ortho
A python package to generate a set of orthogonal functions.
Description
This package generates a set of orthonormal functions, called , based on the set of nonorthonormal functions defined by the inversemonomials
The orthonormalized functions are the linear combination of the functions by
The functions are orthonormal in the interval with respect to the weight function . That is,
where is the Kronecker delta function. The orthogonal functions are generated by GramSchmidt orthogonalization process. This script produces the symbolic functions using Sympy, a Python computer algebraic package. An application of these orthogonal functions can be found in [1].
Build and Test Status
Platform  Arch  Python Version  Continuous Integration  

3.9  3.10  3.11  3.12  
Linux  X8664  ✔  ✔  ✔  ✔  
AARCH64  ✔  ✔  ✔  ✔  
macOS  X8664  ✔  ✔  ✔  ✔  
ARM64  ✔  ✔  ✔  ✔  
Windows  X8664  ✔  ✔  ✔  ✔  
ARM64  ✔  ✔  ✔  ✔ 
Install
Install using either of the following three methods.
1. Install from PyPi
Install using the package available on PyPi by
pip install ortho
2. Install from Anaconda Cloud
Install using Anaconda cloud by
conda install c sameli ortho
3. Install from Source Code
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
>>> 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, . Default is1
. 
num_func
: number of orthogonal functions to generate, . Default is9
. 
end_interval
: the right interval of orthogonality, . Default is1
.
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 executableortho
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 , numfunc[=int]

Number of orthogonal functions to generate. Positive integer. Default is 9. 
s , startfunc[=int]

Starting function index. Nonnegative integer. Default is 1. 
e , endinterval[=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 , , and can be set in the script by the following arguments,
Variable  Variable in script  Option 

start_index 
s , or startfunc


num_func 
n , or numfunc


end_interval 
e , or endinterval

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
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.
Citation
[1]  Ameli, S., and Shadden. S. C. (2022). Interpolating LogDeterminant and Trace of the Powers of Matrix A + t B. Statistics and Computing 32, 108. 
[2]  Ameli, S. (2022). ameli/ortho: (v0.2.0). Zenodo. 