diff/fd: implement Hessian finite difference, and code cleanups. (#109)

* diff/fd: implement Hessian finite difference, and code cleanups.

This commit primarily adds the Hessian function for finding a finite difference approximation to the Hessian. At the same time, it combines duplicated functionality across the difference routines so that the preludes to all the difference routines look similar

diff/fd/simplefunctions_test.go

@@ -0,0 +1,87 @@
+// Copyright ©2017 The gonum Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package fd
+
+import (
+	"gonum.org/v1/gonum/floats"
+	"gonum.org/v1/gonum/mat"
+)
+
+// ConstFunc is a constant function returning the value held by the type.
+type ConstFunc float64
+
+func (c ConstFunc) Func(x []float64) float64 {
+	return float64(c)
+}
+
+func (c ConstFunc) Grad(grad, x []float64) {
+	for i := range grad {
+		grad[i] = 0
+	}
+}
+
+func (c ConstFunc) Hess(dst mat.MutableSymmetric, x []float64) {
+	n := len(x)
+	for i := 0; i < n; i++ {
+		for j := i; j < n; j++ {
+			dst.SetSym(i, j, 0)
+		}
+	}
+}
+
+// LinearFunc is a linear function returning w*x+c.
+type LinearFunc struct {
+	w []float64
+	c float64
+}
+
+func (l LinearFunc) Func(x []float64) float64 {
+	return floats.Dot(l.w, x) + l.c
+}
+
+func (l LinearFunc) Grad(grad, x []float64) {
+	copy(grad, l.w)
+}
+
+func (l LinearFunc) Hess(dst mat.MutableSymmetric, x []float64) {
+	n := len(x)
+	for i := 0; i < n; i++ {
+		for j := i; j < n; j++ {
+			dst.SetSym(i, j, 0)
+		}
+	}
+}
+
+// QuadFunc is a quadratic function returning 0.5*x'*a*x + b*x + c.
+type QuadFunc struct {
+	a *mat.SymDense
+	b *mat.Vector
+	c float64
+}
+
+func (q QuadFunc) Func(x []float64) float64 {
+	v := mat.NewVector(len(x), x)
+	var tmp mat.Vector
+	tmp.MulVec(q.a, v)
+	return 0.5*mat.Dot(&tmp, v) + mat.Dot(q.b, v) + q.c
+}
+
+func (q QuadFunc) Grad(grad, x []float64) {
+	var tmp mat.Vector
+	v := mat.NewVector(len(x), x)
+	tmp.MulVec(q.a, v)
+	for i := range grad {
+		grad[i] = tmp.At(i, 0) + q.b.At(i, 0)
+	}
+}
+
+func (q QuadFunc) Hess(dst mat.MutableSymmetric, x []float64) {
+	n := len(x)
+	for i := 0; i < n; i++ {
+		for j := i; j < n; j++ {
+			dst.SetSym(i, j, q.a.At(i, j))
+		}
+	}
+}