]. Both work. However, the second results in two calls to the __getitem__ method.

4.2 Converting arrays

At times you may need to convert an array from one type to another, for example from a numpy array to a scipy array or the reverse. The array protocol will help. In particular, the asarray() function can convert an array without copying. Examples:

In [8]: import numpy
In [9]: import scipy
In [10]: a1 = zeros((4,6))
In [11]: type(a1)
Out[11]: <type 'scipy.ndarray'>
In [12]: a2 = numpy.asarray(a1)
In [13]: type(a2)
Out[13]: <type 'numpy.ndarray'>
In [14]: a3 = numpy.zeros((3,5))
In [15]: type(a3)
Out[15]: <type 'numpy.ndarray'>
In [16]: a4 = scipy.asarray(a3)
In [17]: type(a4)
Out[17]: <type 'scipy.ndarray'>

5.1 Plain text

The Numpy function genfromtxt is helpful for reading a text file containing tabular data and creating an array from it. See: http://docs.scipy.org/doc/numpy/user/basics.io.genfromtxt.html

5.2 CSV

Here is an example that stores a 2-dimension array of floats into a CSV file, then later reads that CSV file back into a Numpy 2-D array:

import numpy as np
import csv

def test(filename):
    data1 = np.array([[1, 2, 3], [4, 5, 6]], dtype='f')
    print 'data1:', data1
    with open(filename, 'wb') as csvfile:
        spamwriter = csv.writer(csvfile)
        for row in data1:
            spamwriter.writerow(list(row))
    with open(filename, 'rb') as csvfile:
        csvreader = csv.reader(csvfile)
        data2 = []
        for row in csvreader:
            row = [float(x) for x in row]
            data2.append(row)
    data3 = np.array(data2, dtype=np.float_)
    print 'data3:', data3

test('tmp01.csv')

When we run the above code, we see:

$ python test.py
data1: [[ 1.  2.  3.]
 [ 4.  5.  6.]]
data3: [[ 1.  2.  3.]
 [ 4.  5.  6.]]

5.3 h5py

h5py is similar to PyTables (see section PyTables and HDF5) in the sense that it gives Python access to HDF5 files. However, in contrast, the interface to h5py feels a little bit lower level while still being very easy to use. Learn more about h5py here: http://www.h5py.org/

The following is an example that creates a 2-dimensional array, writes that array to an HDF5 file, then reads that array from the file and prints it:

import numpy as np
import h5py

def test(filename, datasetname):
    data1 = np.array([[1, 2, 3], [4, 5, 6]], dtype=np.float_)
    print 'data1:', data1
    hdf5file = h5py.File(filename)
    dataset01 = hdf5file.create_dataset(
        datasetname, data1.shape, dtype=np.float_)
    dataset01[...] = data1
    print 'Dataset names:', list(hdf5file)
    data2 = np.array(hdf5file[datasetname])
    print 'data2:', data2

test('tmp01.hdf5', 'dataset01')

When we run the above code we see:

$ python test.py
data1: [[ 1.  2.  3.]
 [ 4.  5.  6.]]
Dataset names: [u'dataset01']
data2: [[ 1.  2.  3.]
 [ 4.  5.  6.]]

The following is an example that (1) creates several datasets, (2) stores those datasets in an HDF5 file in a structured way (a tree of datasets), and later (3) retrieves those datasets both (a) by walking the entire hdf5 file and (b) by path name:

# Create several datasets/arrays.
# Store the datasets; put some in a sub-group.
# Recursively walk the hdf5 file and retrieve each dataset.
# Retrieve a dataset using a path name to it.

import numpy as np
import h5py

def create_datasets(filename):
    data1 = np.arange(10, 19, 2, dtype=np.float_)
    data2 = np.arange(20, 29, 2, dtype=np.float_)
    data3 = np.arange(30, 39, 2, dtype=np.float_)
    data4 = np.arange(40, 49, 2, dtype=np.float_)
    print 'data1:', data1
    print 'data2:', data2
    print 'data3:', data3
    print 'data4:', data4
    hdf5file = h5py.File(filename)
    group01 = hdf5file.create_group('group01')
    dataset01 = hdf5file.create_dataset(
        'dataset01', data1.shape, dtype=np.float_)
    dataset01[...] = data1
    dataset02 = group01.create_dataset(
        'dataset02', data2.shape, dtype=np.float_)
    dataset02[...] = data2
    dataset03 = group01.create_dataset(
        'dataset03', data2.shape, dtype=np.float_)
    dataset03[...] = data3
    dataset04 = group01.create_dataset(
        'dataset04', data2.shape, dtype=np.float_)
    dataset04[...] = data4
    return hdf5file

def show_group(group):
    print 'group name:', group.name
    print '-' * 40
    for obj in group.values():
        if isinstance(obj, h5py.Group):
            show_group(obj)
        elif isinstance(obj, h5py.Dataset):
            data = np.array(obj)
            print 'dataset name:', obj.name
            print 'dataset data:', data
            print '-' * 20
        else:
            pass

def show_datasets(hdf5file):
    show_group(hdf5file)

def show_dataset_by_name(hdf5file, name):
    print 'dataset by name --'
    obj = hdf5file.get(name)
    data = np.array(obj)
    print 'dataset name:', obj.name
    print 'dataset data:', data

def test(filename):
    hdf5file = create_datasets(filename)
    print '=' * 60
    show_datasets(hdf5file)
    print '=' * 60
    show_dataset_by_name(hdf5file, '/group01/dataset03')

test('tmp01.hdf5')

When we run the above code, we'll see:

$ python test.py
data1: [ 10.  12.  14.  16.  18.]
data2: [ 20.  22.  24.  26.  28.]
data3: [ 30.  32.  34.  36.  38.]
data4: [ 40.  42.  44.  46.  48.]
============================================================
group name: /
----------------------------------------
dataset name: /dataset01
dataset data: [ 10.  12.  14.  16.  18.]
--------------------
group name: /group01
----------------------------------------
dataset name: /group01/dataset02
dataset data: [ 20.  22.  24.  26.  28.]
--------------------
dataset name: /group01/dataset03
dataset data: [ 30.  32.  34.  36.  38.]
--------------------
dataset name: /group01/dataset04
dataset data: [ 40.  42.  44.  46.  48.]
--------------------
============================================================
dataset by name --
dataset name: /group01/dataset03
dataset data: [ 30.  32.  34.  36.  38.]

Notes:

• The hdf5 file itself is a group. It's the root group and its name is "/".

5.4 PyTables and HDF5

PyTables writes and reads HDF5 files. It supports the ability to save and retrieve SciPy arrays into HDF5 files. Multiple arrays and separate datasets can be organized in nested groups (analogous to folders or directories).

You can learn more about PyTables at PyTables -- Hierarchical Datasets in Python.

$ tar xvzf orig/pytables-1.3.2.tar.gz
$ cd pytables-1.3.2/
$ python setup.py build_ext --inplace
$ sudo python setup.py install

numarray_array = numarray.asarray(scipy_array)

scipy_array = scipy.asarray(numarray_array)

In [17]:data1 = s.array([[1.0,2.0],[3.0,4.0],[5.0,6.0]])
In [18]:list1 = data1.to
In [18]:list1 = data1.tolist()
In [19]:print list1
[[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]]

import scipy
import numarray

def test():
    scipyArray = scipy.array([[1.0,2.0],[3.0,4.0],[5.0,6.0]])
    list1 = scipyArray.tolist()
    print 'list1:', list1
    numarrayArray = numarray.array([[1.0,2.0],[3.0,4.0],[5.0,6.0]])
    print 'numarrayArray:\n', numarrayArray
    scipyArray2 = scipy.array(numarrayArray)
    print 'type(scipyArray2):', type(scipyArray2)
    print 'scipyArray2:\n', scipyArray2

test()

list1: [[1.0, 2.0], [3.0, 4.0], [5.0, 6.0]]
numarrayArray:
[[ 1.  2.]
 [ 3.  4.]
 [ 5.  6.]]
type(scipyArray2): <type 'scipy.ndarray'>
scipyArray2:
[[ 1.  2.]
 [ 3.  4.]
 [ 5.  6.]]

#!/usr/bin/env python

import sys
import getopt

import scipy
import tables

Filename = 'testpytables2.h5'
Dataset1 = [[1, 2, 3, 4],[5, 6, 7, 8],[9, 10, 11, 12]]
Dataset2 = [[1.,2., 2.1],[3.,4.,4.1],[5.,6.,6.1]]

def test1():
    """Write out several sample datasets.
    """
    filename = Filename
    print "Creating file:", filename
    #filter = tables.Filters()
    h5file = tables.openFile(filename,
        mode = "w", title = "PyTables test file",
    #    filters=filter
        )
    print '=' * 30
    print h5file
    print '=' * 30
    root = h5file.createGroup(h5file.root, "Datasets", "Test datasets")
    datasets = h5file.createGroup(root, "Phase1", "Test datasets")
    scipy_array = scipy.array(Dataset1)
    h5file.createArray(datasets, 'dataset1', scipy_array, "Test dataset #1")
    scipy_array = scipy.array(Dataset2)
    h5file.createArray(datasets, 'dataset2', scipy_array, "Test dataset #2")
    scipy_array = scipy.zeros((100,100))
    h5file.createArray(datasets, 'dataset3', scipy_array, "Test dataset #3")
    h5file.close()

#
# Read in and display the datasets.
#
def test2():
    filename = Filename
    h5file = tables.openFile(filename, 'r')
    dataset1Obj = h5file.getNode('/Datasets/Phase1', 'dataset1')
    dataset2Obj = h5file.getNode('/Datasets/Phase1', 'dataset2')
    print repr(dataset1Obj)
    print repr(dataset2Obj)
    dataset1Array = dataset1Obj.read()
    dataset2Array = dataset2Obj.read()
    print 'type(dataset1Array):', type(dataset1Array)
    print 'type(dataset2Array):', type(dataset2Array)
    print 'array1:\n', dataset1Array
    print 'array2:\n', dataset2Array
    # print several slices of our array.
    print 'slice [0]:', dataset1Array[0]
    print 'slice [0:2]:',dataset1Array[0:2]
    print 'slice [1, 0:4:2]:',dataset1Array[1, 0:4:2]
    h5file.close()

USAGE_TEXT = """
Usage:
    python test_pytables1.py [options]
Options:
    -h, --help      Display this help message.
    -t n, --test=n  Test number:
                        1: Write file
                        2: Read file
Example:
    python test_pytables1.py -t 1
    python test_pytables1.py -t 2
"""

def usage():
    print USAGE_TEXT
    sys.exit(-1)

def main():
    args = sys.argv[1:]
    try:
        opts, args = getopt.getopt(args, 'ht:', ['help', 'test=', ])
    except:
        usage()
    testno = 0
    for opt, val in opts:
        if opt in ('-h', '--help'):
            usage()
        elif opt in ('-t', '--test'):
            testno = int(val)
    if len(args) != 0:
        usage()
    if testno == 1:
        test1()
    elif testno == 2:
        test2()
    else:
        usage()

if __name__ == '__main__':
    main()

$ python test_pytables2.py -t 1
$ python test_pytables2.py -t 2

6.1 Configuration

For information on using matplotlib in ipython, do:

$ ipython --help

For more information on configuring and running ipython with matplotlib/pylab, see:

• http://matplotlib.org/users/installing.html

• http://matplotlib.org/users/customizing.html

To find out where Python is loading matplotlibrc from, do the following:

In [1]: import matplotlib
In [2]: matplotlib.matplotlib_fname()

6.2 Interactive use

Consider using:

$ ipython --pylab

Here are some convenient and useful commands while using matplotlib interactively:

• clf() -- Clear the current figure.

• cla() -- Clear the current axes.

• draw() -- Redraw the current figure.

• close() -- Close the current figure. close(num) -- Close figure number num. close(h) -- Close figure whose handle is h. close('all') -- Close all figure windows.

Learn the following in order to create and control your plots:

• figure(num) -- Create or activate figure number num.

• subplot() -- Create a sub-plot within the current figure.

• plot()

Learn the following in order to annotate your plots:

• xlabel(s) -- Add a label s to the x axis.

• ylabel(s) -- Add a label s to the y axis.

• title(s) -- Add a title s to the axes.

• text(x, y, s) Add text s to the axes at x, y in data coords.

• figtext(x, y, s) -- Add text to the figure at x, y in relative 0-1 figure coords.

Note that these annotation functions return a matplotlib Text object. You can use this object to get and set properties of the text. Example:

91: cla()
92: plot([1,2,3])
93: t = xlabel('increasing temp')
94: t.set_weight('bold')
95: t.set_color('b')
96: draw()

Notes:

• The call to draw() may be needed in interactive mode (e.g. when in IPython) in order to update or refresh the drawing window.

• Use dir() and help() (or "?" in IPython) to learn what methods are supported by the matplotlib Text object (or any other objects), what they do, what parameters they take, etc. Also, see the matplotlib user guide.

6.3 A simple plot

#!/usr/bin/env python
"""
Synopsis:
    Display a simple plot using pylab.
Usage:
    python simple_plot.py <func_name>
Examples:
    python simple_plot.py sin
    python simple_plot.py cos
    python simple_plot.py tan
"""

import sys
import pylab as pl

def simple(funcName):
    """Create a simple plot and save the plot to a .png file
    @param funcName: The name of a function, e.g. sin, cos, tan, ...
    """
    t = pl.arange(0.0, 1.0+0.01, 0.01)
    funcStr = 'pl.%s(2*2*pl.pi*t)' % (funcName,)
    s = eval(funcStr)
    pl.plot(t, s)
    pl.xlabel('time (s)')
    pl.ylabel('voltage (mV)')
    pl.title('About as simple as it gets, folks')
    pl.grid(True)
    pl.savefig('simple_plot')
    pl.show()

def usage():
    sys.exit(__doc__)

def main():
    args = sys.argv[1:]
    if len(args) != 1:
        usage()
    simple(args[0])

if __name__ == '__main__':
    main()

Notes:

• pl.plot() creates a plot to be displayed.

• pl.xlabel(), pl.ylabel(), and pl.title() add annotations to our plot.

• pl.grid(True) adds a grid.

• pl.savefig() saves the figure. The file name extension determines the type of file generated. On my machine, .png, .svg, and .ps produce Portable Network Graphics (PNG), Scalable Vector Graphics (SVG), and PostScript files, respectively.

• pl.show() shows (makes visible) all the plots we have created. This is typically the last line of your script. Note that in the shell (for example, ipython) when interactive mode is set in your matplotlibrc, show() is not needed.

6.4 Embedding matplotlib in a GUI application

You can display your matplotlib plots inside a GUI application written in Tk, WxPython, ...

There are examples in the examples directory of the matplotlib source distribution. These examples are also available at the matplotlib Web site. Go to matplotlib, then click on "Examples (zip)".

The example files for embedding are named:

embedding_in_gtk.py    embedding_in_tk.py   embedding_in_wx3.py
embedding_in_gtk2.py   embedding_in_tk2.py  embedding_in_wx4.py
embedding_in_gtk3.py   embedding_in_wx.py
embedding_in_qt.py     embedding_in_wx2.py

In general, you can create your plot, possibly testing it interactively in IPython, then use one of the examples for embedding the plot in the GUI tool of your choice.

8.1 scipy (top-level)

SciPy: A scientific computing package for Python

Available subpackages:

• __core_config__

• __scipy_config__

• __svn_version__

• _import_tools

• core_version

• fft, fft2, fftn -- discrete Fourier transform. Also see fftshift, fftfreq.

• fftpack (package)

• integrate (package)

• interpolate (package)

• linalg (package)

• old__init__

• optimize (package)

• scipy_version

• sparse (package)

• special (package)

• stats (package)

8.2 base

scipy provides functions for defining a multi-dimensional array and useful procedures for Numerical computation. Use the following to get a list of the members of scipy:

>>> import scipy
>>> dir(scipy)

Functions:

• array - NumPy Array construction

• zeros - Return an array of all zeros

• empty - Return an unitialized array

• shape - Return shape of sequence or array

• rank - Return number of dimensions

• size - Return number of elements in entire array or a certain dimension

• fromstring - Construct array from (byte) string

• take - Select sub-arrays using sequence of indices

• put - Set sub-arrays using sequence of 1-D indices

• putmask - Set portion of arrays using a mask

• reshape - Return array with new shape

• repeat - Repeat elements of array

• choose - Construct new array from indexed array tuple

• cross_correlate - Correlate two 1-d arrays

• searchsorted - Search for element in 1-d array

• sum - Total sum over a specified dimension

• average - Average, possibly weighted, over axis or array.

• cumsum - Cumulative sum over a specified dimension

• product - Total product over a specified dimension

• cumproduct - Cumulative product over a specified dimension

• alltrue - Logical and over an entire axis

• sometrue - Logical or over an entire axis

• allclose - Tests if sequences are essentially equal

More Functions:

• arrayrange (arange) - Return regularly spaced array

• asarray - Guarantee NumPy array

• sarray - Guarantee a NumPy array that keeps precision

• convolve - Convolve two 1-d arrays

• swapaxes - Exchange axes

• concatenate - Join arrays together

• transpose - Permute axes

• sort - Sort elements of array

• argsort - Indices of sorted array

• argmax - Index of largest value

• argmin - Index of smallest value

• innerproduct - Innerproduct of two arrays

• dot - Dot product (matrix multiplication)

• outerproduct - Outerproduct of two arrays

• resize - Return array with arbitrary new shape

• indices - Tuple of indices

• fromfunction - Construct array from universal function

• diagonal - Return diagonal array

• trace - Trace of array

• dump - Dump array to file object (pickle)

• dumps - Return pickled string representing data

• load - Return array stored in file object

• loads - Return array from pickled string

• ravel - Return array as 1-D

• nonzero - Indices of nonzero elements for 1-D array

• shape - Shape of array

• where - Construct array from binary result

• compress - Elements of array where condition is true

• clip - Clip array between two values

• ones - Array of all ones

• identity - 2-D identity array (matrix)

(Universal) Math Functions -- A universal function (or ufunc for short) is a function that operates on ndarrays in an element-by-element fashion. Note that you can create custom universal functions with np.vectorize. For more information see: http://docs.scipy.org/doc/numpy/reference/ufuncs.html. For example:

In [75]: sp.absolute(np.array([-1, 2, -3, 4, -5, 6, -7]))
Out[75]: array([1, 2, 3, 4, 5, 6, 7])

• absolute

• add

• arccos

• arccosh

• arcsin

• arcsinh

• arctan

• arctan2

• arctanh

• around

• bitwise_and

• bitwise_or

• bitwise_xor

• ceil

• conjugate

• cos

• cosh

• divide

• divide_safe

• equal

• exp

• fabs

• floor

• fmod

• greater

• greater_equal

• hypot

• invert

• left_shift

• less

• less_equal

• log

• log10

• logical_and

• logical_not

• logical_or

• logical_xor

• maximum

• minimum

• multiply

• negative

• not_equal

• power

• right_shift

• sign

• sin

• sinh

• sqrt

• subtract

• tan

• tanh

Basic functions used by several sub-packages and useful to have in the main name-space

Type handling:

• iscomplexobj -- Test for complex object, scalar result

• isrealobj -- Test for real object, scalar result

• iscomplex -- Test for complex elements, array result

• isreal -- Test for real elements, array result

• imag -- Imaginary part

• real -- Real part

• real_if_close -- Turns complex number with tiny imaginary part to real

• isneginf -- Tests for negative infinity

• isposinf -- Tests for positive infinity

• isnan -- Tests for nans

• isinf -- Tests for infinity

• isfinite -- Tests for finite numbers

• isscalar -- True if argument is a scalar

• nan_to_num -- Replaces NaN's with 0 and infinities with large numbers

• cast -- Dictionary of functions to force cast to each type

• common_type -- Determine the 'minimum common type code' for a group of arrays

• mintypecode -- Return minimal allowed common typecode.

Index tricks:

• mgrid -- Method which allows easy construction of N-d 'mesh-grids'

• r_ -- Append and construct arrays: turns slice objects into ranges and concatenates them, for 2d arrays appends rows.

• index_exp -- Konrad Hinsen's index_expression class instance which can be useful for building complicated slicing syntax.

Useful functions:

• select -- Extension of where to multiple conditions and choices

• extract -- Extract 1d array from flattened array according to mask

• insert -- Insert 1d array of values into Nd array according to mask

• linspace -- Evenly spaced samples in linear space

• logspace -- Evenly spaced samples in logarithmic space

• fix -- Round x to nearest integer towards zero

• mod -- Modulo mod(x,y) = x % y except keeps sign of y

• amax -- Array maximum along axis

• amin -- Array minimum along axis

• ptp -- Array max-min along axis

• cumsum -- Cumulative sum along axis

• prod -- Product of elements along axis

• cumprod -- Cumluative product along axis

• diff -- Discrete differences along axis

• angle -- Returns angle of complex argument

• unwrap -- Unwrap phase along given axis (1-d algorithm)

• sort_complex -- Sort a complex-array (based on real, then imaginary)

• trim_zeros -- trim the leading and trailing zeros from 1D array.

• vectorize -- a class that wraps a Python function taking scalar arguments into a generalized function which can handle arrays of arguments using the broadcast rules of numerix Python.

• alter_numeric -- enhance numeric array behavior

• restore_numeric -- restore alterations done by alter_numeric

Shape manipulation:

• squeeze -- Return a with length-one dimensions removed.

• atleast_1d -- Force arrays to be > 1D

• atleast_2d -- Force arrays to be > 2D

• atleast_3d -- Force arrays to be > 3D

• vstack -- Stack arrays vertically (row on row)

• hstack -- Stack arrays horizontally (column on column)

• column_stack -- Stack 1D arrays as columns into 2D array

• dstack -- Stack arrays depthwise (along third dimension)

• split -- Divide array into a list of sub-arrays

• hsplit -- Split into columns

• vsplit -- Split into rows

• dsplit -- Split along third dimension

Matrix (2d array) manipluations:

• fliplr -- 2D array with columns flipped

• flipud -- 2D array with rows flipped

• rot90 -- Rotate a 2D array a multiple of 90 degrees

• eye -- Return a 2D array with ones down a given diagonal

• diag -- Construct a 2D array from a vector, or return a given diagonal from a 2D array.

• mat -- Construct a Matrix

• bmat -- Build a Matrix from blocks

For information on the differences between arrays and matrices, see the section titled "'array' or 'matrix'? Which should I use?" at: http://wiki.scipy.org/NumPy_for_Matlab_Users

Polynomials:

• poly1d -- A one-dimensional polynomial class

• poly -- Return polynomial coefficients from roots

• roots -- Find roots of polynomial given coefficients

• polyint -- Integrate polynomial

• polyder -- Differentiate polynomial

• polyadd -- Add polynomials

• polysub -- Substract polynomials

• polymul -- Multiply polynomials

• polydiv -- Divide polynomials

• polyval -- Evaluate polynomial at given argument

Import tricks:

• ppimport -- Postpone module import until trying to use it

• ppimport_attr -- Postpone module import until trying to use its attribute

• ppresolve -- Import postponed module and return it.

Machine arithmetics:

• machar_single -- MachAr instance storing the parameters of system single precision floating point arithmetics

• machar_double -- MachAr instance storing the parameters of system double precision floating point arithmetics

8.3 Constants

Physical and mathematical constants and units.

Access via:

In [1]: from scipy import constants

Mathematical constants:

• pi -- Pi

• golden -- Golden ratio

Physical constants:

• c -- speed of light in vacuum

• mu_0 -- the magnetic constant mu_0

• epsilon_0 -- the electric constant (vacuum permittivity), epsilon_0

• h -- the Planck constant h

• hbar -- hbar = h/(2pi)

• G -- Newtonian constant of gravitation

• g -- standard acceleration of gravity

• e -- elementary charge

• R -- molar gas constant

• alpha -- fine-structure constant

• N_A -- Avogadro constant

• k -- Boltzmann constant

• sigma -- Stefan-Boltzmann constant sigma

• Wien -- Wien displacement law constant

• Rydberg -- Rydberg constant

• m_e -- electron mass

• m_p -- proton mass

• m_n -- neutron mass

Constants database -- In addition to the above variables, scipy.constants also contains the 2010 CODATA recommended values database containing more physical constants: http://physics.nist.gov/cuu/Constants/index.html

• value(key) -- Value in physical_constants indexed by key

• unit(key) -- Unit in physical_constants indexed by key

• precision(key) -- Relative precision in physical_constants indexed by key

• find([sub, disp]) -- Return list of codata.physical_constant keys containing a given string.

• ConstantWarning -- Accessing a constant no longer in current CODATA data set

scipy.constants.physical_constants -- Dictionary of physical constants, of the format physical_constants[name] = (value, unit, uncertainty)

Available constants:

• alpha particle mass -- 6.64465675e-27 kg

• alpha particle mass energy equivalent -- 5.97191967e-10 J

• alpha particle mass energy equivalent in MeV -- 3727.37924 MeV

• alpha particle mass in u -- 4.00150617913 u

• alpha particle molar mass -- 0.00400150617912 kg mol^-1

• alpha particle-electron mass ratio -- 7294.2995361

• alpha particle-proton mass ratio -- 3.97259968933

• Angstrom star -- 1.00001495e-10 m

• atomic mass constant -- 1.660538921e-27 kg

• atomic mass constant energy equivalent -- 1.492417954e-10 J

• atomic mass constant energy equivalent in MeV -- 931.494061 MeV

• atomic mass unit-electron volt relationship -- 931494061.0 eV

• atomic mass unit-hartree relationship -- 34231776.845 E_h

• atomic mass unit-hertz relationship -- 2.2523427168e+23 Hz

• atomic mass unit-inverse meter relationship -- 7.5130066042e+14 m^-1

• atomic mass unit-joule relationship -- 1.492417954e-10 J

• atomic mass unit-kelvin r -- 1.08095408e+13 K

• atomic mass unit-kilogram relationship -- 1.660538921e-27 kg

• atomic unit of 1st hyperpolarizability -- 3.206361449e-53 C^3 m^3 J^-2

• atomic unit of 2nd hyperpolarizability -- 6.23538054e-65 C^4 m^4 J^-3

• atomic unit of action -- 1.054571726e-34 J s

• atomic unit of charge -- 1.602176565e-19 C

• atomic unit of charge density -- 1.081202338e+12 C m^-3

• atomic unit of current -- 0.00662361795 A

• atomic unit of electric dipole mom. -- 8.47835326e-30 C m

• atomic unit of electric field -- 5.14220652e+11 V m^-1

• atomic unit of electric field gradient -- 9.717362e+21 V m^-2

• atomic unit of electric polarizability -- 1.6487772754e-41 C^2 m^2 J^-1

• atomic unit of electric potential -- 27.21138505 V

• atomic unit of electric quadrupole mom. -- 4.486551331e-40 C m^2

• atomic unit of energy -- 4.35974434e-18 J

• atomic unit of force -- 8.23872278e-08 N

• atomic unit of length -- 5.2917721092e-11 m

• atomic unit of mag. dipole mom. -- 1.854801936e-23 J T^-1

• atomic unit of mag. flux density -- 235051.7464 T

• atomic unit of magnetizability -- 7.891036607e-29 J T^-2

• atomic unit of mass -- 9.10938291e-31 kg

• atomic unit of mom.um -- 1.99285174e-24 kg m s^-1

• atomic unit of permittivity -- 1.11265005605e-10 F m^-1

• atomic unit of time -- 2.4188843265e-17 s

• atomic unit of velocity -- 2187691.26379 m s^-1

• Avogadro constant -- 6.02214129e+23 mol^-1

• Bohr magneton -- 9.27400968e-24 J T^-1

• Bohr magneton in eV/T -- 5.7883818066e-05 eV T^-1

• Bohr magneton in Hz/T -- 13996245550.0 Hz T^-1

• Bohr magneton in inverse meters per tesla -- 46.6864498 m^-1 T^-1

• Bohr magneton in K/T -- 0.67171388 K T^-1

• Bohr radius -- 5.2917721092e-11 m

• Boltzmann constant -- 1.3806488e-23 J K^-1

• Boltzmann constant in eV/K -- 8.6173324e-05 eV K^-1

• Boltzmann constant in Hz/K -- 20836618000.0 Hz K^-1

• Boltzmann constant in inverse meters per kelvin -- 69.503476 m^-1 K^-1

• characteristic impedance of vacuum -- 376.730313462 ohm

• classical electron radius -- 2.8179403267e-15 m

• Compton wavelength -- 2.4263102389e-12 m

• Compton wavelength over 2 pi -- 3.86159268e-13 m

• conductance quantum -- 7.7480917346e-05 S

• conventional value of Josephson constant -- 4.835979e+14 Hz V^-1

• conventional value of von Klitzing constant -- 25812.807 ohm

• Cu x unit -- 1.00207697e-13 m

• deuteron g factor -- 0.8574382308

• deuteron mag. mom. -- 4.33073489e-27 J T^-1

• deuteron mag. mom. to Bohr magneton ratio -- 0.0004669754556

• deuteron mag. mom. to nuclear magneton ratio -- 0.8574382308

• deuteron mass -- 3.34358348e-27 kg

• deuteron mass energy equivalent -- 3.00506297e-10 J

• deuteron mass energy equivalent in MeV -- 1875.612859 MeV

• deuteron mass in u -- 2.01355321271 u

• deuteron molar mass -- 0.00201355321271 kg mol^-1

• deuteron rms charge radius -- 2.1424e-15 m

• deuteron-electron mag. mom. ratio -- -0.0004664345537

• deuteron-electron mass ratio -- 3670.4829652

• deuteron-neutron mag. mom. ratio -- -0.44820652

• deuteron-proton mag. mom. ratio -- 0.307012207

• deuteron-proton mass ratio -- 1.99900750097

• electric constant -- 8.85418781762e-12 F m^-1

• electron charge to mass quotient -- -1.758820088e+11 C kg^-1

• electron g factor -- -2.00231930436

• electron gyromag. ratio -- 1.760859708e+11 s^-1 T^-1

• electron gyromag. ratio over 2 pi -- 28024.95266 MHz T^-1

• electron mag. mom. -- -9.2847643e-24 J T^-1

• electron mag. mom. anomaly -- 0.00115965218076

• electron mag. mom. to Bohr magneton ratio -- -1.00115965218

• electron mag. mom. to nuclear magneton ratio -- -1838.2819709

• electron mass -- 9.10938291e-31 kg

• electron mass energy equivalent -- 8.18710506e-14 J

• electron mass energy equivalent in MeV -- 0.510998928 MeV

• electron mass in u -- 0.00054857990946 u

• electron molar mass -- 5.4857990946e-07 kg mol^-1

• electron to alpha particle mass ratio -- 0.000137093355578

• electron to shielded helion mag. mom. ratio -- 864.058257

• electron to shielded proton mag. mom. ratio -- -658.2275971

• electron volt -- 1.602176565e-19 J

• electron volt-atomic mass unit relationship -- 1.07354415e-09 u

• electron volt-hartree relationship -- 0.03674932379 E_h

• electron volt-hertz relationship -- 2.417989348e+14 Hz

• electron volt-inverse meter relationship -- 806554.429 m^-1

• electron volt-joule relationship -- 1.602176565e-19 J

• electron volt-kelvin relationship -- 11604.519 K

• electron volt-kilogram relationship -- 1.782661845e-36 kg

• electron-deuteron mag. mom. ratio -- -2143.923498

• electron-deuteron mass ratio -- 0.00027244371095

• electron-helion mass ratio -- 0.00018195430761

• electron-muon mag. mom. ratio -- 206.7669896

• electron-muon mass ratio -- 0.00483633166

• electron-neutron mag. mom. ratio -- 960.9205

• electron-neutron mass ratio -- 0.00054386734461

• electron-proton mag. mom. ratio -- -658.2106848

• electron-proton mass ratio -- 0.00054461702178

• electron-tau mass ratio -- 0.000287592

• electron-triton mass ratio -- 0.00018192000653

• elementary charge -- 1.602176565e-19 C

• elementary charge over h -- 2.417989348e+14 A J^-1

• Faraday constant -- 96485.3365 C mol^-1

• Faraday constant for conventional electric current -- 96485.3321 C_90 mol^-1

• Fermi coupling constant -- 1.166364e-05 GeV^-2

• fine-structure constant -- 0.0072973525698

• first radiation constant -- 3.74177153e-16 W m^2

• first radiation constant for spectral radiance -- 1.191042869e-16 W m^2 sr^-1

• Hartree energy -- 4.35974434e-18 J

• Hartree energy in eV -- 27.21138505 eV

• hartree-atomic mass unit relationship -- 2.9212623246e-08 u

• hartree-electron volt relationship -- 27.21138505 eV

• hartree-hertz relationship -- 6.57968392073e+15 Hz

• hartree-inverse meter relationship -- 21947463.1371 m^-1

• hartree-joule relationship -- 4.35974434e-18 J

• hartree-kelvin relationship -- 315775.04 K

• hartree-kilogram relationship -- 4.85086979e-35 kg

• helion g factor -- -4.255250613

• helion mag. mom. -- -1.074617486e-26 J T^-1

• helion mag. mom. to Bohr magneton ratio -- -0.001158740958

• helion mag. mom. to nuclear magneton ratio -- -2.127625306

• helion mass -- 5.00641234e-27 kg

• helion mass energy equivalent -- 4.49953902e-10 J

• helion mass energy equivalent in MeV -- 2808.391482 MeV

• helion mass in u -- 3.0149322468 u

• helion molar mass -- 0.0030149322468 kg mol^-1

• helion-electron mass ratio -- 5495.8852754

• helion-proton mass ratio -- 2.9931526707

• hertz-atomic mass unit relationship -- 4.4398216689e-24 u

• hertz-electron volt relationship -- 4.135667516e-15 eV

• hertz-hartree relationship -- 1.519829846e-16 E_h

• hertz-inverse meter relationship -- 3.33564095198e-09 m^-1

• hertz-joule relationship -- 6.62606957e-34 J

• hertz-kelvin relationship -- 4.7992434e-11 K

• hertz-kilogram relationship -- 7.37249668e-51 kg

• inverse fine-structure constant -- 137.035999074

• inverse meter-atomic mass unit relationship -- 1.3310250512e-15 u

• inverse meter-electron volt relationship -- 1.23984193e-06 eV

• inverse meter-hartree relationship -- 4.55633525276e-08 E_h

• inverse meter-hertz relationship -- 299792458.0 Hz

• inverse meter-joule relationship -- 1.986445684e-25 J

• inverse meter-kelvin relationship -- 0.01438777 K

• inverse meter-kilogram relationship -- 2.210218902e-42 kg

• inverse of conductance quantum -- 12906.4037217 ohm

• Josephson constant -- 4.8359787e+14 Hz V^-1

• joule-atomic mass unit relationship -- 6700535850.0 u

• joule-electron volt relationship -- 6.24150934e+18 eV

• joule-hartree relationship -- 2.29371248e+17 E_h

• joule-hertz relationship -- 1.509190311e+33 Hz

• joule-inverse meter relationship -- 5.03411701e+24 m^-1

• joule-kelvin relationship -- 7.2429716e+22 K

• joule-kilogram relationship -- 1.11265005605e-17 kg

• kelvin-atomic mass unit relationship -- 9.2510868e-14 u

• kelvin-electron volt relationship -- 8.6173324e-05 eV

• kelvin-hartree relationship -- 3.1668114e-06 E_h

• kelvin-hertz relationship -- 20836618000.0 Hz

• kelvin-inverse meter relationship -- 69.503476 m^-1

• kelvin-joule relationship -- 1.3806488e-23 J

• kelvin-kilogram relationship -- 1.536179e-40 kg

• kilogram-atomic mass unit relationship -- 6.02214129e+26 u

• kilogram-electron volt relationship -- 5.60958885e+35 eV

• kilogram-hartree relationship -- 2.061485968e+34 E_h

• kilogram-hertz relationship -- 1.356392608e+50 Hz

• kilogram-inverse meter relationship -- 4.52443873e+41 m^-1

• kilogram-joule relationship -- 8.98755178737e+16 J

• kilogram-kelvin relationship -- 6.5096582e+39 K

• lattice parameter of silicon -- 5.431020504e-10 m

• Loschmidt constant (273.15 K, 100 kPa) -- 2.6516462e+25 m^-3

• Loschmidt constant (273.15 K, 101.325 kPa) -- 2.6867805e+25 m^-3

• mag. constant -- 1.25663706144e-06 N A^-2

• mag. flux quantum -- 2.067833758e-15 Wb

• Mo x unit -- 1.00209952e-13 m

• molar gas constant -- 8.3144621 J mol^-1 K^-1

• molar mass constant -- 0.001 kg mol^-1

• molar mass of carbon-12 -- 0.012 kg mol^-1

• molar Planck constant -- 3.9903127176e-10 J s mol^-1

• molar Planck constant times c -- 0.119626565779 J m mol^-1

• molar volume of ideal gas (273.15 K, 100 kPa) -- 0.022710953 m^3 mol^-1

• molar volume of ideal gas (273.15 K, 101.325 kPa) -- 0.022413968 m^3 mol^-1

• molar volume of silicon -- 1.205883301e-05 m^3 mol^-1

• muon Compton wavelength -- 1.173444103e-14 m

• muon Compton wavelength over 2 pi -- 1.867594294e-15 m

• muon g factor -- -2.0023318418

• muon mag. mom. -- -4.49044807e-26 J T^-1

• muon mag. mom. anomaly -- 0.00116592091

• muon mag. mom. to Bohr magneton ratio -- -0.00484197044

• muon mag. mom. to nuclear magneton ratio -- -8.89059697

• muon mass -- 1.883531475e-28 kg

• muon mass energy equivalent -- 1.692833667e-11 J

• muon mass energy equivalent in MeV -- 105.6583715 MeV

• muon mass in u -- 0.1134289267 u

• muon molar mass -- 0.0001134289267 kg mol^-1

• muon-electron mass ratio -- 206.7682843

• muon-neutron mass ratio -- 0.1124545177

• muon-proton mag. mom. ratio -- -3.183345107

• muon-proton mass ratio -- 0.1126095272

• muon-tau mass ratio -- 0.0594649

• natural unit of action -- 1.054571726e-34 J s

• natural unit of action in eV s -- 6.58211928e-16 eV s

• natural unit of energy -- 8.18710506e-14 J

• natural unit of energy in MeV -- 0.510998928 MeV

• natural unit of length -- 3.86159268e-13 m

• natural unit of mass -- 9.10938291e-31 kg

• natural unit of mom.um -- 2.73092429e-22 kg m s^-1

• natural unit of mom.um in MeV/c -- 0.510998928 MeV/c

• natural unit of time -- 1.28808866833e-21 s

• natural unit of velocity -- 299792458.0 m s^-1

• neutron Compton wavelength -- 1.3195909068e-15 m

• neutron Compton wavelength over 2 pi -- 2.1001941568e-16 m

• neutron g factor -- -3.82608545

• neutron gyromag. ratio -- 183247179.0 s^-1 T^-1

• neutron gyromag. ratio over 2 pi -- 29.1646943 MHz T^-1

• neutron mag. mom. -- -9.6623647e-27 J T^-1

• neutron mag. mom. to Bohr magneton ratio -- -0.00104187563

• neutron mag. mom. to nuclear magneton ratio -- -1.91304272

• neutron mass -- 1.674927351e-27 kg

• neutron mass energy equivalent -- 1.505349631e-10 J

• neutron mass energy equivalent in MeV -- 939.565379 MeV

• neutron mass in u -- 1.008664916 u

• neutron molar mass -- 0.001008664916 kg mol^-1

• neutron to shielded proton mag. mom. ratio -- -0.68499694

• neutron-electron mag. mom. ratio -- 0.00104066882

• neutron-electron mass ratio -- 1838.6836605

• neutron-muon mass ratio -- 8.892484

• neutron-proton mag. mom. ratio -- -0.68497934

• neutron-proton mass difference -- 2.30557392e-30

• neutron-proton mass difference energy equivalent -- 2.0721465e-13

• neutron-proton mass difference energy equivalent in MeV -- 1.29333217

• neutron-proton mass difference in u -- 0.00138844919

• neutron-proton mass ratio -- 1.00137841917

• neutron-tau mass ratio -- 0.52879

• Newtonian constant of gravitation -- 6.67384e-11 m^3 kg^-1 s^-2

• Newtonian constant of gravitation over h-bar c -- 6.70837e-39 (GeV/c^2)^-2

• nuclear magneton -- 5.05078353e-27 J T^-1

• nuclear magneton in eV/T -- 3.1524512605e-08 eV T^-1

• nuclear magneton in inverse meters per tesla -- 0.02542623527 m^-1 T^-1

• nuclear magneton in K/T -- 0.00036582682 K T^-1

• nuclear magneton in MHz/T -- 7.62259357 MHz T^-1

• Planck constant -- 6.62606957e-34 J s

• Planck constant in eV s -- 4.135667516e-15 eV s

• Planck constant over 2 pi -- 1.054571726e-34 J s

• Planck constant over 2 pi in eV s -- 6.58211928e-16 eV s

• Planck constant over 2 pi times c in MeV fm -- 197.3269718 MeV fm

• Planck length -- 1.616199e-35 m

• Planck mass -- 2.17651e-08 kg

• Planck mass energy equivalent in GeV -- 1.220932e+19 GeV

• Planck temperature -- 1.416833e+32 K

• Planck time -- 5.39106e-44 s

• proton charge to mass quotient -- 95788335.8 C kg^-1

• proton Compton wavelength -- 1.32140985623e-15 m

• proton Compton wavelength over 2 pi -- 2.1030891047e-16 m

• proton g factor -- 5.585694713

• proton gyromag. ratio -- 267522200.5 s^-1 T^-1

• proton gyromag. ratio over 2 pi -- 42.5774806 MHz T^-1

• proton mag. mom. -- 1.410606743e-26 J T^-1

• proton mag. mom. to Bohr magneton ratio -- 0.00152103221

• proton mag. mom. to nuclear magneton ratio -- 2.792847356

• proton mag. shielding correction -- 2.5694e-05

• proton mass -- 1.672621777e-27 kg

• proton mass energy equivalent -- 1.503277484e-10 J

• proton mass energy equivalent in MeV -- 938.272046 MeV

• proton mass in u -- 1.00727646681 u

• proton molar mass -- 0.00100727646681 kg mol^-1

• proton rms charge radius -- 8.775e-16 m

• proton-electron mass ratio -- 1836.15267245

• proton-muon mass ratio -- 8.88024331

• proton-neutron mag. mom. ratio -- -1.45989806

• proton-neutron mass ratio -- 0.99862347826

• proton-tau mass ratio -- 0.528063

• quantum of circulation -- 0.0003636947552 m^2 s^-1

• quantum of circulation times 2 -- 0.0007273895104 m^2 s^-1

• Rydberg constant -- 10973731.5685 m^-1

• Rydberg constant times c in Hz -- 3.28984196036e+15 Hz

• Rydberg constant times hc in eV -- 13.60569253 eV

• Rydberg constant times hc in J -- 2.179872171e-18 J

• Sackur-Tetrode constant (1 K, 100 kPa) -- -1.1517078

• Sackur-Tetrode constant (1 K, 101.325 kPa) -- -1.1648708

• second radiation constant -- 0.01438777 m K

• shielded helion gyromag. ratio -- 203789465.9 s^-1 T^-1

• shielded helion gyromag. ratio over 2 pi -- 32.43410084 MHz T^-1

• shielded helion mag. mom. -- -1.074553044e-26 J T^-1

• shielded helion mag. mom. to Bohr magneton ratio -- -0.001158671471

• shielded helion mag. mom. to nuclear magneton ratio -- -2.127497718

• shielded helion to proton mag. mom. ratio -- -0.761766558

• shielded helion to shielded proton mag. mom. ratio -- -0.7617861313

• shielded proton gyromag. ratio -- 267515326.8 s^-1 T^-1

• shielded proton gyromag. ratio over 2 pi -- 42.5763866 MHz T^-1

• shielded proton mag. mom. -- 1.410570499e-26 J T^-1

• shielded proton mag. mom. to Bohr magneton ratio -- 0.001520993128

• shielded proton mag. mom. to nuclear magneton ratio -- 2.792775598

• speed of light in vacuum -- 299792458.0 m s^-1

• standard acceleration of gravity -- 9.80665 m s^-2

• standard atmosphere -- 101325.0 Pa

• standard-state pressure -- 100000.0 Pa

• Stefan-Boltzmann constant -- 5.670373e-08 W m^-2 K^-4

• tau Compton wavelength -- 6.97787e-16 m

• tau Compton wavelength over 2 pi -- 1.11056e-16 m

• tau mass -- 3.16747e-27 kg

• tau mass energy equivalent -- 2.84678e-10 J

• tau mass energy equivalent in MeV -- 1776.82 MeV

• tau mass in u -- 1.90749 u

• tau molar mass -- 0.00190749 kg mol^-1

• tau-electron mass ratio -- 3477.15

• tau-muon mass ratio -- 16.8167

• tau-neutron mass ratio -- 1.89111

• tau-proton mass ratio -- 1.89372

• Thomson cross section -- 6.652458734e-29 m^2

• triton g factor -- 5.957924896

• triton mag. mom. -- 1.504609447e-26 J T^-1

• triton mag. mom. to Bohr magneton ratio -- 0.001622393657

• triton mag. mom. to nuclear magneton ratio -- 2.978962448

• triton mass -- 5.0073563e-27 kg

• triton mass energy equivalent -- 4.50038741e-10 J

• triton mass energy equivalent in MeV -- 2808.921005 MeV

• triton mass in u -- 3.0155007134 u

• triton molar mass -- 0.0030155007134 kg mol^-1

• triton-electron mass ratio -- 5496.9215267

• triton-proton mass ratio -- 2.9937170308

• unified atomic mass unit -- 1.660538921e-27 kg

• von Klitzing constant -- 25812.8074434 ohm

• weak mixing angle -- 0.2223

• Wien frequency displacement law constant -- 58789254000.0 Hz K^-1

• Wien wavelength displacement law constant -- 0.0028977721 m K

• {220} lattice spacing of silicon -- 1.920155714e-10 m

Units:

• SI prefixes:

  • yotta -- 10^{24}

  • zetta -- 10^{21}

  • exa -- 10^{18}

  • peta -- 10^{15}

  • tera -- 10^{12}

  • giga -- 10^{9}

  • mega -- 10^{6}

  • kilo -- 10^{3}

  • hecto -- 10^{2}

  • deka -- 10^{1}

  • deci -- 10^{-1}

  • centi -- 10^{-2}

  • milli -- 10^{-3}

  • micro -- 10^{-6}

  • nano -- 10^{-9}

  • pico -- 10^{-12}

  • femto -- 10^{-15}

  • atto -- 10^{-18}

  • zepto -- 10^{-21}

• Binary prefixes:

  • kibi -- 2^{10}

  • mebi -- 2^{20}

  • gibi -- 2^{30}

  • tebi -- 2^{40}

  • pebi -- 2^{50}

  • exbi -- 2^{60}

  • zebi -- 2^{70}

  • yobi -- 2^{80}

• Weight:

  • gram -- 10^{-3} kg

  • metric_ton -- 10^{3} kg

  • grain -- one grain in kg

  • lb -- one pound (avoirdupous) in kg

  • oz -- one ounce in kg

  • stone -- one stone in kg

  • grain -- one grain in kg

  • long_ton -- one long ton in kg

  • short_ton -- one short ton in kg

  • troy_ounce -- one Troy ounce in kg

  • troy_pound -- one Troy pound in kg

  • carat -- one carat in kg

  • m_u -- atomic mass constant (in kg)

• Angle:

  • degree -- degree in radians

  • arcmin -- arc minute in radians

  • arcsec -- arc second in radians

• Time:

  • minute -- one minute in seconds

  • hour -- one hour in seconds

  • day -- one day in seconds

  • week -- one week in seconds

  • year -- one year (365 days) in seconds

  • Julian_year -- one Julian year (365.25 days) in seconds

• Length:

  • inch -- one inch in meters

  • foot -- one foot in meters

  • yard -- one yard in meters

  • mile -- one mile in meters

  • mil -- one mil in meters

  • pt -- one point in meters

  • survey_foot -- one survey foot in meters

  • survey_mile -- one survey mile in meters

  • nautical_mile -- one nautical mile in meters

  • fermi -- one Fermi in meters

  • angstrom -- one Angstrom in meters

  • micron -- one micron in meters

  • au -- one astronomical unit in meters

  • light_year -- one light year in meters

  • parsec -- one parsec in meters

• Pressure:

  • atm -- standard atmosphere in pascals

  • bar -- one bar in pascals

  • torr -- one torr (mmHg) in pascals

  • psi -- one psi in pascals

• Area:

  • hectare -- one hectare in square meters

  • acre -- one acre in square meters

• Volume:

  • liter -- one liter in cubic meters

  • gallon -- one gallon (US) in cubic meters

  • gallon_imp -- one gallon (UK) in cubic meters

  • fluid_ounce -- one fluid ounce (US) in cubic meters

  • fluid_ounce_imp -- one fluid ounce (UK) in cubic meters

  • bbl -- one barrel in cubic meters

• Speed:

  • kmh -- kilometers per hour in meters per second

  • mph -- miles per hour in meters per second

  • mach -- one Mach (approx., at 15 C, 1 atm) in meters per second

  • knot -- one knot in meters per second

• Temperature:

  • zero_Celsius -- zero of Celsius scale in Kelvin

  • degree_Fahrenheit -- one Fahrenheit (only differences) in Kelvins

  • C2K(C) -- Convert Celsius to Kelvin

  • K2C(K) -- Convert Kelvin to Celsius

  • F2C(F) -- Convert Fahrenheit to Celsius

  • C2F(C) -- Convert Celsius to Fahrenheit

  • F2K(F) -- Convert Fahrenheit to Kelvin

  • K2F(K) -- Convert Kelvin to Fahrenheit

• Energy:

  • eV -- one electron volt in Joules

  • calorie -- one calorie (thermochemical) in Joules

  • calorie_IT -- one calorie (International Steam Table calorie, 1956) in Joules

  • erg -- one erg in Joules

  • Btu -- one British thermal unit (International Steam Table) in Joules

  • Btu_th -- one British thermal unit (thermochemical) in Joules

  • ton_TNT -- one ton of TNT in Joules

• Power:

  • hp -- one horsepower in watts

• Force:

  • dyn -- one dyne in newtons

  • lbf -- one pound force in newtons

  • kgf -- one kilogram force in newtons

• Optics:

  • lambda2nu(lambda\_) -- Convert wavelength to optical frequency

  • nu2lambda(nu) -- Convert optical frequency to wavelength.

References:

• CODATA Recommended Values of the Fundamental Physical Constants 2010. http://physics.nist.gov/cuu/Constants/index.html

8.4 fftpack

Discrete Fourier transforms.

Access via:

In [1]: from scipy import fftpack

Also see: http://docs.scipy.org/doc/scipy/reference/tutorial/fftpack.html

Fast Fourier Transforms (FFTs):

• fft(x[, n, axis, overwrite_x]) -- Return discrete Fourier transform of real or complex sequence.

• ifft(x[, n, axis, overwrite_x]) -- Return discrete inverse Fourier transform of real or complex sequence.

• fft2(x[, shape, axes, overwrite_x]) -- 2-D discrete Fourier transform.

• ifft2(x[, shape, axes, overwrite_x]) -- 2-D discrete inverse Fourier transform of real or complex sequence.

• fftn(x[, shape, axes, overwrite_x]) -- Return multidimensional discrete Fourier transform.

• ifftn(x[, shape, axes, overwrite_x]) -- Return inverse multi-dimensional discrete Fourier transform of arbitrary type sequence x.

• rfft(x[, n, axis, overwrite_x]) -- Discrete Fourier transform of a real sequence.

• irfft(x[, n, axis, overwrite_x]) -- Return inverse discrete Fourier transform of real sequence x.

• dct(x[, type, n, axis, norm, overwrite_x]) -- Return the Discrete Cosine Transform of arbitrary type sequence x.

• idct(x[, type, n, axis, norm, overwrite_x]) -- Return the Inverse Discrete Cosine Transform of an arbitrary type sequence.

Differential and pseudo-differential operators:

• diff(x[, order, period, _cache]) -- Return k-th derivative (or integral) of a periodic sequence x.

• tilbert(x, h[, period, _cache]) -- Return h-Tilbert transform of a periodic sequence x.

• itilbert(x, h[, period, _cache]) -- Return inverse h-Tilbert transform of a periodic sequence x.

• hilbert(x[, _cache]) -- Return Hilbert transform of a periodic sequence x.

• ihilbert(x) -- Return inverse Hilbert transform of a periodic sequence x.

• cs_diff(x, a, b[, period, _cache]) -- Return (a,b)-cosh/sinh pseudo-derivative of a periodic sequence.

• sc_diff(x, a, b[, period, _cache]) -- Return (a,b)-sinh/cosh pseudo-derivative of a periodic sequence x.

• ss_diff(x, a, b[, period, _cache]) -- Return (a,b)-sinh/sinh pseudo-derivative of a periodic sequence x.

• cc_diff(x, a, b[, period, _cache]) -- Return (a,b)-cosh/cosh pseudo-derivative of a periodic sequence.

• shift(x, a[, period, _cache]) -- Shift periodic sequence x by a: y(u) = x(u+a).

Helper functions:

• fftshift(x[, axes]) -- Shift the zero-frequency component to the center of the spectrum.

• ifftshift(x[, axes]) -- The inverse of fftshift.

• fftfreq(n[, d]) -- Return the Discrete Fourier Transform sample frequencies.

• rfftfreq(n[, d]) -- DFT sample frequencies (for usage with rfft, irfft).

Convolutions (scipy.fftpack.convolve):

• convolve(x,omega,[swap_real_imag,overwrite_x]) -- Wrapper for convolve.

• convolve_z(x,omega_real,omega_imag,[overwrite_x]) -- Wrapper for convolve_z.

• init_convolution_kernel(...) -- Wrapper for init_convolution_kernel.

• destroy_convolve_cache() -- Wrapper for destroy_convolve_cache.

Other (scipy.fftpack._fftpack):

• drfft(x,[n,direction,normalize,overwrite_x]) -- Wrapper for drfft.

• zfft(x,[n,direction,normalize,overwrite_x]) -- Wrapper for zfft.

• zrfft(x,[n,direction,normalize,overwrite_x]) -- Wrapper for zrfft.

• zfftnd(x,[s,direction,normalize,overwrite_x]) -- Wrapper for zfftnd.

• destroy_drfft_cache() -- Wrapper for destroy_drfft_cache.

• destroy_zfft_cache() -- Wrapper for destroy_zfft_cache.

• destroy_zfftnd_cache() -- Wrapper for destroy_zfftnd_cache.

8.5 integrate

Integration and ODEs -- Integrating functions, given function object

Access via:

In [1]: from scipy import integrate

Also see: http://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html

• quad(func, a, b[, args, full_output, ...]) -- Compute a definite integral.

• dblquad(func, a, b, gfun, hfun[, args, ...]) -- Compute a double integral.

• tplquad(func, a, b, gfun, hfun, qfun, rfun) -- Compute a triple (definite) integral.

• nquad(func, ranges[, args, opts]) -- Integration over multiple variables.

• fixed_quad(func, a, b[, args, n]) -- Compute a definite integral using fixed-order Gaussian quadrature.

• quadrature(func, a, b[, args, tol, rtol, ...]) -- Compute a definite integral using fixed-tolerance Gaussian quadrature.

• romberg(function, a, b[, args, tol, rtol, ...]) -- Romberg integration of a callable function or method.

Integrating functions, given fixed samples:

• cumtrapz(y[, x, dx, axis, initial]) -- Cumulatively integrate y(x) using the composite trapezoidal rule.

• simps(y[, x, dx, axis, even]) -- Integrate y(x) using samples along the given axis and the composite Simpson’s rule.

• romb(y[, dx, axis, show]) -- Romberg integration using samples of a function.

See also scipy.special for orthogonal polynomials (special) for Gaussian quadrature roots and weights for other weighting factors and regions.

Integrators of ODE systems:

• odeint(func, y0, t[, args, Dfun, col_deriv, ...]) -- Integrate a system of ordinary differential equations.

• ode(f[, jac]) -- A generic interface class to numeric integrators.

• complex_ode(f[, jac]) -- A wrapper of ode for complex systems.

8.6 interpolate

Sub-package for objects used in interpolation.

This sub-package contains spline functions and classes, one-dimensional and multi-dimensional (univariate and multivariate) interpolation classes, Lagrange and Taylor polynomial interpolators, and wrappers for FITPACK and DFITPACK functions.

Access via:

In [2]: from scipy import interpolate

Also see: http://docs.scipy.org/doc/scipy/reference/tutorial/interpolate.html

Univariate interpolation:

• interp1d(x, y[, kind, axis, copy, ...]) -- Interpolate a 1-D function.

• BarycentricInterpolator(xi[, yi, axis]) -- The interpolating polynomial for a set of points

• KroghInterpolator(xi, yi[, axis]) -- Interpolating polynomial for a set of points.

• PiecewisePolynomial(xi, yi[, orders, ...]) -- Piecewise polynomial curve specified by points and derivatives

• PchipInterpolator(x, y[, axis, extrapolate]) -- PCHIP 1-d monotonic cubic interpolation

• barycentric_interpolate(xi, yi, x[, axis]) -- Convenience function for polynomial interpolation.

• krogh_interpolate(xi, yi, x[, der, axis]) -- Convenience function for polynomial interpolation.

• piecewise_polynomial_interpolate(xi, yi, x) -- Convenience function for piecewise polynomial interpolation.

• pchip_interpolate(xi, yi, x[, der, axis]) -- Convenience function for pchip interpolation.

• Akima1DInterpolator(x, y) -- Akima interpolator

• PPoly(c, x[, extrapolate]) -- Piecewise polynomial in terms of coefficients and breakpoints

• BPoly(c, x[, extrapolate]) -- Piecewise polynomial in terms of coefficients and breakpoints

Multivariate interpolation -- Unstructured data:

• griddata(points, values, xi[, method, ...]) -- Interpolate unstructured D-dimensional data.

• LinearNDInterpolator(points, values[, ...]) -- Piecewise linear interpolant in N dimensions.

• NearestNDInterpolator(points, values) -- Nearest-neighbour interpolation in N dimensions.

• CloughTocher2DInterpolator(points, values[, tol]) -- Piecewise cubic, C1 smooth, curvature-minimizing interpolant in 2D.

• Rbf(*args) -- A class for radial basis function approximation/interpolation of n-dimensional scattered data.

• interp2d(x, y, z[, kind, copy, ...]) -- Interpolate over a 2-D grid.

For data on a grid:

• interpn(points, values, xi[, method, ...]) -- Multidimensional interpolation on regular grids.

• RegularGridInterpolator(points, values[, ...]) -- Interpolation on a regular grid in arbitrary dimensions

• RectBivariateSpline(x, y, z[, bbox, kx, ky, s]) -- Bivariate spline approximation over a rectangular mesh.

See also scipy.ndimage.interpolation.map_coordinates

1-D Splines:

• UnivariateSpline(x, y[, w, bbox, k, s]) -- One-dimensional smoothing spline fit to a given set of data points.

• InterpolatedUnivariateSpline(x, y[, w, bbox, k]) -- One-dimensional interpolating spline for a given set of data points.

• LSQUnivariateSpline(x, y, t[, w, bbox, k]) -- One-dimensional spline with explicit internal knots.

The above univariate spline classes have the following methods:

• UnivariateSpline.__call__(x[, nu]) -- Evaluate spline (or its nu-th derivative) at positions x.

• UnivariateSpline.derivatives(x) -- Return all derivatives of the spline at the point x.

• UnivariateSpline.integral(a, b) -- Return definite integral of the spline between two given points.

• UnivariateSpline.roots() -- Return the zeros of the spline.

• UnivariateSpline.derivative([n]) -- Construct a new spline representing the derivative of this spline.

• UnivariateSpline.antiderivative([n]) -- Construct a new spline representing the antiderivative of this spline.

• UnivariateSpline.get_coeffs() -- Return spline coefficients.

• UnivariateSpline.get_knots() -- Return positions of (boundary and interior) knots of the spline.

• UnivariateSpline.get_residual() -- Return weighted sum of squared residuals of the spline approximation: sum((w[i] * (y[i]-s(x[i])))**2, axis=0).

• UnivariateSpline.set_smoothing_factor(s) -- Continue spline computation with the given smoothing factor s and with the knots found at the last call.

Functional interface to FITPACK functions:

• splrep(x, y[, w, xb, xe, k, task, s, t, ...]) -- Find the B-spline representation of 1-D curve.

• splprep(x[, w, u, ub, ue, k, task, s, t, ...]) -- Find the B-spline representation of an N-dimensional curve.

• splev(x, tck[, der, ext]) -- Evaluate a B-spline or its derivatives.

• splint(a, b, tck[, full_output]) -- Evaluate the definite integral of a B-spline.

• sproot(tck[, mest]) -- Find the roots of a cubic B-spline.

• spalde(x, tck) -- Evaluate all derivatives of a B-spline.

• splder(tck[, n]) -- Compute the spline representation of the derivative of a given spline

• splantider(tck[, n]) -- Compute the spline for the antiderivative (integral) of a given spline.

2-D Splines -- For data on a grid:

• RectBivariateSpline(x, y, z[, bbox, kx, ky, s]) -- Bivariate spline approximation over a rectangular mesh.

• RectSphereBivariateSpline(u, v, r[, s, ...]) -- Bivariate spline approximation over a rectangular mesh on a sphere.

For unstructured data -- BivariateSpline Base class for bivariate splines:

• SmoothBivariateSpline(x, y, z[, w, bbox, ...]) -- Smooth bivariate spline approximation.

• SmoothSphereBivariateSpline(theta, phi, r[, ...]) -- Smooth bivariate spline approximation in spherical coordinates.

• LSQBivariateSpline(x, y, z, tx, ty[, w, ...]) -- Weighted least-squares bivariate spline approximation.

• LSQSphereBivariateSpline(theta, phi, r, tt, tp) -- Weighted least-squares bivariate spline approximation in spherical coordinates.

Low-level interface to FITPACK functions:

• bisplrep(x, y, z[, w, xb, xe, yb, ye, kx, ...]) -- Find a bivariate B-spline representation of a surface.

• bisplev(x, y, tck[, dx, dy]) -- Evaluate a bivariate B-spline and its derivatives.

Additional tools:

• lagrange(x, w) -- Return a Lagrange interpolating polynomial.

• approximate_taylor_polynomial(f, x, degree, ...) -- Estimate the Taylor polynomial of f at x by polynomial fitting.

See also -- scipy.ndimage.interpolation.map_coordinates, scipy.ndimage.interpolation.spline_filter, scipy.signal.resample, scipy.signal.bspline, scipy.signal.gauss_spline, scipy.signal.qspline1d, scipy.signal.cspline1d, scipy.signal.qspline1d_eval, scipy.signal.cspline1d_eval, scipy.signal.qspline2d, scipy.signal.cspline2d.

8.7 linalg

Linear algebra functions.

See also numpy.linalg for more linear algebra functions. Note that although scipy.linalg imports most of them, identically named functions from scipy.linalg may offer more or slightly differing functionality.

Access via:

In [3]: from scipy import linalg

Also see: http://docs.scipy.org/doc/scipy/reference/tutorial/linalg.html

Basics:

• inv(a[, overwrite_a, check_finite]) -- Compute the inverse of a matrix.

• solve(a, b[, sym_pos, lower, overwrite_a, ...]) -- Solve the equation a x = b for x.

• solve_banded(l_and_u, ab, b[, overwrite_ab, ...]) -- Solve the equation a x = b for x, assuming a is banded matrix.

• solveh_banded(ab, b[, overwrite_ab, ...]) -- Solve equation a x = b.

• solve_triangular(a, b[, trans, lower, ...]) -- Solve the equation a x = b for x, assuming a is a triangular matrix.

• det(a[, overwrite_a, check_finite]) -- Compute the determinant of a matrix

• norm(a[, ord]) -- Matrix or vector norm.

• lstsq(a, b[, cond, overwrite_a, ...]) -- Compute least-squares solution to equation Ax = b.

• pinv(a[, cond, rcond, return_rank, check_finite]) -- Compute the (Moore-Penrose) pseudo-inverse of a matrix.

• pinv2(a[, cond, rcond, return_rank, ...]) -- Compute the (Moore-Penrose) pseudo-inverse of a matrix.

• pinvh(a[, cond, rcond, lower, return_rank, ...]) -- Compute the (Moore-Penrose) pseudo-inverse of a Hermitian matrix.

• kron(a, b) -- Kronecker product.

• tril(m[, k]) -- Make a copy of a matrix with elements above the k-th diagonal zeroed.

• triu(m[, k]) -- Make a copy of a matrix with elements below the k-th diagonal zeroed.

Eigenvalue problems:

• eig(a[, b, left, right, overwrite_a, ...]) -- Solve an ordinary or generalized eigenvalue problem of a square matrix.

• eigvals(a[, b, overwrite_a, check_finite]) -- Compute eigenvalues from an ordinary or generalized eigenvalue problem.

• eigh(a[, b, lower, eigvals_only, ...]) -- Solve an ordinary or generalized eigenvalue problem for a complex Hermitian or real symmetric matrix.

• eigvalsh(a[, b, lower, overwrite_a, ...]) -- Solve an ordinary or generalized eigenvalue problem for a complex Hermitian or real symmetric matrix.

• eig_banded(a_band[, lower, eigvals_only, ...]) -- Solve real symmetric or complex hermitian band matrix eigenvalue problem.

• eigvals_banded(a_band[, lower, ...]) -- Solve real symmetric or complex hermitian band matrix eigenvalue problem.

Decompositions:

• lu(a[, permute_l, overwrite_a, check_finite]) -- Compute pivoted LU decompostion of a matrix.

• lu_factor(a[, overwrite_a, check_finite]) -- Compute pivoted LU decomposition of a matrix.

• lu_solve(lu_and_piv, b[, trans, ...]) -- Solve an equation system, a x = b, given the LU factorization of a

• svd(a[, full_matrices, compute_uv, ...]) -- Singular Value Decomposition.

• svdvals(a[, overwrite_a, check_finite]) -- Compute singular values of a matrix.

• diagsvd(s, M, N) -- Construct the sigma matrix in SVD from singular values and size M, N.

• orth(A) -- Construct an orthonormal basis for the range of A using SVD

• cholesky(a[, lower, overwrite_a, check_finite]) -- Compute the Cholesky decomposition of a matrix.

• cholesky_banded(ab[, overwrite_ab, lower, ...]) -- Cholesky decompose a banded Hermitian positive-definite matrix

• cho_factor(a[, lower, overwrite_a, check_finite]) -- Compute the Cholesky decomposition of a matrix, to use in cho_solve

• cho_solve(c_and_lower, b[, overwrite_b, ...]) -- Solve the linear equations A x = b, given the Cholesky factorization of A.

• cho_solve_banded(cb_and_lower, b[, ...]) -- Solve the linear equations A x = b, given the Cholesky factorization of A.

• polar(a[, side]) -- Compute the polar decomposition.

• qr(a[, overwrite_a, lwork, mode, pivoting, ...]) -- Compute QR decomposition of a matrix.

• qr_multiply(a, c[, mode, pivoting, ...]) -- Calculate the QR decomposition and multiply Q with a matrix.

• qz(A, B[, output, lwork, sort, overwrite_a, ...]) -- QZ decompostion for generalized eigenvalues of a pair of matrices.

• schur(a[, output, lwork, overwrite_a, sort, ...]) -- Compute Schur decomposition of a matrix.

• rsf2csf(T, Z[, check_finite]) -- Convert real Schur form to complex Schur form.

• hessenberg(a[, calc_q, overwrite_a, ...]) -- Compute Hessenberg form of a matrix.

See also scipy.linalg.interpolative – Interpolative matrix decompositions

Matrix Functions:

• expm(A[, q]) -- Compute the matrix exponential using Pade approximation.

• logm(A[, disp]) -- Compute matrix logarithm.

• cosm(A) -- Compute the matrix cosine.

• sinm(A) -- Compute the matrix sine.

• tanm(A) -- Compute the matrix tangent.

• coshm(A) -- Compute the hyperbolic matrix cosine.

• sinhm(A) -- Compute the hyperbolic matrix sine.

• tanhm(A) -- Compute the hyperbolic matrix tangent.

• signm(A[, disp]) -- Matrix sign function.

• sqrtm(A[, disp, blocksize]) -- Matrix square root.

• funm(A, func[, disp]) -- Evaluate a matrix function specified by a callable.

• expm_frechet(A, E[, method, compute_expm, ...]) -- Frechet derivative of the matrix exponential of A in the direction E.

• expm_cond(A[, check_finite]) -- Relative condition number of the matrix exponential in the Frobenius norm.

• fractional_matrix_power(A, t) --

Matrix Equation Solvers:

• solve_sylvester(a, b, q) -- Computes a solution (X) to the Sylvester equation (AX + XB = Q).

• solve_continuous_are(a, b, q, r) -- Solves the continuous algebraic Riccati equation, or CARE, defined as (A’X + XA - XBR^-1B’X+Q=0) directly using a Schur decomposition method.

• solve_discrete_are(a, b, q, r) -- Solves the disctrete algebraic Riccati equation, or DARE, defined as (X = A’XA-(A’XB)(R+B’XB)^-1(B’XA)+Q), directly using a Schur decomposition method.

• solve_discrete_lyapunov(a, q) -- Solves the Discrete Lyapunov Equation (A’XA-X=-Q) directly.

• solve_lyapunov(a, q) -- Solves the continuous Lyapunov equation (AX + XA^H = Q) given the values of A and Q using the Bartels-Stewart algorithm.

Special Matrices:

• block_diag(*arrs) -- Create a block diagonal matrix from provided arrays.

• circulant(c) -- Construct a circulant matrix.

• companion(a) -- Create a companion matrix.

• dft(n[, scale]) -- Discrete Fourier transform matrix.

• hadamard(n[, dtype]) -- Construct a Hadamard matrix.

• hankel(c[, r]) -- Construct a Hankel matrix.

• hilbert(n) -- Create a Hilbert matrix of order n.

• invhilbert(n[, exact]) -- Compute the inverse of the Hilbert matrix of order n.

• leslie(f, s) -- Create a Leslie matrix.

• pascal(n[, kind, exact]) -- Returns the n x n Pascal matrix.

• toeplitz(c[, r]) -- Construct a Toeplitz matrix.

• tri(N[, M, k, dtype]) -- Construct (N, M) matrix filled with ones at and below the k-th diagonal.

Low-level routines:

• get_blas_funcs(names[, arrays, dtype]) -- Return available BLAS function objects from names.

• get_lapack_funcs(names[, arrays, dtype]) -- Return available LAPACK function objects from names.

• find_best_blas_type([arrays, dtype]) -- Find best-matching BLAS/LAPACK type.

See also:

• scipy.linalg.blas -– Low-level BLAS functions

• scipy.linalg.lapack -– Low-level LAPACK functions

8.8 Multi-dimensional image processing

This package contains various functions for multi-dimensional image processing.

Access via:

In [1]: from scipy import ndimage

Also see: http://docs.scipy.org/doc/scipy/reference/tutorial/ndimage.html

Filters (scipy.ndimage.filters):

• convolve(input, weights[, output, mode, ...]) -- Multidimensional convolution.

• convolve1d(input, weights[, axis, output, ...]) -- Calculate a one-dimensional convolution along the given axis.

• correlate(input, weights[, output, mode, ...]) -- Multi-dimensional correlation.

• correlate1d(input, weights[, axis, output, ...]) -- Calculate a one-dimensional correlation along the given axis.

• gaussian_filter(input, sigma[, order, ...]) -- Multidimensional Gaussian filter.

• gaussian_filter1d(input, sigma[, axis, ...]) -- One-dimensional Gaussian filter.

• gaussian_gradient_magnitude(input, sigma[, ...]) -- Multidimensional gradient magnitude using Gaussian derivatives.

• gaussian_laplace(input, sigma[, output, ...]) -- Multidimensional Laplace filter using gaussian second derivatives.

• generic_filter(input, function[, size, ...]) -- Calculates a multi-dimensional filter using the given function.

• generic_filter1d(input, function, filter_size) -- Calculate a one-dimensional filter along the given axis.

• generic_gradient_magnitude(input, derivative) -- Gradient magnitude using a provided gradient function.

• generic_laplace(input, derivative2[, ...]) -- N-dimensional Laplace filter using a provided second derivative function

• laplace(input[, output, mode, cval]) -- N-dimensional Laplace filter based on approximate second derivatives.

• maximum_filter(input[, size, footprint, ...]) -- Calculates a multi-dimensional maximum filter.

• maximum_filter1d(input, size[, axis, ...]) -- Calculate a one-dimensional maximum filter along the given axis.

• median_filter(input[, size, footprint, ...]) -- Calculates a multidimensional median filter.

• minimum_filter(input[, size, footprint, ...]) -- Calculates a multi-dimensional minimum filter.

• minimum_filter1d(input, size[, axis, ...]) -- Calculate a one-dimensional minimum filter along the given axis.

• percentile_filter(input, percentile[, size, ...]) -- Calculates a multi-dimensional percentile filter.

• prewitt(input[, axis, output, mode, cval]) -- Calculate a Prewitt filter.

• rank_filter(input, rank[, size, footprint, ...]) -- Calculates a multi-dimensional rank filter.

• sobel(input[, axis, output, mode, cval]) -- Calculate a Sobel filter.

• uniform_filter(input[, size, output, mode, ...]) -- Multi-dimensional uniform filter.

• uniform_filter1d(input, size[, axis, ...]) -- Calculate a one-dimensional uniform filter along the given axis.

Fourier filters (scipy.ndimage.fourier):

• fourier_ellipsoid(input, size[, n, axis, output]) -- Multi-dimensional ellipsoid fourier filter.

• fourier_gaussian(input, sigma[, n, axis, output]) -- Multi-dimensional Gaussian fourier filter.

• fourier_shift(input, shift[, n, axis, output]) -- Multi-dimensional fourier shift filter.

• fourier_uniform(input, size[, n, axis, output]) -- Multi-dimensional uniform fourier filter.

Interpolation (scipy.ndimage.interpolation):

• affine_transform(input, matrix[, offset, ...]) -- Apply an affine transformation.

• geometric_transform(input, mapping[, ...]) -- Apply an arbritrary geometric transform.

• map_coordinates(input, coordinates[, ...]) -- Map the input array to new coordinates by interpolation.

• rotate(input, angle[, axes, reshape, ...]) -- Rotate an array.

• shift(input, shift[, output, order, mode, ...]) -- Shift an array.

• spline_filter(input[, order, output]) -- Multi-dimensional spline filter.

• spline_filter1d(input[, order, axis, output]) -- Calculates a one-dimensional spline filter along the given axis.

• zoom(input, zoom[, output, order, mode, ...]) -- Zoom an array.

Measurements (scipy.ndimage.measurements):

• center_of_mass(input[, labels, index]) -- Calculate the center of mass of the values of an array at labels.

• extrema(input[, labels, index]) -- Calculate the minimums and maximums of the values of an array at labels, along with their positions.

• find_objects(input[, max_label]) -- Find objects in a labeled array.

• histogram(input, min, max, bins[, labels, index]) -- Calculate the histogram of the values of an array, optionally at labels.

• label(input[, structure, output]) -- Label features in an array.

• labeled_comprehension(input, labels, index, ...) -- Roughly equivalent to [func(input[labels == i]) for i in index].

• maximum(input[, labels, index]) -- Calculate the maximum of the values of an array over labeled regions.

• maximum_position(input[, labels, index]) -- Find the positions of the maximums of the values of an array at labels.

• mean(input[, labels, index]) -- Calculate the mean of the values of an array at labels.

• minimum(input[, labels, index]) -- Calculate the minimum of the values of an array over labeled regions.

• minimum_position(input[, labels, index]) -- Find the positions of the minimums of the values of an array at labels.

• standard_deviation(input[, labels, index]) -- Calculate the standard deviation of the values of an n-D image array, optionally at specified sub-regions.

• sum(input[, labels, index]) -- Calculate the sum of the values of the array.

• variance(input[, labels, index]) -- Calculate the variance of the values of an n-D image array, optionally at specified sub-regions.

• watershed_ift(input, markers[, structure, ...]) -- Apply watershed from markers using an iterative forest transform algorithm.

Morphology (scipy.ndimage.morphology):

• binary_closing(input[, structure, ...]) -- Multi-dimensional binary closing with the given structuring element.

• binary_dilation(input[, structure, ...]) -- Multi-dimensional binary dilation with the given structuring element.

• binary_erosion(input[, structure, ...]) -- Multi-dimensional binary erosion with a given structuring element.

• binary_fill_holes(input[, structure, ...]) -- Fill the holes in binary objects.

• binary_hit_or_miss(input[, structure1, ...]) -- Multi-dimensional binary hit-or-miss transform.

• binary_opening(input[, structure, ...]) -- Multi-dimensional binary opening with the given structuring element.

• binary_propagation(input[, structure, mask, ...]) -- Multi-dimensional binary propagation with the given structuring element.

• black_tophat(input[, size, footprint, ...]) -- Multi-dimensional black tophat filter.

• distance_transform_bf(input[, metric, ...]) -- Distance transform function by a brute force algorithm.

• distance_transform_cdt(input[, metric, ...]) -- Distance transform for chamfer type of transforms.

• distance_transform_edt(input[, sampling, ...]) -- Exact euclidean distance transform.

• generate_binary_structure(rank, connectivity) -- Generate a binary structure for binary morphological operations.

• grey_closing(input[, size, footprint, ...]) -- Multi-dimensional greyscale closing.

• grey_dilation(input[, size, footprint, ...]) -- Calculate a greyscale dilation, using either a structuring element, or a footprint corresponding to a flat structuring element.

• grey_erosion(input[, size, footprint, ...]) -- Calculate a greyscale erosion, using either a structuring element, or a footprint corresponding to a flat structuring element.

• grey_opening(input[, size, footprint, ...]) -- Multi-dimensional greyscale opening.

• iterate_structure(structure, iterations[, ...]) -- Iterate a structure by dilating it with itself.

• morphological_gradient(input[, size, ...]) -- Multi-dimensional morphological gradient.

• morphological_laplace(input[, size, ...]) -- Multi-dimensional morphological laplace.

• white_tophat(input[, size, footprint, ...]) -- Multi-dimensional white tophat filter.

Utility:

• imread(fname[, flatten, mode]) -- Read an image from a file as an array.

8.9 optimize

Optimization and root finding

Access with:

In [4]: from scipy import optimize

Also see: http://docs.scipy.org/doc/scipy/reference/tutorial/optimize.html

Local Optimization:

• minimize(fun, x0[, args, method, jac, hess, ...]) -- Minimization of scalar function of one or more variables.

• minimize_scalar(fun[, bracket, bounds, ...]) -- Minimization of scalar function of one variable.

OptimizeResult -- Represents the optimization result.

The specific optimization method interfaces below in this subsection are not recommended for use in new scripts; all of these methods are accessible via a newer, more consistent interface provided by the functions above.

General-purpose multivariate methods:

• fmin(func, x0[, args, xtol, ftol, maxiter, ...]) -- Minimize a function using the downhill simplex algorithm.

• fmin_powell(func, x0[, args, xtol, ftol, ...]) -- Minimize a function using modified Powell’s method.

• fmin_cg(f, x0[, fprime, args, gtol, norm, ...]) -- Minimize a function using a nonlinear conjugate gradient algorithm.

• fmin_bfgs(f, x0[, fprime, args, gtol, norm, ...]) -- Minimize a function using the BFGS algorithm.

• fmin_ncg(f, x0, fprime[, fhess_p, fhess, ...]) -- Unconstrained minimization of a function using the Newton-CG method.

Constrained multivariate methods:

• fmin_l_bfgs_b(func, x0[, fprime, args, ...]) -- Minimize a function func using the L-BFGS-B algorithm.

• fmin_tnc(func, x0[, fprime, args, ...]) -- Minimize a function with variables subject to bounds, using gradient information in a truncated Newton algorithm.

• fmin_cobyla(func, x0, cons[, args, ...]) -- Minimize a function using the Constrained Optimization BY Linear Approximation (COBYLA) method.

• fmin_slsqp(func, x0[, eqcons, f_eqcons, ...]) -- Minimize a function using Sequential Least SQuares Programming

Univariate (scalar) minimization methods:

• fminbound(func, x1, x2[, args, xtol, ...]) -- Bounded minimization for scalar functions.

• brent(func[, args, brack, tol, full_output, ...]) -- Given a function of one-variable and a possible bracketing interval, return the minimum of the function isolated to a fractional precision of tol.

• golden(func[, args, brack, tol, full_output]) -- Return the minimum of a function of one variable.

Equation (Local) Minimizers:

• leastsq(func, x0[, args, Dfun, full_output, ...]) -- Minimize the sum of squares of a set of equations.

• nnls(A, b) -- Solve argmin_x || Ax - b ||_2 for x>=0.

Global Optimization:

• anneal(*args, **kwds) -- anneal is deprecated!

• basinhopping(func, x0[, niter, T, stepsize, ...]) -- Find the global minimum of a function using the basin-hopping algorithm

• brute(func, ranges[, args, Ns, full_output, ...]) -- Minimize a function over a given range by brute force.

Rosenbrock functions:

• rosen(x) -- The Rosenbrock function.

• rosen_der(x) -- The derivative (i.e.

• rosen_hess(x) -- The Hessian matrix of the Rosenbrock function.

• rosen_hess_prod(x, p) -- Product of the Hessian matrix of the Rosenbrock function with a vector.

Fitting:

• curve_fit(f, xdata, ydata[, p0, sigma, ...]) -- Use non-linear least squares to fit a function, f, to data.

Root finding

Scalar functions:

• brentq(f, a, b[, args, xtol, rtol, maxiter, ...]) -- Find a root of a function in given interval.

• brenth(f, a, b[, args, xtol, rtol, maxiter, ...]) -- Find root of f in [a,b].

• ridder(f, a, b[, args, xtol, rtol, maxiter, ...]) -- Find a root of a function in an interval.

• bisect(f, a, b[, args, xtol, rtol, maxiter, ...]) -- Find root of a function within an interval.

• newton(func, x0[, fprime, args, tol, ...]) -- Find a zero using the Newton-Raphson or secant method.

Fixed point finding:

• fixed_point(func, x0[, args, xtol, maxiter]) -- Find a fixed point of the function.

Multidimensional

General nonlinear solvers:

• root(fun, x0[, args, method, jac, tol, ...]) -- Find a root of a vector function.

• fsolve(func, x0[, args, fprime, ...]) -- Find the roots of a function.

• broyden1(F, xin[, iter, alpha, ...]) -- Find a root of a function, using Broyden’s first Jacobian approximation.

• broyden2(F, xin[, iter, alpha, ...]) -- Find a root of a function, using Broyden’s second Jacobian approximation.

Large-scale nonlinear solvers:

• newton_krylov(F, xin[, iter, rdiff, method, ...]) -- Find a root of a function, using Krylov approximation for inverse Jacobian.

• anderson(F, xin[, iter, alpha, w0, M, ...]) -- Find a root of a function, using (extended) Anderson mixing.

Simple iterations:

• excitingmixing(F, xin[, iter, alpha, ...]) -- Find a root of a function, using a tuned diagonal Jacobian approximation.

• linearmixing(F, xin[, iter, alpha, verbose, ...]) -- Find a root of a function, using a scalar Jacobian approximation.

• diagbroyden(F, xin[, iter, alpha, verbose, ...]) -- Find a root of a function, using diagonal Broyden Jacobian approximation.

Additional information on the nonlinear solvers

Utility Functions:

• approx_fprime(xk, f, epsilon, *args) -- Finite-difference approximation of the gradient of a scalar function.

• bracket(func[, xa, xb, args, grow_limit, ...]) -- Bracket the minimum of the function.

• check_grad(func, grad, x0, *args) -- Check the correctness of a gradient function by comparing it against a (forward) finite-difference approximation of the gradient.

• line_search(f, myfprime, xk, pk[, gfk, ...]) -- Find alpha that satisfies strong Wolfe conditions.

• show_options([solver, method]) -- Show documentation for additional options of optimization solvers.i

8.10 Orthogonal distance regression

Access via:

In [1]: from scipy import odr

See: http://docs.scipy.org/doc/scipy/reference/odr.html

• Data(x[, y, we, wd, fix, meta]) -- The data to fit.

• RealData(x[, y, sx, sy, covx, covy, fix, meta]) -- The data, with weightings as actual standard deviations and/or covariances.

• Model(fcn[, fjacb, fjacd, extra_args, ...]) -- The Model class stores information about the function you wish to fit.

• ODR(data, model[, beta0, delta0, ifixb, ...]) -- The ODR class gathers all information and coordinates the running of the

• Output(output) -- The Output class stores the output of an ODR run.

• odr(fcn, beta0, y, x[, we, wd, fjacb, ...]) -- Low-level function for ODR.

• odr_error -- Exception indicating an error in fitting.

• odr_stop -- Exception stopping fitting.

8.11 Signal processing

Access via:

In [1]: from scipy import signal

Also see: http://docs.scipy.org/doc/scipy/reference/tutorial/signal.html

Convolution:

• convolve(in1, in2[, mode]) -- Convolve two N-dimensional arrays.

• correlate(in1, in2[, mode]) -- Cross-correlate two N-dimensional arrays.

• fftconvolve(in1, in2[, mode]) -- Convolve two N-dimensional arrays using FFT.

• convolve2d(in1, in2[, mode, boundary, fillvalue]) -- Convolve two 2-dimensional arrays.

• correlate2d(in1, in2[, mode, boundary, ...]) -- Cross-correlate two 2-dimensional arrays.

• sepfir2d((input, hrow, hcol) -> output) --Convolve the rank-2 input array with the separable filter defined by the rank-1 arrays hrow, and hcol.

B-splines:

• bspline(x, n) -- B-spline basis function of order n.

• cubic(x) -- A cubic B-spline.

• quadratic(x) -- A quadratic B-spline.

• gauss_spline(x, n) -- Gaussian approximation to B-spline basis function of order n.

• cspline1d(signal[, lamb]) -- Compute cubic spline coefficients for rank-1 array.

• qspline1d(signal[, lamb]) -- Compute quadratic spline coefficients for rank-1 array.

• cspline2d((input {, lambda, precision}) -- -> ck) -- Return the third-order B-spline coefficients over a regularly spaced input grid for the two-dimensional input image.

• qspline2d((input {, lambda, precision}) -- -> qk) -- Return the second-order B-spline coefficients over a regularly spaced input grid for the two-dimensional input image.

• cspline1d_eval(cj, newx[, dx, x0]) -- Evaluate a spline at the new set of points.

• qspline1d_eval(cj, newx[, dx, x0]) -- Evaluate a quadratic spline at the new set of points.

• spline_filter(Iin[, lmbda]) -- Smoothing spline (cubic) filtering of a rank-2 array.

Filtering:

• order_filter(a, domain, rank) -- Perform an order filter on an N-dimensional array.

• medfilt(volume[, kernel_size]) -- Perform a median filter on an N-dimensional array.

• medfilt2d(input[, kernel_size]) -- Median filter a 2-dimensional array.

• wiener(im[, mysize, noise]) -- Perform a Wiener filter on an N-dimensional array.

• symiirorder1((input, c0, z1 {, ...) -- Implement a smoothing IIR filter with mirror-symmetric boundary conditions using a cascade of first-order sections.

• symiirorder2((input, r, omega {, ...) -- Implement a smoothing IIR filter with mirror-symmetric boundary conditions using a cascade of second-order sections.

• lfilter(b, a, x[, axis, zi]) -- Filter data along one-dimension with an IIR or FIR filter.

• lfiltic(b, a, y[, x]) -- Construct initial conditions for lfilter.

• lfilter_zi(b, a) -- Compute an initial state zi for the lfilter function that corresponds to the steady state of the step response.

• filtfilt(b, a, x[, axis, padtype, padlen]) -- A forward-backward filter.

• savgol_filter(x, window_length, polyorder[, ...]) -- Apply a Savitzky-Golay filter to an array.

• deconvolve(signal, divisor) -- Deconvolves divisor out of signal.

• hilbert(x[, N, axis]) -- Compute the analytic signal, using the Hilbert transform.

• get_window(window, Nx[, fftbins]) -- Return a window.

• decimate(x, q[, n, ftype, axis]) -- Downsample the signal by using a filter.

• detrend(data[, axis, type, bp]) -- Remove linear trend along axis from data.

• resample(x, num[, t, axis, window]) -- Resample x to num samples using Fourier method along the given axis.

Filter design:

• bilinear(b, a[, fs]) -- Return a digital filter from an analog one using a bilinear transform.

• firwin(numtaps, cutoff[, width, window, ...]) -- FIR filter design using the window method.

• firwin2(numtaps, freq, gain[, nfreqs, ...]) -- FIR filter design using the window method.

• freqs(b, a[, worN, plot]) -- Compute frequency response of analog filter.

• freqz(b[, a, worN, whole, plot]) -- Compute the frequency response of a digital filter.

• iirdesign(wp, ws, gpass, gstop[, analog, ...]) -- Complete IIR digital and analog filter design.

• iirfilter(N, Wn[, rp, rs, btype, analog, ...]) -- IIR digital and analog filter design given order and critical points.

• kaiser_atten(numtaps, width) -- Compute the attenuation of a Kaiser FIR filter.

• kaiser_beta(a) -- Compute the Kaiser parameter beta, given the attenuation a.

• kaiserord(ripple, width) -- Design a Kaiser window to limit ripple and width of transition region.

• savgol_coeffs(window_length, polyorder[, ...]) -- Compute the coefficients for a 1-d Savitzky-Golay FIR filter.

• remez(numtaps, bands, desired[, weight, Hz, ...]) -- Calculate the minimax optimal filter using the Remez exchange algorithm.

• unique_roots(p[, tol, rtype]) -- Determine unique roots and their multiplicities from a list of roots.

• residue(b, a[, tol, rtype]) -- Compute partial-fraction expansion of b(s) / a(s).

• residuez(b, a[, tol, rtype]) -- Compute partial-fraction expansion of b(z) / a(z).

• invres(r, p, k[, tol, rtype]) -- Compute b(s) and a(s) from partial fraction expansion: r,p,k

Matlab-style IIR filter design:

• butter(N, Wn[, btype, analog, output]) -- Butterworth digital and analog filter design.

• buttord(wp, ws, gpass, gstop[, analog]) -- Butterworth filter order selection.

• cheby1(N, rp, Wn[, btype, analog, output]) -- Chebyshev type I digital and analog filter design.

• cheb1ord(wp, ws, gpass, gstop[, analog]) -- Chebyshev type I filter order selection.

• cheby2(N, rs, Wn[, btype, analog, output]) -- Chebyshev type II digital and analog filter design.

• cheb2ord(wp, ws, gpass, gstop[, analog]) -- Chebyshev type II filter order selection.

• ellip(N, rp, rs, Wn[, btype, analog, output]) -- Elliptic (Cauer) digital and analog filter design.

• ellipord(wp, ws, gpass, gstop[, analog]) -- Elliptic (Cauer) filter order selection.

• bessel(N, Wn[, btype, analog, output]) -- Bessel/Thomson digital and analog filter design.

Continuous-Time Linear Systems:

• freqresp(system[, w, n]) -- Calculate the frequency response of a continuous-time system.

• lti(*args, **kwords) -- Linear Time Invariant class which simplifies representation.

• lsim(system, U, T[, X0, interp]) -- Simulate output of a continuous-time linear system.

• lsim2(system[, U, T, X0]) -- Simulate output of a continuous-time linear system, by using the ODE solver scipy.integrate.odeint.

• impulse(system[, X0, T, N]) -- Impulse response of continuous-time system.

• impulse2(system[, X0, T, N]) -- Impulse response of a single-input, continuous-time linear system.

• step(system[, X0, T, N]) -- Step response of continuous-time system.

• step2(system[, X0, T, N]) -- Step response of continuous-time system.

• bode(system[, w, n]) -- Calculate Bode magnitude and phase data of a continuous-time system.

Discrete-Time Linear Systems:

• dlsim(system, u[, t, x0]) -- Simulate output of a discrete-time linear system.

• dimpulse(system[, x0, t, n]) -- Impulse response of discrete-time system.

• dstep(system[, x0, t, n]) -- Step response of discrete-time system.

LTI Representations:

• tf2zpk(b, a) -- Return zero, pole, gain (z,p,k) representation from a numerator, denominator representation of a linear filter.

• zpk2tf(z, p, k) -- Return polynomial transfer function representation from zeros

• tf2ss(num, den) -- Transfer function to state-space representation.

• ss2tf(A, B, C, D[, input]) -- State-space to transfer function.

• zpk2ss(z, p, k) -- Zero-pole-gain representation to state-space representation

• ss2zpk(A, B, C, D[, input]) -- State-space representation to zero-pole-gain representation.

• cont2discrete(sys, dt[, method, alpha]) -- Transform a continuous to a discrete state-space system.

Waveforms:

• chirp(t, f0, t1, f1[, method, phi, vertex_zero]) -- Frequency-swept cosine generator.

• gausspulse(t[, fc, bw, bwr, tpr, retquad, ...]) -- Return a Gaussian modulated sinusoid: exp(-a t^2) exp(1j*2*pi*fc*t)

• sawtooth(t[, width]) -- Return a periodic sawtooth or triangle waveform.

• square(t[, duty]) -- Return a periodic square-wave waveform.

• sweep_poly(t, poly[, phi]) -- Frequency-swept cosine generator, with a time-dependent frequency.

Window functions:

• get_window(window, Nx[, fftbins]) -- Return a window.

• barthann(M[, sym]) -- Return a modified Bartlett-Hann window.

• bartlett(M[, sym]) -- Return a Bartlett window.

• blackman(M[, sym]) -- Return a Blackman window.

• blackmanharris(M[, sym]) -- Return a minimum 4-term Blackman-Harris window.

• bohman(M[, sym]) -- Return a Bohman window.

• boxcar(M[, sym]) -- Return a boxcar or rectangular window.

• chebwin(M, at[, sym]) -- Return a Dolph-Chebyshev window.

• cosine(M[, sym]) -- Return a window with a simple cosine shape.

• flattop(M[, sym]) -- Return a flat top window.

• gaussian(M, std[, sym]) -- Return a Gaussian window.

• general_gaussian(M, p, sig[, sym]) -- Return a window with a generalized Gaussian shape.

• hamming(M[, sym]) -- Return a Hamming window.

• hann(M[, sym]) -- Return a Hann window.

• kaiser(M, beta[, sym]) -- Return a Kaiser window.

• nuttall(M[, sym]) -- Return a minimum 4-term Blackman-Harris window according to Nuttall.

• parzen(M[, sym]) -- Return a Parzen window.

• slepian(M, width[, sym]) -- Return a digital Slepian (DPSS) window.

• triang(M[, sym]) -- Return a triangular window.

Wavelets:

• cascade(hk[, J]) -- Return (x, phi, psi) at dyadic points K/2**J from filter coefficients.

• daub(p) -- The coefficients for the FIR low-pass filter producing Daubechies wavelets.

• morlet(M[, w, s, complete]) -- Complex Morlet wavelet.

• qmf(hk) -- Return high-pass qmf filter from low-pass

• ricker(points, a) -- Return a Ricker wavelet, also known as the “Mexican hat wavelet”.

• cwt(data, wavelet, widths) -- Continuous wavelet transform.

Peak finding:

• find_peaks_cwt(vector, widths[, wavelet, ...]) -- Attempt to find the peaks in a 1-D array.

• argrelmin(data[, axis, order, mode]) -- Calculate the relative minima of data.

• argrelmax(data[, axis, order, mode]) -- Calculate the relative maxima of data.

• argrelextrema(data, comparator[, axis, ...]) -- Calculate the relative extrema of data.

Spectral Analysis:

• periodogram(x[, fs, window, nfft, detrend, ...]) -- Estimate power spectral density using a periodogram.

• welch(x[, fs, window, nperseg, noverlap, ...]) -- Estimate power spectral density using Welch’s method.

• lombscargle(x, y, freqs) -- Computes the Lomb-Scargle periodogram.

• vectorstrength(events, period) -- Determine the vector strength of the events corresponding to the given period.

8.12 sparse

SciPy 2-D sparse matrix package for numeric data.

Access via:

In [1]: from scipy import sparse

See: http://docs.scipy.org/doc/scipy/reference/sparse.html

Sparse matrix classes:

• bsr_matrix(arg1[, shape, dtype, copy, blocksize]) -- Block Sparse Row matrix

• coo_matrix(arg1[, shape, dtype, copy]) -- A sparse matrix in COOrdinate format.

• csc_matrix(arg1[, shape, dtype, copy]) -- Compressed Sparse Column matrix

• csr_matrix(arg1[, shape, dtype, copy]) -- Compressed Sparse Row matrix

• dia_matrix(arg1[, shape, dtype, copy]) -- Sparse matrix with DIAgonal storage

• dok_matrix(arg1[, shape, dtype, copy]) -- Dictionary Of Keys based sparse matrix.

• lil_matrix(arg1[, shape, dtype, copy]) -- Row-based linked list sparse matrix

Functions:

Building sparse matrices:

• eye(m[, n, k, dtype, format]) -- Sparse matrix with ones on diagonal

• identity(n[, dtype, format]) -- Identity matrix in sparse format

• kron(A, B[, format]) -- kronecker product of sparse matrices A and B

• kronsum(A, B[, format]) -- kronecker sum of sparse matrices A and B

• diags(diagonals, offsets[, shape, format, dtype]) -- Construct a sparse matrix from diagonals.

• spdiags(data, diags, m, n[, format]) -- Return a sparse matrix from diagonals.

• block_diag(mats[, format, dtype]) -- Build a block diagonal sparse matrix from provided matrices.

• tril(A[, k, format]) -- Return the lower triangular portion of a matrix in sparse format

• triu(A[, k, format]) -- Return the upper triangular portion of a matrix in sparse format

• bmat(blocks[, format, dtype]) -- Build a sparse matrix from sparse sub-blocks

• hstack(blocks[, format, dtype]) -- Stack sparse matrices horizontally (column wise)

• vstack(blocks[, format, dtype]) -- Stack sparse matrices vertically (row wise)

• rand(m, n[, density, format, dtype, ...]) -- Generate a sparse matrix of the given shape and density with uniformly distributed values.

Identifying sparse matrices:

• issparse(x)

• isspmatrix(x)

• isspmatrix_csc(x)

• isspmatrix_csr(x)

• isspmatrix_bsr(x)

• isspmatrix_lil(x)

• isspmatrix_dok(x)

• isspmatrix_coo(x)

• isspmatrix_dia(x)

Submodules:

• csgraph

• linalg

Exceptions:

• SparseEfficiencyWarning

• SparseWarning

8.13 special

Special Functions.

Access via:

In [1]: from scipy import special

Also see: http://docs.scipy.org/doc/scipy/reference/tutorial/special.html

Airy functions:

• airy(z) -- Airy functions and their derivatives.

• airye(z) -- Exponentially scaled Airy functions and their derivatives.

• ai_zeros(nt) -- Compute the zeros of Airy Functions Ai(x) and Ai’(x), a and a’ respectively, and the associated values of Ai(a’) and Ai’(a).

• bi_zeros(nt) -- Compute the zeros of Airy Functions Bi(x) and Bi’(x), b and b’ respectively, and the associated values of Ai(b’) and Ai’(b).

Elliptic Functions and Integrals:

• ellipj(u, m) -- Jacobian elliptic functions

• ellipk(m) -- Computes the complete elliptic integral of the first kind.

• ellipkm1(p) -- The complete elliptic integral of the first kind around m=1.

• ellipkinc(phi, m) -- Incomplete elliptic integral of the first kind

• ellipe(m) -- Complete elliptic integral of the second kind

• ellipeinc(phi,m) -- Incomplete elliptic integral of the second kind

Bessel Functions:

• jn(v, z) -- Bessel function of the first kind of real order v

• jv(v, z) -- Bessel function of the first kind of real order v

• jve(v, z) -- Exponentially scaled Bessel function of order v

• yn(n,x) -- Bessel function of the second kind of integer order

• yv(v,z) -- Bessel function of the second kind of real order

• yve(v,z) -- Exponentially scaled Bessel function of the second kind of real order

• kn(n, x) -- Modified Bessel function of the second kind of integer order n

• kv(v,z) -- Modified Bessel function of the second kind of real order v

• kve(v,z) -- Exponentially scaled modified Bessel function of the second kind.

• iv(v,z) -- Modified Bessel function of the first kind of real order

• ive(v,z) -- Exponentially scaled modified Bessel function of the first kind

• hankel1(v, z) -- Hankel function of the first kind

• hankel1e(v, z) -- Exponentially scaled Hankel function of the first kind

• hankel2(v, z) -- Hankel function of the second kind

• hankel2e(v, z) -- Exponentially scaled Hankel function of the second kind

The following is not an universal function:

• lmbda(v, x) -- Compute sequence of lambda functions with arbitrary order v and their derivatives.

Zeros of Bessel Functions -- These are not universal functions:

• jnjnp_zeros(nt) -- Compute nt (<=1200) zeros of the Bessel functions Jn and Jn’ and arange them in order of their magnitudes.

• jnyn_zeros(n, nt) -- Compute nt zeros of the Bessel functions Jn(x), Jn’(x), Yn(x), and Yn’(x), respectively.

• jn_zeros(n, nt) -- Compute nt zeros of the Bessel function Jn(x).

• jnp_zeros(n, nt) -- Compute nt zeros of the Bessel function Jn’(x).

• yn_zeros(n, nt) -- Compute nt zeros of the Bessel function Yn(x).

• ynp_zeros(n, nt) -- Compute nt zeros of the Bessel function Yn’(x).

• y0_zeros(nt[, complex]) -- Returns nt (complex or real) zeros of Y0(z), z0, and the value of Y0’(z0) = -Y1(z0) at each zero.

• y1_zeros(nt[, complex]) -- Returns nt (complex or real) zeros of Y1(z), z1, and the value of Y1’(z1) = Y0(z1) at each zero.

• y1p_zeros(nt[, complex]) -- Returns nt (complex or real) zeros of Y1’(z), z1’, and the value of Y1(z1’) at each zero.

Faster versions of common Bessel Functions:

• j0(x) -- Bessel function the first kind of order 0

• j1(x) -- Bessel function of the first kind of order 1

• y1(x) -- Bessel function of the second kind of order 1

• y0(x) -- Bessel function of the second kind of order 0

• i0(x) -- Modified Bessel function of order 0

• i0e(x) -- Exponentially scaled modified Bessel function of order 0.

• i1(x) -- Modified Bessel function of order 1

• i1e(x) -- Exponentially scaled modified Bessel function of order 0.

• k0(x) -- Modified Bessel function K of order 0

• k0e(x) -- Exponentially scaled modified Bessel function K of order 0

• k1(x) -- Modified Bessel function of the first kind of order 1

• k1e(x) -- Exponentially scaled modified Bessel function K of order 1

Integrals of Bessel Functions:

• itj0y0(x) -- Integrals of Bessel functions of order 0

• it2j0y0(x) -- Integrals related to Bessel functions of order 0

• iti0k0(x) -- Integrals of modified Bessel functions of order 0

• it2i0k0(x) -- Integrals related to modified Bessel functions of order 0

• besselpoly(a, lmb, nu) -- Weighed integral of a Bessel function.

Derivatives of Bessel Functions:

• jvp(v, z[, n]) -- Return the nth derivative of Jv(z) with respect to z.

• yvp(v, z[, n]) -- Return the nth derivative of Yv(z) with respect to z.

• kvp(v, z[, n]) -- Return the nth derivative of Kv(z) with respect to z.

• ivp(v, z[, n]) -- Return the nth derivative of Iv(z) with respect to z.

• h1vp(v, z[, n]) -- Return the nth derivative of H1v(z) with respect to z.

• h2vp(v, z[, n]) -- Return the nth derivative of H2v(z) with respect to z.

Spherical Bessel Functions -- These are not universal functions:

• sph_jn(n, z) -- Compute the spherical Bessel function jn(z) and its derivative for all orders up to and including n.

• sph_yn(n, z) -- Compute the spherical Bessel function yn(z) and its derivative for all orders up to and including n.

• sph_jnyn(n, z) -- Compute the spherical Bessel functions, jn(z) and yn(z) and their derivatives for all orders up to and including n.

• sph_in(n, z) -- Compute the spherical Bessel function in(z) and its derivative for all orders up to and including n.

• sph_kn(n, z) -- Compute the spherical Bessel function kn(z) and its derivative for all orders up to and including n.

• sph_inkn(n, z) -- Compute the spherical Bessel functions, in(z) and kn(z) and their derivatives for all orders up to and including n.

Riccati-Bessel Functions -- These are not universal functions:

• riccati_jn(n, x) -- Compute the Ricatti-Bessel function of the first kind and its derivative for all orders up to and including n.

• riccati_yn(n, x) -- Compute the Ricatti-Bessel function of the second kind and its derivative for all orders up to and including n.

Struve Functions:

• struve(v,x) -- Struve function

• modstruve(v, x) -- Modified Struve function

• itstruve0(x) -- Integral of the Struve function of order 0

• it2struve0(x) -- Integral related to Struve function of order 0

• itmodstruve0(x) -- Integral of the modified Struve function of order 0

Raw Statistical Functions -- See also scipy.stats: Friendly versions of these functions.

• bdtr(k, n, p) -- Binomial distribution cumulative distribution function.

• bdtrc(k, n, p) -- Binomial distribution survival function.

• bdtri(k, n, y) -- Inverse function to bdtr vs.

• btdtr(a,b,x) -- Cumulative beta distribution.

• btdtri(a,b,p) -- p-th quantile of the beta distribution.

• fdtr(dfn, dfd, x) -- F cumulative distribution function

• fdtrc(dfn, dfd, x) -- F survival function

• fdtri(dfn, dfd, p) -- Inverse to fdtr vs x

• gdtr(a,b,x) -- Gamma distribution cumulative density function.

• gdtrc(a,b,x) -- Gamma distribution survival function.

• gdtria(p, b, x[, out]) -- Inverse of gdtr vs a.

• gdtrib(a, p, x[, out]) -- Inverse of gdtr vs b.

• gdtrix(a, b, p[, out]) -- Inverse of gdtr vs x.

• nbdtr(k, n, p) -- Negative binomial cumulative distribution function

• nbdtrc(k,n,p) -- Negative binomial survival function

• nbdtri(k, n, y) -- Inverse of nbdtr vs p

• pdtr(k, m) -- Poisson cumulative distribution function

• pdtrc(k, m) -- Poisson survival function

• pdtri(k,y) -- Inverse to pdtr vs m

• stdtr(df,t) -- Student t distribution cumulative density function

• stdtridf(p,t) -- Inverse of stdtr vs df

• stdtrit(df,p) -- Inverse of stdtr vs t

• chdtr(v, x) -- Chi square cumulative distribution function

• chdtrc(v,x) -- Chi square survival function

• chdtri(v,p) -- Inverse to chdtrc

• ndtr(x) -- Gaussian cumulative distribution function

• ndtri(y) -- Inverse of ndtr vs x

• smirnov(n,e) -- Kolmogorov-Smirnov complementary cumulative distribution function

• smirnovi(n,y) -- Inverse to smirnov

• kolmogorov(y) -- Complementary cumulative distribution function of Kolmogorov distribution

• kolmogi(p) -- Inverse function to kolmogorov

• tklmbda(x, lmbda) -- Tukey-Lambda cumulative distribution function

• logit(x) -- Logit ufunc for ndarrays.

• expit(x) -- Expit ufunc for ndarrays.

• boxcox(x, lmbda) -- Compute the Box-Cox transformation.

• boxcox1p(x, lmbda) -- Compute the Box-Cox transformation of 1 + x.

Gamma and Related Functions:

• gamma(z) -- Gamma function

• gammaln(z) -- Logarithm of absolute value of gamma function

• gammasgn(x) -- Sign of the gamma function.

• gammainc(a, x) -- Incomplete gamma function

• gammaincinv(a, y) -- Inverse to gammainc

• gammaincc(a,x) -- Complemented incomplete gamma integral

• gammainccinv(a,y) -- Inverse to gammaincc

• beta(a, b) -- Beta function.

• betaln(a, b) -- Natural logarithm of absolute value of beta function.

• betainc(a, b, x) -- Incomplete beta integral.

• betaincinv(a, b, y) -- Inverse function to beta integral.

• psi(z) -- Digamma function

• rgamma(z) -- Gamma function inverted

• polygamma(n, x) -- Polygamma function which is the nth derivative of the digamma (psi) function.

• multigammaln(a, d) -- Returns the log of multivariate gamma, also sometimes called the generalized gamma.

Error Function and Fresnel Integrals:

• erf(z) -- Returns the error function of complex argument.

• erfc(x) -- Complementary error function, 1 - erf(x).

• erfcx(x) -- Scaled complementary error function, exp(x^2) erfc(x).

• erfi(z) -- Imaginary error function, -i erf(i z).

• erfinv(y) -- Inverse function for erf

• erfcinv(y) -- Inverse function for erfc

• wofz(z) -- Faddeeva function

• dawsn(x) -- Dawson’s integral.

• fresnel(z) -- Fresnel sin and cos integrals

• fresnel_zeros(nt) -- Compute nt complex zeros of the sine and cosine Fresnel integrals S(z) and C(z).

• modfresnelp(x) -- Modified Fresnel positive integrals

• modfresnelm(x) -- Modified Fresnel negative integrals

These are not universal functions:

• erf_zeros(nt) -- Compute nt complex zeros of the error function erf(z).

• fresnelc_zeros(nt) -- Compute nt complex zeros of the cosine Fresnel integral C(z).

• fresnels_zeros(nt) -- Compute nt complex zeros of the sine Fresnel integral S(z).

Legendre Functions:

• lpmv(m, v, x) -- Associated legendre function of integer order.

• sph_harm Compute spherical harmonics.

These are not universal functions:

• clpmn(m, n, z[, type]) -- Associated Legendre function of the first kind, Pmn(z)

• lpn(n, z) -- Compute sequence of Legendre functions of the first kind (polynomials), Pn(z) and derivatives for all degrees from 0 to n (inclusive).

• lqn(n, z) -- Compute sequence of Legendre functions of the second kind, Qn(z) and derivatives for all degrees from 0 to n (inclusive).

• lpmn(m, n, z) -- Associated Legendre function of the first kind, Pmn(z)

• lqmn(m, n, z) -- Associated Legendre functions of the second kind, Qmn(z) and its derivative, Qmn'(z) of order m and degree n.

Orthogonal polynomials -- The following functions evaluate values of orthogonal polynomials:

• eval_legendre(n, x[, out]) -- Evaluate Legendre polynomial at a point.

• eval_chebyt(n, x[, out]) -- Evaluate Chebyshev T polynomial at a point.

• eval_chebyu(n, x[, out]) -- Evaluate Chebyshev U polynomial at a point.

• eval_chebyc(n, x[, out]) -- Evaluate Chebyshev C polynomial at a point.

• eval_chebys(n, x[, out]) -- Evaluate Chebyshev S polynomial at a point.

• eval_jacobi(n, alpha, beta, x[, out]) -- Evaluate Jacobi polynomial at a point.

• eval_laguerre(n, x[, out]) -- Evaluate Laguerre polynomial at a point.

• eval_genlaguerre(n, alpha, x[, out]) -- Evaluate generalized Laguerre polynomial at a point.

• eval_hermite(n, x[, out]) -- Evaluate Hermite polynomial at a point.

• eval_hermitenorm(n, x[, out]) -- Evaluate normalized Hermite polynomial at a point.

• eval_gegenbauer(n, alpha, x[, out]) -- Evaluate Gegenbauer polynomial at a point.

• eval_sh_legendre(n, x[, out]) -- Evaluate shifted Legendre polynomial at a point.

• eval_sh_chebyt(n, x[, out]) -- Evaluate shifted Chebyshev T polynomial at a point.

• eval_sh_chebyu(n, x[, out]) -- Evaluate shifted Chebyshev U polynomial at a point.

• eval_sh_jacobi(n, p, q, x[, out]) -- Evaluate shifted Jacobi polynomial at a point.

The functions below, in turn, return orthopoly1d objects, which functions similarly as numpy.poly1d. The orthopoly1d class also has an attribute weights which returns the roots, weights, and total weights for the appropriate form of Gaussian quadrature. These are returned in an n x 3 array with roots in the first column, weights in the second column, and total weights in the final column:

• legendre(n[, monic]) -- Returns the nth order Legendre polynomial, P_n(x), orthogonal over [-1,1] with weight function 1.

• chebyt(n[, monic]) -- Return nth order Chebyshev polynomial of first kind, Tn(x).

• chebyu(n[, monic]) -- Return nth order Chebyshev polynomial of second kind, Un(x).

• chebyc(n[, monic]) -- Return nth order Chebyshev polynomial of first kind, Cn(x).

• chebys(n[, monic]) -- Return nth order Chebyshev polynomial of second kind, Sn(x).

• jacobi(n, alpha, beta[, monic]) -- Returns the nth order Jacobi polynomial, P^(alpha,beta)_n(x) orthogonal over [-1,1] with weighting function (1-x)**alpha (1+x)**beta with alpha,beta > -1.

• laguerre(n[, monic]) -- Return the nth order Laguerre polynoimal, L_n(x), orthogonal over

• genlaguerre(n, alpha[, monic]) -- Returns the nth order generalized (associated) Laguerre polynomial,

• hermite(n[, monic]) -- Return the nth order Hermite polynomial, H_n(x), orthogonal over

• hermitenorm(n[, monic]) -- Return the nth order normalized Hermite polynomial, He_n(x), orthogonal

• gegenbauer(n, alpha[, monic]) -- Return the nth order Gegenbauer (ultraspherical) polynomial,

• sh_legendre(n[, monic]) -- Returns the nth order shifted Legendre polynomial, P^*_n(x), orthogonal over [0,1] with weighting function 1.

• sh_chebyt(n[, monic]) -- Return nth order shifted Chebyshev polynomial of first kind, Tn(x).

• sh_chebyu(n[, monic]) -- Return nth order shifted Chebyshev polynomial of second kind, Un(x).

• sh_jacobi(n, p, q[, monic]) -- Returns the nth order Jacobi polynomial, G_n(p,q,x) orthogonal over [0,1] with weighting function (1-x)**(p-q) (x)**(q-1) with p>q-1 and q > 0.

Warning: Large-order polynomials obtained from these functions are numerically unstable. orthopoly1d objects are converted to poly1d, when doing arithmetic. numpy.poly1d works in power basis and cannot represent high-order polynomials accurately, which can cause significant inaccuracy.

Hypergeometric Functions:

• hyp2f1(a, b, c, z) -- Gauss hypergeometric function 2F1(a, b; c; z).

• hyp1f1(a, b, x) -- Confluent hypergeometric function 1F1(a, b; x)

• hyperu(a, b, x) -- Confluent hypergeometric function U(a, b, x) of the second kind

• hyp0f1(v, z) -- Confluent hypergeometric limit function 0F1.

• hyp2f0(a, b, x, type) -- Hypergeometric function 2F0 in y and an error estimate

• hyp1f2(a, b, c, x) -- Hypergeometric function 1F2 and error estimate

• hyp3f0(a, b, c, x) -- Hypergeometric function 3F0 in y and an error estimate

Parabolic Cylinder Functions:

• pbdv(v, x) -- Parabolic cylinder function D

• pbvv(v,x) -- Parabolic cylinder function V

• pbwa(a,x) -- Parabolic cylinder function W

These are not universal functions:

• pbdv_seq(v, x) -- Compute sequence of parabolic cylinder functions Dv(x) and their derivatives for Dv0(x)..Dv(x) with v0=v-int(v).

• pbvv_seq(v, x) -- Compute sequence of parabolic cylinder functions Dv(x) and their derivatives for Dv0(x)..Dv(x) with v0=v-int(v).

• pbdn_seq(n, z) -- Compute sequence of parabolic cylinder functions Dn(z) and their derivatives for D0(z)..Dn(z).

Mathieu and Related Functions:

• mathieu_a(m,q) -- Characteristic value of even Mathieu functions

• mathieu_b(m,q) -- Characteristic value of odd Mathieu functions

These are not universal functions:

• mathieu_even_coef(m, q) -- Compute expansion coefficients for even Mathieu functions and modified Mathieu functions.

• mathieu_odd_coef(m, q) -- Compute expansion coefficients for even Mathieu functions and modified Mathieu functions.

The following return both function and first derivative:

• mathieu_cem(m,q,x) -- Even Mathieu function and its derivative

• mathieu_sem(m, q, x) -- Odd Mathieu function and its derivative

• mathieu_modcem1(m, q, x) -- Even modified Mathieu function of the first kind and its derivative

• mathieu_modcem2(m, q, x) -- Even modified Mathieu function of the second kind and its derivative

• mathieu_modsem1(m,q,x) -- Odd modified Mathieu function of the first kind and its derivative

• mathieu_modsem2(m, q, x) -- Odd modified Mathieu function of the second kind and its derivative

Spheroidal Wave Functions:

• pro_ang1(m,n,c,x) -- Prolate spheroidal angular function of the first kind and its derivative

• pro_rad1(m,n,c,x) -- Prolate spheroidal radial function of the first kind and its derivative

• pro_rad2(m,n,c,x) -- Prolate spheroidal radial function of the secon kind and its derivative

• obl_ang1(m, n, c, x) -- Oblate spheroidal angular function of the first kind and its derivative

• obl_rad1(m,n,c,x) -- Oblate spheroidal radial function of the first kind and its derivative

• obl_rad2(m,n,c,x) -- Oblate spheroidal radial function of the second kind and its derivative.

• pro_cv(m,n,c) -- Characteristic value of prolate spheroidal function

• obl_cv(m, n, c) -- Characteristic value of oblate spheroidal function

• pro_cv_seq(m, n, c) -- Compute a sequence of characteristic values for the prolate spheroidal wave functions for mode m and n’=m..n and spheroidal parameter c.

• obl_cv_seq(m, n, c) -- Compute a sequence of characteristic values for the oblate spheroidal wave functions for mode m and n’=m..n and spheroidal parameter c.

The following functions require pre-computed characteristic value:

• pro_ang1_cv(m,n,c,cv,x) -- Prolate sheroidal angular function pro_ang1 for precomputed characteristic value

• pro_rad1_cv(m,n,c,cv,x) -- Prolate sheroidal radial function pro_rad1 for precomputed characteristic value

• pro_rad2_cv(m,n,c,cv,x) -- Prolate sheroidal radial function pro_rad2 for precomputed characteristic value

• obl_ang1_cv(m, n, c, cv, x) -- Oblate sheroidal angular function obl_ang1 for precomputed characteristic value

• obl_rad1_cv(m,n,c,cv,x) -- Oblate sheroidal radial function obl_rad1 for precomputed characteristic value

• obl_rad2_cv(m,n,c,cv,x) -- Oblate sheroidal radial function obl_rad2 for precomputed characteristic value

Kelvin Functions:

• kelvin(x) -- Kelvin functions as complex numbers

• kelvin_zeros(nt) -- Compute nt zeros of all the Kelvin functions returned in a length 8 tuple of arrays of length nt.

• ber(x) -- Kelvin function ber.

• bei(x) -- Kelvin function bei

• berp(x) -- Derivative of the Kelvin function ber

• beip(x) -- Derivative of the Kelvin function bei

• ker(x) -- Kelvin function ker

• kei(x) -- Kelvin function ker

• kerp(x) -- Derivative of the Kelvin function ker

• keip(x) -- Derivative of the Kelvin function kei

These are not universal functions:

• ber_zeros(nt) -- Compute nt zeros of the Kelvin function ber x

• bei_zeros(nt) -- Compute nt zeros of the Kelvin function bei x

• berp_zeros(nt) -- Compute nt zeros of the Kelvin function ber’ x

• beip_zeros(nt) -- Compute nt zeros of the Kelvin function bei’ x

• ker_zeros(nt) -- Compute nt zeros of the Kelvin function ker x

• kei_zeros(nt) -- Compute nt zeros of the Kelvin function kei x

• kerp_zeros(nt) -- Compute nt zeros of the Kelvin function ker’ x

• keip_zeros(nt) -- Compute nt zeros of the Kelvin function kei’ x

Combinatorics:

• comb(N, k[, exact, repetition]) -- The number of combinations of N things taken k at a time.

• perm(N, k[, exact]) -- Permutations of N things taken k at a time, i.e., k-permutations of N.

Other Special Functions:

• binom(n, k) -- Binomial coefficient

• expn(n, x) -- Exponential integral E_n

• exp1(z) -- Exponential integral E_1 of complex argument z

• expi(x) -- Exponential integral Ei

• factorial(n[, exact]) -- The factorial function, n! = special.gamma(n+1).

• factorial2(n[, exact]) -- Double factorial.

• factorialk(n, k[, exact]) -- n(!!...!) = multifactorial of order k

• shichi(x) -- Hyperbolic sine and cosine integrals

• sici(x) -- Sine and cosine integrals

• spence(x) -- Dilogarithm integral

• lambertw(z[, k, tol]) -- Lambert W function [R416].

• zeta(x, q) -- Hurwitz zeta function

• zetac(x) -- Riemann zeta function minus 1.

Convenience Functions:

• cbrt(x) -- Cube root of x

• exp10(x) -- 10**x

• exp2(x) -- 2**x

• radian(d, m, s) -- Convert from degrees to radians

• cosdg(x) -- Cosine of the angle x given in degrees.

• sindg(x) -- Sine of angle given in degrees

• tandg(x) -- Tangent of angle x given in degrees.

• cotdg(x) -- Cotangent of the angle x given in degrees.

• log1p(x) -- Calculates log(1+x) for use when x is near zero

• expm1(x) -- exp(x) - 1 for use when x is near zero.

• cosm1(x) -- cos(x) - 1 for use when x is near zero.

• round(x) -- Round to nearest integer

• xlogy(x, y) -- Compute x*log(y) so that the result is 0 if x = 0.

• xlog1py(x, y) -- Compute x*log1p(y) so that the result is 0 if x = 0.

8.14 stats

Statistical functions -- This module contains a large number of probability distributions as well as a growing library of statistical functions.

Access via:

In [1]: from scipy import stats

Also see: http://docs.scipy.org/doc/scipy/reference/tutorial/stats.html

For an overview, in IPython, do:

In [1]: from scipy import stats
In [2]: stats?

Each included distribution is an instance of the class rv_continous: For each given name the following methods are available:

• rv_continuous([momtype, a, b, xtol, ...]) -- A generic continuous random variable class meant for subclassing.

• rv_continuous.pdf(x, *args, **kwds) -- Probability density function at x of the given RV.

• rv_continuous.logpdf(x, *args, **kwds) -- Log of the probability density function at x of the given RV.

• rv_continuous.cdf(x, *args, **kwds) -- Cumulative distribution function of the given RV.

• rv_continuous.logcdf(x, *args, **kwds) -- Log of the cumulative distribution function at x of the given RV.

• rv_continuous.sf(x, *args, **kwds) -- Survival function (1-cdf) at x of the given RV.

• rv_continuous.logsf(x, *args, **kwds) -- Log of the survival function of the given RV.

• rv_continuous.ppf(q, *args, **kwds) -- Percent point function (inverse of cdf) at q of the given RV.

• rv_continuous.isf(q, *args, **kwds) -- Inverse survival function at q of the given RV.

• rv_continuous.moment(n, *args, **kwds) -- n’th order non-central moment of distribution.

• rv_continuous.stats(*args, **kwds) -- Some statistics of the given RV

• rv_continuous.entropy(*args, **kwds) -- Differential entropy of the RV.

• rv_continuous.fit(data, *args, **kwds) -- Return MLEs for shape, location, and scale parameters from data.

• rv_continuous.expect([func, args, loc, ...]) -- Calculate expected value of a function with respect to the distribution.

Calling the instance as a function returns a frozen pdf whose shape, location, and scale parameters are fixed.

Similarly, each discrete distribution is an instance of the class rv_discrete:

• rv_discrete([a, b, name, badvalue, ...]) -- A generic discrete random variable class meant for subclassing.

• rv_discrete.rvs(*args, **kwargs) -- Random variates of given type.

• rv_discrete.pmf(k, *args, **kwds) -- Probability mass function at k of the given RV.

• rv_discrete.logpmf(k, *args, **kwds) -- Log of the probability mass function at k of the given RV.

• rv_discrete.cdf(k, *args, **kwds) -- Cumulative distribution function of the given RV.

• rv_discrete.logcdf(k, *args, **kwds) -- Log of the cumulative distribution function at k of the given RV

• rv_discrete.sf(k, *args, **kwds) -- Survival function (1-cdf) at k of the given RV.

• rv_discrete.logsf(k, *args, **kwds) -- Log of the survival function of the given RV.

• rv_discrete.ppf(q, *args, **kwds) -- Percent point function (inverse of cdf) at q of the given RV

• rv_discrete.isf(q, *args, **kwds) -- Inverse survival function (1-sf) at q of the given RV.

• rv_discrete.stats(*args, **kwds) -- Some statistics of the given RV

• rv_discrete.moment(n, *args, **kwds) -- n’th order non-central moment of distribution.

• rv_discrete.entropy(*args, **kwds) -- Differential entropy of the RV.

• rv_discrete.expect([func, args, loc, lb, ...]) -- Calculate expected value of a function with respect to the distribution

Continuous distributions:

• alpha -- An alpha continuous random variable.

• anglit -- An anglit continuous random variable.

• arcsine -- An arcsine continuous random variable.

• beta -- A beta continuous random variable.

• betaprime -- A beta prime continuous random variable.

• bradford -- A Bradford continuous random variable.

• burr -- A Burr continuous random variable.

• cauchy -- A Cauchy continuous random variable.

• chi -- A chi continuous random variable.

• chi2 -- A chi-squared continuous random variable.

• cosine -- A cosine continuous random variable.

• dgamma -- A double gamma continuous random variable.

• dweibull -- A double Weibull continuous random variable.

• erlang -- An Erlang continuous random variable.

• expon -- An exponential continuous random variable.

• exponweib -- An exponentiated Weibull continuous random variable.

• exponpow -- An exponential power continuous random variable.

• f -- An F continuous random variable.

• fatiguelife -- A fatigue-life (Birnbaum-Sanders) continuous random variable.

• fisk -- A Fisk continuous random variable.

• foldcauchy -- A folded Cauchy continuous random variable.

• foldnorm -- A folded normal continuous random variable.

• frechet_r -- A Frechet right (or Weibull minimum) continuous random variable.

• frechet_l -- A Frechet left (or Weibull maximum) continuous random variable.

• genlogistic -- A generalized logistic continuous random variable.

• genpareto -- A generalized Pareto continuous random variable.

• genexpon -- A generalized exponential continuous random variable.

• genextreme -- A generalized extreme value continuous random variable.

• gausshyper -- A Gauss hypergeometric continuous random variable.

• gamma -- A gamma continuous random variable.

• gengamma -- A generalized gamma continuous random variable.

• genhalflogistic -- A generalized half-logistic continuous random variable.

• gilbrat -- A Gilbrat continuous random variable.

• gompertz -- A Gompertz (or truncated Gumbel) continuous random variable.

• gumbel_r -- A right-skewed Gumbel continuous random variable.

• gumbel_l -- A left-skewed Gumbel continuous random variable.

• halfcauchy -- A Half-Cauchy continuous random variable.

• halflogistic -- A half-logistic continuous random variable.

• halfnorm -- A half-normal continuous random variable.

• hypsecant -- A hyperbolic secant continuous random variable.

• invgamma -- An inverted gamma continuous random variable.

• invgauss -- An inverse Gaussian continuous random variable.

• invweibull -- An inverted Weibull continuous random variable.

• johnsonsb -- A Johnson SB continuous random variable.

• johnsonsu -- A Johnson SU continuous random variable.

• ksone -- General Kolmogorov-Smirnov one-sided test.

• kstwobign -- Kolmogorov-Smirnov two-sided test for large N.

• laplace -- A Laplace continuous random variable.

• logistic -- A logistic (or Sech-squared) continuous random variable.

• loggamma -- A log gamma continuous random variable.

• loglaplace -- A log-Laplace continuous random variable.

• lognorm -- A lognormal continuous random variable.

• lomax -- A Lomax (Pareto of the second kind) continuous random variable.

• maxwell -- A Maxwell continuous random variable.

• mielke -- A Mielke’s Beta-Kappa continuous random variable.

• nakagami -- A Nakagami continuous random variable.

• ncx2 -- A non-central chi-squared continuous random variable.

• ncf -- A non-central F distribution continuous random variable.

• nct -- A non-central Student’s T continuous random variable.

• norm -- A normal continuous random variable.

• pareto -- A Pareto continuous random variable.

• pearson3 -- A pearson type III continuous random variable.

• powerlaw -- A power-function continuous random variable.

• powerlognorm -- A power log-normal continuous random variable.

• powernorm -- A power normal continuous random variable.

• rdist -- An R-distributed continuous random variable.

• reciprocal -- A reciprocal continuous random variable.

• rayleigh -- A Rayleigh continuous random variable.

• rice -- A Rice continuous random variable.

• recipinvgauss -- A reciprocal inverse Gaussian continuous random variable.

• semicircular -- A semicircular continuous random variable.

• t -- A Student’s T continuous random variable.

• triang -- A triangular continuous random variable.

• truncexpon -- A truncated exponential continuous random variable.

• truncnorm -- A truncated normal continuous random variable.

• tukeylambda -- A Tukey-Lamdba continuous random variable.

• uniform -- A uniform continuous random variable.

• vonmises -- A Von Mises continuous random variable.

• wald -- A Wald continuous random variable.

• weibull_min -- A Frechet right (or Weibull minimum) continuous random variable.

• weibull_max -- A Frechet left (or Weibull maximum) continuous random variable.

• wrapcauchy -- A wrapped Cauchy continuous random variable.

• Multivariate -- distributions

• multivariate_normal -- A multivariate normal random variable.

• Discrete -- distributions

• bernoulli -- A Bernoulli discrete random variable.

• binom -- A binomial discrete random variable.

• boltzmann -- A Boltzmann (Truncated Discrete Exponential) random variable.

• dlaplace -- A Laplacian discrete random variable.

• geom -- A geometric discrete random variable.

• hypergeom -- A hypergeometric discrete random variable.

• logser -- A Logarithmic (Log-Series, Series) discrete random variable.

• nbinom -- A negative binomial discrete random variable.

• planck -- A Planck discrete exponential random variable.

• poisson -- A Poisson discrete random variable.

• randint -- A uniform discrete random variable.

• skellam -- A Skellam discrete random variable.

• zipf -- A Zipf discrete random variable.

Statistical functions -- Several of these functions have a similar version in scipy.stats.mstats which work for masked arrays.

• describe(a[, axis]) -- Computes several descriptive statistics of the passed array.

• gmean(a[, axis, dtype]) -- Compute the geometric mean along the specified axis.

• hmean(a[, axis, dtype]) -- Calculates the harmonic mean along the specified axis.

• kurtosis(a[, axis, fisher, bias]) -- Computes the kurtosis (Fisher or Pearson) of a dataset.

• kurtosistest(a[, axis]) -- Tests whether a dataset has normal kurtosis

• mode(a[, axis]) -- Returns an array of the modal (most common) value in the passed array.

• moment(a[, moment, axis]) -- Calculates the nth moment about the mean for a sample.

• normaltest(a[, axis]) -- Tests whether a sample differs from a normal distribution.

• skew(a[, axis, bias]) -- Computes the skewness of a data set.

• skewtest(a[, axis]) -- Tests whether the skew is different from the normal distribution.

• tmean(a[, limits, inclusive]) -- Compute the trimmed mean.

• tvar(a[, limits, inclusive]) -- Compute the trimmed variance

• tmin(a[, lowerlimit, axis, inclusive]) -- Compute the trimmed minimum

• tmax(a[, upperlimit, axis, inclusive]) -- Compute the trimmed maximum

• tstd(a[, limits, inclusive]) -- Compute the trimmed sample standard deviation

• tsem(a[, limits, inclusive]) -- Compute the trimmed standard error of the mean.

• nanmean(x[, axis]) -- Compute the mean over the given axis ignoring nans.

• nanstd(x[, axis, bias]) -- Compute the standard deviation over the given axis, ignoring nans.

• nanmedian(x[, axis]) -- Compute the median along the given axis ignoring nan values.

• variation(a[, axis]) -- Computes the coefficient of variation, the ratio of the biased standard deviation to the mean.

• cumfreq(a[, numbins, defaultreallimits, weights]) -- Returns a cumulative frequency histogram, using the histogram function.

• histogram2(a, bins) -- Compute histogram using divisions in bins.

• histogram(a[, numbins, defaultlimits, ...]) -- Separates the range into several bins and returns the number of instances in each bin.

• itemfreq(a) -- Returns a 2-D array of item frequencies.

• percentileofscore(a, score[, kind]) -- The percentile rank of a score relative to a list of scores.

• scoreatpercentile(a, per[, limit, ...]) -- Calculate the score at a given percentile of the input sequence.

• relfreq(a[, numbins, defaultreallimits, weights]) -- Returns a relative frequency histogram, using the histogram function.

• binned_statistic(x, values[, statistic, ...]) -- Compute a binned statistic for a set of data.

• binned_statistic_2d(x, y, values[, ...]) -- Compute a bidimensional binned statistic for a set of data.

• binned_statistic_dd(sample, values[, ...]) -- Compute a multidimensional binned statistic for a set of data.

• obrientransform(*args) -- Computes the O’Brien transform on input data (any number of arrays).

• signaltonoise(a[, axis, ddof]) -- The signal-to-noise ratio of the input data.

• bayes_mvs(data[, alpha]) -- Bayesian confidence intervals for the mean, var, and std.

• sem(a[, axis, ddof]) -- Calculates the standard error of the mean (or standard error of measurement) of the values in the input array.

• zmap(scores, compare[, axis, ddof]) -- Calculates the relative z-scores.

• zscore(a[, axis, ddof]) -- Calculates the z score of each value in the sample, relative to the sample mean and standard deviation.

• sigmaclip(a[, low, high]) -- Iterative sigma-clipping of array elements.

• threshold(a[, threshmin, threshmax, newval]) -- Clip array to a given value.

• trimboth(a, proportiontocut[, axis]) -- Slices off a proportion of items from both ends of an array.

• trim1(a, proportiontocut[, tail]) -- Slices off a proportion of items from ONE end of the passed array distribution.

• f_oneway(*args) -- Performs a 1-way ANOVA.

• pearsonr(x, y) -- Calculates a Pearson correlation coefficient and the p-value for testing non-correlation.

• spearmanr(a[, b, axis]) -- Calculates a Spearman rank-order correlation coefficient and the p-value to test for non-correlation.

• pointbiserialr(x, y) -- Calculates a point biserial correlation coefficient and the associated p-value.

• kendalltau(x, y[, initial_lexsort]) -- Calculates Kendall’s tau, a correlation measure for ordinal data.

• linregress(x[, y]) -- Calculate a regression line

• ttest_1samp(a, popmean[, axis]) -- Calculates the T-test for the mean of ONE group of scores.

• ttest_ind(a, b[, axis, equal_var]) -- Calculates the T-test for the means of TWO INDEPENDENT samples of scores.

• ttest_rel(a, b[, axis]) -- Calculates the T-test on TWO RELATED samples of scores, a and b.

• kstest(rvs, cdf[, args, N, alternative, mode]) -- Perform the Kolmogorov-Smirnov test for goodness of fit.

• chisquare(f_obs[, f_exp, ddof, axis]) -- Calculates a one-way chi square test.

• power_divergence(f_obs[, f_exp, ddof, axis, ...]) -- Cressie-Read power divergence statistic and goodness of fit test.

• ks_2samp(data1, data2) -- Computes the Kolmogorov-Smirnov statistic on 2 samples.

• mannwhitneyu(x, y[, use_continuity]) -- Computes the Mann-Whitney rank test on samples x and y.

• tiecorrect(rankvals) -- Tie correction factor for ties in the Mann-Whitney U and Kruskal-Wallis H tests.

• rankdata(a[, method]) -- Assign ranks to data, dealing with ties appropriately.

• ranksums(x, y) -- Compute the Wilcoxon rank-sum statistic for two samples.

• wilcoxon(x[, y, zero_method, correction]) -- Calculate the Wilcoxon signed-rank test.

• kruskal(*args) -- Compute the Kruskal-Wallis H-test for independent samples

• friedmanchisquare(*args) -- Computes the Friedman test for repeated measurements

• ansari(x, y) -- Perform the Ansari-Bradley test for equal scale parameters

• bartlett(*args) -- Perform Bartlett’s test for equal variances

• levene(*args, **kwds) -- Perform Levene test for equal variances.

• shapiro(x[, a, reta]) -- Perform the Shapiro-Wilk test for normality.

• anderson(x[, dist]) -- Anderson-Darling test for data coming from a particular distribution

• anderson_ksamp(samples[, midrank]) -- The Anderson-Darling test for k-samples.

• binom_test(x[, n, p]) -- Perform a test that the probability of success is p.

• fligner(*args, **kwds) -- Perform Fligner’s test for equal variances.

• mood(x, y[, axis]) -- Perform Mood’s test for equal scale parameters.

• boxcox(x[, lmbda, alpha]) -- Return a positive dataset transformed by a Box-Cox power transformation.

• boxcox_normmax(x[, brack, method]) -- Compute optimal Box-Cox transform parameter for input data.

• boxcox_llf(lmb, data) -- The boxcox log-likelihood function.

• entropy(pk[, qk, base]) -- Calculate the entropy of a distribution for given probability values.

Contingency table functions:

• chi2_contingency(observed[, correction, lambda_]) -- Chi-square test of independence of variables in a contingency table.

• contingency.expected_freq(observed) -- Compute the expected frequencies from a contingency table.

• contingency.margins(a) -- Return a list of the marginal sums of the array a.

• fisher_exact(table[, alternative]) -- Performs a Fisher exact test on a 2x2 contingency table.

Plot-tests:

• ppcc_max(x[, brack, dist]) -- Returns the shape parameter that maximizes the probability plot correlation coefficient for the given data to a one-parameter family of distributions.

• ppcc_plot(x, a, b[, dist, plot, N]) -- Returns (shape, ppcc), and optionally plots shape vs.

• probplot(x[, sparams, dist, fit, plot]) -- Calculate quantiles for a probability plot, and optionally show the plot.

• boxcox_normplot(x, la, lb[, plot, N]) -- Compute parameters for a Box-Cox normality plot, optionally show it.

Masked statistics functions:

Statistical functions for masked arrays (scipy.stats.mstats):

• scipy.stats.mstats.argstoarray

• scipy.stats.mstats.betai

• scipy.stats.mstats.chisquare

• scipy.stats.mstats.count_tied_groups

• scipy.stats.mstats.describe

• scipy.stats.mstats.f_oneway

• scipy.stats.mstats.f_value_wilks_lambda

• scipy.stats.mstats.find_repeats

• scipy.stats.mstats.friedmanchisquare

• scipy.stats.mstats.kendalltau

• scipy.stats.mstats.kendalltau_seasonal

• scipy.stats.mstats.kruskalwallis

• scipy.stats.mstats.kruskalwallis

• scipy.stats.mstats.ks_twosamp

• scipy.stats.mstats.ks_twosamp

• scipy.stats.mstats.kurtosis

• scipy.stats.mstats.kurtosistest

• scipy.stats.mstats.linregress

• scipy.stats.mstats.mannwhitneyu

• scipy.stats.mstats.plotting_positions

• scipy.stats.mstats.mode

• scipy.stats.mstats.moment

• scipy.stats.mstats.mquantiles

• scipy.stats.mstats.msign

• scipy.stats.mstats.normaltest

• scipy.stats.mstats.obrientransform

• scipy.stats.mstats.pearsonr

• scipy.stats.mstats.plotting_positions

• scipy.stats.mstats.pointbiserialr

• scipy.stats.mstats.rankdata

• scipy.stats.mstats.scoreatpercentile

• scipy.stats.mstats.sem

• scipy.stats.mstats.signaltonoise

• scipy.stats.mstats.skew

• scipy.stats.mstats.skewtest

• scipy.stats.mstats.spearmanr

• scipy.stats.mstats.theilslopes

• scipy.stats.mstats.threshold

• scipy.stats.mstats.tmax

• scipy.stats.mstats.tmean

• scipy.stats.mstats.tmin

• scipy.stats.mstats.trim

• scipy.stats.mstats.trima

• scipy.stats.mstats.trimboth

• scipy.stats.mstats.trimmed_stde

• scipy.stats.mstats.trimr

• scipy.stats.mstats.trimtail

• scipy.stats.mstats.tsem

• scipy.stats.mstats.ttest_onesamp

• scipy.stats.mstats.ttest_ind

• scipy.stats.mstats.ttest_onesamp

• scipy.stats.mstats.ttest_rel

• scipy.stats.mstats.tvar

• scipy.stats.mstats.variation

• scipy.stats.mstats.winsorize

• scipy.stats.mstats.zmap

• scipy.stats.mstats.zscore

Univariate and multivariate kernel density estimation (scipy.stats.kde):

• gaussian_kde(dataset[, bw_method]) -- Representation of a kernel-density estimate using Gaussian kernels.

For many more stat related functions install the software R and the interface package rpy.

8.15 Miscellaneous routines

Miscellaneous routines

Access via:

In [1]: from scipy import misc

Note that the Python Imaging Library (PIL) is not a dependency of SciPy and therefore the pilutil module is not available on systems that don’t have PIL installed.

• bytescale(data[, cmin, cmax, high, low]) -- Byte scales an array (image).

• central_diff_weights(Np[, ndiv]) -- Return weights for an Np-point central derivative.

• comb(N, k[, exact, repetition]) -- The number of combinations of N things taken k at a time.

• derivative(func, x0[, dx, n, args, order]) -- Find the n-th derivative of a function at a point.

• factorial(n[, exact]) -- The factorial function, n! = special.gamma(n+1).

• factorial2(n[, exact]) -- Double factorial.

• factorialk(n, k[, exact]) -- n(!!...!) = multifactorial of order k

• fromimage(im[, flatten]) -- Return a copy of a PIL image as a numpy array.

• imfilter(arr, ftype) -- Simple filtering of an image.

• imread(name[, flatten]) -- Read an image from a file as an array.

• imresize(arr, size[, interp, mode]) -- Resize an image.

• imrotate(arr, angle[, interp]) -- Rotate an image counter-clockwise by angle degrees.

• imsave(name, arr[, format]) -- Save an array as an image.

• imshow(arr) -- Simple showing of an image through an external viewer.

• info([object, maxwidth, output, toplevel]) -- Get help information for a function, class, or module.

• lena() -- Get classic image processing example image, Lena, at 8-bit grayscale bit-depth, 512 x 512 size.

• logsumexp(a[, axis, b]) -- Compute the log of the sum of exponentials of input elements.

• pade(an, m) -- Return Pade approximation to a polynomial as the ratio of two polynomials.

• toimage(arr[, high, low, cmin, cmax, pal, ...]) -- Takes a numpy array and returns a PIL image.

• who([vardict]) -- Print the Numpy arrays in the given dictionary.