dual

Documentation

Overview

Package dual provides the dual numeric type and functions. Dual numbers are an extension of the real numbers in the form a+bϵ where ϵ^2=0, but ϵ≠0.

See https://en.wikipedia.org/wiki/Dual_number for details of their properties and uses.

Index

• type Number
  • func Abs(d Number) Number
  • func Acos(d Number) Number
  • func Acosh(d Number) Number
  • func Add(x, y Number) Number
  • func Asin(d Number) Number
  • func Asinh(d Number) Number
  • func Atan(d Number) Number
  • func Atanh(d Number) Number
  • func Cos(d Number) Number
  • func Cosh(d Number) Number
  • func Exp(d Number) Number
  • func Inv(d Number) Number
  • func Log(d Number) Number
  • func Mul(x, y Number) Number
  • func Pow(d, p Number) Number
  • func PowReal(d Number, p float64) Number
  • func Scale(f float64, d Number) Number
  • func Sin(d Number) Number
  • func Sinh(d Number) Number
  • func Sqrt(d Number) Number
  • func Sub(x, y Number) Number
  • func Tan(d Number) Number
  • func Tanh(d Number) Number
  • func (d Number) Format(fs fmt.State, c rune)

Examples

• Number (Fike)

Types

type Number

type Number struct {
	Real, Emag float64
}

Number is a float64 precision dual number.

Example (Fike)

package main

import (
	"fmt"

	"gonum.org/v1/gonum/num/dual"
)

func main() {
	// Calculate the value and derivative of the function
	// e^x/(sqrt(sin(x)^3 + cos(x)^3)).
	fn := func(x dual.Number) dual.Number {
		return dual.Mul(
			dual.Exp(x),
			dual.Inv(dual.Sqrt(
				dual.Add(
					dual.PowReal(dual.Sin(x), 3),
					dual.PowReal(dual.Cos(x), 3)))))
	}

	v := fn(dual.Number{Real: 1.5, Emag: 1})
	fmt.Printf("v=%.4f\n", v)
	fmt.Printf("fn(1.5)=%.4f\nfn'(1.5)=%.4f\n", v.Real, v.Emag)

}

Output:


v=(4.4978+4.0534ϵ)
fn(1.5)=4.4978
fn'(1.5)=4.0534

func Abs

func Abs(d Number) Number

Abs returns the absolute value of d.

func Acos

func Acos(d Number) Number

Acos returns the inverse cosine of d.

Special cases are:

Acos(-1) = (Pi-Infϵ)
Acos(1) = (0-Infϵ)
Acos(x) = NaN if x < -1 or x > 1

func Acosh

func Acosh(d Number) Number

Acosh returns the inverse hyperbolic cosine of d.

Special cases are:

Acosh(+Inf) = +Inf
Acosh(1) = (0+Infϵ)
Acosh(x) = NaN if x < 1
Acosh(NaN) = NaN

func Add

func Add(x, y Number) Number

Add returns the sum of x and y.

func Asin

func Asin(d Number) Number

Asin returns the inverse sine of d.

Special cases are:

Asin(±0) = (±0+Nϵ)
Asin(±1) = (±Inf+Infϵ)
Asin(x) = NaN if x < -1 or x > 1

func Asinh

func Asinh(d Number) Number

Asinh returns the inverse hyperbolic sine of d.

Special cases are:

Asinh(±0) = (±0+Nϵ)
Asinh(±Inf) = ±Inf
Asinh(NaN) = NaN

func Atan

func Atan(d Number) Number

Atan returns the inverse tangent of d.

Special cases are:

Atan(±0) = (±0+Nϵ)
Atan(±Inf) = (±Pi/2+0ϵ)

func Atanh

func Atanh(d Number) Number

Atanh returns the inverse hyperbolic tangent of d.

Special cases are:

Atanh(1) = +Inf
Atanh(±0) = (±0+Nϵ)
Atanh(-1) = -Inf
Atanh(x) = NaN if x < -1 or x > 1
Atanh(NaN) = NaN

func Cos

func Cos(d Number) Number

Cos returns the cosine of d.

Special cases are:

Cos(±Inf) = NaN
Cos(NaN) = NaN

func Cosh

func Cosh(d Number) Number

Cosh returns the hyperbolic cosine of d.

Special cases are:

Cosh(±0) = 1
Cosh(±Inf) = +Inf
Cosh(NaN) = NaN

func Exp

func Exp(d Number) Number

Exp returns e**q, the base-e exponential of d.

Special cases are:

Exp(+Inf) = +Inf
Exp(NaN) = NaN

Very large values overflow to 0 or +Inf. Very small values underflow to 1.

func Inv

func Inv(d Number) Number

Inv returns the dual inverse of d.

Special cases are:

Inv(±Inf) = ±0-0ϵ
Inv(±0) = ±Inf-Infϵ

func Log

func Log(d Number) Number

Log returns the natural logarithm of d.

Special cases are:

Log(+Inf) = (+Inf+0ϵ)
Log(0) = (-Inf±Infϵ)
Log(x < 0) = NaN
Log(NaN) = NaN

func Mul

func Mul(x, y Number) Number

Mul returns the dual product of x and y.

func Pow

func Pow(d, p Number) Number

Pow returns d**r, the base-d exponential of r.

func PowReal

func PowReal(d Number, p float64) Number

PowReal returns x**p, the base-x exponential of p.

Special cases are (in order):

PowReal(NaN+xϵ, ±0) = 1+NaNϵ for any x
PowReal(x, ±0) = 1 for any x
PowReal(1+xϵ, y) = 1+xyϵ for any y
PowReal(x, 1) = x for any x
PowReal(NaN+xϵ, y) = NaN+NaNϵ
PowReal(x, NaN) = NaN+NaNϵ
PowReal(±0, y) = ±Inf for y an odd integer < 0
PowReal(±0, -Inf) = +Inf
PowReal(±0, +Inf) = +0
PowReal(±0, y) = +Inf for finite y < 0 and not an odd integer
PowReal(±0, y) = ±0 for y an odd integer > 0
PowReal(±0, y) = +0 for finite y > 0 and not an odd integer
PowReal(-1, ±Inf) = 1
PowReal(x+0ϵ, +Inf) = +Inf+NaNϵ for |x| > 1
PowReal(x+yϵ, +Inf) = +Inf for |x| > 1
PowReal(x, -Inf) = +0+NaNϵ for |x| > 1
PowReal(x, +Inf) = +0+NaNϵ for |x| < 1
PowReal(x+0ϵ, -Inf) = +Inf+NaNϵ for |x| < 1
PowReal(x, -Inf) = +Inf-Infϵ for |x| < 1
PowReal(+Inf, y) = +Inf for y > 0
PowReal(+Inf, y) = +0 for y < 0
PowReal(-Inf, y) = Pow(-0, -y)
PowReal(x, y) = NaN+NaNϵ for finite x < 0 and finite non-integer y

func Scale

func Scale(f float64, d Number) Number

Scale returns d scaled by f.

func Sin

func Sin(d Number) Number

Sin returns the sine of d.

Special cases are:

Sin(±0) = (±0+Nϵ)
Sin(±Inf) = NaN
Sin(NaN) = NaN

func Sinh

func Sinh(d Number) Number

Sinh returns the hyperbolic sine of d.

Special cases are:

Sinh(±0) = (±0+Nϵ)
Sinh(±Inf) = ±Inf
Sinh(NaN) = NaN

func Sqrt

func Sqrt(d Number) Number

Sqrt returns the square root of d.

Special cases are:

Sqrt(+Inf) = +Inf
Sqrt(±0) = (±0+Infϵ)
Sqrt(x < 0) = NaN
Sqrt(NaN) = NaN

func Sub

func Sub(x, y Number) Number

Sub returns the difference of x and y, x-y.

func Tan

func Tan(d Number) Number

Tan returns the tangent of d.

Special cases are:

Tan(±0) = (±0+Nϵ)
Tan(±Inf) = NaN
Tan(NaN) = NaN

func Tanh

func Tanh(d Number) Number

Tanh returns the hyperbolic tangent of d.

Special cases are:

Tanh(±0) = (±0+Nϵ)
Tanh(±Inf) = (±1+0ϵ)
Tanh(NaN) = NaN

func (Number) Format

func (d Number) Format(fs fmt.State, c rune)

Format implements fmt.Formatter.