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// Copyright ©2019 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 testquad provides integrals for testing quadrature algorithms.
package testquad
import (
"fmt"
"math"
)
// Integral is a definite integral
//
// ∫_a^b f(x)dx
//
// with a known value.
type Integral struct {
Name string
A, B float64 // Integration limits
F func(float64) float64 // Integrand
Value float64
}
// Constant returns the integral of a constant function
//
// ∫_{-1}^2 alpha dx
func Constant(alpha float64) Integral {
return Integral{
Name: fmt.Sprintf("∫_{-1}^{2} %vdx", alpha),
A: -1,
B: 2,
F: func(float64) float64 {
return alpha
},
Value: 3 * alpha,
}
}
// Poly returns the integral of a polynomial
//
// ∫_{-1}^2 x^degree dx
func Poly(degree int) Integral {
d := float64(degree)
return Integral{
Name: fmt.Sprintf("∫_{-1}^{2} x^%vdx", degree),
A: -1,
B: 2,
F: func(x float64) float64 {
return math.Pow(x, d)
},
Value: (math.Pow(2, d+1) - math.Pow(-1, d+1)) / (d + 1),
}
}
// Sin returns the integral
//
// ∫_0^1 sin(x)dx
func Sin() Integral {
return Integral{
Name: "∫_0^1 sin(x)dx",
A: 0,
B: 1,
F: func(x float64) float64 {
return math.Sin(x)
},
Value: 1 - math.Cos(1),
}
}
// XExpMinusX returns the integral
//
// ∫_0^1 x*exp(-x)dx
func XExpMinusX() Integral {
return Integral{
Name: "∫_0^1 x*exp(-x)dx",
A: 0,
B: 1,
F: func(x float64) float64 {
return x * math.Exp(-x)
},
Value: (math.E - 2) / math.E,
}
}
// Sqrt returns the integral
//
// ∫_0^1 sqrt(x)dx
func Sqrt() Integral {
return Integral{
Name: "∫_0^1 sqrt(x)dx",
A: 0,
B: 1,
F: func(x float64) float64 {
return math.Sqrt(x)
},
Value: 2 / 3.0,
}
}
// ExpOverX2Plus1 returns the integral
//
// ∫_0^1 exp(x)/(x*x+1)dx
func ExpOverX2Plus1() Integral {
return Integral{
Name: "∫_0^1 exp(x)/(x*x+1)dx",
A: 0,
B: 1,
F: func(x float64) float64 {
return math.Exp(x) / (x*x + 1)
},
Value: 1.270724139833620220138,
}
}
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