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Add hyperlink to additional linear algebra methods
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mhostetter committed Feb 1, 2023
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129 changes: 22 additions & 107 deletions docs/basic-usage/array-arithmetic.rst
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Expand Up @@ -15,9 +15,10 @@ In the sections below, the finite field :math:`\mathrm{GF}(3^5)` and arrays :mat
Standard arithmetic
-------------------

`NumPy ufuncs <https://numpy.org/devdocs/reference/ufuncs.html>`_ are universal functions that operate on scalars. Unary ufuncs operate on
a single scalar and binary ufuncs operate on two scalars. NumPy extends the scalar operation of ufuncs to operate on arrays in various ways.
This extensibility enables `NumPy broadcasting <https://numpy.org/doc/stable/user/basics.broadcasting.html>`_.
`NumPy ufuncs <https://numpy.org/devdocs/reference/ufuncs.html>`_ are universal functions that operate on scalars.
Unary ufuncs operate on a single scalar and binary ufuncs operate on two scalars. NumPy extends the scalar operation
of ufuncs to operate on arrays in various ways. This extensibility enables
`NumPy broadcasting <https://numpy.org/doc/stable/user/basics.broadcasting.html>`_.

Expand any section for more details.

Expand Down Expand Up @@ -161,7 +162,8 @@ Expand any section for more details.
z = np.sqrt(x); z
z ** 2 == x

See also :func:`~galois.FieldArray.is_square`, :func:`~galois.FieldArray.squares`, and :func:`~galois.FieldArray.non_squares`.
See also :func:`~galois.FieldArray.is_square`, :func:`~galois.FieldArray.squares`, and
:func:`~galois.FieldArray.non_squares`.

.. example:: Logarithm: `np.log(x)` or `x.log()`
:collapsible:
Expand All @@ -175,8 +177,8 @@ Expand any section for more details.
alpha = GF.primitive_element; alpha
alpha ** z == y

Compute the logarithm base :math:`\beta`, a different primitive element of the field. See :func:`FieldArray.log` for more
details.
Compute the logarithm base :math:`\beta`, a different primitive element of the field. See :func:`FieldArray.log`
for more details.

.. ipython-with-reprs:: int,poly,power

Expand All @@ -188,15 +190,17 @@ Expand any section for more details.
Ufunc methods
-------------

:obj:`~galois.FieldArray` instances support `NumPy ufunc methods <https://numpy.org/devdocs/reference/ufuncs.html#methods>`_. Ufunc methods allow
a user to apply a NumPy ufunc in a unique way across the target array. All arithmetic ufuncs are supported.
:obj:`~galois.FieldArray` instances support `NumPy ufunc methods
<https://numpy.org/devdocs/reference/ufuncs.html#methods>`_. Ufunc methods allow a user to apply a NumPy ufunc in a
unique way across the target array. All arithmetic ufuncs are supported.

Expand any section for more details.

.. example:: `reduce()`
:collapsible:

The :obj:`~numpy.ufunc.reduce` methods reduce the input array's dimension by one, applying the ufunc across one axis.
The :obj:`~numpy.ufunc.reduce` methods reduce the input array's dimension by one, applying the ufunc across one
axis.

.. ipython-with-reprs:: int,poly,power

Expand Down Expand Up @@ -228,8 +232,8 @@ Expand any section for more details.
.. example:: `reduceat()`
:collapsible:

The :obj:`~numpy.ufunc.reduceat` methods reduces the input array's dimension by one, applying the ufunc across one axis
in-between certain indices.
The :obj:`~numpy.ufunc.reduceat` methods reduces the input array's dimension by one, applying the ufunc across one
axis in-between certain indices.

.. ipython-with-reprs:: int,poly,power

Expand Down Expand Up @@ -287,8 +291,8 @@ Advanced arithmetic
.. example:: FFT: `np.fft.fft(x)`
:collapsible:

The Discrete Fourier Transform (DFT) of size :math:`n` over the finite field :math:`\mathrm{GF}(p^m)` exists when there
exists a primitive :math:`n`-th root of unity. This occurs when :math:`n\ |\ p^m - 1`.
The Discrete Fourier Transform (DFT) of size :math:`n` over the finite field :math:`\mathrm{GF}(p^m)` exists when
there exists a primitive :math:`n`-th root of unity. This occurs when :math:`n\ |\ p^m - 1`.

.. ipython-with-reprs:: int,poly,power

Expand All @@ -305,8 +309,8 @@ Advanced arithmetic
.. example:: Inverse FFT: `np.fft.ifft(X)`
:collapsible:

The inverse Discrete Fourier Transform (DFT) of size :math:`n` over the finite field :math:`\mathrm{GF}(p^m)` exists when there
exists a primitive :math:`n`-th root of unity. This occurs when :math:`n\ |\ p^m - 1`.
The inverse Discrete Fourier Transform (DFT) of size :math:`n` over the finite field :math:`\mathrm{GF}(p^m)`
exists when there exists a primitive :math:`n`-th root of unity. This occurs when :math:`n\ |\ p^m - 1`.

.. ipython-with-reprs:: int,poly,power

Expand All @@ -323,8 +327,9 @@ Advanced arithmetic
Linear algebra
--------------

Linear algebra on :obj:`~galois.FieldArray` arrays/matrices is supported through both native NumPy linear algebra functions
in :obj:`numpy.linalg` and additional linear algebra routines not included in NumPy.
Linear algebra on :obj:`~galois.FieldArray` arrays/matrices is supported through both native NumPy linear algebra
functions in :obj:`numpy.linalg` and additional `linear algebra methods
<https://mhostetter.github.io/galois/latest/api/galois.FieldArray/#linear-algebra>`_ not included in NumPy.

Expand any section for more details.

Expand Down Expand Up @@ -438,93 +443,3 @@ Expand any section for more details.
A = GF([[14, 21, 14, 28], [24, 22, 23, 23], [16, 30, 26, 18], [4, 23, 18, 3]]); A
A_inv = np.linalg.inv(A); A_inv
A @ A_inv
Additional linear algebra
-------------------------

Below are additional linear algebra routines provided for :obj:`~galois.FieldArray` arrays/matrices that are
not included in NumPy.

.. example:: Row space: `A.row_space()`
:collapsible:

.. ipython:: python
GF = galois.GF(31)
A = GF([[23, 11, 3, 3], [13, 6, 16, 4], [12, 10, 5, 3], [17, 23, 15, 28]]); A
A.row_space()
See :func:`~galois.FieldArray.row_space` for more details.

.. example:: Column space: `A.column_space()`
:collapsible:

.. ipython:: python
GF = galois.GF(31)
A = GF([[23, 11, 3, 3], [13, 6, 16, 4], [12, 10, 5, 3], [17, 23, 15, 28]]); A
A.column_space()
See :func:`~galois.FieldArray.column_space` for more details.

.. example:: Left null space: `A.left_null_space()`
:collapsible:

.. ipython:: python
GF = galois.GF(31)
A = GF([[23, 11, 3, 3], [13, 6, 16, 4], [12, 10, 5, 3], [17, 23, 15, 28]]); A
A.left_null_space()
See :func:`~galois.FieldArray.left_null_space` for more details.

.. example:: Null space: `A.null_space()`
:collapsible:

.. ipython:: python
GF = galois.GF(31)
A = GF([[23, 11, 3, 3], [13, 6, 16, 4], [12, 10, 5, 3], [17, 23, 15, 28]]); A
A.null_space()
See :func:`~galois.FieldArray.null_space` for more details.

.. example:: Gaussian elimination: `A.row_reduce()`
:collapsible:

.. ipython:: python
GF = galois.GF(31)
A = GF([[23, 11, 3, 3], [13, 6, 16, 4], [12, 10, 5, 3], [17, 23, 15, 28]]); A
A.row_reduce()
See :func:`~galois.FieldArray.row_reduce` for more details.

.. example:: LU decomposition: `A.lu_decompose()`
:collapsible:

.. ipython:: python
GF = galois.GF(31)
A = GF([[4, 1, 24], [7, 6, 1], [11, 20, 2]]); A
L, U = A.lu_decompose()
L
U
np.array_equal(L @ U, A)
See :func:`~galois.FieldArray.lu_decompose` for more details.

.. example:: PLU decomposition: `A.plu_decompose()`
:collapsible:

.. ipython:: python
GF = galois.GF(31)
A = GF([[15, 4, 11], [7, 6, 1], [11, 20, 2]]); A
P, L, U = A.plu_decompose()
P
L
U
np.array_equal(P @ L @ U, A)
See :func:`~galois.FieldArray.plu_decompose` for more details.

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