From 0d89696569e247e9fc2e5e11445bafb2f4563d7f Mon Sep 17 00:00:00 2001
From: Toshiki Teramura <toshiki.teramura@gmail.com>
Date: Thu, 8 Sep 2022 14:43:50 +0900
Subject: [PATCH] Update document of `Eig_` and `Eigh_` traits

---
 lax/src/eig.rs  | 15 +++++++++++----
 lax/src/eigh.rs | 24 ++++++++++++++++++++----
 lax/src/lib.rs  | 19 ++++---------------
 3 files changed, 35 insertions(+), 23 deletions(-)

diff --git a/lax/src/eig.rs b/lax/src/eig.rs
index d3c07222..deafba2f 100644
--- a/lax/src/eig.rs
+++ b/lax/src/eig.rs
@@ -1,12 +1,19 @@
-//! Eigenvalue decomposition for general matrices
-
 use crate::{error::*, layout::MatrixLayout, *};
 use cauchy::*;
 use num_traits::{ToPrimitive, Zero};
 
-/// Wraps `*geev` for general matrices
+#[cfg_attr(doc, katexit::katexit)]
+/// Eigenvalue problem for general matrix
 pub trait Eig_: Scalar {
-    /// Calculate Right eigenvalue
+    /// Compute right eigenvalue and eigenvectors $Ax = \lambda x$
+    ///
+    /// LAPACK correspondance
+    /// ----------------------
+    ///
+    /// | f32   | f64   | c32   | c64   |
+    /// |:------|:------|:------|:------|
+    /// | sgeev | dgeev | cgeev | zgeev |
+    ///
     fn eig(
         calc_v: bool,
         l: MatrixLayout,
diff --git a/lax/src/eigh.rs b/lax/src/eigh.rs
index a9406ee6..055b2f47 100644
--- a/lax/src/eigh.rs
+++ b/lax/src/eigh.rs
@@ -1,12 +1,20 @@
-//! Eigenvalue decomposition for Symmetric/Hermite matrices
-
 use super::*;
 use crate::{error::*, layout::MatrixLayout};
 use cauchy::*;
 use num_traits::{ToPrimitive, Zero};
 
+#[cfg_attr(doc, katexit::katexit)]
+/// Eigenvalue problem for symmetric/hermite matrix
 pub trait Eigh_: Scalar {
-    /// Wraps `*syev` for real and `*heev` for complex
+    /// Compute right eigenvalue and eigenvectors $Ax = \lambda x$
+    ///
+    /// LAPACK correspondance
+    /// ----------------------
+    ///
+    /// | f32   | f64   | c32   | c64   |
+    /// |:------|:------|:------|:------|
+    /// | ssyev | dsyev | cheev | zheev |
+    ///
     fn eigh(
         calc_eigenvec: bool,
         layout: MatrixLayout,
@@ -14,7 +22,15 @@ pub trait Eigh_: Scalar {
         a: &mut [Self],
     ) -> Result<Vec<Self::Real>>;
 
-    /// Wraps `*sygv` for real and `*hegv` for complex
+    /// Compute generalized right eigenvalue and eigenvectors $Ax = \lambda B x$
+    ///
+    /// LAPACK correspondance
+    /// ----------------------
+    ///
+    /// | f32   | f64   | c32   | c64   |
+    /// |:------|:------|:------|:------|
+    /// | ssygv | dsygv | chegv | zhegv |
+    ///
     fn eigh_generalized(
         calc_eigenvec: bool,
         layout: MatrixLayout,
diff --git a/lax/src/lib.rs b/lax/src/lib.rs
index d7b1be4a..9a8bf1cf 100644
--- a/lax/src/lib.rs
+++ b/lax/src/lib.rs
@@ -55,22 +55,11 @@
 //! Eigenvalue Problem
 //! -------------------
 //!
-//! Solve eigenvalue problem for a matrix $A$
+//! According to the property input metrix,
+//! there are several types of eigenvalue problem API
 //!
-//! $$ Av_i = \lambda_i v_i $$
-//!
-//! or generalized eigenvalue problem
-//!
-//! $$ Av_i = \lambda_i B v_i $$
-//!
-//! | matrix type                     | Eigenvalue (EV) | Generalized Eigenvalue Problem (EG) |
-//! |:--------------------------------|:----------------|:------------------------------------|
-//! | General (GE)                    |[eig]            | -                                   |
-//! | Symmetric (SY) / Hermitian (HE) |[eigh]           |[eigh_generalized]                   |
-//!
-//! [eig]:              eig/trait.Eig_.html#tymethod.eig
-//! [eigh]:             eigh/trait.Eigh_.html#tymethod.eigh
-//! [eigh_generalized]: eigh/trait.Eigh_.html#tymethod.eigh_generalized
+//! - [Eig_] trait provides methods for eigenvalue problem for general matrix.
+//! - [Eigh_] trait provides methods for eigenvalue problem for symmetric/hermite matrix.
 //!
 //! Singular Value Decomposition (SVD), Least square problem
 //! ----------------------------------------------------------