From 15619d8f358e8535ef8d0211856796be77ae8146 Mon Sep 17 00:00:00 2001
From: Benjamin Cohen-Stead <benwcs@gmail.com>
Date: Thu, 19 Sep 2024 17:02:38 -0400
Subject: [PATCH] Minor Typo Corrections

---
 examples/usage.jl |  9 ++++-----
 src/lanczos.jl    | 18 ++++++++++++++++++
 2 files changed, 22 insertions(+), 5 deletions(-)

diff --git a/examples/usage.jl b/examples/usage.jl
index a395d78..0a1a2c8 100644
--- a/examples/usage.jl
+++ b/examples/usage.jl
@@ -12,7 +12,7 @@
 # ```math
 # f(\epsilon) = \frac{1}{1+e^{\beta (\epsilon-\mu)}} = \frac{1}{2} \left[ 1 + \tanh\left(\tfrac{\beta(\epsilon-\mu)}{2}\right) \right]
 # ```
-# is the Fermi function, with ``\beta = 1/T`` is the inverse temperature and ``\mu`` is the chemical potential.
+# is the Fermi function, with ``\beta = 1/T`` the inverse temperature and ``\mu`` the chemical potential.
 # Here we will applying the kernel polynomial method (KPM) algorithm to approximate the density matrix ``\rho`` by
 # a Chebyshev polynomial expansion.
 
@@ -58,10 +58,9 @@ fermi(ϵ, μ, β) = (1+tanh(β*(ϵ-μ)/2))/2
 ρ = U * Diagonal(fermi.(ϵ, μ, β)) * adjoint(U)
 
 # Now let use the KPM to approximate ``\rho``. We will need to define the order ``M`` of the expansion
-# and give approximate bounds for the egenspectrum of ``H``, making sure to overestimate the the true
-# interval spanned by the egienvalues of ``H``. Note that because we are considering the simple
-# non-interacting model here, the exact eigenspectrum of ``H`` is known and spans
-# the interval ``\epsilon \in [-2t, 2t].``
+# and give approximate bounds for the eigenspectrum of ``H``, making sure to overestimate the the true
+# interval spanned by the eigenvalues of ``H``. Because we are considering the simple non-interacting model,
+# the exact eigenspectrum of ``H`` is known and spans the interval ``\epsilon \in [-2t, 2t].``
 
 ## Define eigenspectrum bounds.
 bounds = (-3.0t, 3.0t)
diff --git a/src/lanczos.jl b/src/lanczos.jl
index 9334df2..6bef86c 100644
--- a/src/lanczos.jl
+++ b/src/lanczos.jl
@@ -34,6 +34,15 @@ Note that the [`eigmin`](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#
 and [`eigmax`](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#LinearAlgebra.eigmax) routines have
 [specialized implementations](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#Elementary-operations) for a
 [`SymTridiagonal`](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#LinearAlgebra.SymTridiagonal) matrix type.
+
+# Similar generalizations of Lanczos have been considered in [2] and [3].
+#
+# 1. C. C. Paige, IMA J. Appl. Math., 373-381 (1972),
+# https://doi.org/10.1093%2Fimamat%2F10.3.373.
+# 2. H. A. van der Vorst, Math. Comp. 39, 559-561 (1982),
+# https://doi.org/10.1090/s0025-5718-1982-0669648-0
+# 3. M. Grüning, A. Marini, X. Gonze, Comput. Mater. Sci. 50, 2148-2156 (2011),
+# https://doi.org/10.1016/j.commatsci.2011.02.021.
 """
 function lanczos(niters, v, A, S = I, rng = Random.default_rng())
 
@@ -67,6 +76,15 @@ based on the contents of the vectors `αs` and `βs`. Note that the [`eigmin`](h
 and [`eigmax`](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#LinearAlgebra.eigmax) routines have
 [specialized implementations](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#Elementary-operations) for a
 [`SymTridiagonal`](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#LinearAlgebra.SymTridiagonal) matrix type.
+
+# Similar generalizations of Lanczos have been considered in [2] and [3].
+#
+# 1. C. C. Paige, IMA J. Appl. Math., 373-381 (1972),
+# https://doi.org/10.1093%2Fimamat%2F10.3.373.
+# 2. H. A. van der Vorst, Math. Comp. 39, 559-561 (1982),
+# https://doi.org/10.1090/s0025-5718-1982-0669648-0
+# 3. M. Grüning, A. Marini, X. Gonze, Comput. Mater. Sci. 50, 2148-2156 (2011),
+# https://doi.org/10.1016/j.commatsci.2011.02.021.
 """
 function lanczos!(
     αs::AbstractVector, βs::AbstractVector, v::AbstractVector,