From c7ae96fbd74fbd803dfdc11f5395354091241915 Mon Sep 17 00:00:00 2001
From: ShawnL00 <63332547+ShawnL00@users.noreply.github.com>
Date: Sat, 28 Mar 2026 08:20:42 -0500
Subject: [PATCH 1/2] indexing in the formula for phi_j

---
 notes/notes.org | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/notes/notes.org b/notes/notes.org
index ef575f1..518fdd0 100644
--- a/notes/notes.org
+++ b/notes/notes.org
@@ -4990,7 +4990,7 @@ Write down the Lagrange interpolant for nodes $(x_i)_{i=1}^m$ and values $(y_i)_
 Find a basis so that \(V\) is triangular.
 #+LATEX: \begin{hidden}
 Easier to build than Lagrange, but: coefficient finding costs \(O (n^2)\).
-\[\varphi _j (x) = \prod _{k = 1}^{j - 1} (x - x_k) . \]
+\[\varphi _j (x) = \prod _{k = 0}^{j - 1} (x - x_k) . \]
 (At least) two possibilities for coefficient finding:
 
 - Set up \(V\), run forward substitution.

From 7080930b02d6ec14928d5ec8dee061ce6e483c10 Mon Sep 17 00:00:00 2001
From: ShawnL00 <63332547+ShawnL00@users.noreply.github.com>
Date: Sat, 28 Mar 2026 13:27:40 +0000
Subject: [PATCH 2/2] Use k=1..j indexing for phi_j in Newton interpolation

---
 notes/notes.org | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/notes/notes.org b/notes/notes.org
index 518fdd0..7111e7c 100644
--- a/notes/notes.org
+++ b/notes/notes.org
@@ -4990,7 +4990,7 @@ Write down the Lagrange interpolant for nodes $(x_i)_{i=1}^m$ and values $(y_i)_
 Find a basis so that \(V\) is triangular.
 #+LATEX: \begin{hidden}
 Easier to build than Lagrange, but: coefficient finding costs \(O (n^2)\).
-\[\varphi _j (x) = \prod _{k = 0}^{j - 1} (x - x_k) . \]
+\[\varphi _j (x) = \prod _{k = 1}^{j} (x - x_k) . \]
 (At least) two possibilities for coefficient finding:
 
 - Set up \(V\), run forward substitution.