From c9a3f5ef7cd9bcda70b377b563c0391d6f09b83a Mon Sep 17 00:00:00 2001
From: David Sanders <dpsanders@gmail.com>
Date: Sun, 10 Feb 2019 10:53:11 -0500
Subject: [PATCH 1/5] Add centered forms for scalar functions

---
 src/IntervalOptimisation.jl | 9 +++++++--
 1 file changed, 7 insertions(+), 2 deletions(-)

diff --git a/src/IntervalOptimisation.jl b/src/IntervalOptimisation.jl
index 65cd52c..2a84ea0 100644
--- a/src/IntervalOptimisation.jl
+++ b/src/IntervalOptimisation.jl
@@ -2,16 +2,21 @@
 
 module IntervalOptimisation
 
+using IntervalArithmetic, IntervalRootFinding
+
+
 export minimise, maximise,
        minimize, maximize
 
+export mean_value_form_scalar, third_order_taylor_form_scalar
+
+
 include("SortedVectors.jl")
 using .SortedVectors
 
-using IntervalArithmetic, IntervalRootFinding
-
 
 include("optimise.jl")
+include("centered_forms.jl")
 
 const minimize = minimise
 const maximize = maximise

From 7bcf11c461efbb20450ea86ff8eba346952737bf Mon Sep 17 00:00:00 2001
From: David Sanders <dpsanders@gmail.com>
Date: Sun, 10 Feb 2019 10:59:29 -0500
Subject: [PATCH 2/5] Fix centered forms

---
 src/IntervalOptimisation.jl | 2 +-
 src/optimise.jl             | 5 ++++-
 2 files changed, 5 insertions(+), 2 deletions(-)

diff --git a/src/IntervalOptimisation.jl b/src/IntervalOptimisation.jl
index 2a84ea0..3de776a 100644
--- a/src/IntervalOptimisation.jl
+++ b/src/IntervalOptimisation.jl
@@ -3,7 +3,7 @@
 module IntervalOptimisation
 
 using IntervalArithmetic, IntervalRootFinding
-
+using LinearAlgebra
 
 export minimise, maximise,
        minimize, maximize
diff --git a/src/optimise.jl b/src/optimise.jl
index 4629811..659b4c3 100644
--- a/src/optimise.jl
+++ b/src/optimise.jl
@@ -1,4 +1,7 @@
 
+interval_mid(X::Interval) = Interval(mid(X))
+interval_mid(X::IntervalBox) = IntervalBox(mid(X))
+
 """
     minimise(f, X, tol=1e-3)
 
@@ -27,7 +30,7 @@ function minimise(f, X::T, tol=1e-3) where {T}
         end
 
         # find candidate for upper bound of global minimum by just evaluating a point in the interval:
-        m = sup(f(Interval.(mid.(X))))   # evaluate at midpoint of current interval
+        m = sup(f(interval_mid(X)))   # evaluate at midpoint of current interval
 
         if m < global_min
             global_min = m

From 86012d1995252e9ae20fb5034ba2627f7ad00493 Mon Sep 17 00:00:00 2001
From: David Sanders <dpsanders@gmail.com>
Date: Sun, 10 Feb 2019 22:19:16 -0500
Subject: [PATCH 3/5] Add missing file

---
 src/centered_forms.jl | 34 ++++++++++++++++++++++++++++++++++
 1 file changed, 34 insertions(+)
 create mode 100644 src/centered_forms.jl

diff --git a/src/centered_forms.jl b/src/centered_forms.jl
new file mode 100644
index 0000000..689fe24
--- /dev/null
+++ b/src/centered_forms.jl
@@ -0,0 +1,34 @@
+
+import ForwardDiff: gradient, jacobian, hessian
+
+gradient(f, X::IntervalBox) = gradient(f, X.v)
+# jacobian(f, X::IntervalBox) = jacobian(f, X.v)
+hessian(f, X::IntervalBox) = hessian(f, X.v)
+
+"""
+Calculate the mean-value form of a function ``f:\\mathbb{R}^n \\to \\mathbb{R}``
+using the gradient ``\nabla f``;
+this gives a second-order approximation.
+"""
+function mean_value_form_scalar(f, X)
+    m = IntervalBox(mid(X))
+
+    return f(m) + gradient(f, X) ⋅ (X - m)
+end
+
+mean_value_form_scalar(f) = X -> mean_value_form_scalar(f, X)
+
+
+"""
+Calculate a third-order Taylor form of ``f:\\mathbb{R}^n \\to \\mathbb{R}`` using the Hessian.
+"""
+function third_order_taylor_form_scalar(f, X)
+    m = IntervalBox(mid(X))
+
+    H = ForwardDiff.hessian(f, X)
+    δ = X - m
+
+    return f(m) + gradient(f, m) ⋅ δ + 0.5 * sum(δ[i]*H[i,j]*δ[j] for i in 1:length(X) for j in 1:length(X))
+end
+
+third_order_taylor_form_scalar(f) = X -> third_order_taylor_form_scalar(f, X)

From 3da36ffe8857bafe1d5728658d755167bbfe8a1a Mon Sep 17 00:00:00 2001
From: David Sanders <dpsanders@gmail.com>
Date: Sun, 10 Feb 2019 22:54:58 -0500
Subject: [PATCH 4/5] Fix 3rd order

---
 src/centered_forms.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/centered_forms.jl b/src/centered_forms.jl
index 689fe24..23fb87e 100644
--- a/src/centered_forms.jl
+++ b/src/centered_forms.jl
@@ -25,7 +25,7 @@ Calculate a third-order Taylor form of ``f:\\mathbb{R}^n \\to \\mathbb{R}`` usin
 function third_order_taylor_form_scalar(f, X)
     m = IntervalBox(mid(X))
 
-    H = ForwardDiff.hessian(f, X)
+    H = hessian(f, X)
     δ = X - m
 
     return f(m) + gradient(f, m) ⋅ δ + 0.5 * sum(δ[i]*H[i,j]*δ[j] for i in 1:length(X) for j in 1:length(X))

From dc2694b0d88d661c0e53de27a5e706e31310b8da Mon Sep 17 00:00:00 2001
From: David Sanders <dpsanders@gmail.com>
Date: Sat, 23 Feb 2019 17:25:23 -0600
Subject: [PATCH 5/5] Try directly acting on vector part

---
 src/centered_forms.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/centered_forms.jl b/src/centered_forms.jl
index 23fb87e..8116503 100644
--- a/src/centered_forms.jl
+++ b/src/centered_forms.jl
@@ -13,7 +13,7 @@ this gives a second-order approximation.
 function mean_value_form_scalar(f, X)
     m = IntervalBox(mid(X))
 
-    return f(m) + gradient(f, X) ⋅ (X - m)
+    return f(m) + gradient(f, X.v) ⋅ (X - m)
 end
 
 mean_value_form_scalar(f) = X -> mean_value_form_scalar(f, X)