Polishing documentation

doc/index.md

@@ -58,9 +58,15 @@ Higher are the dimensions weirder are these algebras
    - quaternion   is a non-commutative field
    - octonion     is a non associative (but alternative) division algebra
 
+The functions [commutator](@ref kyosu::commutator ) (resp. [associator](@ref kyosu::associator ))
+can be used to see if two (resp. three)  Cayley-Dickson value commute (resp. associate).
+
 Greater dimensions are not even alternative but keep power-associativity which allows
 to define most elementary functions.
 
+@note Let us recall that alternative means that every subalgrebra generated by
+      two elements is associative.
+
 
 What does this implementation provide
 ======================================
@@ -69,32 +75,48 @@ All operators and functions implemented can receive a mix of scalar or simd of c
 dimensionnality and are defined in the namespace kyosu.
 
 Of course the algebra operation +, -, * and / are provided, but as \ is not an usable **C++**
-character as an operator, the left division a\b is provided as the call ldiv(a,b).
+character as an operator, the left division `a \ b` is provided as the call ldiv(a,b).
 
 Constructors
 ------------
 
 complex and  quaternion can be constructed using callables facilities `complex` and `quaternion`.
 
-complex can also be constructed from their polar representation
+complex can also be constructed from their polar representation\n
 quaternion from various parametrizations of \f$\mathbb{R}^4\f$ or from \f$\mathbb{R}^3\f$ rotations:
 
-  * angle and axis
-  * cylindrical
-  * cylindricospherical
-  * euler
-  * multipolar
-  * rotation_matrix
-  * semipolar
-  * spherical
+  |                         |                                                                 |                                                             |
+  |-------------------------|-----------------------------------------------------------------|-------------------------------------------------------------|
+  | angle and axis          | [from_angle_axis](@ref kyosu::from_angle_axis )                 | [to_angle_axis](@ref kyosu::to_angle_axis )                 |
+  | cylindrical             | [from_cylindrical](@ref kyosu::from_cylindrical )               | [to_cylindrical](@ref kyosu::to_cylindrical )               |
+  | cylindrospherical       | [from_cylindrospherical](@ref kyosu::from_cylindrospherical )   | [to_cylindrospherical](@ref kyosu::to_cylindrospherical )   |
+  | euler                   | [from_euler](@ref kyosu::from_euler )                           | [to_euler](@ref kyosu::to_euler )                           |
+  | multipolar              | [from_multipolar](@ref kyosu::from_multipolar )                 | [to_multipolar](@ref kyosu::to_multipolar )                 |
+  | rotation matrix         | [from_rotation_matrix](@ref kyosu::from_rotation_matrix )       | [to_rotation_matrix](@ref kyosu::to_rotation_matrix )       |
+  | semipolar               | [from_semipolar](@ref kyosu::from_semipolar )                   | [to_semipolar](@ref kyosu::to_semipolar )                   |
+  | spherical               | [from_spherical](@ref kyosu::from_spherical )                   | [to_spherical](@ref kyosu::to_spherical )                   |
+  | two vectors             | [align](@ref kyosu::align )                                     |                                                             |
+
+The third column references to the corresponding to_xxx version that gives back the
+chosen representation from a quaternion input.
+
+TODO cayley_dickson<N> construction by a function.
+
+Operators
+---------
 
-TODO cayley_dickson<N>
+Operators (as said before) `+`, `-`, `*` and `/` can be used in infix form and can mix cayley-dickson values of
+ different dimensinalities. Of course the biggest dimensionlity is recovered in the output.
+
+Prefix forms are also provided as `add`, `sub`, `multiply` and `div`. Also plus and minus for unary versions.
+
+The left division sometimes necessary if the dimensionality is greater than 2 is given as `ldiv`.
 
 Functions
 ---------
 
 Most **KYOSU** callables are usable with all cayley_dickson types. The exception being mainly special
- complex functions and rotation related quaternion usage.
+complex functions and rotation related quaternion usage.
 
 @warning: **EVE** callables that correspond to mathematical functions that
           are only defined on a proper part of the real axis as, for example, `acos` DOES NOT ever provide the same result
@@ -107,10 +129,48 @@ Most **KYOSU** callables are usable with all cayley_dickson types. The exception
 
   * callables usable with all cayley_dickson types
 
-    Most **EVE** arithmetic functions are provided.
-
-  * callables usable with complex only
-
-  * callables usable with quaternion (and complex) only
-
-Most **EVE** arithmetic functions are provided.
+    Most **EVE** arithmetic and math functions are provided.
+
+   |              |               |                 |            |                 |
+   |--------------|---------------|-----------------|------------|-----------------|
+   | [abs](@ref kyosu::abs )                   | [acos](@ref kyosu::acos )               | [acosh](@ref kyosu::acosh )           | [acospi](@ref kyosu::acospi )  | [acot](@ref kyosu::acot )            |
+   | [acotpi](@ref kyosu::acotpi )             | [atanpi](@ref kyosu::atanpi )           | [acoth](@ref kyosu::acoth )           | [arg](@ref kyosu::arg )        | [acsc](@ref kyosu::acsc )            |
+   | [acscpi](@ref kyosu::acscpi )             | [acsch](@ref kyosu::acsch )             | [asec](@ref kyosu::asec )             | [asecpi](@ref kyosu::asecpi )  | [asech](@ref kyosu::asech )          |
+   | [asin](@ref kyosu::asin )                 | [asinpi](@ref kyosu::asinpi )           | [asinh](@ref kyosu::asinh )           | [atan](@ref kyosu::atan )      | [atanh](@ref kyosu::atanh )          |
+   | [average](@ref kyosu::average )           | [ceil](@ref kyosu::ceil )               | [conj](@ref kyosu::conj )             | [cos](@ref kyosu::cos )        | [cosh](@ref kyosu::cosh )            |
+   | [cospi](@ref kyosu::cospi )               | [cot](@ref kyosu::cot )                 | [cotpi](@ref kyosu::cotpi )           | [coth](@ref kyosu::coth )      | [convert](@ref kyosu::convert )      |
+   | [csc](@ref kyosu::csc )                   | [cscpi](@ref kyosu::cscpi )             | [csch](@ref kyosu::csch )             | [dec](@ref kyosu::dec )        | [dist](@ref kyosu::dist )            |
+   | [dot](@ref kyosu::dot )                   | [exp](@ref kyosu::exp )                 | [exp10](@ref kyosu::exp10 )           | [exp2](@ref kyosu::exp2 )      | [exp_i](@ref kyosu::exp_i )          |
+   | [exp_ipi](@ref kyosu::exp_ipi )           | [expm1](@ref kyosu::expm1 )             | [expmx2](@ref kyosu::expmx2 )         | [expx2](@ref kyosu::expx2 )    | [floor](@ref kyosu::floor )          |
+   | [frac](@ref kyosu::frac )                 | [from_polar](@ref kyosu::from_polar )   | [hypot](@ref kyosu::hypot )           | [if_else](@ref kyosu::if_else )| [inc](@ref kyosu::inc )              |
+   | [ipart](@ref kyosu::ipart )               | [is_denormal](@ref kyosu::is_denormal ) | [is_equal](@ref kyosu::is_equal )     | [is_eqz](@ref kyosu::is_eqz )  | [is_finite](@ref kyosu::is_finite )  |
+   | [is_infinite](@ref kyosu::is_infinite )   | [is_imag](@ref kyosu::is_imag )         | [is_nan](@ref kyosu::is_nan )         | [is_nez](@ref kyosu::is_nez )  | [is_not_denormal](@ref kyosu::is_not_denormal ) |
+   | [is_not_equal](@ref kyosu::is_not_equal ) | [is_not_finite](@ref kyosu::is_not_finite ) | [is_not_infinite](@ref kyosu::is_not_finite ) | [is_not_nan](@ref kyosu::is_not_nan ) | [is_not_real](@ref kyosu::is_not_real )     |
+   | [is_pure](@ref kyosu::is_pure )           | [is_real](@ref kyosu::is_real )         | [is_unitary](@ref kyosu::is_unitary ) | [jpart](@ref kyosu::jpart )    | [kpart](@ref kyosu::kpart )          |
+   | [ldiv](@ref kyosu::ldiv )                 | [lerp](@ref kyosu::lerp )               | [log](@ref kyosu::log )               | [log10](@ref kyosu::log10 )    | [log1p](@ref kyosu::log1p )          |
+   | [log_abs](@ref kyosu::log_abs )           | [log2](@ref kyosu::log2 )               | [lpnorm](@ref kyosu::lpnorm )         | [manhattan](@ref kyosu::manhattan ) | [minus](@ref kyosu::minus )     |
+   | [lpart](@ref kyosu::lpart )               | [lipart](@ref kyosu::lipart )           | [ljpart](@ref kyosu::ljpart )         | [lkpart](@ref kyosu::lkpart )  |                                      |
+   | [nearest](@ref kyosu::nearest )           | [oneminus](@ref kyosu::oneminus )       | [pow](@ref kyosu::pow )               | [pow1p](@ref kyosu::pow1p )    | [pow_abs](@ref kyosu::pow_abs )      |
+   | [powm1](@ref kyosu::powm1 )               | [proj](@ref kyosu::proj )               | [pure](@ref kyosu::imag )             | [radinpi](@ref kyosu::radinpi )| [real](@ref kyosu::real )            |
+   | [rec](@ref kyosu::rec )                   | [reldist](@ref kyosu::reldist )         | [sec](@ref kyosu::sec )               | [secpi](@ref kyosu::secpi )    | [sech](@ref kyosu::sech )            |
+   | [sign](@ref kyosu::sign )                 | [sin](@ref kyosu::sin )                 | [sinc](@ref kyosu::sinc )             | [sincos](@ref kyosu::sincos )  | [sinpi](@ref kyosu::sinpi )          |
+   | [sinpicospi](@ref kyosu::sinpicospi )     | [sinh](@ref kyosu::sinh )               | [sinhcosh](@ref kyosu::sinhcosh )     | [slerp](@ref kyosu::slerp )    | [sqr](@ref kyosu::sqr )              |
+   | [sqr_abs](@ref kyosu::sqr_abs )           | [sqrt](@ref kyosu::sqrt )               | [tan](@ref kyosu::tan )               | [tanpi](@ref kyosu::tanpi )    | [tanh](@ref kyosu::tanh )            |
+   | [to_polar](@ref kyosu::to_polar )         | [trunc](@ref kyosu::trunc )             |                                       |                                |                                      |
+
+  * callables usable with complex or real only. These are mainly implementation of some classical meromorphic functions.
+
+    |                  |                  |                    |                  |               |
+    |------------------|------------------|--------------------|------------------|---------------|
+    | [beta](@ref kyosu::beta ) | [deta](@ref kyosu::deta )    | [digamma](@ref kyosu::digamma )  | [erf](@ref kyosu::erf )  | [erfcx](@ref kyosu::erfcx )  |
+    | [erfi](@ref kyosu::erfi ) | [eta](@ref kyosu::eta )      | [faddeeva](@ref kyosu::faddeeva ) | [lambda](@ref kyosu::lambda )  | [lbeta](@ref kyosu::lbeta )  |
+    | [log_abs_gamma](@ref kyosu::log_abs_gamma )    | [log_gamma](@ref kyosu::log_gamma )  | [lrising_factorial](@ref kyosu::lrising_factorial ) | [rising_factorial](@ref kyosu::rising_factorial ) | [tgamma](@ref kyosu::tgamma )        |
+    | [zeta](@ref kyosu::zeta ) |                  |                    |                  |               |
+
+  * callables usable with quaternion complex and real only
+
+    These functions are related to \f$\mathbb{R}^3\f$ rotations
+
+    |           |            |            |
+    |-----------|------------|------------|
+    | [rot_angle](@ref kyosu::rot_angle) |  [rot_axis](@ref kyosu::rot_axis) | [rotate_vec](@ref kyosu::rotate_vec) |

include/kyosu/complex/beta.hpp

@@ -23,7 +23,8 @@ namespace kyosu::tags
       return fn(complex(v), w); }
 
     template<typename T1, typename T2>
-    KYOSU_FORCEINLINE auto operator()(T1 const& target1, T2 const& target2) const noexcept -> decltype(eve::tag_invoke(*this, target1, target2))
+    KYOSU_FORCEINLINE auto operator()(T1 const& target1, T2 const& target2) const noexcept
+    -> decltype(eve::tag_invoke(*this, target1, target2))
     {
       return eve::tag_invoke(*this, target1, target2);
     }
@@ -40,7 +41,8 @@ namespace kyosu
 //! @addtogroup functions
 //! @{
 //!   @var beta
-//!   @brief Computes the beta function: \f$\displaystyle \mathbf{B}(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\f$.
+//!   @brief Computes the beta function: \f$\displaystyle \mathbf{B}(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\f$
+//!   for real or complex entries.
 //!
 //!   **Defined in Header**
 //!
@@ -53,28 +55,18 @@ namespace kyosu
 //!   @code
 //!   namespace kyosu
 //!   {
-//!      template< eve::floating_ordered_value T, eve::floating_ordered_value U >
-//!      auto beta(T x,U y) noexcept;                                        //1
-//!
-//!      template< eve::floating_value T, eve::floating_value U >
-//!      auto beta(eve::complex_t<T> x, U y) noexcept;                    //2
-//!
-//!      template< eve::floating_value T, eve::floating_value U >
-//!      auto beta(T x, eve::complex_t<U> y) noexcept;                    //2
-//!
-//!      template< eve::floating_value T, eve::floating_value U >
-//!      auto beta(eve::complex_t<T> x, eve::complex_t<U> y) noexcept; //2
+//!      auto beta(auto x, auto y) noexcept;
 //!   }
 //!   @endcode
 //!
 //!   **Parameters**
 //!
-//!     * `x`,`y` : Values to process.
+//!     * `x`,`y` : Values to process. Can be a mix of complex and real floating values and complex values.
 //!
 //!   **Return value**
 //!
-//!     1.  \f$\displaystyle \mathbf{B}(x,y) = \int_0^1 t^{x-1}(1-t)^{y-1}\mbox{d}t\f$
-//!     2.  The complex \f$\displaystyle  \mathbb{B}(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\f$ is returned.
+//!     1.  If x and y are real typed values returns \f$\displaystyle \mathbf{B}(x,y) = \int_0^1 t^{x-1}(1-t)^{y-1}\mbox{d}t\f$
+//!     2.  The complex value \f$\displaystyle  \mathbb{B}(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\f$ is returned.
 //!
 //!  @groupheader{Example}
 //!

include/kyosu/complex/deta.hpp

@@ -43,7 +43,7 @@ namespace kyosu
 //! @addtogroup functions
 //! @{
 //!   @var deta
-//!   @brief Computes the Dirichlet sums \f$ \displaystyle \sum_0^\infty \frac{(-1)^n}{(kn+1)^z}\f$.
+//!   @brief Computes the Dirichlet sums \f$ \displaystyle \sum_{n = 0}^\infty \frac{(-1)^n}{(kn+1)^z}\f$.
 //!
 //!   **Defined in Header**
 //!
@@ -56,19 +56,19 @@ namespace kyosu
 //!   @code
 //!   namespace kyosu
 //!   {
-//!      template<unsigned_scalar_value K, eve::ordered_value T>       constexpr auto deta(K, k, T z) noexcept;  //1
-//!      template<unsigned_scalar_value K, kyosu::concepts::complex T> constexpr auto deta(K, k, T z) noexcept;  //2
+//!      template<unsigned_scalar_value K, eve::ordered_value T>       constexpr auto deta(K k, T z) noexcept;  //1
+//!      template<unsigned_scalar_value K, kyosu::concepts::complex T> constexpr auto deta(K k, T z) noexcept;  //2
 //!   }
 //!   @endcode
 //!
 //!   **Parameters**
 //!
 //!     * `k` : scalar unsigned value,  parameter of the sum.
-//!     * `z` : Vcomplex or real value to process.
+//!     * `z` : complex or real value to process.
 //!
 //! **Return value**
 //!
-//!   Returns the Dirichlet sum \f$  \displaystyle \sum_0^\infty \frac{(-1)^n}{(kn+1)^z}\f$
+//!   Returns the Dirichlet sum \f$  \displaystyle \sum_{n = 0}^\infty \frac{(-1)^n}{(kn+1)^z}\f$
 //!
 //!  @groupheader{Example}
 //!

include/kyosu/complex/digamma.hpp

@@ -39,7 +39,7 @@ namespace kyosu
 //! @addtogroup functions
 //! @{
 //!   @var digamma
-//!   @brief Computes the Digamma function i.e. the logarithmic derivative of the \f$\Gamma\f$
+//!   @brief Computes the Digamma function i.e. the logarithmic derivative of the \f$\Gamma\f$ function
 //!
 //!   **Defined in Header**
 //!
@@ -53,13 +53,13 @@ namespace kyosu
 //!   namespace kyosu
 //!   {
 //!      template<kyosu::concepts::complex T>   constexpr T  digamma(T z) noexcept;
-//!      template<eve::floatingordered_value T> constexpr T  digamma(T z) noexcept;
+//!      template<eve::floating_ordered_value T> constexpr T  digamma(T z) noexcept;
 //!   }
 //!   @endcode
 //!
 //!   **Parameters**
 //!
-//!     * `z` : Value to process.
+//!     * `z` : real or complex value to process.
 //!
 //!   **Return value**
 //!

include/kyosu/complex/erf.hpp

@@ -68,7 +68,7 @@ namespace kyosu
 //!
 //! **Return value**
 //!
-//!   1. a real input z is treated as if complex(z) was entered.
+//!   1. a real input z returns eve::erf(z).
 //!
 //!   2.  The value of the error function in the complex plane is returned
 //!

include/kyosu/complex/erfcx.hpp

@@ -67,7 +67,7 @@ namespace kyosu
 //!
 //! **Return value**
 //!
-//!   1. a real input z is treated as if complex(z) was entered.
+//!   1. a real input z return eve::erfcx(z).
 //!
 //!   2. The value of the normalized complementary error function is returned.
 //!

include/kyosu/complex/erfi.hpp

@@ -65,7 +65,7 @@ namespace kyosu
 //!
 //!   **Parameters**
 //!
-//!     * `z` : Vcomplex or real value to process.
+//!     * `z` : complex or real value to process.
 //!
 //! **Return value**
 //!

include/kyosu/complex/eta.hpp

@@ -64,11 +64,11 @@ namespace kyosu
 //!
 //!   **Parameters**
 //!
-//!     * `z` : Vcomplex or real value to process.
+//!     * `z` : complex or real value to process.
 //!
 //! **Return value**
 //!
-//!   Returns the Dirichlet alternating zeta function: sum \f$  \displaystyle \sum_0^\infty \frac{(-1)^n}{(n+1)^z}\f$
+//!   Returns the Dirichlet alternating zeta function: \f$  \displaystyle \sum_0^\infty \frac{(-1)^n}{(n+1)^z}\f$
 //!
 //!  @groupheader{Example}
 //!

include/kyosu/complex/lambda.hpp

@@ -67,7 +67,7 @@ namespace kyosu
 //!
 //!   **Parameters**
 //!
-//!     * `z` : Vcomplex or real value to process.
+//!     * `z` : complex or real value to process.
 //!
 //! **Return value**
 //!

include/kyosu/complex/log_abs_gamma.hpp

@@ -56,7 +56,7 @@ namespace kyosu
 //!   namespace kyosu
 //!   {
 //!      template<kyosu::concepts::complex T>   constexpr T  log_abs_gamma(T z) noexcept;
-//!      template<eve::floatingordered_value T> constexpr T  log_abs_gamma(T z) noexcept;
+//!      template<eve::floating_ordered_value T> constexpr T  log_abs_gamma(T z) noexcept;
 //!   }
 //!   @endcode
 //!

include/kyosu/complex/log_gamma.hpp

@@ -54,7 +54,7 @@ namespace kyosu
 //!   namespace kyosu
 //!   {
 //!      template<kyosu::concepts::complex T>   constexpr T  log_gamma(T z) noexcept;
-//!      template<eve::floatingordered_value T> constexpr T  log_gamma(T z) noexcept;
+//!      template<eve::floating_ordered_value T> constexpr T  log_gamma(T z) noexcept;
 //!   }
 //!   @endcode
 //!

include/kyosu/complex/lrising_factorial.hpp

@@ -73,7 +73,7 @@ namespace kyosu
 //!
 //!   **Return value**
 //!
-//!   @brief Computes the Rising Factorial function i.e. \f$\log\frac{\Gamma(x+y)}{\Gamma(x)}\f$.
+//!   @brief Computes the logarithm Rising Factorial function i.e. \f$\log\frac{\Gamma(x+y)}{\Gamma(x)}\f$.
 //!
 //!  @groupheader{Example}
 //!

include/kyosu/complex/tgamma.hpp

@@ -55,7 +55,7 @@ namespace kyosu
 //!   namespace kyosu
 //!   {
 //!      template<kyosu::concepts::complex T>   constexpr T  tgamma(T z) noexcept;
-//!      template<eve::floatingordered_value T> constexpr T  tgamma(T z) noexcept;
+//!      template<eve::floating_ordered_value T> constexpr T  tgamma(T z) noexcept;
 //!   }
 //!   @endcode
 //!
@@ -65,7 +65,7 @@ namespace kyosu
 //!
 //!   **Return value**
 //!
-//!     Returns \f$\Gamma(z)\f$. If z is floating the result is as if complex(z) was used in the call.
+//!     Returns \f$\Gamma(z)\f$.
 //!
 //!  @groupheader{Example}
 //!

include/kyosu/complex/zeta.hpp

@@ -43,7 +43,7 @@ namespace kyosu
 //! @addtogroup functions
 //! @{
 //!   @var zeta
-//!   @brief Computes the Riemann \f$\zeta\f$  \f$ \displaystyle \sum_0^\infty \frac{1}{(n+1)^z}\f$.
+//!   @brief Computes the Riemann \f$ \displaystyle\zeta(z)=\sum_0^\infty \frac{1}{(n+1)^z}\f$.
 //!
 //!   **Defined in Header**
 //!
@@ -56,19 +56,18 @@ namespace kyosu
 //!   @code
 //!   namespace kyosu
 //!   {
-//!      template<unsigned_scalar_value K, eve::ordered_value T>       constexpr auto zeta(K, k, T z) noexcept;  //1
-//!      template<unsigned_scalar_value K, kyosu::concepts::complex T> constexpr auto zeta(K, k, T z) noexcept;  //2
+//!      template<eve::floating_ordered_value T> constexpr auto zeta(T z) noexcept;
+//!      template<kyosu::concepts::complex T>  constexpr auto zeta(T z) noexcept;
 //!   }
 //!   @endcode
 //!
 //!   **Parameters**
 //!
-//!     * `k` : scalar unsigned value,  parameter of the sum.
-//!     * `z` : Vcomplex or real value to process.
+//!     * `z` : complex or real value to process.
 //!
 //! **Return value**
 //!
-//!   Returns the Dirichlet alternating zzeta function: sum \f$  \displaystyle \sum_0^\infty \frac{1}{(n+1)^z}\f$
+//!   Returns the Dirichlet zeta function: \f$  \displaystyle \sum_0^\infty \frac{1}{(n+1)^z}\f$
 //!
 //!  @groupheader{Example}
 //!