all: convert ' to ′ and '' to ′′

This is necessary because gofmt in go1.19 imposes smart quotes on comments
that use pairs of single quotes. Doubled-up single tick, U+2032, is chosen
over double tick, U+2033, since the latter is harder to distinguish in many
fonts at normally used font sizes, sometimes being indistinguishable from
other superscript marks such as asterisk. Comparison: ′ ″ *

diff/fd/example_test.go

@@ -17,14 +17,14 @@ func ExampleDerivative() {
 		return math.Sin(x)
 	}
 	// Compute the first derivative of f at 0 using the default settings.
-	fmt.Println("f'(0) ≈", fd.Derivative(f, 0, nil))
+	fmt.Println("f′(0) ≈", fd.Derivative(f, 0, nil))
 	// Compute the first derivative of f at 0 using the forward approximation
 	// with a custom step size.
 	df := fd.Derivative(f, 0, &fd.Settings{
 		Formula: fd.Forward,
 		Step:    1e-3,
 	})
-	fmt.Println("f'(0) ≈", df)
+	fmt.Println("f′(0) ≈", df)
 
 	f = func(x float64) float64 {
 		return math.Pow(math.Cos(x), 3)
@@ -38,12 +38,12 @@ func ExampleDerivative() {
 		OriginKnown: true,
 		OriginValue: f(0),
 	})
-	fmt.Println("f''(0) ≈", df)
+	fmt.Println("f′′(0) ≈", df)
 
 	// Output:
-	// f'(0) ≈ 1
-	// f'(0) ≈ 0.9999998333333416
-	// f''(0) ≈ -2.999999981767587
+	// f′(0) ≈ 1
+	// f′(0) ≈ 0.9999998333333416
+	// f′′(0) ≈ -2.999999981767587
 }
 
 func ExampleJacobian() {

interp/cubic.go

@@ -371,7 +371,7 @@ func makeCubicSplineSecondDerivativeEquations(a mat.MutableBanded, b mat.Mutable
 // NaturalCubic is a piecewise cubic 1-dimensional interpolator with
 // continuous value, first and second derivatives, which can be fitted to (X, Y)
 // value pairs without providing derivatives. It uses the boundary conditions
-// Y''(left end ) = Y''(right end) = 0.
+// Y′′(left end ) = Y′′(right end) = 0.
 // See e.g. https://www.math.drexel.edu/~tolya/cubicspline.pdf for details.
 type NaturalCubic struct {
 	cubic PiecewiseCubic
@@ -396,7 +396,7 @@ func (nc *NaturalCubic) Fit(xs, ys []float64) error {
 	a := mat.NewTridiag(n, nil, nil, nil)
 	b := mat.NewVecDense(n, nil)
 	makeCubicSplineSecondDerivativeEquations(a, b, xs, ys)
-	// Add boundary conditions y''(left) = y''(right) = 0:
+	// Add boundary conditions y′′(left) = y′′(right) = 0:
 	b.SetVec(0, 0)
 	b.SetVec(n-1, 0)
 	a.SetBand(0, 0, 1)
@@ -412,7 +412,7 @@ func (nc *NaturalCubic) Fit(xs, ys []float64) error {
 // ClampedCubic is a piecewise cubic 1-dimensional interpolator with
 // continuous value, first and second derivatives, which can be fitted to (X, Y)
 // value pairs without providing derivatives. It uses the boundary conditions
-// Y'(left end ) = Y'(right end) = 0.
+// Y′(left end ) = Y′(right end) = 0.
 type ClampedCubic struct {
 	cubic PiecewiseCubic
 }
@@ -436,13 +436,13 @@ func (cc *ClampedCubic) Fit(xs, ys []float64) error {
 	a := mat.NewTridiag(n, nil, nil, nil)
 	b := mat.NewVecDense(n, nil)
 	makeCubicSplineSecondDerivativeEquations(a, b, xs, ys)
-	// Add boundary conditions y''(left) = y''(right) = 0:
-	// Condition Y'(left end) = 0:
+	// Add boundary conditions y′′(left) = y′′(right) = 0:
+	// Condition Y′(left end) = 0:
 	dxL := xs[1] - xs[0]
 	b.SetVec(0, (ys[1]-ys[0])/dxL)
 	a.SetBand(0, 0, dxL/3)
 	a.SetBand(0, 1, dxL/6)
-	// Condition Y'(right end) = 0:
+	// Condition Y′(right end) = 0:
 	m := n - 1
 	dxR := xs[m] - xs[m-1]
 	b.SetVec(m, (ys[m]-ys[m-1])/dxR)

mathext/airy.go

@@ -8,7 +8,7 @@ import "gonum.org/v1/gonum/mathext/internal/amos"
 
 // AiryAi returns the value of the Airy function at z. The Airy function here,
 // Ai(z), is one of the two linearly independent solutions to
-//  y'' - y*z = 0.
+//  y′′ - y*z = 0.
 // See http://mathworld.wolfram.com/AiryFunctions.html for more detailed information.
 func AiryAi(z complex128) complex128 {
 	// id specifies the order of the derivative to compute,
@@ -23,7 +23,7 @@ func AiryAi(z complex128) complex128 {
 
 // AiryAiDeriv returns the value of the derivative of the Airy function at z. The
 // Airy function here, Ai(z), is one of the two linearly independent solutions to
-//  y'' - y*z = 0.
+//  y′′ - y*z = 0.
 // See http://mathworld.wolfram.com/AiryFunctions.html for more detailed information.
 func AiryAiDeriv(z complex128) complex128 {
 	// id specifies the order of the derivative to compute,

num/hyperdual/hyperdual_example_test.go

@@ -24,12 +24,12 @@ func ExampleNumber_fike() {
 
 	v := fn(hyperdual.Number{Real: 1.5, E1mag: 1, E2mag: 1})
 	fmt.Printf("v=%.4f\n", v)
-	fmt.Printf("fn(1.5)=%.4f\nfn'(1.5)=%.4f\nfn''(1.5)=%.4f\n", v.Real, v.E1mag, v.E1E2mag)
+	fmt.Printf("fn(1.5)=%.4f\nfn′(1.5)=%.4f\nfn′′(1.5)=%.4f\n", v.Real, v.E1mag, v.E1E2mag)
 
 	// Output:
 	//
 	// v=(4.4978+4.0534ϵ₁+4.0534ϵ₂+9.4631ϵ₁ϵ₂)
 	// fn(1.5)=4.4978
-	// fn'(1.5)=4.0534
-	// fn''(1.5)=9.4631
+	// fn′(1.5)=4.0534
+	// fn′′(1.5)=9.4631
 }

num/hyperdual/hyperdual_test.go

@@ -301,13 +301,13 @@ func TestHyperdual(t *testing.T) {
 				t.Errorf("unexpected %s(%v): got:%v want:%v", test.name, x, fxHyperdual.Real, fx)
 			}
 			if !same(fxHyperdual.E1mag, dFx, tol) {
-				t.Errorf("unexpected %s'(%v) (ϵ₁): got:%v want:%v", test.name, x, fxHyperdual.E1mag, dFx)
+				t.Errorf("unexpected %s′(%v) (ϵ₁): got:%v want:%v", test.name, x, fxHyperdual.E1mag, dFx)
 			}
 			if !same(fxHyperdual.E1mag, fxHyperdual.E2mag, tol) {
 				t.Errorf("mismatched ϵ₁ and ϵ₂ for %s(%v): ϵ₁:%v ϵ₂:%v", test.name, x, fxHyperdual.E1mag, fxHyperdual.E2mag)
 			}
 			if !same(fxHyperdual.E1E2mag, d2Fx, tol) {
-				t.Errorf("unexpected %s''(%v): got:%v want:%v", test.name, x, fxHyperdual.E1E2mag, d2Fx)
+				t.Errorf("unexpected %s′′(%v): got:%v want:%v", test.name, x, fxHyperdual.E1E2mag, d2Fx)
 			}
 		}
 	}