From aa7baaf8d12542043564d3f47c67ad0557d7b36e Mon Sep 17 00:00:00 2001
From: Vladimir Chalupecky <vladimir.chalupecky@gmail.com>
Date: Fri, 15 Apr 2016 11:58:58 +0900
Subject: [PATCH] native: zero index elementary reflectors

---
 native/dgebrd.go | 10 ++++++----
 native/dgelq2.go |  2 +-
 native/dlabrd.go |  8 ++++----
 native/dlarft.go | 12 ++++++------
 4 files changed, 17 insertions(+), 15 deletions(-)

diff --git a/native/dgebrd.go b/native/dgebrd.go
index 0ec2ab60..ecf4d420 100644
--- a/native/dgebrd.go
+++ b/native/dgebrd.go
@@ -19,11 +19,13 @@ import (
 //
 // The remaining elements of A store the data needed to construct Q and P.
 // The matrices Q and P are products of elementary reflectors
-//  Q = H_1 * H_2 * ... * H_nb
-//  P = G_1 * G_2 * ... * G_nb
+//  if m >= n, Q = H(0) * H(1) * ... * H(n-1),
+//             P = G(0) * G(1) * ... * G(n-2),
+//  if m < n,  Q = H(0) * H(1) * ... * H(m-2),
+//             P = G(0) * G(1) * ... * G(m-1),
 // where
-//  H_i = I - tauQ[i] * v_i * v_i^T
-//  G_i = I - tauP[i] * u_i * u_i^T
+//  H(i) = I - tauQ[i] * v_i * v_i^T,
+//  G(i) = I - tauP[i] * u_i * u_i^T.
 //
 // As an example, on exit the entries of A when m = 6, and n = 5
 //  [ d   e  u1  u1  u1]
diff --git a/native/dgelq2.go b/native/dgelq2.go
index e2012457..2521ef52 100644
--- a/native/dgelq2.go
+++ b/native/dgelq2.go
@@ -19,7 +19,7 @@ import "github.com/gonum/blas"
 //
 // See Dgeqr2 for a description of the elementary reflectors and orthonormal
 // matrix Q. Q is constructed as a product of these elementary reflectors,
-// Q = H_k ... H_2*H_1.
+// Q = H(k-1) ... H(1) * H(0).
 //
 // work is temporary storage of length at least m and this function will panic otherwise.
 //
diff --git a/native/dlabrd.go b/native/dlabrd.go
index 03cd4233..037d9ae1 100644
--- a/native/dlabrd.go
+++ b/native/dlabrd.go
@@ -22,11 +22,11 @@ import (
 // elements, and e are the off-diagonal elements.
 //
 // The matrices Q and P are products of elementary reflectors
-//  Q = H_1 * H_2 * ... * H_nb
-//  P = G_1 * G_2 * ... * G_nb
+//  Q = H(0) * H(1) ... H(nb-1)
+//  P = G(0) * G(1) ... G(nb-1)
 // where
-//  H_i = I - tauQ[i] * v_i * v_i^T
-//  G_i = I - tauP[i] * u_i * u_i^T
+//  H(i) = I - tauQ[i] * v_i * v_i^T
+//  G(i) = I - tauP[i] * u_i * u_i^T
 //
 // As an example, on exit the entries of A when m = 6, n = 5, and nb = 2
 //  [ 1   1  u1  u1  u1]
diff --git a/native/dlarft.go b/native/dlarft.go
index ca0a5c16..b3e69c16 100644
--- a/native/dlarft.go
+++ b/native/dlarft.go
@@ -10,13 +10,13 @@ import (
 	"github.com/gonum/lapack"
 )
 
-// Dlarft forms the triangular factor t of a block reflector, storing the answer
+// Dlarft forms the triangular factor T of a block reflector H, storing the answer
 // in t.
-//  H = 1 - V * T * V^T if store == lapack.ColumnWise
-//  H = 1 - V^T * T * V if store == lapack.RowWise
+//  H = I - V * T * V^T  if store == lapack.ColumnWise
+//  H = I - V^T * T * V  if store == lapack.RowWise
 // H is defined by a product of the elementary reflectors where
-//  H = H_1 * H_2 * ... * H_k if direct == lapack.Forward
-//  H = H_k * H_k-1 * ... * H_1 if direct == lapack.Backward
+//  H = H(0) * H(1) ... H(k-1)  if direct == lapack.Forward
+//  H = H(k-1) ... H(1) * H(0)  if direct == lapack.Backward
 //
 // t is a k×k triangular matrix. t is upper triangular if direct = lapack.Forward
 // and lower triangular otherwise. This function will panic if t is not of
@@ -25,7 +25,7 @@ import (
 // store describes the storage of the elementary reflectors in v. Please see
 // Dlarfb for a description of layout.
 //
-// tau contains the scalar factor of the elementary reflectors h.
+// tau contains the scalar factors of the elementary reflectors H(i).
 //
 // Dlarft is an internal routine. It is exported for testing purposes.
 func (Implementation) Dlarft(direct lapack.Direct, store lapack.StoreV, n, k int,