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 non-orthonormal functions defined by the inverse-monomials

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 Gram-Schmidt 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
Platform	Arch	3.9	3.10	3.11	3.12	Continuous Integration
Linux	X86-64	✔	✔	✔	✔
Linux	AARCH-64	✔	✔	✔	✔
macOS	X86-64	✔	✔	✔	✔
macOS	ARM-64	✔	✔	✔	✔
Windows	X86-64	✔	✔	✔	✔
Windows	ARM-64	✔	✔	✔	✔

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 s-ameli 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 is 1.
num_func: number of orthogonal functions to generate, . Default is 9.
end_interval: the right interval of orthogonality, . 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 , , and can be set in the script by the following arguments,

Variable	Variable in script	Option
	`start_index`	`-s`, or `--start-func`
	`num_func`	`-n`, or `--num-func`
	`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, and 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.

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.

[2]	Ameli, S. (2022). ameli/ortho: (v0.2.0). Zenodo.