Convert .. math:: to $$...$$

docs/basic-usage/compilation-modes.rst

@@ -33,36 +33,36 @@ during the call to :func:`~galois.GF`.
 The exponential and logarithm lookup tables map every finite field element to a power of the primitive element
 $\alpha$.
 
-.. math::
-    x = \alpha^i
+$$x = \alpha^i$$
 
-.. math::
-    \textrm{log}_{\alpha}(x) = i
+$$\textrm{log}_{\alpha}(x) = i$$
 
 With these lookup tables, many arithmetic operations are simplified. For instance, multiplication of two finite field
 elements is reduced to three lookups and one integer addition.
 
-.. math::
-    x \cdot y &= \alpha^m \cdot \alpha^n \\
-              &= \alpha^{m + n}
+$$
+x \cdot y
+&= \alpha^m \cdot \alpha^n \\
+&= \alpha^{m + n}
+$$
 
 The `Zech logarithm <https://en.wikipedia.org/wiki/Zech%27s_logarithm>`_ is defined below. A similar lookup table is
 created for it.
 
-.. math::
-    1 + \alpha^i = \alpha^{Z(i)}
+$$1 + \alpha^i = \alpha^{Z(i)}$$
 
-.. math::
-    Z(i) = \textrm{log}_{\alpha}(1 + \alpha^i)
+$$Z(i) = \textrm{log}_{\alpha}(1 + \alpha^i)$$
 
 With Zech logarithms, addition of two finite field elements becomes three lookups, one integer addition, and one
 integer subtraction.
 
-.. math::
-    x + y &= \alpha^m + \alpha^n \\
-          &= \alpha^m (1 + \alpha^{n - m}) \\
-          &= \alpha^m \alpha^{Z(n - m)} \\
-          &= \alpha^{m + Z(n - m)}
+$$
+x + y
+&= \alpha^m + \alpha^n \\
+&= \alpha^m (1 + \alpha^{n - m}) \\
+&= \alpha^m \alpha^{Z(n - m)} \\
+&= \alpha^{m + Z(n - m)}
+$$
 
 Finite fields with order less than $2^{20}$ use lookup tables by default. In the limited cases where explicit calculation
 is faster than table lookup, the explicit calculation is used.

src/galois/_codes/_bch.py

@@ -41,9 +41,7 @@ class BCH(_CyclicCode):
     evaluated in $\mathrm{GF}(q^m)$. The element $\alpha$ is a primitive $n$-th root of unity in
     $\mathrm{GF}(q^m)$.
 
-    .. math::
-
-        g(x) = \textrm{LCM}(m_{\alpha^c}(x), \dots, m_{\alpha^{c+d-2}}(x))
+    $$g(x) = \textrm{LCM}(m_{\alpha^c}(x), \dots, m_{\alpha^{c+d-2}}(x))$$
 
     Examples:
         Construct a binary $\textrm{BCH}(15, 7)$ code.
@@ -507,11 +505,9 @@ def decode(
         operation computes the remainder of $r(x)$ by $g(x)$ in the extension field
         $\mathrm{GF}(q^m)$.
 
-        .. math::
-            \mathbf{s} = [r(\alpha^c),\ \dots,\ r(\alpha^{c+d-2})] \in \mathrm{GF}(q^m)^{d-1}
+        $$\mathbf{s} = [r(\alpha^c),\ \dots,\ r(\alpha^{c+d-2})] \in \mathrm{GF}(q^m)^{d-1}$$
 
-        .. math::
-            s(x) = r(x)\ \textrm{mod}\ g(x) \in \mathrm{GF}(q^m)[x]
+        $$s(x) = r(x)\ \textrm{mod}\ g(x) \in \mathrm{GF}(q^m)[x]$$
 
         A syndrome of zeros indicates the received codeword is a valid codeword and there are no errors. If the
         syndrome is non-zero, the decoder will find an error-locator polynomial $\sigma(x)$ and the corresponding

src/galois/_codes/_cyclic.py

@@ -61,30 +61,24 @@ def __init__(
             The message vector $\mathbf{m}$ is a member of $\mathrm{GF}(q)^k$. The corresponding
             message polynomial $m(x)$ is a degree-$k$ polynomial over $\mathrm{GF}(q)$.
 
-            .. math::
-                \mathbf{m} = [m_{k-1},\ \dots,\ m_1,\ m_0] \in \mathrm{GF}(q)^k
+            $$\mathbf{m} = [m_{k-1},\ \dots,\ m_1,\ m_0] \in \mathrm{GF}(q)^k$$
 
-            .. math::
-                m(x) = m_{k-1} x^{k-1} + \dots + m_1 x + m_0 \in \mathrm{GF}(q)[x]
+            $$m(x) = m_{k-1} x^{k-1} + \dots + m_1 x + m_0 \in \mathrm{GF}(q)[x]$$
 
             The codeword vector $\mathbf{c}$ is a member of $\mathrm{GF}(q)^n$. The corresponding
             codeword polynomial $c(x)$ is a degree-$n$ polynomial over $\mathrm{GF}(q)$.
 
-            .. math::
-                \mathbf{c} = [c_{n-1},\ \dots,\ c_1,\ c_0] \in \mathrm{GF}(q)^n
+            $$\mathbf{c} = [c_{n-1},\ \dots,\ c_1,\ c_0] \in \mathrm{GF}(q)^n$$
 
-            .. math::
-                c(x) = c_{n-1} x^{n-1} + \dots + c_1 x + c_0 \in \mathrm{GF}(q)[x]
+            $$c(x) = c_{n-1} x^{n-1} + \dots + c_1 x + c_0 \in \mathrm{GF}(q)[x]$$
 
             The codeword vector is computed by matrix multiplication of the message vector with the generator matrix.
             The equivalent polynomial operation is multiplication of the message polynomial with the generator
             polynomial.
 
-            .. math::
-                \mathbf{c} = \mathbf{m} \mathbf{G}
+            $$\mathbf{c} = \mathbf{m} \mathbf{G}$$
 
-            .. math::
-                c(x) = m(x) g(x)
+            $$c(x) = m(x) g(x)$$
         """,
     )
     def encode(self, message: ArrayLike, output: Literal["codeword", "parity"] = "codeword") -> FieldArray:
@@ -116,21 +110,17 @@ def decode(
             The message vector $\mathbf{m}$ is a member of $\mathrm{GF}(q)^k$. The corresponding
             message polynomial $m(x)$ is a degree-$k$ polynomial over $\mathrm{GF}(q)$.
 
-            .. math::
-                \mathbf{m} = [m_{k-1},\ \dots,\ m_1,\ m_0] \in \mathrm{GF}(q)^k
+            $$\mathbf{m} = [m_{k-1},\ \dots,\ m_1,\ m_0] \in \mathrm{GF}(q)^k$$
 
-            .. math::
-                m(x) = m_{k-1} x^{k-1} + \dots + m_1 x + m_0 \in \mathrm{GF}(q)[x]
+            $$m(x) = m_{k-1} x^{k-1} + \dots + m_1 x + m_0 \in \mathrm{GF}(q)[x]$$
 
             The codeword vector $\mathbf{c}$ is a member of $\mathrm{GF}(q)^n$. The corresponding
             codeword polynomial $c(x)$ is a degree-$n$ polynomial over $\mathrm{GF}(q)$.
             Each codeword polynomial $c(x)$ is divisible by the generator polynomial $g(x)$.
 
-            .. math::
-                \mathbf{c} = [c_{n-1},\ \dots,\ c_1,\ c_0] \in \mathrm{GF}(q)^n
+            $$\mathbf{c} = [c_{n-1},\ \dots,\ c_1,\ c_0] \in \mathrm{GF}(q)^n$$
 
-            .. math::
-                c(x) = c_{n-1} x^{n-1} + \dots + c_1 x + c_0 \in \mathrm{GF}(q)[x]
+            $$c(x) = c_{n-1} x^{n-1} + \dots + c_1 x + c_0 \in \mathrm{GF}(q)[x]$$
         """,
     )
     def decode(self, codeword, output="message", errors=False):

src/galois/_codes/_linear.py

@@ -337,8 +337,7 @@ def t(self) -> int:
         Notes:
             The code can correct $t$ symbol errors in a codeword.
 
-            .. math::
-                t = \bigg\lfloor \frac{d - 1}{2} \bigg\rfloor
+            $$t = \bigg\lfloor \frac{d - 1}{2} \bigg\rfloor$$
         """
         return (self.d - 1) // 2
 

src/galois/_codes/_reed_solomon.py

@@ -37,9 +37,7 @@ class ReedSolomon(_CyclicCode):
     generator polynomial has $d-1$ roots $\alpha^c, \dots, \alpha^{c+d-2}$. The element $\alpha$ is
     a primitive $n$-th root of unity in $\mathrm{GF}(q)$.
 
-    .. math::
-
-        g(x) = (x - \alpha^c) \dots (x - \alpha^{c+d-2})
+    $$g(x) = (x - \alpha^c) \dots (x - \alpha^{c+d-2})$$
 
     Examples:
         Construct a $\textrm{RS}(15, 9)$ code.
@@ -461,11 +459,9 @@ def decode(
         $\mathbf{r}$ at the roots $\alpha^c, \dots, \alpha^{c+d-2}$ of the generator polynomial
         $g(x)$. The equivalent polynomial operation computes the remainder of $r(x)$ by $g(x)$.
 
-        .. math::
-            \mathbf{s} = [r(\alpha^c),\ \dots,\ r(\alpha^{c+d-2})] \in \mathrm{GF}(q)^{d-1}
+        $$\mathbf{s} = [r(\alpha^c),\ \dots,\ r(\alpha^{c+d-2})] \in \mathrm{GF}(q)^{d-1}$$
 
-        .. math::
-            s(x) = r(x)\ \textrm{mod}\ g(x) \in \mathrm{GF}(q)[x]
+        $$s(x) = r(x)\ \textrm{mod}\ g(x) \in \mathrm{GF}(q)[x]$$
 
         A syndrome of zeros indicates the received codeword is a valid codeword and there are no errors. If the
         syndrome is non-zero, the decoder will find an error-locator polynomial $\sigma(x)$ and the corresponding

src/galois/_fields/_array.py

@@ -1474,8 +1474,7 @@ def field_trace(self) -> FieldArray:
             For finite fields, since $L$ is a Galois extension of $K$, the field trace of $x$ is
             defined as a sum of the Galois conjugates of $x$.
 
-            .. math::
-                \mathrm{Tr}_{L / K}(x) = \sum_{i=0}^{m-1} x^{p^i}
+            $$\mathrm{Tr}_{L / K}(x) = \sum_{i=0}^{m-1} x^{p^i}$$
 
         References:
             - https://en.wikipedia.org/wiki/Field_trace
@@ -1520,8 +1519,7 @@ def field_norm(self) -> FieldArray:
             For finite fields, since $L$ is a Galois extension of $K$, the field norm of $x$ is
             defined as a product of the Galois conjugates of $x$.
 
-            .. math::
-                \mathrm{N}_{L / K}(x) = \prod_{i=0}^{m-1} x^{p^i} = x^{(p^m - 1) / (p - 1)}
+            $$\mathrm{N}_{L / K}(x) = \prod_{i=0}^{m-1} x^{p^i} = x^{(p^m - 1) / (p - 1)}$$
 
         References:
             - https://en.wikipedia.org/wiki/Field_norm

src/galois/_lfsr.py

@@ -183,17 +183,14 @@ class FLFSR(_LFSR):
     Notes:
         A Fibonacci LFSR is defined by its feedback polynomial $f(x)$.
 
-        .. math::
-            f(x) = -c_{0}x^{n} - c_{1}x^{n-1} - \dots - c_{n-2}x^{2} - c_{n-1}x + 1 = x^n c(x^{-1})
+        $$f(x) = -c_{0}x^{n} - c_{1}x^{n-1} - \dots - c_{n-2}x^{2} - c_{n-1}x + 1 = x^n c(x^{-1})$$
 
         The feedback polynomial is the reciprocal of the characteristic polynomial $c(x)$ of the linear recurrent
         sequence $y$ produced by the Fibonacci LFSR.
 
-        .. math::
-            c(x) = x^{n} - c_{n-1}x^{n-1} - c_{n-2}x^{n-2} - \dots - c_{1}x - c_{0}
+        $$c(x) = x^{n} - c_{n-1}x^{n-1} - c_{n-2}x^{n-2} - \dots - c_{1}x - c_{0}$$
 
-        .. math::
-            y_t = c_{n-1}y_{t-1} + c_{n-2}y_{t-2} + \dots + c_{1}y_{t-n+2} + c_{0}y_{t-n+1}
+        $$y_t = c_{n-1}y_{t-1} + c_{n-2}y_{t-2} + \dots + c_{1}y_{t-n+2} + c_{0}y_{t-n+1}$$
 
         .. code-block:: text
            :caption: Fibonacci LFSR Configuration
@@ -816,17 +813,14 @@ class GLFSR(_LFSR):
     Notes:
         A Galois LFSR is defined by its feedback polynomial $f(x)$.
 
-        .. math::
-            f(x) = -c_{0}x^{n} - c_{1}x^{n-1} - \dots - c_{n-2}x^{2} - c_{n-1}x + 1 = x^n c(x^{-1})
+        $$f(x) = -c_{0}x^{n} - c_{1}x^{n-1} - \dots - c_{n-2}x^{2} - c_{n-1}x + 1 = x^n c(x^{-1})$$
 
         The feedback polynomial is the reciprocal of the characteristic polynomial $c(x)$ of the linear recurrent
         sequence $y$ produced by the Galois LFSR.
 
-        .. math::
-            c(x) = x^{n} - c_{n-1}x^{n-1} - c_{n-2}x^{n-2} - \dots - c_{1}x - c_{0}
+        $$c(x) = x^{n} - c_{n-1}x^{n-1} - c_{n-2}x^{n-2} - \dots - c_{1}x - c_{0}$$
 
-        .. math::
-            y_t = c_{n-1}y_{t-1} + c_{n-2}y_{t-2} + \dots + c_{1}y_{t-n+2} + c_{0}y_{t-n+1}
+        $$y_t = c_{n-1}y_{t-1} + c_{n-2}y_{t-2} + \dots + c_{1}y_{t-n+2} + c_{0}y_{t-n+1}$$
 
         .. code-block:: text
            :caption: Galois LFSR Configuration
@@ -1460,11 +1454,9 @@ def berlekamp_massey(sequence, output="minimal"):
         The minimal polynomial is the characteristic polynomial $c(x)$ of minimal degree that produces the
         linear recurrent sequence $y$.
 
-        .. math::
-            c(x) = x^{n} - c_{n-1}x^{n-1} - c_{n-2}x^{n-2} - \dots - c_{1}x - c_{0}
+        $$c(x) = x^{n} - c_{n-1}x^{n-1} - c_{n-2}x^{n-2} - \dots - c_{1}x - c_{0}$$
 
-        .. math::
-            y_t = c_{n-1}y_{t-1} + c_{n-2}y_{t-2} + \dots + c_{1}y_{t-n+2} + c_{0}y_{t-n+1}
+        $$y_t = c_{n-1}y_{t-1} + c_{n-2}y_{t-2} + \dots + c_{1}y_{t-n+2} + c_{0}y_{t-n+1}$$
 
         For a linear sequence with order $n$, at least $2n$ output symbols are required to determine the
         minimal polynomial.

src/galois/_modular.py

@@ -81,8 +81,7 @@ def euler_phi(n: int) -> int:
     Notes:
         This function implements the Euler totient function
 
-        .. math::
-            \phi(n) = n \prod_{p\ |\ n} \bigg(1 - \frac{1}{p}\bigg) = \prod_{i=1}^{k} p_i^{e_i-1} \big(p_i - 1\big)
+        $$\phi(n) = n \prod_{p\ |\ n} \bigg(1 - \frac{1}{p}\bigg) = \prod_{i=1}^{k} p_i^{e_i-1} \big(p_i - 1\big)$$
 
         for prime $p$ and the prime factorization $n = p_1^{e_1} \dots p_k^{e_k}$.
 
@@ -743,8 +742,7 @@ def is_primitive_root(g: int, n: int) -> bool:
         Alternatively said, $g$ is a primitive root modulo $n$ if and only if $g$ is a generator of
         the multiplicative group of integers modulo $n$,
 
-        .. math::
-            (\mathbb{Z}/n\mathbb{Z}){^\times} = \{1, g, g^2, \dots, g^{\phi(n)-1}\}
+        $$(\mathbb{Z}/n\mathbb{Z}){^\times} = \{1, g, g^2, \dots, g^{\phi(n)-1}\}$$
 
         where $\phi(n)$ is order of the group.
 

src/galois/_ntt.py

@@ -53,8 +53,7 @@ def ntt(
 
         The $k$-th value of the $N$-point NTT $X = \mathrm{NTT}(x)$ is
 
-        .. math::
-            X_k = \sum_{j=0}^{N-1} x_j \omega_N^{jk} ,
+        $$X_k = \sum_{j=0}^{N-1} x_j \omega_N^{jk} ,$$
 
         with all arithmetic performed in $\mathrm{GF}(p)$.
 
@@ -163,8 +162,7 @@ def intt(
 
         The $j$-th value of the scaled $N$-point INTT $x = \mathrm{INTT}(X)$ is
 
-        .. math::
-            x_j = \frac{1}{N} \sum_{k=0}^{N-1} X_k \omega_N^{-kj} ,
+        $$x_j = \frac{1}{N} \sum_{k=0}^{N-1} X_k \omega_N^{-kj} ,$$
 
         with all arithmetic performed in $\mathrm{GF}(p)$. The scaled INTT has the property that
         $x = \mathrm{INTT}(\mathrm{NTT}(x))$.

src/galois/_polymorphic.py

@@ -409,11 +409,12 @@ def crt(remainders, moduli):
     Notes:
         This function implements the Chinese Remainder Theorem.
 
-        .. math::
-            x &\equiv a_1\ (\textrm{mod}\ m_1) \\
-            x &\equiv a_2\ (\textrm{mod}\ m_2) \\
-            x &\equiv \ldots \\
-            x &\equiv a_n\ (\textrm{mod}\ m_n)
+        $$
+        x &\equiv a_1\ (\textrm{mod}\ m_1) \\
+        x &\equiv a_2\ (\textrm{mod}\ m_2) \\
+        x &\equiv \ldots \\
+        x &\equiv a_n\ (\textrm{mod}\ m_n)
+        $$
 
     References:
         - Section 14.5 from https://cacr.uwaterloo.ca/hac/about/chap14.pdf
@@ -684,16 +685,14 @@ def is_square_free(value):
                 A square-free integer $n$ is divisible by no perfect squares. As a consequence, the prime
                 factorization of a square-free integer $n$ is
 
-                .. math::
-                    n = \prod_{i=1}^{k} p_i^{e_i} = \prod_{i=1}^{k} p_i .
+                $$n = \prod_{i=1}^{k} p_i^{e_i} = \prod_{i=1}^{k} p_i .$$
 
             .. md-tab-item:: Polynomials
 
                 A square-free polynomial $f(x)$ has no irreducible factors with multiplicity greater than one.
                 Therefore, its canonical factorization is
 
-                .. math::
-                    f(x) = \prod_{i=1}^{k} g_i(x)^{e_i} = \prod_{i=1}^{k} g_i(x) .
+                $$f(x) = \prod_{i=1}^{k} g_i(x)^{e_i} = \prod_{i=1}^{k} g_i(x) .$$
 
                 This test is also available in :func:`Poly.is_square_free`.
 

src/galois/_polys/_factor.py

@@ -27,8 +27,7 @@ def is_square_free(f) -> bool:
         A square-free polynomial $f(x)$ has no irreducible factors with multiplicity greater than one.
         Therefore, its canonical factorization is
 
-        .. math::
-            f(x) = \prod_{i=1}^{k} g_i(x)^{e_i} = \prod_{i=1}^{k} g_i(x) .
+        $$f(x) = \prod_{i=1}^{k} g_i(x)^{e_i} = \prod_{i=1}^{k} g_i(x) .$$
 
     Examples:
         Generate irreducible polynomials over $\mathrm{GF}(3)$.
@@ -74,8 +73,7 @@ def square_free_factors(f: Poly) -> tuple[list[Poly], list[int]]:
         The Square-Free Factorization algorithm factors $f(x)$ into a product of $m$ square-free
         polynomials $h_j(x)$ with multiplicity $j$.
 
-        .. math::
-            f(x) = \prod_{j=1}^{m} h_j(x)^j
+        $$f(x) = \prod_{j=1}^{m} h_j(x)^j$$
 
         Some $h_j(x) = 1$, but those polynomials are not returned by this function.
 
@@ -175,17 +173,17 @@ def distinct_degree_factors(f: Poly) -> tuple[list[Poly], list[int]]:
         $d$ into a product of $d$ polynomials $f_i(x)$, where $f_i(x)$ is the product of
         all irreducible factors of $f(x)$ with degree $i$.
 
-        .. math::
-            f(x) = \prod_{i=1}^{d} f_i(x)
+        $$f(x) = \prod_{i=1}^{d} f_i(x)$$
 
         For example, suppose $f(x) = x(x + 1)(x^2 + x + 1)(x^3 + x + 1)(x^3 + x^2 + 1)$ over
         $\mathrm{GF}(2)$, then the distinct-degree factorization is
 
-        .. math::
-            f_1(x) &= x(x + 1) = x^2 + x \\
-            f_2(x) &= x^2 + x + 1 \\
-            f_3(x) &= (x^3 + x + 1)(x^3 + x^2 + 1) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 \\
-            f_i(x) &= 1\ \textrm{for}\ i = 4, \dots, 10.
+        $$
+        f_1(x) &= x(x + 1) = x^2 + x \\
+        f_2(x) &= x^2 + x + 1 \\
+        f_3(x) &= (x^3 + x + 1)(x^3 + x^2 + 1) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 \\
+        f_i(x) &= 1\ \textrm{for}\ i = 4, \dots, 10.
+        $$
 
         Some $f_i(x) = 1$, but those polynomials are not returned by this function. In this example,
         the function returns $\{f_1(x), f_2(x), f_3(x)\}$ and $\{1, 2, 3\}$.

src/galois/_polys/_lagrange.py

@@ -31,11 +31,9 @@ def lagrange_poly(x: Array, y: Array) -> Poly:
     Notes:
         The Lagrange interpolating polynomial is defined as
 
-        .. math::
-            L(x) = \sum_{j=0}^{k-1} y_j \ell_j(x)
+        $$L(x) = \sum_{j=0}^{k-1} y_j \ell_j(x)$$
 
-        .. math::
-            \ell_j(x) = \prod_{\substack{0 \le m < k \\ m \ne j}} \frac{x - x_m}{x_j - x_m} .
+        $$\ell_j(x) = \prod_{\substack{0 \le m < k \\ m \ne j}} \frac{x - x_m}{x_j - x_m} .$$
 
         It is the polynomial of minimal degree that satisfies $L(x_i) = y_i$.