Packaging (numpy.distutils) NumPy provides enhanced distutils functionality to make it easier to build and install sub-packages, auto-generate code, and extension modules that use Fortran-compiled libraries. To use features of NumPy distutils, use the setup command from numpy.distutils.core. A useful Configuration class is also provided in numpy.distutils.misc_util that can make it easier to construct keyword arguments to pass to the setup function (by passing the dictionary obtained from the todict() method of the class). More information is available in the NumPy Distutils - Users Guide. The choice and location of linked libraries such as BLAS and LAPACK as well as include paths and other such build options can be specified in a site.cfg file located in the NumPy root repository or a .numpy-site.cfg file in your home directory. See the site.cfg.example example file included in the NumPy repository or sdist for documentation.  Modules in numpy.distutils  distutils.misc_util
ccompiler
ccompiler_opt Provides the CCompilerOpt class, used for handling the CPU/hardware optimization, starting from parsing the command arguments, to managing the relation between the CPU baseline and dispatch-able features, also generating the required C headers and ending with compiling the sources with proper compiler's flags.
cpuinfo.cpu
core.Extension(name, sources[, ...])
 Parameters
exec_command exec_command
log.set_verbosity(v[, force])
system_info.get_info(name[, notfound_action]) notfound_action:
system_info.get_standard_file(fname) Returns a list of files named 'fname' from 1) System-wide directory (directory-location of this module) 2) Users HOME directory (os.environ['HOME']) 3) Local directory     Configuration class   class numpy.distutils.misc_util.Configuration(package_name=None, parent_name=None, top_path=None, package_path=None, **attrs)[source]

Construct a configuration instance for the given package name. If parent_name is not None, then construct the package as a sub-package of the parent_name package. If top_path and package_path are None then they are assumed equal to the path of the file this instance was created in. The setup.py files in the numpy distribution are good examples of how to use the Configuration instance.   todict()[source]

Return a dictionary compatible with the keyword arguments of distutils setup function. Examples >>> setup(**config.todict())

   get_distribution()[source]

Return the distutils distribution object for self.
   get_subpackage(subpackage_name, subpackage_path=None, parent_name=None, caller_level=1)[source]

Return list of subpackage configurations.  Parameters

subpackage_namestr or None

Name of the subpackage to get the configuration. ‘*’ in subpackage_name is handled as a wildcard.
subpackage_pathstr

If None, then the path is assumed to be the local path plus the subpackage_name. If a setup.py file is not found in the subpackage_path, then a default configuration is used.
parent_namestr

Parent name.
   add_subpackage(subpackage_name, subpackage_path=None, standalone=False)[source]

Add a sub-package to the current Configuration instance. This is useful in a setup.py script for adding sub-packages to a package.  Parameters

subpackage_namestr

name of the subpackage
subpackage_pathstr

if given, the subpackage path such as the subpackage is in subpackage_path / subpackage_name. If None,the subpackage is assumed to be located in the local path / subpackage_name.
standalonebool

   add_data_files(*files)[source]

Add data files to configuration data_files.  Parameters

filessequence

Argument(s) can be either  2-sequence (<datadir prefix>,<path to data file(s)>) paths to data files where python datadir prefix defaults to package dir.      Notes The form of each element of the files sequence is very flexible allowing many combinations of where to get the files from the package and where they should ultimately be installed on the system. The most basic usage is for an element of the files argument sequence to be a simple filename. This will cause that file from the local path to be installed to the installation path of the self.name package (package path). The file argument can also be a relative path in which case the entire relative path will be installed into the package directory. Finally, the file can be an absolute path name in which case the file will be found at the absolute path name but installed to the package path. This basic behavior can be augmented by passing a 2-tuple in as the file argument. The first element of the tuple should specify the relative path (under the package install directory) where the remaining sequence of files should be installed to (it has nothing to do with the file-names in the source distribution). The second element of the tuple is the sequence of files that should be installed. The files in this sequence can be filenames, relative paths, or absolute paths. For absolute paths the file will be installed in the top-level package installation directory (regardless of the first argument). Filenames and relative path names will be installed in the package install directory under the path name given as the first element of the tuple. Rules for installation paths:  file.txt -> (., file.txt)-> parent/file.txt foo/file.txt -> (foo, foo/file.txt) -> parent/foo/file.txt /foo/bar/file.txt -> (., /foo/bar/file.txt) -> parent/file.txt
*.txt -> parent/a.txt, parent/b.txt foo/*.txt`` -> parent/foo/a.txt, parent/foo/b.txt
*/*.txt -> (*, */*.txt) -> parent/c/a.txt, parent/d/b.txt (sun, file.txt) -> parent/sun/file.txt (sun, bar/file.txt) -> parent/sun/file.txt (sun, /foo/bar/file.txt) -> parent/sun/file.txt (sun, *.txt) -> parent/sun/a.txt, parent/sun/b.txt (sun, bar/*.txt) -> parent/sun/a.txt, parent/sun/b.txt (sun/*, */*.txt) -> parent/sun/c/a.txt, parent/d/b.txt  An additional feature is that the path to a data-file can actually be a function that takes no arguments and returns the actual path(s) to the data-files. This is useful when the data files are generated while building the package. Examples Add files to the list of data_files to be included with the package. >>> self.add_data_files('foo.dat',
...     ('fun', ['gun.dat', 'nun/pun.dat', '/tmp/sun.dat']),
...     'bar/cat.dat',
...     '/full/path/to/can.dat')
 will install these data files to: <package install directory>/
 foo.dat
 fun/
   gun.dat
   nun/
     pun.dat
 sun.dat
 bar/
   car.dat
 can.dat
 where <package install directory> is the package (or sub-package) directory such as ‘/usr/lib/python2.4/site-packages/mypackage’ (‘C: Python2.4 Lib site-packages mypackage’) or ‘/usr/lib/python2.4/site- packages/mypackage/mysubpackage’ (‘C: Python2.4 Lib site-packages mypackage mysubpackage’).
   add_data_dir(data_path)[source]

Recursively add files under data_path to data_files list. Recursively add files under data_path to the list of data_files to be installed (and distributed). The data_path can be either a relative path-name, or an absolute path-name, or a 2-tuple where the first argument shows where in the install directory the data directory should be installed to.  Parameters

data_pathseq or str

Argument can be either  2-sequence (<datadir suffix>, <path to data directory>) path to data directory where python datadir suffix defaults to package dir.      Notes Rules for installation paths: foo/bar -> (foo/bar, foo/bar) -> parent/foo/bar
(gun, foo/bar) -> parent/gun
foo/* -> (foo/a, foo/a), (foo/b, foo/b) -> parent/foo/a, parent/foo/b
(gun, foo/*) -> (gun, foo/a), (gun, foo/b) -> gun
(gun/*, foo/*) -> parent/gun/a, parent/gun/b
/foo/bar -> (bar, /foo/bar) -> parent/bar
(gun, /foo/bar) -> parent/gun
(fun/*/gun/*, sun/foo/bar) -> parent/fun/foo/gun/bar
 Examples For example suppose the source directory contains fun/foo.dat and fun/bar/car.dat: >>> self.add_data_dir('fun')
>>> self.add_data_dir(('sun', 'fun'))
>>> self.add_data_dir(('gun', '/full/path/to/fun'))
 Will install data-files to the locations: <package install directory>/
  fun/
    foo.dat
    bar/
      car.dat
  sun/
    foo.dat
    bar/
      car.dat
  gun/
    foo.dat
    car.dat

   add_include_dirs(*paths)[source]

Add paths to configuration include directories. Add the given sequence of paths to the beginning of the include_dirs list. This list will be visible to all extension modules of the current package.
   add_headers(*files)[source]

Add installable headers to configuration. Add the given sequence of files to the beginning of the headers list. By default, headers will be installed under <python- include>/<self.name.replace(‘.’,’/’)>/ directory. If an item of files is a tuple, then its first argument specifies the actual installation location relative to the <python-include> path.  Parameters

filesstr or seq

Argument(s) can be either:  2-sequence (<includedir suffix>,<path to header file(s)>) path(s) to header file(s) where python includedir suffix will default to package name.
   add_extension(name, sources, **kw)[source]

Add extension to configuration. Create and add an Extension instance to the ext_modules list. This method also takes the following optional keyword arguments that are passed on to the Extension constructor.  Parameters

namestr

name of the extension
sourcesseq

list of the sources. The list of sources may contain functions (called source generators) which must take an extension instance and a build directory as inputs and return a source file or list of source files or None. If None is returned then no sources are generated. If the Extension instance has no sources after processing all source generators, then no extension module is built.  include_dirs :
define_macros :
undef_macros :
library_dirs :
libraries :
runtime_library_dirs :
extra_objects :
extra_compile_args :
extra_link_args :
extra_f77_compile_args :
extra_f90_compile_args :
export_symbols :
swig_opts :
depends :

The depends list contains paths to files or directories that the sources of the extension module depend on. If any path in the depends list is newer than the extension module, then the module will be rebuilt.  language :
f2py_options :
module_dirs :

extra_infodict or list

dict or list of dict of keywords to be appended to keywords.     Notes The self.paths(…) method is applied to all lists that may contain paths.
   add_library(name, sources, **build_info)[source]

Add library to configuration.  Parameters

namestr

Name of the extension.
sourcessequence

List of the sources. The list of sources may contain functions (called source generators) which must take an extension instance and a build directory as inputs and return a source file or list of source files or None. If None is returned then no sources are generated. If the Extension instance has no sources after processing all source generators, then no extension module is built.
build_infodict, optional

The following keys are allowed:  depends macros include_dirs extra_compiler_args extra_f77_compile_args extra_f90_compile_args f2py_options language
   add_scripts(*files)[source]

Add scripts to configuration. Add the sequence of files to the beginning of the scripts list. Scripts will be installed under the <prefix>/bin/ directory.
   add_installed_library(name, sources, install_dir, build_info=None)[source]

Similar to add_library, but the specified library is installed. Most C libraries used with distutils are only used to build python extensions, but libraries built through this method will be installed so that they can be reused by third-party packages.  Parameters

namestr

Name of the installed library.
sourcessequence

List of the library’s source files. See add_library for details.
install_dirstr

Path to install the library, relative to the current sub-package.
build_infodict, optional

The following keys are allowed:  depends macros include_dirs extra_compiler_args extra_f77_compile_args extra_f90_compile_args f2py_options language     Returns
 None
    See also
add_library, add_npy_pkg_config, get_info

  Notes The best way to encode the options required to link against the specified C libraries is to use a “libname.ini” file, and use get_info to retrieve the required options (see add_npy_pkg_config for more information).
   add_npy_pkg_config(template, install_dir, subst_dict=None)[source]

Generate and install a npy-pkg config file from a template. The config file generated from template is installed in the given install directory, using subst_dict for variable substitution.  Parameters

templatestr

The path of the template, relatively to the current package path.
install_dirstr

Where to install the npy-pkg config file, relatively to the current package path.
subst_dictdict, optional

If given, any string of the form @key@ will be replaced by subst_dict[key] in the template file when installed. The install prefix is always available through the variable @prefix@, since the install prefix is not easy to get reliably from setup.py.      See also
add_installed_library, get_info

  Notes This works for both standard installs and in-place builds, i.e. the @prefix@ refer to the source directory for in-place builds. Examples config.add_npy_pkg_config('foo.ini.in', 'lib', {'foo': bar})
 Assuming the foo.ini.in file has the following content: [meta]
Name=@foo@
Version=1.0
Description=dummy description

[default]
Cflags=-I@prefix@/include
Libs=
 The generated file will have the following content: [meta]
Name=bar
Version=1.0
Description=dummy description

[default]
Cflags=-Iprefix_dir/include
Libs=
 and will be installed as foo.ini in the ‘lib’ subpath. When cross-compiling with numpy distutils, it might be necessary to use modified npy-pkg-config files. Using the default/generated files will link with the host libraries (i.e. libnpymath.a). For cross-compilation you of-course need to link with target libraries, while using the host Python installation. You can copy out the numpy/core/lib/npy-pkg-config directory, add a pkgdir value to the .ini files and set NPY_PKG_CONFIG_PATH environment variable to point to the directory with the modified npy-pkg-config files. Example npymath.ini modified for cross-compilation: [meta]
Name=npymath
Description=Portable, core math library implementing C99 standard
Version=0.1

[variables]
pkgname=numpy.core
pkgdir=/build/arm-linux-gnueabi/sysroot/usr/lib/python3.7/site-packages/numpy/core
prefix=${pkgdir}
libdir=${prefix}/lib
includedir=${prefix}/include

[default]
Libs=-L${libdir} -lnpymath
Cflags=-I${includedir}
Requires=mlib

[msvc]
Libs=/LIBPATH:${libdir} npymath.lib
Cflags=/INCLUDE:${includedir}
Requires=mlib

   paths(*paths, **kws)[source]

Apply glob to paths and prepend local_path if needed. Applies glob.glob(…) to each path in the sequence (if needed) and pre-pends the local_path if needed. Because this is called on all source lists, this allows wildcard characters to be specified in lists of sources for extension modules and libraries and scripts and allows path-names be relative to the source directory.
   get_config_cmd()[source]

Returns the numpy.distutils config command instance.
   get_build_temp_dir()[source]

Return a path to a temporary directory where temporary files should be placed.
   have_f77c()[source]

Check for availability of Fortran 77 compiler. Use it inside source generating function to ensure that setup distribution instance has been initialized. Notes True if a Fortran 77 compiler is available (because a simple Fortran 77 code was able to be compiled successfully).
   have_f90c()[source]

Check for availability of Fortran 90 compiler. Use it inside source generating function to ensure that setup distribution instance has been initialized. Notes True if a Fortran 90 compiler is available (because a simple Fortran 90 code was able to be compiled successfully)
   get_version(version_file=None, version_variable=None)[source]

Try to get version string of a package. Return a version string of the current package or None if the version information could not be detected. Notes This method scans files named __version__.py, <packagename>_version.py, version.py, and __svn_version__.py for string variables version, __version__, and <packagename>_version, until a version number is found.
   make_svn_version_py(delete=True)[source]

Appends a data function to the data_files list that will generate __svn_version__.py file to the current package directory. Generate package __svn_version__.py file from SVN revision number, it will be removed after python exits but will be available when sdist, etc commands are executed. Notes If __svn_version__.py existed before, nothing is done. This is intended for working with source directories that are in an SVN repository.
   make_config_py(name='__config__')[source]

Generate package __config__.py file containing system_info information used during building the package. This file is installed to the package installation directory.
   get_info(*names)[source]

Get resources information. Return information (from system_info.get_info) for all of the names in the argument list in a single dictionary.

   Building Installable C libraries Conventional C libraries (installed through add_library) are not installed, and are just used during the build (they are statically linked). An installable C library is a pure C library, which does not depend on the python C runtime, and is installed such that it may be used by third-party packages. To build and install the C library, you just use the method add_installed_library instead of add_library, which takes the same arguments except for an additional install_dir argument: .. hidden in a comment so as to be included in refguide but not rendered documentation
  >>> import numpy.distutils.misc_util
  >>> config = np.distutils.misc_util.Configuration(None, '', '.')
  >>> with open('foo.c', 'w') as f: pass

>>> config.add_installed_library('foo', sources=['foo.c'], install_dir='lib')
  npy-pkg-config files To make the necessary build options available to third parties, you could use the npy-pkg-config mechanism implemented in numpy.distutils. This mechanism is based on a .ini file which contains all the options. A .ini file is very similar to .pc files as used by the pkg-config unix utility: [meta]
Name: foo
Version: 1.0
Description: foo library

[variables]
prefix = /home/user/local
libdir = ${prefix}/lib
includedir = ${prefix}/include

[default]
cflags = -I${includedir}
libs = -L${libdir} -lfoo
 Generally, the file needs to be generated during the build, since it needs some information known at build time only (e.g. prefix). This is mostly automatic if one uses the Configuration method add_npy_pkg_config. Assuming we have a template file foo.ini.in as follows: [meta]
Name: foo
Version: @version@
Description: foo library

[variables]
prefix = @prefix@
libdir = ${prefix}/lib
includedir = ${prefix}/include

[default]
cflags = -I${includedir}
libs = -L${libdir} -lfoo
 and the following code in setup.py: >>> config.add_installed_library('foo', sources=['foo.c'], install_dir='lib')
>>> subst = {'version': '1.0'}
>>> config.add_npy_pkg_config('foo.ini.in', 'lib', subst_dict=subst)
 This will install the file foo.ini into the directory package_dir/lib, and the foo.ini file will be generated from foo.ini.in, where each @version@ will be replaced by subst_dict['version']. The dictionary has an additional prefix substitution rule automatically added, which contains the install prefix (since this is not easy to get from setup.py). npy-pkg-config files can also be installed at the same location as used for numpy, using the path returned from get_npy_pkg_dir function.   Reusing a C library from another package Info are easily retrieved from the get_info function in numpy.distutils.misc_util: >>> info = np.distutils.misc_util.get_info('npymath')
>>> config.add_extension('foo', sources=['foo.c'], extra_info=info)
 An additional list of paths to look for .ini files can be given to get_info.    Conversion of .src files NumPy distutils supports automatic conversion of source files named <somefile>.src. This facility can be used to maintain very similar code blocks requiring only simple changes between blocks. During the build phase of setup, if a template file named <somefile>.src is encountered, a new file named <somefile> is constructed from the template and placed in the build directory to be used instead. Two forms of template conversion are supported. The first form occurs for files named <file>.ext.src where ext is a recognized Fortran extension (f, f90, f95, f77, for, ftn, pyf). The second form is used for all other cases. See Conversion of .src files using Templates.


paths(*paths, **kws)[source]

Apply glob to paths and prepend local_path if needed. Applies glob.glob(…) to each path in the sequence (if needed) and pre-pends the local_path if needed. Because this is called on all source lists, this allows wildcard characters to be specified in lists of sources for extension modules and libraries and scripts and allows path-names be relative to the source directory.


Performance  Recommendation The recommended generator for general use is PCG64 or its upgraded variant PCG64DXSM for heavily-parallel use cases. They are statistically high quality, full-featured, and fast on most platforms, but somewhat slow when compiled for 32-bit processes. See Upgrading PCG64 with PCG64DXSM for details on when heavy parallelism would indicate using PCG64DXSM. Philox is fairly slow, but its statistical properties have very high quality, and it is easy to get an assuredly-independent stream by using unique keys. If that is the style you wish to use for parallel streams, or you are porting from another system that uses that style, then Philox is your choice. SFC64 is statistically high quality and very fast. However, it lacks jumpability. If you are not using that capability and want lots of speed, even on 32-bit processes, this is your choice. MT19937 fails some statistical tests and is not especially fast compared to modern PRNGs. For these reasons, we mostly do not recommend using it on its own, only through the legacy RandomState for reproducing old results. That said, it has a very long history as a default in many systems.   Timings The timings below are the time in ns to produce 1 random value from a specific distribution. The original MT19937 generator is much slower since it requires 2 32-bit values to equal the output of the faster generators. Integer performance has a similar ordering. The pattern is similar for other, more complex generators. The normal performance of the legacy RandomState generator is much lower than the other since it uses the Box-Muller transform rather than the Ziggurat method. The performance gap for Exponentials is also large due to the cost of computing the log function to invert the CDF. The column labeled MT19973 uses the same 32-bit generator as RandomState but produces random variates using Generator.
 MT19937 PCG64 PCG64DXSM Philox SFC64 RandomState
32-bit Unsigned Ints 3.3 1.9 2.0 3.3 1.8 3.1
64-bit Unsigned Ints 5.6 3.2 2.9 4.9 2.5 5.5
Uniforms 5.9 3.1 2.9 5.0 2.6 6.0
Normals 13.9 10.8 10.5 12.0 8.3 56.8
Exponentials 9.1 6.0 5.8 8.1 5.4 63.9
Gammas 37.2 30.8 28.9 34.0 27.5 77.0
Binomials 21.3 17.4 17.6 19.3 15.6 21.4
Laplaces 73.2 72.3 76.1 73.0 72.3 82.5
Poissons 111.7 103.4 100.5 109.4 90.7 115.2   The next table presents the performance in percentage relative to values generated by the legacy generator, RandomState(MT19937()). The overall performance was computed using a geometric mean.
 MT19937 PCG64 PCG64DXSM Philox SFC64
32-bit Unsigned Ints 96 162 160 96 175
64-bit Unsigned Ints 97 171 188 113 218
Uniforms 102 192 206 121 233
Normals 409 526 541 471 684
Exponentials 701 1071 1101 784 1179
Gammas 207 250 266 227 281
Binomials 100 123 122 111 138
Laplaces 113 114 108 113 114
Poissons 103 111 115 105 127
Overall 159 219 225 174 251    Note All timings were taken using Linux on an AMD Ryzen 9 3900X processor.    Performance on different Operating Systems Performance differs across platforms due to compiler and hardware availability (e.g., register width) differences. The default bit generator has been chosen to perform well on 64-bit platforms. Performance on 32-bit operating systems is very different. The values reported are normalized relative to the speed of MT19937 in each table. A value of 100 indicates that the performance matches the MT19937. Higher values indicate improved performance. These values cannot be compared across tables.  64-bit Linux
Distribution MT19937 PCG64 PCG64DXSM Philox SFC64
32-bit Unsigned Ints 100 168 166 100 182
64-bit Unsigned Ints 100 176 193 116 224
Uniforms 100 188 202 118 228
Normals 100 128 132 115 167
Exponentials 100 152 157 111 168
Overall 100 161 168 112 192     64-bit Windows The relative performance on 64-bit Linux and 64-bit Windows is broadly similar with the notable exception of the Philox generator.
Distribution MT19937 PCG64 PCG64DXSM Philox SFC64
32-bit Unsigned Ints 100 155 131 29 150
64-bit Unsigned Ints 100 157 143 25 154
Uniforms 100 151 144 24 155
Normals 100 129 128 37 150
Exponentials 100 150 145 28 159
Overall 100 148 138 28 154     32-bit Windows The performance of 64-bit generators on 32-bit Windows is much lower than on 64-bit operating systems due to register width. MT19937, the generator that has been in NumPy since 2005, operates on 32-bit integers.
Distribution MT19937 PCG64 PCG64DXSM Philox SFC64
32-bit Unsigned Ints 100 24 34 14 57
64-bit Unsigned Ints 100 21 32 14 74
Uniforms 100 21 34 16 73
Normals 100 36 57 28 101
Exponentials 100 28 44 20 88
Overall 100 25 39 18 77    Note Linux timings used Ubuntu 20.04 and GCC 9.3.0. Windows timings were made on Windows 10 using Microsoft C/C++ Optimizing Compiler Version 19 (Visual Studio 2019). All timings were produced on an AMD Ryzen 9 3900X processor.


Poly1d  Basics
poly1d(c_or_r[, r, variable]) A one-dimensional polynomial class.
polyval(p, x) Evaluate a polynomial at specific values.
poly(seq_of_zeros) Find the coefficients of a polynomial with the given sequence of roots.
roots(p) Return the roots of a polynomial with coefficients given in p.     Fitting
polyfit(x, y, deg[, rcond, full, w, cov]) Least squares polynomial fit.     Calculus
polyder(p[, m]) Return the derivative of the specified order of a polynomial.
polyint(p[, m, k]) Return an antiderivative (indefinite integral) of a polynomial.     Arithmetic
polyadd(a1, a2) Find the sum of two polynomials.
polydiv(u, v) Returns the quotient and remainder of polynomial division.
polymul(a1, a2) Find the product of two polynomials.
polysub(a1, a2) Difference (subtraction) of two polynomials.     Warnings
RankWarning Issued by polyfit when the Vandermonde matrix is rank deficient.


numpy.poly1d.__call__ method   poly1d.__call__(val)[source]

Call self as a function.


numpy.poly1d.deriv method   poly1d.deriv(m=1)[source]

Return a derivative of this polynomial. Refer to polyder for full documentation.  See also  polyder

equivalent function


numpy.poly1d.integ method   poly1d.integ(m=1, k=0)[source]

Return an antiderivative (indefinite integral) of this polynomial. Refer to polyint for full documentation.  See also  polyint

equivalent function


numpy.polynomial.chebyshev.cheb2poly   polynomial.chebyshev.cheb2poly(c)[source]

Convert a Chebyshev series to a polynomial. Convert an array representing the coefficients of a Chebyshev series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the “standard” basis) ordered from lowest to highest degree.  Parameters

carray_like

1-D array containing the Chebyshev series coefficients, ordered from lowest order term to highest.    Returns

polndarray

1-D array containing the coefficients of the equivalent polynomial (relative to the “standard” basis) ordered from lowest order term to highest.      See also  poly2cheb
  Notes The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples >>> from numpy import polynomial as P
>>> c = P.Chebyshev(range(4))
>>> c
Chebyshev([0., 1., 2., 3.], domain=[-1,  1], window=[-1,  1])
>>> p = c.convert(kind=P.Polynomial)
>>> p
Polynomial([-2., -8.,  4., 12.], domain=[-1.,  1.], window=[-1.,  1.])
>>> P.chebyshev.cheb2poly(range(4))
array([-2.,  -8.,   4.,  12.])


numpy.polynomial.chebyshev.chebadd   polynomial.chebyshev.chebadd(c1, c2)[source]

Add one Chebyshev series to another. Returns the sum of two Chebyshev series c1 + c2. The arguments are sequences of coefficients ordered from lowest order term to highest, i.e., [1,2,3] represents the series T_0 + 2*T_1 + 3*T_2.  Parameters

c1, c2array_like

1-D arrays of Chebyshev series coefficients ordered from low to high.    Returns

outndarray

Array representing the Chebyshev series of their sum.      See also
chebsub, chebmulx, chebmul, chebdiv, chebpow

  Notes Unlike multiplication, division, etc., the sum of two Chebyshev series is a Chebyshev series (without having to “reproject” the result onto the basis set) so addition, just like that of “standard” polynomials, is simply “component-wise.” Examples >>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebadd(c1,c2)
array([4., 4., 4.])


numpy.polynomial.chebyshev.chebcompanion   polynomial.chebyshev.chebcompanion(c)[source]

Return the scaled companion matrix of c. The basis polynomials are scaled so that the companion matrix is symmetric when c is a Chebyshev basis polynomial. This provides better eigenvalue estimates than the unscaled case and for basis polynomials the eigenvalues are guaranteed to be real if numpy.linalg.eigvalsh is used to obtain them.  Parameters

carray_like

1-D array of Chebyshev series coefficients ordered from low to high degree.    Returns

matndarray

Scaled companion matrix of dimensions (deg, deg).     Notes  New in version 1.7.0.


numpy.polynomial.chebyshev.chebder   polynomial.chebyshev.chebder(c, m=1, scl=1, axis=0)[source]

Differentiate a Chebyshev series. Returns the Chebyshev series coefficients c differentiated m times along axis. At each iteration the result is multiplied by scl (the scaling factor is for use in a linear change of variable). The argument c is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series 1*T_0 + 2*T_1 + 3*T_2 while [[1,2],[1,2]] represents 1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) +
2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y) if axis=0 is x and axis=1 is y.  Parameters

carray_like

Array of Chebyshev series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index.
mint, optional

Number of derivatives taken, must be non-negative. (Default: 1)
sclscalar, optional

Each differentiation is multiplied by scl. The end result is multiplication by scl**m. This is for use in a linear change of variable. (Default: 1)
axisint, optional

Axis over which the derivative is taken. (Default: 0).  New in version 1.7.0.     Returns

derndarray

Chebyshev series of the derivative.      See also  chebint
  Notes In general, the result of differentiating a C-series needs to be “reprojected” onto the C-series basis set. Thus, typically, the result of this function is “unintuitive,” albeit correct; see Examples section below. Examples >>> from numpy.polynomial import chebyshev as C
>>> c = (1,2,3,4)
>>> C.chebder(c)
array([14., 12., 24.])
>>> C.chebder(c,3)
array([96.])
>>> C.chebder(c,scl=-1)
array([-14., -12., -24.])
>>> C.chebder(c,2,-1)
array([12.,  96.])


numpy.polynomial.chebyshev.chebdiv   polynomial.chebyshev.chebdiv(c1, c2)[source]

Divide one Chebyshev series by another. Returns the quotient-with-remainder of two Chebyshev series c1 / c2. The arguments are sequences of coefficients from lowest order “term” to highest, e.g., [1,2,3] represents the series T_0 + 2*T_1 + 3*T_2.  Parameters

c1, c2array_like

1-D arrays of Chebyshev series coefficients ordered from low to high.    Returns

[quo, rem]ndarrays

Of Chebyshev series coefficients representing the quotient and remainder.      See also
chebadd, chebsub, chebmulx, chebmul, chebpow

  Notes In general, the (polynomial) division of one C-series by another results in quotient and remainder terms that are not in the Chebyshev polynomial basis set. Thus, to express these results as C-series, it is typically necessary to “reproject” the results onto said basis set, which typically produces “unintuitive” (but correct) results; see Examples section below. Examples >>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebdiv(c1,c2) # quotient "intuitive," remainder not
(array([3.]), array([-8., -4.]))
>>> c2 = (0,1,2,3)
>>> C.chebdiv(c2,c1) # neither "intuitive"
(array([0., 2.]), array([-2., -4.]))


numpy.polynomial.chebyshev.chebdomain   polynomial.chebyshev.chebdomain = array([-1, 1])

An array object represents a multidimensional, homogeneous array of fixed-size items. An associated data-type object describes the format of each element in the array (its byte-order, how many bytes it occupies in memory, whether it is an integer, a floating point number, or something else, etc.) Arrays should be constructed using array, zeros or empty (refer to the See Also section below). The parameters given here refer to a low-level method (ndarray(…)) for instantiating an array. For more information, refer to the numpy module and examine the methods and attributes of an array.  Parameters
 (for the __new__ method; see Notes below)

shapetuple of ints

Shape of created array.
dtypedata-type, optional

Any object that can be interpreted as a numpy data type.
bufferobject exposing buffer interface, optional

Used to fill the array with data.
offsetint, optional

Offset of array data in buffer.
stridestuple of ints, optional

Strides of data in memory.
order{‘C’, ‘F’}, optional

Row-major (C-style) or column-major (Fortran-style) order.      See also  array

Construct an array.  zeros

Create an array, each element of which is zero.  empty

Create an array, but leave its allocated memory unchanged (i.e., it contains “garbage”).  dtype

Create a data-type.  numpy.typing.NDArray

An ndarray alias generic w.r.t. its dtype.type.    Notes There are two modes of creating an array using __new__:  If buffer is None, then only shape, dtype, and order are used. If buffer is an object exposing the buffer interface, then all keywords are interpreted.  No __init__ method is needed because the array is fully initialized after the __new__ method. Examples These examples illustrate the low-level ndarray constructor. Refer to the See Also section above for easier ways of constructing an ndarray. First mode, buffer is None: >>> np.ndarray(shape=(2,2), dtype=float, order='F')
array([[0.0e+000, 0.0e+000], # random
       [     nan, 2.5e-323]])
 Second mode: >>> np.ndarray((2,), buffer=np.array([1,2,3]),
...            offset=np.int_().itemsize,
...            dtype=int) # offset = 1*itemsize, i.e. skip first element
array([2, 3])
  Attributes

Tndarray

Transpose of the array.
databuffer

The array’s elements, in memory.
dtypedtype object

Describes the format of the elements in the array.
flagsdict

Dictionary containing information related to memory use, e.g., ‘C_CONTIGUOUS’, ‘OWNDATA’, ‘WRITEABLE’, etc.
flatnumpy.flatiter object

Flattened version of the array as an iterator. The iterator allows assignments, e.g., x.flat = 3 (See ndarray.flat for assignment examples; TODO).
imagndarray

Imaginary part of the array.
realndarray

Real part of the array.
sizeint

Number of elements in the array.
itemsizeint

The memory use of each array element in bytes.
nbytesint

The total number of bytes required to store the array data, i.e., itemsize * size.
ndimint

The array’s number of dimensions.
shapetuple of ints

Shape of the array.
stridestuple of ints

The step-size required to move from one element to the next in memory. For example, a contiguous (3, 4) array of type int16 in C-order has strides (8, 2). This implies that to move from element to element in memory requires jumps of 2 bytes. To move from row-to-row, one needs to jump 8 bytes at a time (2 * 4).
ctypesctypes object

Class containing properties of the array needed for interaction with ctypes.
basendarray

If the array is a view into another array, that array is its base (unless that array is also a view). The base array is where the array data is actually stored.


numpy.polynomial.chebyshev.chebfit   polynomial.chebyshev.chebfit(x, y, deg, rcond=None, full=False, w=None)[source]

Least squares fit of Chebyshev series to data. Return the coefficients of a Chebyshev series of degree deg that is the least squares fit to the data values y given at points x. If y is 1-D the returned coefficients will also be 1-D. If y is 2-D multiple fits are done, one for each column of y, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form  \[p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x),\] where n is deg.  Parameters

xarray_like, shape (M,)

x-coordinates of the M sample points (x[i], y[i]).
yarray_like, shape (M,) or (M, K)

y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column.
degint or 1-D array_like

Degree(s) of the fitting polynomials. If deg is a single integer, all terms up to and including the deg’th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead.
rcondfloat, optional

Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases.
fullbool, optional

Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned.
warray_like, shape (M,), optional

Weights. If not None, the weight w[i] applies to the unsquared residual y[i] - y_hat[i] at x[i]. Ideally the weights are chosen so that the errors of the products w[i]*y[i] all have the same variance. When using inverse-variance weighting, use w[i] = 1/sigma(y[i]). The default value is None.  New in version 1.5.0.     Returns

coefndarray, shape (M,) or (M, K)

Chebyshev coefficients ordered from low to high. If y was 2-D, the coefficients for the data in column k of y are in column k.
[residuals, rank, singular_values, rcond]list

These values are only returned if full == True  residuals – sum of squared residuals of the least squares fit rank – the numerical rank of the scaled Vandermonde matrix singular_values – singular values of the scaled Vandermonde matrix rcond – value of rcond.  For more details, see numpy.linalg.lstsq.    Warns
 RankWarning

The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if full == False. The warnings can be turned off by >>> import warnings
>>> warnings.simplefilter('ignore', np.RankWarning)
      See also  numpy.polynomial.polynomial.polyfit
numpy.polynomial.legendre.legfit
numpy.polynomial.laguerre.lagfit
numpy.polynomial.hermite.hermfit
numpy.polynomial.hermite_e.hermefit
chebval

Evaluates a Chebyshev series.  chebvander

Vandermonde matrix of Chebyshev series.  chebweight

Chebyshev weight function.  numpy.linalg.lstsq

Computes a least-squares fit from the matrix.  scipy.interpolate.UnivariateSpline

Computes spline fits.    Notes The solution is the coefficients of the Chebyshev series p that minimizes the sum of the weighted squared errors  \[E = \sum_j w_j^2 * |y_j - p(x_j)|^2,\] where \(w_j\) are the weights. This problem is solved by setting up as the (typically) overdetermined matrix equation  \[V(x) * c = w * y,\] where V is the weighted pseudo Vandermonde matrix of x, c are the coefficients to be solved for, w are the weights, and y are the observed values. This equation is then solved using the singular value decomposition of V. If some of the singular values of V are so small that they are neglected, then a RankWarning will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The rcond parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error. Fits using Chebyshev series are usually better conditioned than fits using power series, but much can depend on the distribution of the sample points and the smoothness of the data. If the quality of the fit is inadequate splines may be a good alternative. References  1
Wikipedia, “Curve fitting”, https://en.wikipedia.org/wiki/Curve_fitting


numpy.polynomial.chebyshev.chebfromroots   polynomial.chebyshev.chebfromroots(roots)[source]

Generate a Chebyshev series with given roots. The function returns the coefficients of the polynomial  \[p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),\] in Chebyshev form, where the r_n are the roots specified in roots. If a zero has multiplicity n, then it must appear in roots n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then roots looks something like [2, 2, 2, 3, 3]. The roots can appear in any order. If the returned coefficients are c, then  \[p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x)\] The coefficient of the last term is not generally 1 for monic polynomials in Chebyshev form.  Parameters

rootsarray_like

Sequence containing the roots.    Returns

outndarray

1-D array of coefficients. If all roots are real then out is a real array, if some of the roots are complex, then out is complex even if all the coefficients in the result are real (see Examples below).      See also  numpy.polynomial.polynomial.polyfromroots
numpy.polynomial.legendre.legfromroots
numpy.polynomial.laguerre.lagfromroots
numpy.polynomial.hermite.hermfromroots
numpy.polynomial.hermite_e.hermefromroots
  Examples >>> import numpy.polynomial.chebyshev as C
>>> C.chebfromroots((-1,0,1)) # x^3 - x relative to the standard basis
array([ 0.  , -0.25,  0.  ,  0.25])
>>> j = complex(0,1)
>>> C.chebfromroots((-j,j)) # x^2 + 1 relative to the standard basis
array([1.5+0.j, 0. +0.j, 0.5+0.j])


numpy.polynomial.chebyshev.chebgauss   polynomial.chebyshev.chebgauss(deg)[source]

Gauss-Chebyshev quadrature. Computes the sample points and weights for Gauss-Chebyshev quadrature. These sample points and weights will correctly integrate polynomials of degree \(2*deg - 1\) or less over the interval \([-1, 1]\) with the weight function \(f(x) = 1/\sqrt{1 - x^2}\).  Parameters

degint

Number of sample points and weights. It must be >= 1.    Returns

xndarray

1-D ndarray containing the sample points.
yndarray

1-D ndarray containing the weights.     Notes  New in version 1.7.0.  The results have only been tested up to degree 100, higher degrees may be problematic. For Gauss-Chebyshev there are closed form solutions for the sample points and weights. If n = deg, then  \[x_i = \cos(\pi (2 i - 1) / (2 n))\]  \[w_i = \pi / n\]


numpy.polynomial.chebyshev.chebgrid2d   polynomial.chebyshev.chebgrid2d(x, y, c)[source]

Evaluate a 2-D Chebyshev series on the Cartesian product of x and y. This function returns the values:  \[p(a,b) = \sum_{i,j} c_{i,j} * T_i(a) * T_j(b),\] where the points (a, b) consist of all pairs formed by taking a from x and b from y. The resulting points form a grid with x in the first dimension and y in the second. The parameters x and y are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either x and y or their elements must support multiplication and addition both with themselves and with the elements of c. If c has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape + y.shape.  Parameters

x, yarray_like, compatible objects

The two dimensional series is evaluated at the points in the Cartesian product of x and y. If x or y is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn’t an ndarray, it is treated as a scalar.
carray_like

Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in c[i,j]. If c has dimension greater than two the remaining indices enumerate multiple sets of coefficients.    Returns

valuesndarray, compatible object

The values of the two dimensional Chebyshev series at points in the Cartesian product of x and y.      See also
chebval, chebval2d, chebval3d, chebgrid3d

  Notes  New in version 1.7.0.


numpy.polynomial.chebyshev.chebgrid3d   polynomial.chebyshev.chebgrid3d(x, y, z, c)[source]

Evaluate a 3-D Chebyshev series on the Cartesian product of x, y, and z. This function returns the values:  \[p(a,b,c) = \sum_{i,j,k} c_{i,j,k} * T_i(a) * T_j(b) * T_k(c)\] where the points (a, b, c) consist of all triples formed by taking a from x, b from y, and c from z. The resulting points form a grid with x in the first dimension, y in the second, and z in the third. The parameters x, y, and z are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either x, y, and z or their elements must support multiplication and addition both with themselves and with the elements of c. If c has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape.  Parameters

x, y, zarray_like, compatible objects

The three dimensional series is evaluated at the points in the Cartesian product of x, y, and z. If x,`y`, or z is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn’t an ndarray, it is treated as a scalar.
carray_like

Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in c[i,j]. If c has dimension greater than two the remaining indices enumerate multiple sets of coefficients.    Returns

valuesndarray, compatible object

The values of the two dimensional polynomial at points in the Cartesian product of x and y.      See also
chebval, chebval2d, chebgrid2d, chebval3d

  Notes  New in version 1.7.0.


numpy.polynomial.chebyshev.chebint   polynomial.chebyshev.chebint(c, m=1, k=[], lbnd=0, scl=1, axis=0)[source]

Integrate a Chebyshev series. Returns the Chebyshev series coefficients c integrated m times from lbnd along axis. At each iteration the resulting series is multiplied by scl and an integration constant, k, is added. The scaling factor is for use in a linear change of variable. (“Buyer beware”: note that, depending on what one is doing, one may want scl to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument c is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series T_0 + 2*T_1 + 3*T_2 while [[1,2],[1,2]] represents 1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) +
2*T_1(x)*T_1(y) if axis=0 is x and axis=1 is y.  Parameters

carray_like

Array of Chebyshev series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index.
mint, optional

Order of integration, must be positive. (Default: 1)
k{[], list, scalar}, optional

Integration constant(s). The value of the first integral at zero is the first value in the list, the value of the second integral at zero is the second value, etc. If k == [] (the default), all constants are set to zero. If m == 1, a single scalar can be given instead of a list.
lbndscalar, optional

The lower bound of the integral. (Default: 0)
sclscalar, optional

Following each integration the result is multiplied by scl before the integration constant is added. (Default: 1)
axisint, optional

Axis over which the integral is taken. (Default: 0).  New in version 1.7.0.     Returns

Sndarray

C-series coefficients of the integral.    Raises
 ValueError

If m < 1, len(k) > m, np.ndim(lbnd) != 0, or np.ndim(scl) != 0.      See also  chebder
  Notes Note that the result of each integration is multiplied by scl. Why is this important to note? Say one is making a linear change of variable \(u = ax + b\) in an integral relative to x. Then \(dx = du/a\), so one will need to set scl equal to \(1/a\)- perhaps not what one would have first thought. Also note that, in general, the result of integrating a C-series needs to be “reprojected” onto the C-series basis set. Thus, typically, the result of this function is “unintuitive,” albeit correct; see Examples section below. Examples >>> from numpy.polynomial import chebyshev as C
>>> c = (1,2,3)
>>> C.chebint(c)
array([ 0.5, -0.5,  0.5,  0.5])
>>> C.chebint(c,3)
array([ 0.03125   , -0.1875    ,  0.04166667, -0.05208333,  0.01041667, # may vary
    0.00625   ])
>>> C.chebint(c, k=3)
array([ 3.5, -0.5,  0.5,  0.5])
>>> C.chebint(c,lbnd=-2)
array([ 8.5, -0.5,  0.5,  0.5])
>>> C.chebint(c,scl=-2)
array([-1.,  1., -1., -1.])


numpy.polynomial.chebyshev.chebinterpolate   polynomial.chebyshev.chebinterpolate(func, deg, args=())[source]

Interpolate a function at the Chebyshev points of the first kind. Returns the Chebyshev series that interpolates func at the Chebyshev points of the first kind in the interval [-1, 1]. The interpolating series tends to a minmax approximation to func with increasing deg if the function is continuous in the interval.  New in version 1.14.0.   Parameters

funcfunction

The function to be approximated. It must be a function of a single variable of the form f(x, a, b, c...), where a, b, c... are extra arguments passed in the args parameter.
degint

Degree of the interpolating polynomial
argstuple, optional

Extra arguments to be used in the function call. Default is no extra arguments.    Returns

coefndarray, shape (deg + 1,)

Chebyshev coefficients of the interpolating series ordered from low to high.     Notes The Chebyshev polynomials used in the interpolation are orthogonal when sampled at the Chebyshev points of the first kind. If it is desired to constrain some of the coefficients they can simply be set to the desired value after the interpolation, no new interpolation or fit is needed. This is especially useful if it is known apriori that some of coefficients are zero. For instance, if the function is even then the coefficients of the terms of odd degree in the result can be set to zero. Examples >>> import numpy.polynomial.chebyshev as C
>>> C.chebfromfunction(lambda x: np.tanh(x) + 0.5, 8)
array([  5.00000000e-01,   8.11675684e-01,  -9.86864911e-17,
        -5.42457905e-02,  -2.71387850e-16,   4.51658839e-03,
         2.46716228e-17,  -3.79694221e-04,  -3.26899002e-16])


numpy.polynomial.chebyshev.chebline   polynomial.chebyshev.chebline(off, scl)[source]

Chebyshev series whose graph is a straight line.  Parameters

off, sclscalars

The specified line is given by off + scl*x.    Returns

yndarray

This module’s representation of the Chebyshev series for off + scl*x.      See also  numpy.polynomial.polynomial.polyline
numpy.polynomial.legendre.legline
numpy.polynomial.laguerre.lagline
numpy.polynomial.hermite.hermline
numpy.polynomial.hermite_e.hermeline
  Examples >>> import numpy.polynomial.chebyshev as C
>>> C.chebline(3,2)
array([3, 2])
>>> C.chebval(-3, C.chebline(3,2)) # should be -3
-3.0


numpy.polynomial.chebyshev.chebmul   polynomial.chebyshev.chebmul(c1, c2)[source]

Multiply one Chebyshev series by another. Returns the product of two Chebyshev series c1 * c2. The arguments are sequences of coefficients, from lowest order “term” to highest, e.g., [1,2,3] represents the series T_0 + 2*T_1 + 3*T_2.  Parameters

c1, c2array_like

1-D arrays of Chebyshev series coefficients ordered from low to high.    Returns

outndarray

Of Chebyshev series coefficients representing their product.      See also
chebadd, chebsub, chebmulx, chebdiv, chebpow

  Notes In general, the (polynomial) product of two C-series results in terms that are not in the Chebyshev polynomial basis set. Thus, to express the product as a C-series, it is typically necessary to “reproject” the product onto said basis set, which typically produces “unintuitive live” (but correct) results; see Examples section below. Examples >>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebmul(c1,c2) # multiplication requires "reprojection"
array([  6.5,  12. ,  12. ,   4. ,   1.5])


numpy.polynomial.chebyshev.chebmulx   polynomial.chebyshev.chebmulx(c)[source]

Multiply a Chebyshev series by x. Multiply the polynomial c by x, where x is the independent variable.  Parameters

carray_like

1-D array of Chebyshev series coefficients ordered from low to high.    Returns

outndarray

Array representing the result of the multiplication.     Notes  New in version 1.5.0.  Examples >>> from numpy.polynomial import chebyshev as C
>>> C.chebmulx([1,2,3])
array([1. , 2.5, 1. , 1.5])


numpy.polynomial.chebyshev.chebone   polynomial.chebyshev.chebone = array([1])

An array object represents a multidimensional, homogeneous array of fixed-size items. An associated data-type object describes the format of each element in the array (its byte-order, how many bytes it occupies in memory, whether it is an integer, a floating point number, or something else, etc.) Arrays should be constructed using array, zeros or empty (refer to the See Also section below). The parameters given here refer to a low-level method (ndarray(…)) for instantiating an array. For more information, refer to the numpy module and examine the methods and attributes of an array.  Parameters
 (for the __new__ method; see Notes below)

shapetuple of ints

Shape of created array.
dtypedata-type, optional

Any object that can be interpreted as a numpy data type.
bufferobject exposing buffer interface, optional

Used to fill the array with data.
offsetint, optional

Offset of array data in buffer.
stridestuple of ints, optional

Strides of data in memory.
order{‘C’, ‘F’}, optional

Row-major (C-style) or column-major (Fortran-style) order.      See also  array

Construct an array.  zeros

Create an array, each element of which is zero.  empty

Create an array, but leave its allocated memory unchanged (i.e., it contains “garbage”).  dtype

Create a data-type.  numpy.typing.NDArray

An ndarray alias generic w.r.t. its dtype.type.    Notes There are two modes of creating an array using __new__:  If buffer is None, then only shape, dtype, and order are used. If buffer is an object exposing the buffer interface, then all keywords are interpreted.  No __init__ method is needed because the array is fully initialized after the __new__ method. Examples These examples illustrate the low-level ndarray constructor. Refer to the See Also section above for easier ways of constructing an ndarray. First mode, buffer is None: >>> np.ndarray(shape=(2,2), dtype=float, order='F')
array([[0.0e+000, 0.0e+000], # random
       [     nan, 2.5e-323]])
 Second mode: >>> np.ndarray((2,), buffer=np.array([1,2,3]),
...            offset=np.int_().itemsize,
...            dtype=int) # offset = 1*itemsize, i.e. skip first element
array([2, 3])
  Attributes

Tndarray

Transpose of the array.
databuffer

The array’s elements, in memory.
dtypedtype object

Describes the format of the elements in the array.
flagsdict

Dictionary containing information related to memory use, e.g., ‘C_CONTIGUOUS’, ‘OWNDATA’, ‘WRITEABLE’, etc.
flatnumpy.flatiter object

Flattened version of the array as an iterator. The iterator allows assignments, e.g., x.flat = 3 (See ndarray.flat for assignment examples; TODO).
imagndarray

Imaginary part of the array.
realndarray

Real part of the array.
sizeint

Number of elements in the array.
itemsizeint

The memory use of each array element in bytes.
nbytesint

The total number of bytes required to store the array data, i.e., itemsize * size.
ndimint

The array’s number of dimensions.
shapetuple of ints

Shape of the array.
stridestuple of ints

The step-size required to move from one element to the next in memory. For example, a contiguous (3, 4) array of type int16 in C-order has strides (8, 2). This implies that to move from element to element in memory requires jumps of 2 bytes. To move from row-to-row, one needs to jump 8 bytes at a time (2 * 4).
ctypesctypes object

Class containing properties of the array needed for interaction with ctypes.
basendarray

If the array is a view into another array, that array is its base (unless that array is also a view). The base array is where the array data is actually stored.


numpy.polynomial.chebyshev.chebpow   polynomial.chebyshev.chebpow(c, pow, maxpower=16)[source]

Raise a Chebyshev series to a power. Returns the Chebyshev series c raised to the power pow. The argument c is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series T_0 + 2*T_1 + 3*T_2.  Parameters

carray_like

1-D array of Chebyshev series coefficients ordered from low to high.
powinteger

Power to which the series will be raised
maxpowerinteger, optional

Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16    Returns

coefndarray

Chebyshev series of power.      See also
chebadd, chebsub, chebmulx, chebmul, chebdiv

  Examples >>> from numpy.polynomial import chebyshev as C
>>> C.chebpow([1, 2, 3, 4], 2)
array([15.5, 22. , 16. , ..., 12.5, 12. ,  8. ])


numpy.polynomial.chebyshev.chebpts1   polynomial.chebyshev.chebpts1(npts)[source]

Chebyshev points of the first kind. The Chebyshev points of the first kind are the points cos(x), where x = [pi*(k + .5)/npts for k in range(npts)].  Parameters

nptsint

Number of sample points desired.    Returns

ptsndarray

The Chebyshev points of the first kind.      See also  chebpts2
  Notes  New in version 1.5.0.


numpy.polynomial.chebyshev.chebpts2   polynomial.chebyshev.chebpts2(npts)[source]

Chebyshev points of the second kind. The Chebyshev points of the second kind are the points cos(x), where x = [pi*k/(npts - 1) for k in range(npts)].  Parameters

nptsint

Number of sample points desired.    Returns

ptsndarray

The Chebyshev points of the second kind.     Notes  New in version 1.5.0.


numpy.polynomial.chebyshev.chebroots   polynomial.chebyshev.chebroots(c)[source]

Compute the roots of a Chebyshev series. Return the roots (a.k.a. “zeros”) of the polynomial  \[p(x) = \sum_i c[i] * T_i(x).\]  Parameters

c1-D array_like

1-D array of coefficients.    Returns

outndarray

Array of the roots of the series. If all the roots are real, then out is also real, otherwise it is complex.      See also  numpy.polynomial.polynomial.polyroots
numpy.polynomial.legendre.legroots
numpy.polynomial.laguerre.lagroots
numpy.polynomial.hermite.hermroots
numpy.polynomial.hermite_e.hermeroots
  Notes The root estimates are obtained as the eigenvalues of the companion matrix, Roots far from the origin of the complex plane may have large errors due to the numerical instability of the series for such values. Roots with multiplicity greater than 1 will also show larger errors as the value of the series near such points is relatively insensitive to errors in the roots. Isolated roots near the origin can be improved by a few iterations of Newton’s method. The Chebyshev series basis polynomials aren’t powers of x so the results of this function may seem unintuitive. Examples >>> import numpy.polynomial.chebyshev as cheb
>>> cheb.chebroots((-1, 1,-1, 1)) # T3 - T2 + T1 - T0 has real roots
array([ -5.00000000e-01,   2.60860684e-17,   1.00000000e+00]) # may vary


numpy.polynomial.chebyshev.chebsub   polynomial.chebyshev.chebsub(c1, c2)[source]

Subtract one Chebyshev series from another. Returns the difference of two Chebyshev series c1 - c2. The sequences of coefficients are from lowest order term to highest, i.e., [1,2,3] represents the series T_0 + 2*T_1 + 3*T_2.  Parameters

c1, c2array_like

1-D arrays of Chebyshev series coefficients ordered from low to high.    Returns

outndarray

Of Chebyshev series coefficients representing their difference.      See also
chebadd, chebmulx, chebmul, chebdiv, chebpow

  Notes Unlike multiplication, division, etc., the difference of two Chebyshev series is a Chebyshev series (without having to “reproject” the result onto the basis set) so subtraction, just like that of “standard” polynomials, is simply “component-wise.” Examples >>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebsub(c1,c2)
array([-2.,  0.,  2.])
>>> C.chebsub(c2,c1) # -C.chebsub(c1,c2)
array([ 2.,  0., -2.])


numpy.polynomial.chebyshev.chebtrim   polynomial.chebyshev.chebtrim(c, tol=0)[source]

Remove “small” “trailing” coefficients from a polynomial. “Small” means “small in absolute value” and is controlled by the parameter tol; “trailing” means highest order coefficient(s), e.g., in [0, 1, 1, 0, 0] (which represents 0 + x + x**2 + 0*x**3 + 0*x**4) both the 3-rd and 4-th order coefficients would be “trimmed.”  Parameters

carray_like

1-d array of coefficients, ordered from lowest order to highest.
tolnumber, optional

Trailing (i.e., highest order) elements with absolute value less than or equal to tol (default value is zero) are removed.    Returns

trimmedndarray

1-d array with trailing zeros removed. If the resulting series would be empty, a series containing a single zero is returned.    Raises
 ValueError

If tol < 0      See also  trimseq
  Examples >>> from numpy.polynomial import polyutils as pu
>>> pu.trimcoef((0,0,3,0,5,0,0))
array([0.,  0.,  3.,  0.,  5.])
>>> pu.trimcoef((0,0,1e-3,0,1e-5,0,0),1e-3) # item == tol is trimmed
array([0.])
>>> i = complex(0,1) # works for complex
>>> pu.trimcoef((3e-4,1e-3*(1-i),5e-4,2e-5*(1+i)), 1e-3)
array([0.0003+0.j   , 0.001 -0.001j])


numpy.polynomial.chebyshev.chebval   polynomial.chebyshev.chebval(x, c, tensor=True)[source]

Evaluate a Chebyshev series at points x. If c is of length n + 1, this function returns the value:  \[p(x) = c_0 * T_0(x) + c_1 * T_1(x) + ... + c_n * T_n(x)\] The parameter x is converted to an array only if it is a tuple or a list, otherwise it is treated as a scalar. In either case, either x or its elements must support multiplication and addition both with themselves and with the elements of c. If c is a 1-D array, then p(x) will have the same shape as x. If c is multidimensional, then the shape of the result depends on the value of tensor. If tensor is true the shape will be c.shape[1:] + x.shape. If tensor is false the shape will be c.shape[1:]. Note that scalars have shape (,). Trailing zeros in the coefficients will be used in the evaluation, so they should be avoided if efficiency is a concern.  Parameters

xarray_like, compatible object

If x is a list or tuple, it is converted to an ndarray, otherwise it is left unchanged and treated as a scalar. In either case, x or its elements must support addition and multiplication with with themselves and with the elements of c.
carray_like

Array of coefficients ordered so that the coefficients for terms of degree n are contained in c[n]. If c is multidimensional the remaining indices enumerate multiple polynomials. In the two dimensional case the coefficients may be thought of as stored in the columns of c.
tensorboolean, optional

If True, the shape of the coefficient array is extended with ones on the right, one for each dimension of x. Scalars have dimension 0 for this action. The result is that every column of coefficients in c is evaluated for every element of x. If False, x is broadcast over the columns of c for the evaluation. This keyword is useful when c is multidimensional. The default value is True.  New in version 1.7.0.     Returns

valuesndarray, algebra_like

The shape of the return value is described above.      See also
chebval2d, chebgrid2d, chebval3d, chebgrid3d

  Notes The evaluation uses Clenshaw recursion, aka synthetic division.


numpy.polynomial.chebyshev.chebval2d   polynomial.chebyshev.chebval2d(x, y, c)[source]

Evaluate a 2-D Chebyshev series at points (x, y). This function returns the values:  \[p(x,y) = \sum_{i,j} c_{i,j} * T_i(x) * T_j(y)\] The parameters x and y are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either x and y or their elements must support multiplication and addition both with themselves and with the elements of c. If c is a 1-D array a one is implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape.  Parameters

x, yarray_like, compatible objects

The two dimensional series is evaluated at the points (x, y), where x and y must have the same shape. If x or y is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn’t an ndarray it is treated as a scalar.
carray_like

Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in c[i,j]. If c has dimension greater than 2 the remaining indices enumerate multiple sets of coefficients.    Returns

valuesndarray, compatible object

The values of the two dimensional Chebyshev series at points formed from pairs of corresponding values from x and y.      See also
chebval, chebgrid2d, chebval3d, chebgrid3d

  Notes  New in version 1.7.0.


numpy.polynomial.chebyshev.chebval3d   polynomial.chebyshev.chebval3d(x, y, z, c)[source]

Evaluate a 3-D Chebyshev series at points (x, y, z). This function returns the values:  \[p(x,y,z) = \sum_{i,j,k} c_{i,j,k} * T_i(x) * T_j(y) * T_k(z)\] The parameters x, y, and z are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either x, y, and z or their elements must support multiplication and addition both with themselves and with the elements of c. If c has fewer than 3 dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape.  Parameters

x, y, zarray_like, compatible object

The three dimensional series is evaluated at the points (x, y, z), where x, y, and z must have the same shape. If any of x, y, or z is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn’t an ndarray it is treated as a scalar.
carray_like

Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in c[i,j,k]. If c has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients.    Returns

valuesndarray, compatible object

The values of the multidimensional polynomial on points formed with triples of corresponding values from x, y, and z.      See also
chebval, chebval2d, chebgrid2d, chebgrid3d

  Notes  New in version 1.7.0.


numpy.polynomial.chebyshev.chebvander   polynomial.chebyshev.chebvander(x, deg)[source]

Pseudo-Vandermonde matrix of given degree. Returns the pseudo-Vandermonde matrix of degree deg and sample points x. The pseudo-Vandermonde matrix is defined by  \[V[..., i] = T_i(x),\] where 0 <= i <= deg. The leading indices of V index the elements of x and the last index is the degree of the Chebyshev polynomial. If c is a 1-D array of coefficients of length n + 1 and V is the matrix V = chebvander(x, n), then np.dot(V, c) and chebval(x, c) are the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of Chebyshev series of the same degree and sample points.  Parameters

xarray_like

Array of points. The dtype is converted to float64 or complex128 depending on whether any of the elements are complex. If x is scalar it is converted to a 1-D array.
degint

Degree of the resulting matrix.    Returns

vanderndarray

The pseudo Vandermonde matrix. The shape of the returned matrix is x.shape + (deg + 1,), where The last index is the degree of the corresponding Chebyshev polynomial. The dtype will be the same as the converted x.


numpy.polynomial.chebyshev.chebvander2d   polynomial.chebyshev.chebvander2d(x, y, deg)[source]

Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees deg and sample points (x, y). The pseudo-Vandermonde matrix is defined by  \[V[..., (deg[1] + 1)*i + j] = T_i(x) * T_j(y),\] where 0 <= i <= deg[0] and 0 <= j <= deg[1]. The leading indices of V index the points (x, y) and the last index encodes the degrees of the Chebyshev polynomials. If V = chebvander2d(x, y, [xdeg, ydeg]), then the columns of V correspond to the elements of a 2-D coefficient array c of shape (xdeg + 1, ydeg + 1) in the order  \[c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...\] and np.dot(V, c.flat) and chebval2d(x, y, c) will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D Chebyshev series of the same degrees and sample points.  Parameters

x, yarray_like

Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays.
deglist of ints

List of maximum degrees of the form [x_deg, y_deg].    Returns

vander2dndarray

The shape of the returned matrix is x.shape + (order,), where \(order = (deg[0]+1)*(deg[1]+1)\). The dtype will be the same as the converted x and y.      See also
chebvander, chebvander3d, chebval2d, chebval3d

  Notes  New in version 1.7.0.


numpy.polynomial.chebyshev.chebvander3d   polynomial.chebyshev.chebvander3d(x, y, z, deg)[source]

Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees deg and sample points (x, y, z). If l, m, n are the given degrees in x, y, z, then The pseudo-Vandermonde matrix is defined by  \[V[..., (m+1)(n+1)i + (n+1)j + k] = T_i(x)*T_j(y)*T_k(z),\] where 0 <= i <= l, 0 <= j <= m, and 0 <= j <= n. The leading indices of V index the points (x, y, z) and the last index encodes the degrees of the Chebyshev polynomials. If V = chebvander3d(x, y, z, [xdeg, ydeg, zdeg]), then the columns of V correspond to the elements of a 3-D coefficient array c of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order  \[c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...\] and np.dot(V, c.flat) and chebval3d(x, y, z, c) will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D Chebyshev series of the same degrees and sample points.  Parameters

x, y, zarray_like

Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays.
deglist of ints

List of maximum degrees of the form [x_deg, y_deg, z_deg].    Returns

vander3dndarray

The shape of the returned matrix is x.shape + (order,), where \(order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)\). The dtype will be the same as the converted x, y, and z.      See also
chebvander, chebvander3d, chebval2d, chebval3d

  Notes  New in version 1.7.0.


numpy.polynomial.chebyshev.chebweight   polynomial.chebyshev.chebweight(x)[source]

The weight function of the Chebyshev polynomials. The weight function is \(1/\sqrt{1 - x^2}\) and the interval of integration is \([-1, 1]\). The Chebyshev polynomials are orthogonal, but not normalized, with respect to this weight function.  Parameters

xarray_like

Values at which the weight function will be computed.    Returns

wndarray

The weight function at x.     Notes  New in version 1.7.0.


numpy.polynomial.chebyshev.chebx   polynomial.chebyshev.chebx = array([0, 1])

An array object represents a multidimensional, homogeneous array of fixed-size items. An associated data-type object describes the format of each element in the array (its byte-order, how many bytes it occupies in memory, whether it is an integer, a floating point number, or something else, etc.) Arrays should be constructed using array, zeros or empty (refer to the See Also section below). The parameters given here refer to a low-level method (ndarray(…)) for instantiating an array. For more information, refer to the numpy module and examine the methods and attributes of an array.  Parameters
 (for the __new__ method; see Notes below)

shapetuple of ints

Shape of created array.
dtypedata-type, optional

Any object that can be interpreted as a numpy data type.
bufferobject exposing buffer interface, optional

Used to fill the array with data.
offsetint, optional

Offset of array data in buffer.
stridestuple of ints, optional

Strides of data in memory.
order{‘C’, ‘F’}, optional

Row-major (C-style) or column-major (Fortran-style) order.      See also  array

Construct an array.  zeros

Create an array, each element of which is zero.  empty

Create an array, but leave its allocated memory unchanged (i.e., it contains “garbage”).  dtype

Create a data-type.  numpy.typing.NDArray

An ndarray alias generic w.r.t. its dtype.type.    Notes There are two modes of creating an array using __new__:  If buffer is None, then only shape, dtype, and order are used. If buffer is an object exposing the buffer interface, then all keywords are interpreted.  No __init__ method is needed because the array is fully initialized after the __new__ method. Examples These examples illustrate the low-level ndarray constructor. Refer to the See Also section above for easier ways of constructing an ndarray. First mode, buffer is None: >>> np.ndarray(shape=(2,2), dtype=float, order='F')
array([[0.0e+000, 0.0e+000], # random
       [     nan, 2.5e-323]])
 Second mode: >>> np.ndarray((2,), buffer=np.array([1,2,3]),
...            offset=np.int_().itemsize,
...            dtype=int) # offset = 1*itemsize, i.e. skip first element
array([2, 3])
  Attributes

Tndarray

Transpose of the array.
databuffer

The array’s elements, in memory.
dtypedtype object

Describes the format of the elements in the array.
flagsdict

Dictionary containing information related to memory use, e.g., ‘C_CONTIGUOUS’, ‘OWNDATA’, ‘WRITEABLE’, etc.
flatnumpy.flatiter object

Flattened version of the array as an iterator. The iterator allows assignments, e.g., x.flat = 3 (See ndarray.flat for assignment examples; TODO).
imagndarray

Imaginary part of the array.
realndarray

Real part of the array.
sizeint

Number of elements in the array.
itemsizeint

The memory use of each array element in bytes.
nbytesint

The total number of bytes required to store the array data, i.e., itemsize * size.
ndimint

The array’s number of dimensions.
shapetuple of ints

Shape of the array.
stridestuple of ints

The step-size required to move from one element to the next in memory. For example, a contiguous (3, 4) array of type int16 in C-order has strides (8, 2). This implies that to move from element to element in memory requires jumps of 2 bytes. To move from row-to-row, one needs to jump 8 bytes at a time (2 * 4).
ctypesctypes object

Class containing properties of the array needed for interaction with ctypes.
basendarray

If the array is a view into another array, that array is its base (unless that array is also a view). The base array is where the array data is actually stored.


numpy.polynomial.chebyshev.Chebyshev.__call__ method   polynomial.chebyshev.Chebyshev.__call__(arg)[source]

Call self as a function.


numpy.polynomial.chebyshev.Chebyshev.basis method   classmethod polynomial.chebyshev.Chebyshev.basis(deg, domain=None, window=None)[source]

Series basis polynomial of degree deg. Returns the series representing the basis polynomial of degree deg.  New in version 1.7.0.   Parameters

degint

Degree of the basis polynomial for the series. Must be >= 0.
domain{None, array_like}, optional

If given, the array must be of the form [beg, end], where beg and end are the endpoints of the domain. If None is given then the class domain is used. The default is None.
window{None, array_like}, optional

If given, the resulting array must be if the form [beg, end], where beg and end are the endpoints of the window. If None is given then the class window is used. The default is None.    Returns

new_seriesseries

A series with the coefficient of the deg term set to one and all others zero.


numpy.polynomial.chebyshev.Chebyshev.cast method   classmethod polynomial.chebyshev.Chebyshev.cast(series, domain=None, window=None)[source]

Convert series to series of this class. The series is expected to be an instance of some polynomial series of one of the types supported by by the numpy.polynomial module, but could be some other class that supports the convert method.  New in version 1.7.0.   Parameters

seriesseries

The series instance to be converted.
domain{None, array_like}, optional

If given, the array must be of the form [beg, end], where beg and end are the endpoints of the domain. If None is given then the class domain is used. The default is None.
window{None, array_like}, optional

If given, the resulting array must be if the form [beg, end], where beg and end are the endpoints of the window. If None is given then the class window is used. The default is None.    Returns

new_seriesseries

A series of the same kind as the calling class and equal to series when evaluated.      See also  convert

similar instance method


numpy.polynomial.chebyshev.Chebyshev.convert method   polynomial.chebyshev.Chebyshev.convert(domain=None, kind=None, window=None)[source]

Convert series to a different kind and/or domain and/or window.  Parameters

domainarray_like, optional

The domain of the converted series. If the value is None, the default domain of kind is used.
kindclass, optional

The polynomial series type class to which the current instance should be converted. If kind is None, then the class of the current instance is used.
windowarray_like, optional

The window of the converted series. If the value is None, the default window of kind is used.    Returns

new_seriesseries

The returned class can be of different type than the current instance and/or have a different domain and/or different window.     Notes Conversion between domains and class types can result in numerically ill defined series.


numpy.polynomial.chebyshev.Chebyshev.copy method   polynomial.chebyshev.Chebyshev.copy()[source]

Return a copy.  Returns

new_seriesseries

Copy of self.


numpy.polynomial.chebyshev.Chebyshev.cutdeg method   polynomial.chebyshev.Chebyshev.cutdeg(deg)[source]

Truncate series to the given degree. Reduce the degree of the series to deg by discarding the high order terms. If deg is greater than the current degree a copy of the current series is returned. This can be useful in least squares where the coefficients of the high degree terms may be very small.  New in version 1.5.0.   Parameters

degnon-negative int

The series is reduced to degree deg by discarding the high order terms. The value of deg must be a non-negative integer.    Returns

new_seriesseries

New instance of series with reduced degree.


numpy.polynomial.chebyshev.Chebyshev.degree method   polynomial.chebyshev.Chebyshev.degree()[source]

The degree of the series.  New in version 1.5.0.   Returns

degreeint

Degree of the series, one less than the number of coefficients.


numpy.polynomial.chebyshev.Chebyshev.deriv method   polynomial.chebyshev.Chebyshev.deriv(m=1)[source]

Differentiate. Return a series instance of that is the derivative of the current series.  Parameters

mnon-negative int

Find the derivative of order m.    Returns

new_seriesseries

A new series representing the derivative. The domain is the same as the domain of the differentiated series.


numpy.polynomial.chebyshev.Chebyshev.domain attribute   polynomial.chebyshev.Chebyshev.domain = array([-1, 1])


numpy.polynomial.chebyshev.Chebyshev.fit method   classmethod polynomial.chebyshev.Chebyshev.fit(x, y, deg, domain=None, rcond=None, full=False, w=None, window=None)[source]

Least squares fit to data. Return a series instance that is the least squares fit to the data y sampled at x. The domain of the returned instance can be specified and this will often result in a superior fit with less chance of ill conditioning.  Parameters

xarray_like, shape (M,)

x-coordinates of the M sample points (x[i], y[i]).
yarray_like, shape (M,)

y-coordinates of the M sample points (x[i], y[i]).
degint or 1-D array_like

Degree(s) of the fitting polynomials. If deg is a single integer all terms up to and including the deg’th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead.
domain{None, [beg, end], []}, optional

Domain to use for the returned series. If None, then a minimal domain that covers the points x is chosen. If [] the class domain is used. The default value was the class domain in NumPy 1.4 and None in later versions. The [] option was added in numpy 1.5.0.
rcondfloat, optional

Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases.
fullbool, optional

Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned.
warray_like, shape (M,), optional

Weights. If not None, the weight w[i] applies to the unsquared residual y[i] - y_hat[i] at x[i]. Ideally the weights are chosen so that the errors of the products w[i]*y[i] all have the same variance. When using inverse-variance weighting, use w[i] = 1/sigma(y[i]). The default value is None.  New in version 1.5.0.
window{[beg, end]}, optional

Window to use for the returned series. The default value is the default class domain  New in version 1.6.0.     Returns

new_seriesseries

A series that represents the least squares fit to the data and has the domain and window specified in the call. If the coefficients for the unscaled and unshifted basis polynomials are of interest, do new_series.convert().coef.
[resid, rank, sv, rcond]list

These values are only returned if full == True  resid – sum of squared residuals of the least squares fit rank – the numerical rank of the scaled Vandermonde matrix sv – singular values of the scaled Vandermonde matrix rcond – value of rcond.  For more details, see linalg.lstsq.


numpy.polynomial.chebyshev.Chebyshev.fromroots method   classmethod polynomial.chebyshev.Chebyshev.fromroots(roots, domain=[], window=None)[source]

Return series instance that has the specified roots. Returns a series representing the product (x - r[0])*(x - r[1])*...*(x - r[n-1]), where r is a list of roots.  Parameters

rootsarray_like

List of roots.
domain{[], None, array_like}, optional

Domain for the resulting series. If None the domain is the interval from the smallest root to the largest. If [] the domain is the class domain. The default is [].
window{None, array_like}, optional

Window for the returned series. If None the class window is used. The default is None.    Returns

new_seriesseries

Series with the specified roots.


numpy.polynomial.chebyshev.Chebyshev.has_samecoef method   polynomial.chebyshev.Chebyshev.has_samecoef(other)[source]

Check if coefficients match.  New in version 1.6.0.   Parameters

otherclass instance

The other class must have the coef attribute.    Returns

boolboolean

True if the coefficients are the same, False otherwise.


numpy.polynomial.chebyshev.Chebyshev.has_samedomain method   polynomial.chebyshev.Chebyshev.has_samedomain(other)[source]

Check if domains match.  New in version 1.6.0.   Parameters

otherclass instance

The other class must have the domain attribute.    Returns

boolboolean

True if the domains are the same, False otherwise.


numpy.polynomial.chebyshev.Chebyshev.has_sametype method   polynomial.chebyshev.Chebyshev.has_sametype(other)[source]

Check if types match.  New in version 1.7.0.   Parameters

otherobject

Class instance.    Returns

boolboolean

True if other is same class as self


numpy.polynomial.chebyshev.Chebyshev.has_samewindow method   polynomial.chebyshev.Chebyshev.has_samewindow(other)[source]

Check if windows match.  New in version 1.6.0.   Parameters

otherclass instance

The other class must have the window attribute.    Returns

boolboolean

True if the windows are the same, False otherwise.


numpy.polynomial.chebyshev.Chebyshev.identity method   classmethod polynomial.chebyshev.Chebyshev.identity(domain=None, window=None)[source]

Identity function. If p is the returned series, then p(x) == x for all values of x.  Parameters

domain{None, array_like}, optional

If given, the array must be of the form [beg, end], where beg and end are the endpoints of the domain. If None is given then the class domain is used. The default is None.
window{None, array_like}, optional

If given, the resulting array must be if the form [beg, end], where beg and end are the endpoints of the window. If None is given then the class window is used. The default is None.    Returns

new_seriesseries

Series of representing the identity.


numpy.polynomial.chebyshev.Chebyshev.integ method   polynomial.chebyshev.Chebyshev.integ(m=1, k=[], lbnd=None)[source]

Integrate. Return a series instance that is the definite integral of the current series.  Parameters

mnon-negative int

The number of integrations to perform.
karray_like

Integration constants. The first constant is applied to the first integration, the second to the second, and so on. The list of values must less than or equal to m in length and any missing values are set to zero.
lbndScalar

The lower bound of the definite integral.    Returns

new_seriesseries

A new series representing the integral. The domain is the same as the domain of the integrated series.


numpy.polynomial.chebyshev.Chebyshev.interpolate method   classmethod polynomial.chebyshev.Chebyshev.interpolate(func, deg, domain=None, args=())[source]

Interpolate a function at the Chebyshev points of the first kind. Returns the series that interpolates func at the Chebyshev points of the first kind scaled and shifted to the domain. The resulting series tends to a minmax approximation of func when the function is continuous in the domain.  New in version 1.14.0.   Parameters

funcfunction

The function to be interpolated. It must be a function of a single variable of the form f(x, a, b, c...), where a, b, c... are extra arguments passed in the args parameter.
degint

Degree of the interpolating polynomial.
domain{None, [beg, end]}, optional

Domain over which func is interpolated. The default is None, in which case the domain is [-1, 1].
argstuple, optional

Extra arguments to be used in the function call. Default is no extra arguments.    Returns

polynomialChebyshev instance

Interpolating Chebyshev instance.     Notes See numpy.polynomial.chebfromfunction for more details.


numpy.polynomial.chebyshev.Chebyshev.linspace method   polynomial.chebyshev.Chebyshev.linspace(n=100, domain=None)[source]

Return x, y values at equally spaced points in domain. Returns the x, y values at n linearly spaced points across the domain. Here y is the value of the polynomial at the points x. By default the domain is the same as that of the series instance. This method is intended mostly as a plotting aid.  New in version 1.5.0.   Parameters

nint, optional

Number of point pairs to return. The default value is 100.
domain{None, array_like}, optional

If not None, the specified domain is used instead of that of the calling instance. It should be of the form [beg,end]. The default is None which case the class domain is used.    Returns

x, yndarray

x is equal to linspace(self.domain[0], self.domain[1], n) and y is the series evaluated at element of x.


numpy.polynomial.chebyshev.Chebyshev.mapparms method   polynomial.chebyshev.Chebyshev.mapparms()[source]

Return the mapping parameters. The returned values define a linear map off + scl*x that is applied to the input arguments before the series is evaluated. The map depends on the domain and window; if the current domain is equal to the window the resulting map is the identity. If the coefficients of the series instance are to be used by themselves outside this class, then the linear function must be substituted for the x in the standard representation of the base polynomials.  Returns

off, sclfloat or complex

The mapping function is defined by off + scl*x.     Notes If the current domain is the interval [l1, r1] and the window is [l2, r2], then the linear mapping function L is defined by the equations: L(l1) = l2
L(r1) = r2


numpy.polynomial.chebyshev.Chebyshev.roots method   polynomial.chebyshev.Chebyshev.roots()[source]

Return the roots of the series polynomial. Compute the roots for the series. Note that the accuracy of the roots decrease the further outside the domain they lie.  Returns

rootsndarray

Array containing the roots of the series.


numpy.polynomial.chebyshev.Chebyshev.trim method   polynomial.chebyshev.Chebyshev.trim(tol=0)[source]

Remove trailing coefficients Remove trailing coefficients until a coefficient is reached whose absolute value greater than tol or the beginning of the series is reached. If all the coefficients would be removed the series is set to [0]. A new series instance is returned with the new coefficients. The current instance remains unchanged.  Parameters

tolnon-negative number.

All trailing coefficients less than tol will be removed.    Returns

new_seriesseries

New instance of series with trimmed coefficients.


numpy.polynomial.chebyshev.Chebyshev.truncate method   polynomial.chebyshev.Chebyshev.truncate(size)[source]

Truncate series to length size. Reduce the series to length size by discarding the high degree terms. The value of size must be a positive integer. This can be useful in least squares where the coefficients of the high degree terms may be very small.  Parameters

sizepositive int

The series is reduced to length size by discarding the high degree terms. The value of size must be a positive integer.    Returns

new_seriesseries

New instance of series with truncated coefficients.


numpy.polynomial.chebyshev.chebzero   polynomial.chebyshev.chebzero = array([0])

An array object represents a multidimensional, homogeneous array of fixed-size items. An associated data-type object describes the format of each element in the array (its byte-order, how many bytes it occupies in memory, whether it is an integer, a floating point number, or something else, etc.) Arrays should be constructed using array, zeros or empty (refer to the See Also section below). The parameters given here refer to a low-level method (ndarray(…)) for instantiating an array. For more information, refer to the numpy module and examine the methods and attributes of an array.  Parameters
 (for the __new__ method; see Notes below)

shapetuple of ints

Shape of created array.
dtypedata-type, optional

Any object that can be interpreted as a numpy data type.
bufferobject exposing buffer interface, optional

Used to fill the array with data.
offsetint, optional

Offset of array data in buffer.
stridestuple of ints, optional

Strides of data in memory.
order{‘C’, ‘F’}, optional

Row-major (C-style) or column-major (Fortran-style) order.      See also  array

Construct an array.  zeros

Create an array, each element of which is zero.  empty

Create an array, but leave its allocated memory unchanged (i.e., it contains “garbage”).  dtype

Create a data-type.  numpy.typing.NDArray

An ndarray alias generic w.r.t. its dtype.type.    Notes There are two modes of creating an array using __new__:  If buffer is None, then only shape, dtype, and order are used. If buffer is an object exposing the buffer interface, then all keywords are interpreted.  No __init__ method is needed because the array is fully initialized after the __new__ method. Examples These examples illustrate the low-level ndarray constructor. Refer to the See Also section above for easier ways of constructing an ndarray. First mode, buffer is None: >>> np.ndarray(shape=(2,2), dtype=float, order='F')
array([[0.0e+000, 0.0e+000], # random
       [     nan, 2.5e-323]])
 Second mode: >>> np.ndarray((2,), buffer=np.array([1,2,3]),
...            offset=np.int_().itemsize,
...            dtype=int) # offset = 1*itemsize, i.e. skip first element
array([2, 3])
  Attributes

Tndarray

Transpose of the array.
databuffer

The array’s elements, in memory.
dtypedtype object

Describes the format of the elements in the array.
flagsdict

Dictionary containing information related to memory use, e.g., ‘C_CONTIGUOUS’, ‘OWNDATA’, ‘WRITEABLE’, etc.
flatnumpy.flatiter object

Flattened version of the array as an iterator. The iterator allows assignments, e.g., x.flat = 3 (See ndarray.flat for assignment examples; TODO).
imagndarray

Imaginary part of the array.
realndarray

Real part of the array.
sizeint

Number of elements in the array.
itemsizeint

The memory use of each array element in bytes.
nbytesint

The total number of bytes required to store the array data, i.e., itemsize * size.
ndimint

The array’s number of dimensions.
shapetuple of ints

Shape of the array.
stridestuple of ints

The step-size required to move from one element to the next in memory. For example, a contiguous (3, 4) array of type int16 in C-order has strides (8, 2). This implies that to move from element to element in memory requires jumps of 2 bytes. To move from row-to-row, one needs to jump 8 bytes at a time (2 * 4).
ctypesctypes object

Class containing properties of the array needed for interaction with ctypes.
basendarray

If the array is a view into another array, that array is its base (unless that array is also a view). The base array is where the array data is actually stored.


numpy.polynomial.chebyshev.poly2cheb   polynomial.chebyshev.poly2cheb(pol)[source]

Convert a polynomial to a Chebyshev series. Convert an array representing the coefficients of a polynomial (relative to the “standard” basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Chebyshev series, ordered from lowest to highest degree.  Parameters

polarray_like

1-D array containing the polynomial coefficients    Returns

cndarray

1-D array containing the coefficients of the equivalent Chebyshev series.      See also  cheb2poly
  Notes The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples >>> from numpy import polynomial as P
>>> p = P.Polynomial(range(4))
>>> p
Polynomial([0., 1., 2., 3.], domain=[-1,  1], window=[-1,  1])
>>> c = p.convert(kind=P.Chebyshev)
>>> c
Chebyshev([1.  , 3.25, 1.  , 0.75], domain=[-1.,  1.], window=[-1.,  1.])
>>> P.chebyshev.poly2cheb(range(4))
array([1.  , 3.25, 1.  , 0.75])


numpy.polynomial.hermite.herm2poly   polynomial.hermite.herm2poly(c)[source]

Convert a Hermite series to a polynomial. Convert an array representing the coefficients of a Hermite series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the “standard” basis) ordered from lowest to highest degree.  Parameters

carray_like

1-D array containing the Hermite series coefficients, ordered from lowest order term to highest.    Returns

polndarray

1-D array containing the coefficients of the equivalent polynomial (relative to the “standard” basis) ordered from lowest order term to highest.      See also  poly2herm
  Notes The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples >>> from numpy.polynomial.hermite import herm2poly
>>> herm2poly([ 1.   ,  2.75 ,  0.5  ,  0.375])
array([0., 1., 2., 3.])


numpy.polynomial.hermite.hermadd   polynomial.hermite.hermadd(c1, c2)[source]

Add one Hermite series to another. Returns the sum of two Hermite series c1 + c2. The arguments are sequences of coefficients ordered from lowest order term to highest, i.e., [1,2,3] represents the series P_0 + 2*P_1 + 3*P_2.  Parameters

c1, c2array_like

1-D arrays of Hermite series coefficients ordered from low to high.    Returns

outndarray

Array representing the Hermite series of their sum.      See also
hermsub, hermmulx, hermmul, hermdiv, hermpow

  Notes Unlike multiplication, division, etc., the sum of two Hermite series is a Hermite series (without having to “reproject” the result onto the basis set) so addition, just like that of “standard” polynomials, is simply “component-wise.” Examples >>> from numpy.polynomial.hermite import hermadd
>>> hermadd([1, 2, 3], [1, 2, 3, 4])
array([2., 4., 6., 4.])


numpy.polynomial.hermite.hermcompanion   polynomial.hermite.hermcompanion(c)[source]

Return the scaled companion matrix of c. The basis polynomials are scaled so that the companion matrix is symmetric when c is an Hermite basis polynomial. This provides better eigenvalue estimates than the unscaled case and for basis polynomials the eigenvalues are guaranteed to be real if numpy.linalg.eigvalsh is used to obtain them.  Parameters

carray_like

1-D array of Hermite series coefficients ordered from low to high degree.    Returns

matndarray

Scaled companion matrix of dimensions (deg, deg).     Notes  New in version 1.7.0.


numpy.polynomial.hermite.hermder   polynomial.hermite.hermder(c, m=1, scl=1, axis=0)[source]

Differentiate a Hermite series. Returns the Hermite series coefficients c differentiated m times along axis. At each iteration the result is multiplied by scl (the scaling factor is for use in a linear change of variable). The argument c is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series 1*H_0 + 2*H_1 + 3*H_2 while [[1,2],[1,2]] represents 1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) +
2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y) if axis=0 is x and axis=1 is y.  Parameters

carray_like

Array of Hermite series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index.
mint, optional

Number of derivatives taken, must be non-negative. (Default: 1)
sclscalar, optional

Each differentiation is multiplied by scl. The end result is multiplication by scl**m. This is for use in a linear change of variable. (Default: 1)
axisint, optional

Axis over which the derivative is taken. (Default: 0).  New in version 1.7.0.     Returns

derndarray

Hermite series of the derivative.      See also  hermint
  Notes In general, the result of differentiating a Hermite series does not resemble the same operation on a power series. Thus the result of this function may be “unintuitive,” albeit correct; see Examples section below. Examples >>> from numpy.polynomial.hermite import hermder
>>> hermder([ 1. ,  0.5,  0.5,  0.5])
array([1., 2., 3.])
>>> hermder([-0.5,  1./2.,  1./8.,  1./12.,  1./16.], m=2)
array([1., 2., 3.])