=====
ortho
=====
.. |image01| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/phi_i_perp.svg
.. |image02| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/phi_i.svg
.. |image03| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/interval.svg
.. |image04| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/w.svg
.. |image05| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/delta.svg
.. |image06| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/i_0.svg
.. |image07| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/n.svg
.. |image08| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/L.svg
.. |image09| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/alpha_i.svg
.. |image10| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/a_ij.svg
.. |image11| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/Equation_phi_i.svg
.. |image12| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/Equation_phi_i_perp.svg
.. |image13| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/Equation_orthogonality.svg
.. include_after_this_line
A python package to generate a set of orthogonal functions.
* `Documentation <https://ameli.github.io/ortho/index.html>`__
* `API Reference <https://ameli.github.io/ortho/api.html>`__
* `Github <https://ameli.github.io/ortho>`__
-----------
Description
-----------
This package generates a set of orthonormal functions, called |image01|, based on the set of non-orthonormal functions |image02| defined by the inverse-monomials
|image11|
The orthonormalized functions |image01| are the linear combination of the functions |image02| by
|image12|
The functions |image01| are orthonormal in the interval |image03| with respect to the weight function |image04|. That is,
|image13|
where |image05| is the Kronecker delta function. The orthogonal functions are generated by `Gram-Schmidt orthogonalization process <https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process>`__. This script produces the symbolic functions using `Sympy <https://www.sympy.org>`__, a Python computer algebraic package. An application of these orthogonal functions can be found in [1]_.
---------------------
Build and Test Status
---------------------
|codecov-devel|
.. |y| unicode:: U+2714
.. |n| unicode:: U+2716
+----------+----------+-------+-------+-------+-------+-----------------+
| Platform | Arch | Python Version | Continuous |
+ | +-------+-------+-------+-------+ Integration +
| | | 3.9 | 3.10 | 3.11 | 3.12 | |
+==========+==========+=======+=======+=======+=======+=================+
| Linux | X86-64 | |y| | |y| | |y| | |y| | |build-linux| |
+ +----------+-------+-------+-------+-------+ +
| | AARCH-64 | |y| | |y| | |y| | |y| | |
+----------+----------+-------+-------+-------+-------+-----------------+
| macOS | X86-64 | |y| | |y| | |y| | |y| | |build-macos| |
+ +----------+-------+-------+-------+-------+ +
| | ARM-64 | |y| | |y| | |y| | |y| | |
+----------+----------+-------+-------+-------+-------+-----------------+
| Windows | X86-64 | |y| | |y| | |y| | |y| | |build-windows| |
+ +----------+-------+-------+-------+-------+ +
| | ARM-64 | |y| | |y| | |y| | |y| | |
+----------+----------+-------+-------+-------+-------+-----------------+
.. |build-linux| image:: https://img.shields.io/github/actions/workflow/status/ameli/ortho/build-linux.yml
:target: https://github.com/ameli/ortho/actions?query=workflow%3Abuild-linux
.. |build-macos| image:: https://img.shields.io/github/actions/workflow/status/ameli/ortho/build-macos.yml
:target: https://github.com/ameli/ortho/actions?query=workflow%3Abuild-macos
.. |build-windows| image:: https://img.shields.io/github/actions/workflow/status/ameli/ortho/build-windows.yml
:target: https://github.com/ameli/ortho/actions?query=workflow%3Abuild-windows
-------
Install
-------
Install using either of the following three methods.
~~~~~~~~~~~~~~~~~~~~
1. Install from PyPi
~~~~~~~~~~~~~~~~~~~~
|pypi| |pyversions| |format|
Install using the package available on `PyPi <https://pypi.org/project/ortho>`__ by
::
pip install ortho
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
2. Install from Anaconda Cloud
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
|conda-version| |conda-platform|
Install using `Anaconda cloud <https://anaconda.org/s-ameli/traceinv>`__ by
::
conda install -c s-ameli ortho
~~~~~~~~~~~~~~~~~~~~~~~~~~~
3. Install from Source Code
~~~~~~~~~~~~~~~~~~~~~~~~~~~
|release| |licence| |implementation|
Install directly from the source code by
::
git clone https://github.com/ameli/ortho.git
cd ortho
pip install .
-------
Testing
-------
To test the package, download the source code and use one of the following methods in the directory of the source code:
- *Method 1*: test locally by:
::
python setup.py test
- *Method 2*: test in a virtual environment using ``tox``:
::
pip install tox
tox
-----
Usage
-----
The package can be used in two ways:
~~~~~~~~~~~~~~~~~~~~~
1. Import as a Module
~~~~~~~~~~~~~~~~~~~~~
.. code-block:: python
>>> from ortho import OrthogonalFunctions
>>> # Generate object of orthogonal functions
>>> OF = OrthogonalFunctions(
... start_index=1,
... num_func=9,
... end_interval=1,
... verbose=True)
>>> # Get numeric coefficients alpha[i] and a[i][j]
>>> alpha = OF.alpha
>>> a = OF.coeffs
>>> # Get symbolic coefficients alpha[i] and a[i][j]
>>> sym_alpha = OF.sym_alpha
>>> sym_a = OF.sym_coeffs
>>> # Get symbolic functions phi[i]
>>> sym_phi = OF.sym_phi
>>> # Print Functions
>>> OF.print()
>>> # Check mutual orthogonality of Functions
>>> status = OF.check(verbose=True)
>>> # Plot Functions
>>> OF.plot()
The parameters are:
- ``start_index``: the index of the starting function, |image06|. Default is ``1``.
- ``num_func``: number of orthogonal functions to generate, |image07|. Default is ``9``.
- ``end_interval``: the right interval of orthogonality, |image08|. Default is ``1``.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
2. Use As Standalone Application
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The standalone application can be executed in the terminal in two ways:
#. If you have installed the package, call ``ortho`` executable in terminal:
::
ortho [options]
The optional argument ``[options]`` will be explained in the next section. When the package *ortho* is installed, the executable ``ortho`` is located in the ``/bin`` directory of the python.
#. Without installing the package, the main script of the package can be executed directly from the source code by
::
# Download the package
git clone https://github.com/ameli/ortho.git
# Go to the package source directory
cd ortho
# Execute the main script of the package
python -m ortho [options]
~~~~~~~~~~~~~~~~~~
Optional arguments
~~~~~~~~~~~~~~~~~~
When the *standalone application* (the second method in the above) is called, the executable accepts some optional arguments as follows.
+--------------------------------------+------------------------------------------------------------------------------------------+
| Option | Description |
+======================================+==========================================================================================+
| ``-h``, ``--help`` | Prints a help message. |
+--------------------------------------+------------------------------------------------------------------------------------------+
| ``-v``, ``--version`` | Prints version. |
+--------------------------------------+------------------------------------------------------------------------------------------+
| ``-l``, ``--license`` | Prints author info, citation and license. |
+--------------------------------------+------------------------------------------------------------------------------------------+
| ``-n``, ``--num-func[=int]`` | Number of orthogonal functions to generate. Positive integer. Default is 9. |
+--------------------------------------+------------------------------------------------------------------------------------------+
| ``-s``, ``--start-func[=int]`` | Starting function index. Non-negative integer. Default is 1. |
+--------------------------------------+------------------------------------------------------------------------------------------+
| ``-e``, ``--end-interval[=float]`` | End of the interval of functions domains. A real number greater than zero. Default is 1. |
+--------------------------------------+------------------------------------------------------------------------------------------+
| ``-c``,\ ``--check`` | Checks orthogonality of generated functions. |
+--------------------------------------+------------------------------------------------------------------------------------------+
| ``-p``, ``--plot`` | Plots generated functions, also saves the plot as pdf file in the current directory. |
+--------------------------------------+------------------------------------------------------------------------------------------+
~~~~~~~~~~
Parameters
~~~~~~~~~~
The variables |image06|, |image07|, and |image08| can be set in the script by the following arguments,
+-------------+----------------------+---------------------------------+
| Variable | Variable in script | Option |
+=============+======================+=================================+
| |image06| | ``start_index`` | ``-s``, or ``--start-func`` |
+-------------+----------------------+---------------------------------+
| |image07| | ``num_func`` | ``-n``, or ``--num-func`` |
+-------------+----------------------+---------------------------------+
| |image08| | ``end_interval`` | ``-e``, or ``--end-interval`` |
+-------------+----------------------+---------------------------------+
~~~~~~~~
Examples
~~~~~~~~
#. Generate nine orthogonal functions from index 1 to 9 (defaults)
::
ortho
#. Generate eight orthogonal functions from index 1 to 8
::
ortho -n 8
#. Generate nine orthogonal functions from index 0 to 8
::
ortho -s 0
#. Generate nine orthogonal functions that are orthonormal in the interval [0,10]
::
ortho -e 10
#. Check orthogonality of each two functions, plot the orthonormal functions and save the plot to pdf
::
ortho -c -p
#. A complete example:
::
ortho -n 9 -s 1 -e 1 -c -p
------
Output
------
- Displays the orthogonal functions as computer algebraic symbolic functions. An example a set of generated functions is shown below.
::
phi_1(t) = sqrt(x)
phi_2(t) = sqrt(6)*(5*x**(1/3) - 6*sqrt(x))/3
phi_3(t) = sqrt(2)*(21*x**(1/4) - 40*x**(1/3) + 20*sqrt(x))/2
phi_4(t) = sqrt(10)*(84*x**(1/5) - 210*x**(1/4) + 175*x**(1/3) - 50*sqrt(x))/5
phi_5(t) = sqrt(3)*(330*x**(1/6) - 1008*x**(1/5) + 1134*x**(1/4) - 560*x**(1/3) + 105*sqrt(x))/3
phi_6(t) = sqrt(14)*(1287*x**(1/7) - 4620*x**(1/6) + 6468*x**(1/5) - 4410*x**(1/4) + 1470*x**(1/3) - 196*sqrt(x))/7
phi_7(t) = 5005*x**(1/8)/2 - 10296*x**(1/7) + 17160*x**(1/6) - 14784*x**(1/5) + 6930*x**(1/4) - 1680*x**(1/3) + 168*sqrt(x)
phi_8(t) = sqrt(2)*(19448*x**(1/9) - 90090*x**(1/8) + 173745*x**(1/7) - 180180*x**(1/6) + 108108*x**(1/5) - 37422*x**(1/4) + 6930*x**(1/3) - 540*sqrt(x))/3
phi_9(t) = sqrt(5)*(75582*x**(1/10) - 388960*x**(1/9) + 850850*x**(1/8) - 1029600*x**(1/7) + 750750*x**(1/6) - 336336*x**(1/5) + 90090*x**(1/4) - 13200*x**(1/3) + 825*sqrt(x))/5
- Displays readable coefficients, |image09| and |image10| of the functions. For instance,
::
i alpha_i a_[ij]
------ ---------- -----------------------------------------------------------------------
i = 1: +sqrt(2/2) [1 ]
i = 2: -sqrt(2/3) [6, -5 ]
i = 3: +sqrt(2/4) [20, -40, 21 ]
i = 4: -sqrt(2/5) [50, -175, 210, -84 ]
i = 5: +sqrt(2/6) [105, -560, 1134, -1008, 330 ]
i = 6: -sqrt(2/7) [196, -1470, 4410, -6468, 4620, -1287 ]
i = 7: +sqrt(2/8) [336, -3360, 13860, -29568, 34320, -20592, 5005 ]
i = 8: -sqrt(2/9) [540, -6930, 37422, -108108, 180180, -173745, 90090, -19448 ]
i = 9: +sqrt(2/10) [825, -13200, 90090, -336336, 750750, -1029600, 850850, -388960, 75582]
- Displays the matrix of the mutual inner product of functions to check orthogonality (using option ``-c``). An example of the generated matrix of the mutual inner product of functions is shown below.
::
[[1 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0]
[0 0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 0 1]]
- Plots the set of functions (using option ``-p``) and saves the plot in the current directory. An example of a generated plot is shown below.
.. image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/orthogonal_functions.svg
:align: center
--------
Citation
--------
.. [1] Ameli, S., and Shadden. S. C. (2022). Interpolating Log-Determinant and Trace of the Powers of Matrix **A** + t **B**. *Statistics and Computing* 32, 108. |paper-doi| |paper-arxiv|
.. [2] Ameli, S. (2022). ameli/ortho: (v0.2.0). Zenodo. |code-doi|
.. |travis-devel| image:: https://img.shields.io/travis/com/ameli/ortho
:target: https://travis-ci.com/github/ameli/ortho
.. |codecov-devel| image:: https://img.shields.io/codecov/c/github/ameli/ortho
:target: https://codecov.io/gh/ameli/ortho
.. |licence| image:: https://img.shields.io/github/license/ameli/ortho
:target: https://opensource.org/licenses/BSD-3-Clause
.. |travis-devel-linux| image:: https://img.shields.io/travis/com/ameli/ortho?env=BADGE=linux&label=build&branch=main
:target: https://travis-ci.com/github/ameli/ortho
.. |travis-devel-osx| image:: https://img.shields.io/travis/com/ameli/ortho?env=BADGE=osx&label=build&branch=main
:target: https://travis-ci.com/github/ameli/ortho
.. |travis-devel-windows| image:: https://img.shields.io/travis/com/ameli/ortho?env=BADGE=windows&label=build&branch=main
:target: https://travis-ci.com/github/ameli/ortho
.. |implementation| image:: https://img.shields.io/pypi/implementation/ortho
.. |pyversions| image:: https://img.shields.io/pypi/pyversions/ortho
.. |format| image:: https://img.shields.io/pypi/format/ortho
.. |pypi| image:: https://img.shields.io/pypi/v/ortho
.. |conda| image:: https://anaconda.org/s-ameli/ortho/badges/installer/conda.svg
:target: https://anaconda.org/s-ameli/ortho
.. |platforms| image:: https://img.shields.io/conda/pn/s-ameli/ortho?color=orange?label=platforms
:target: https://anaconda.org/s-ameli/ortho
.. |conda-version| image:: https://img.shields.io/conda/v/s-ameli/ortho
:target: https://anaconda.org/s-ameli/ortho
.. |release| image:: https://img.shields.io/github/v/tag/ameli/ortho
:target: https://github.com/ameli/ortho/releases/
.. |conda-platform| image:: https://anaconda.org/s-ameli/ortho/badges/platforms.svg
:target: https://anaconda.org/s-ameli/ortho
.. |code-doi| image:: https://zenodo.org/badge/DOI/10.5281/zenodo.6395268.svg
:target: https://doi.org/10.5281/zenodo.6395268
.. |paper-doi| image:: https://img.shields.io/badge/doi-10.1007%2Fs11222--022--10173--4-green
:target: https://doi.org/10.1007/s11222-022-10173-4
.. |paper-arxiv| image:: https://img.shields.io/badge/arXiv-2009.07385 [math.NA]-%23B31B1B
:target: https://doi.org/10.48550/arXiv.2009.07385
Raw data
{
"_id": null,
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"keywords": "orthogonal-functions regression sympy computer-algebragram-schmidt",
"author": "Siavash Ameli",
"author_email": "sameli@berkeley.edu",
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"platform": "Linux",
"description": "=====\northo\n=====\n\n.. |image01| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/phi_i_perp.svg\n.. |image02| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/phi_i.svg\n.. |image03| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/interval.svg\n.. |image04| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/w.svg\n.. |image05| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/delta.svg\n.. |image06| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/i_0.svg\n.. |image07| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/n.svg\n.. |image08| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/L.svg\n.. |image09| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/alpha_i.svg\n.. |image10| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/a_ij.svg\n.. |image11| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/Equation_phi_i.svg\n.. |image12| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/Equation_phi_i_perp.svg\n.. |image13| image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/Equation_orthogonality.svg\n\n.. include_after_this_line\n\nA python package to generate a set of orthogonal functions.\n\n* `Documentation <https://ameli.github.io/ortho/index.html>`__\n* `API Reference <https://ameli.github.io/ortho/api.html>`__\n* `Github <https://ameli.github.io/ortho>`__\n\n-----------\nDescription\n-----------\n\nThis package generates a set of orthonormal functions, called |image01|, based on the set of non-orthonormal functions |image02| defined by the inverse-monomials\n\n|image11|\n\nThe orthonormalized functions |image01| are the linear combination of the functions |image02| by\n\n|image12|\n\nThe functions |image01| are orthonormal in the interval |image03| with respect to the weight function |image04|. That is,\n\n|image13|\n\nwhere |image05| is the Kronecker delta function. The orthogonal functions are generated by `Gram-Schmidt orthogonalization process <https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process>`__. This script produces the symbolic functions using `Sympy <https://www.sympy.org>`__, a Python computer algebraic package. An application of these orthogonal functions can be found in [1]_.\n\n---------------------\nBuild and Test Status\n---------------------\n\n|codecov-devel| \n\n.. |y| unicode:: U+2714\n.. |n| unicode:: U+2716\n\n+----------+----------+-------+-------+-------+-------+-----------------+\n| Platform | Arch | Python Version | Continuous |\n+ | +-------+-------+-------+-------+ Integration +\n| | | 3.9 | 3.10 | 3.11 | 3.12 | |\n+==========+==========+=======+=======+=======+=======+=================+\n| Linux | X86-64 | |y| | |y| | |y| | |y| | |build-linux| |\n+ +----------+-------+-------+-------+-------+ +\n| | AARCH-64 | |y| | |y| | |y| | |y| | |\n+----------+----------+-------+-------+-------+-------+-----------------+\n| macOS | X86-64 | |y| | |y| | |y| | |y| | |build-macos| |\n+ +----------+-------+-------+-------+-------+ +\n| | ARM-64 | |y| | |y| | |y| | |y| | |\n+----------+----------+-------+-------+-------+-------+-----------------+\n| Windows | X86-64 | |y| | |y| | |y| | |y| | |build-windows| |\n+ +----------+-------+-------+-------+-------+ +\n| | ARM-64 | |y| | |y| | |y| | |y| | |\n+----------+----------+-------+-------+-------+-------+-----------------+\n\n.. |build-linux| image:: https://img.shields.io/github/actions/workflow/status/ameli/ortho/build-linux.yml\n :target: https://github.com/ameli/ortho/actions?query=workflow%3Abuild-linux \n.. |build-macos| image:: https://img.shields.io/github/actions/workflow/status/ameli/ortho/build-macos.yml\n :target: https://github.com/ameli/ortho/actions?query=workflow%3Abuild-macos\n.. |build-windows| image:: https://img.shields.io/github/actions/workflow/status/ameli/ortho/build-windows.yml\n :target: https://github.com/ameli/ortho/actions?query=workflow%3Abuild-windows\n\n-------\nInstall\n-------\n\nInstall using either of the following three methods.\n\n~~~~~~~~~~~~~~~~~~~~\n1. Install from PyPi\n~~~~~~~~~~~~~~~~~~~~\n\n|pypi| |pyversions| |format| \n\nInstall using the package available on `PyPi <https://pypi.org/project/ortho>`__ by\n\n::\n\n pip install ortho\n\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n2. Install from Anaconda Cloud\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n|conda-version| |conda-platform|\n\nInstall using `Anaconda cloud <https://anaconda.org/s-ameli/traceinv>`__ by\n\n::\n\n conda install -c s-ameli ortho\n\n~~~~~~~~~~~~~~~~~~~~~~~~~~~\n3. Install from Source Code\n~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\n|release| |licence| |implementation|\n\nInstall directly from the source code by\n\n::\n\n git clone https://github.com/ameli/ortho.git\n cd ortho\n pip install .\n\n-------\nTesting\n-------\n\nTo test the package, download the source code and use one of the following methods in the directory of the source code:\n\n- *Method 1*: test locally by:\n\n ::\n \n python setup.py test\n\n- *Method 2*: test in a virtual environment using ``tox``:\n\n ::\n\n pip install tox\n tox\n\n-----\nUsage\n-----\n\nThe package can be used in two ways:\n\n~~~~~~~~~~~~~~~~~~~~~\n1. Import as a Module\n~~~~~~~~~~~~~~~~~~~~~\n\n.. code-block:: python\n\n >>> from ortho import OrthogonalFunctions\n \n >>> # Generate object of orthogonal functions\n >>> OF = OrthogonalFunctions(\n ... start_index=1,\n ... num_func=9,\n ... end_interval=1,\n ... verbose=True)\n \n >>> # Get numeric coefficients alpha[i] and a[i][j]\n >>> alpha = OF.alpha\n >>> a = OF.coeffs\n\n >>> # Get symbolic coefficients alpha[i] and a[i][j]\n >>> sym_alpha = OF.sym_alpha\n >>> sym_a = OF.sym_coeffs\n\n >>> # Get symbolic functions phi[i]\n >>> sym_phi = OF.sym_phi\n \n >>> # Print Functions\n >>> OF.print()\n \n >>> # Check mutual orthogonality of Functions\n >>> status = OF.check(verbose=True)\n \n >>> # Plot Functions\n >>> OF.plot()\n\nThe parameters are:\n\n- ``start_index``: the index of the starting function, |image06|. Default is ``1``.\n- ``num_func``: number of orthogonal functions to generate, |image07|. Default is ``9``.\n- ``end_interval``: the right interval of orthogonality, |image08|. Default is ``1``.\n\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n2. Use As Standalone Application\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\nThe standalone application can be executed in the terminal in two ways:\n\n#. If you have installed the package, call ``ortho`` executable in terminal:\n\n ::\n\n ortho [options]\n\n The optional argument ``[options]`` will be explained in the next section. When the package *ortho* is installed, the executable ``ortho`` is located in the ``/bin`` directory of the python.\n\n#. Without installing the package, the main script of the package can be executed directly from the source code by\n\n ::\n\n # Download the package\n git clone https://github.com/ameli/ortho.git\n\n # Go to the package source directory\n cd ortho\n\n # Execute the main script of the package\n python -m ortho [options]\n\n~~~~~~~~~~~~~~~~~~\nOptional arguments\n~~~~~~~~~~~~~~~~~~\n\nWhen the *standalone application* (the second method in the above) is called, the executable accepts some optional arguments as follows.\n\n+--------------------------------------+------------------------------------------------------------------------------------------+\n| Option | Description |\n+======================================+==========================================================================================+\n| ``-h``, ``--help`` | Prints a help message. |\n+--------------------------------------+------------------------------------------------------------------------------------------+\n| ``-v``, ``--version`` | Prints version. |\n+--------------------------------------+------------------------------------------------------------------------------------------+\n| ``-l``, ``--license`` | Prints author info, citation and license. |\n+--------------------------------------+------------------------------------------------------------------------------------------+\n| ``-n``, ``--num-func[=int]`` | Number of orthogonal functions to generate. Positive integer. Default is 9. |\n+--------------------------------------+------------------------------------------------------------------------------------------+\n| ``-s``, ``--start-func[=int]`` | Starting function index. Non-negative integer. Default is 1. |\n+--------------------------------------+------------------------------------------------------------------------------------------+\n| ``-e``, ``--end-interval[=float]`` | End of the interval of functions domains. A real number greater than zero. Default is 1. |\n+--------------------------------------+------------------------------------------------------------------------------------------+\n| ``-c``,\\ ``--check`` | Checks orthogonality of generated functions. |\n+--------------------------------------+------------------------------------------------------------------------------------------+\n| ``-p``, ``--plot`` | Plots generated functions, also saves the plot as pdf file in the current directory. |\n+--------------------------------------+------------------------------------------------------------------------------------------+\n\n~~~~~~~~~~\nParameters\n~~~~~~~~~~\n\nThe variables |image06|, |image07|, and |image08| can be set in the script by the following arguments,\n\n+-------------+----------------------+---------------------------------+\n| Variable | Variable in script | Option |\n+=============+======================+=================================+\n| |image06| | ``start_index`` | ``-s``, or ``--start-func`` |\n+-------------+----------------------+---------------------------------+\n| |image07| | ``num_func`` | ``-n``, or ``--num-func`` |\n+-------------+----------------------+---------------------------------+\n| |image08| | ``end_interval`` | ``-e``, or ``--end-interval`` |\n+-------------+----------------------+---------------------------------+\n\n~~~~~~~~\nExamples\n~~~~~~~~\n\n#. Generate nine orthogonal functions from index 1 to 9 (defaults)\n\n ::\n\n ortho\n\n#. Generate eight orthogonal functions from index 1 to 8\n\n ::\n\n ortho -n 8\n\n#. Generate nine orthogonal functions from index 0 to 8\n\n ::\n\n ortho -s 0\n\n#. Generate nine orthogonal functions that are orthonormal in the interval [0,10]\n\n ::\n\n ortho -e 10\n\n#. Check orthogonality of each two functions, plot the orthonormal functions and save the plot to pdf\n\n ::\n\n ortho -c -p\n\n#. A complete example:\n\n ::\n\n ortho -n 9 -s 1 -e 1 -c -p\n\n------\nOutput\n------\n\n- Displays the orthogonal functions as computer algebraic symbolic functions. An example a set of generated functions is shown below.\n\n::\n\n phi_1(t) = sqrt(x)\n phi_2(t) = sqrt(6)*(5*x**(1/3) - 6*sqrt(x))/3\n phi_3(t) = sqrt(2)*(21*x**(1/4) - 40*x**(1/3) + 20*sqrt(x))/2\n phi_4(t) = sqrt(10)*(84*x**(1/5) - 210*x**(1/4) + 175*x**(1/3) - 50*sqrt(x))/5\n phi_5(t) = sqrt(3)*(330*x**(1/6) - 1008*x**(1/5) + 1134*x**(1/4) - 560*x**(1/3) + 105*sqrt(x))/3\n phi_6(t) = sqrt(14)*(1287*x**(1/7) - 4620*x**(1/6) + 6468*x**(1/5) - 4410*x**(1/4) + 1470*x**(1/3) - 196*sqrt(x))/7\n phi_7(t) = 5005*x**(1/8)/2 - 10296*x**(1/7) + 17160*x**(1/6) - 14784*x**(1/5) + 6930*x**(1/4) - 1680*x**(1/3) + 168*sqrt(x)\n phi_8(t) = sqrt(2)*(19448*x**(1/9) - 90090*x**(1/8) + 173745*x**(1/7) - 180180*x**(1/6) + 108108*x**(1/5) - 37422*x**(1/4) + 6930*x**(1/3) - 540*sqrt(x))/3\n phi_9(t) = sqrt(5)*(75582*x**(1/10) - 388960*x**(1/9) + 850850*x**(1/8) - 1029600*x**(1/7) + 750750*x**(1/6) - 336336*x**(1/5) + 90090*x**(1/4) - 13200*x**(1/3) + 825*sqrt(x))/5\n\n- Displays readable coefficients, |image09| and |image10| of the functions. For instance,\n\n::\n\n i alpha_i a_[ij]\n ------ ---------- -----------------------------------------------------------------------\n i = 1: +sqrt(2/2) [1 ]\n i = 2: -sqrt(2/3) [6, -5 ]\n i = 3: +sqrt(2/4) [20, -40, 21 ]\n i = 4: -sqrt(2/5) [50, -175, 210, -84 ]\n i = 5: +sqrt(2/6) [105, -560, 1134, -1008, 330 ]\n i = 6: -sqrt(2/7) [196, -1470, 4410, -6468, 4620, -1287 ]\n i = 7: +sqrt(2/8) [336, -3360, 13860, -29568, 34320, -20592, 5005 ]\n i = 8: -sqrt(2/9) [540, -6930, 37422, -108108, 180180, -173745, 90090, -19448 ]\n i = 9: +sqrt(2/10) [825, -13200, 90090, -336336, 750750, -1029600, 850850, -388960, 75582]\n\n- Displays the matrix of the mutual inner product of functions to check orthogonality (using option ``-c``). An example of the generated matrix of the mutual inner product of functions is shown below.\n\n::\n\n [[1 0 0 0 0 0 0 0 0]\n [0 1 0 0 0 0 0 0 0]\n [0 0 1 0 0 0 0 0 0]\n [0 0 0 1 0 0 0 0 0]\n [0 0 0 0 1 0 0 0 0]\n [0 0 0 0 0 1 0 0 0]\n [0 0 0 0 0 0 1 0 0]\n [0 0 0 0 0 0 0 1 0]\n [0 0 0 0 0 0 0 0 1]]\n\n- Plots the set of functions (using option ``-p``) and saves the plot in the current directory. An example of a generated plot is shown below.\n\n.. image:: https://raw.githubusercontent.com/ameli/ortho/main/docs/source/images/orthogonal_functions.svg\n :align: center\n\n--------\nCitation\n--------\n\n.. [1] Ameli, S., and Shadden. S. C. (2022). Interpolating Log-Determinant and Trace of the Powers of Matrix **A** + t **B**. *Statistics and Computing* 32, 108. |paper-doi| |paper-arxiv|\n.. [2] Ameli, S. (2022). ameli/ortho: (v0.2.0). Zenodo. |code-doi|\n\n.. |travis-devel| image:: https://img.shields.io/travis/com/ameli/ortho\n :target: https://travis-ci.com/github/ameli/ortho\n.. |codecov-devel| image:: https://img.shields.io/codecov/c/github/ameli/ortho\n :target: https://codecov.io/gh/ameli/ortho\n.. |licence| image:: https://img.shields.io/github/license/ameli/ortho\n :target: https://opensource.org/licenses/BSD-3-Clause\n.. |travis-devel-linux| image:: https://img.shields.io/travis/com/ameli/ortho?env=BADGE=linux&label=build&branch=main\n :target: https://travis-ci.com/github/ameli/ortho\n.. |travis-devel-osx| image:: https://img.shields.io/travis/com/ameli/ortho?env=BADGE=osx&label=build&branch=main\n :target: https://travis-ci.com/github/ameli/ortho\n.. |travis-devel-windows| image:: https://img.shields.io/travis/com/ameli/ortho?env=BADGE=windows&label=build&branch=main\n :target: https://travis-ci.com/github/ameli/ortho\n.. |implementation| image:: https://img.shields.io/pypi/implementation/ortho\n.. |pyversions| image:: https://img.shields.io/pypi/pyversions/ortho\n.. |format| image:: https://img.shields.io/pypi/format/ortho\n.. |pypi| image:: https://img.shields.io/pypi/v/ortho\n.. |conda| image:: https://anaconda.org/s-ameli/ortho/badges/installer/conda.svg\n :target: https://anaconda.org/s-ameli/ortho\n.. |platforms| image:: https://img.shields.io/conda/pn/s-ameli/ortho?color=orange?label=platforms\n :target: https://anaconda.org/s-ameli/ortho\n.. |conda-version| image:: https://img.shields.io/conda/v/s-ameli/ortho\n :target: https://anaconda.org/s-ameli/ortho\n.. |release| image:: https://img.shields.io/github/v/tag/ameli/ortho\n :target: https://github.com/ameli/ortho/releases/\n.. |conda-platform| image:: https://anaconda.org/s-ameli/ortho/badges/platforms.svg\n :target: https://anaconda.org/s-ameli/ortho\n.. |code-doi| image:: https://zenodo.org/badge/DOI/10.5281/zenodo.6395268.svg\n :target: https://doi.org/10.5281/zenodo.6395268\n.. |paper-doi| image:: https://img.shields.io/badge/doi-10.1007%2Fs11222--022--10173--4-green\n :target: https://doi.org/10.1007/s11222-022-10173-4\n.. |paper-arxiv| image:: https://img.shields.io/badge/arXiv-2009.07385 [math.NA]-%23B31B1B\n :target: https://doi.org/10.48550/arXiv.2009.07385\n",
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