hyperdual

Documentation

Overview

Package hyperdual provides the hyperdual numeric type and functions. Hyperdual numbers are an extension of the real numbers in the form a+bϵ₁+bϵ₂+dϵ₁ϵ₂ where ϵ₁^2=0 and ϵ₂^2=0, but ϵ₁≠0, ϵ₂≠0 and ϵ₁ϵ₂≠0.

See https://doi.org/10.2514/6.2011-886 and http://adl.stanford.edu/hyperdual/ 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)

Types

type Number

type Number struct {
	Real, E1mag, E2mag, E1E2mag float64
}

Number is a float64 precision hyperdual number.

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-Infϵ₁-Infϵ₂+Infϵ₁ϵ₂)
Acos(1) = (0-Infϵ₁-Infϵ₂-Infϵ₁ϵ₂)
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+Infϵ₁+Infϵ₂-Infϵ₁ϵ₂)
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+Nϵ₁+Nϵ₂±0ϵ₁ϵ₂)
Asin(±1) = (±Inf+Infϵ₁+Infϵ₂±Infϵ₁ϵ₂)
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+Nϵ₁+Nϵ₂∓0ϵ₁ϵ₂)
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+Nϵ₁+Nϵ₂∓0ϵ₁ϵ₂)
Atan(±Inf) = (±Pi/2+0ϵ₁+0ϵ₂∓0ϵ₁ϵ₂)

func Atanh

func Atanh(d Number) Number

Atanh returns the inverse hyperbolic tangent of d.

Special cases are:

Atanh(1) = +Inf
Atanh(±0) = (±0+Nϵ₁+Nϵ₂±0ϵ₁ϵ₂)
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 hyperdual inverse of d.

Special cases are:

Inv(±Inf) = ±0-0ϵ₁-0ϵ₂±0ϵ₁ϵ₂
Inv(±0) = ±Inf-Infϵ₁-Infϵ₂±Infϵ₁ϵ₂

func Log

func Log(d Number) Number

Log returns the natural logarithm of d.

Special cases are:

Log(+Inf) = (+Inf+0ϵ₁+0ϵ₂-0ϵ₁ϵ₂)
Log(0) = (-Inf±Infϵ₁±Infϵ₂-Infϵ₁ϵ₂)
Log(x < 0) = NaN
Log(NaN) = NaN

func Mul

func Mul(x, y Number) Number

Mul returns the hyperdual product of x and y.

func Pow

func Pow(d, p Number) Number

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

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+xϵ₁+yϵ₂, ±0) = 1+NaNϵ₁+NaNϵ₂+NaNϵ₁ϵ₂ for any x and y
PowReal(x, ±0) = 1 for any x
PowReal(1+xϵ₁+yϵ₂, z) = 1+xzϵ₁+yzϵ₂+2xyzϵ₁ϵ₂ for any z
PowReal(NaN+xϵ₁+yϵ₂, 1) = NaN+xϵ₁+yϵ₂+NaNϵ₁ϵ₂ for any x
PowReal(x, 1) = x for any x
PowReal(NaN+xϵ₁+xϵ₂, y) = NaN+NaNϵ₁+NaNϵ₂+NaNϵ₁ϵ₂
PowReal(x, NaN) = NaN+NaNϵ₁+NaNϵ₂+NaNϵ₁ϵ₂
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+0ϵ₁+0ϵ₂, +Inf) = +Inf+NaNϵ₁+NaNϵ₂+NaNϵ₁ϵ₂ for |x| > 1
PowReal(x+xϵ₁+yϵ₂, +Inf) = +Inf+Infϵ₁+Infϵ₂+NaNϵ₁ϵ₂ for |x| > 1
PowReal(x, -Inf) = +0+NaNϵ₁+NaNϵ₂+NaNϵ₁ϵ₂ for |x| > 1
PowReal(x+yϵ₁+zϵ₂, +Inf) = +0+NaNϵ₁+NaNϵ₂+NaNϵ₁ϵ₂ for |x| < 1
PowReal(x+0ϵ₁+0ϵ₂, -Inf) = +Inf+NaNϵ₁+NaNϵ₂+NaNϵ₁ϵ₂ for |x| < 1
PowReal(x, -Inf) = +Inf-Infϵ₁-Infϵ₂+NaNϵ₁ϵ₂ 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+NaNϵ₁+NaNϵ₂+NaNϵ₁ϵ₂ 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+Nϵ₁+Nϵ₂∓0ϵ₁ϵ₂)
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+Nϵ₁+Nϵ₂±0ϵ₁ϵ₂)
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+Infϵ₁+Infϵ₂-Infϵ₁ϵ₂)
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+Nϵ₁+Nϵ₂±0ϵ₁ϵ₂)
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+Nϵ₁+Nϵ₂∓0ϵ₁ϵ₂)
Tanh(±Inf) = (±1+0ϵ₁+0ϵ₂∓0ϵ₁ϵ₂)
Tanh(NaN) = NaN

func (Number) Format

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

Format implements fmt.Formatter.