| // Copyright ©2018 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. |
| |
| // Derived from code by Jeffrey A. Fike at http://adl.stanford.edu/hyperdual/ |
| |
| // The MIT License (MIT) |
| // |
| // Copyright (c) 2006 Jeffrey A. Fike |
| // |
| // Permission is hereby granted, free of charge, to any person obtaining a copy |
| // of this software and associated documentation files (the "Software"), to deal |
| // in the Software without restriction, including without limitation the rights |
| // to use, copy, modify, merge, publish, distribute, sublicense, and/or sell |
| // copies of the Software, and to permit persons to whom the Software is |
| // furnished to do so, subject to the following conditions: |
| // |
| // The above copyright notice and this permission notice shall be included in |
| // all copies or substantial portions of the Software. |
| // |
| // THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR |
| // IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, |
| // FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE |
| // AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER |
| // LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, |
| // OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN |
| // THE SOFTWARE. |
| |
| package dualquat |
| |
| import "gonum.org/v1/gonum/num/quat" |
| |
| // 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 PowReal(d Number, p quat.Number) Number { |
| deriv := quat.Mul(p, quat.Pow(d.Real, quat.Number{Real: p.Real - 1, Imag: p.Imag, Jmag: p.Jmag, Kmag: p.Kmag})) |
| return Number{ |
| Real: quat.Pow(d.Real, p), |
| Dual: quat.Mul(d.Dual, deriv), |
| } |
| } |
| |
| // Pow return d**r, the base-d exponential of r. |
| func Pow(d, p Number) Number { |
| return Exp(Mul(p, Log(d))) |
| } |
| |
| // 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 Sqrt(d Number) Number { |
| return PowReal(d, quat.Number{Real: 0.5}) |
| } |
| |
| // 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 Exp(d Number) Number { |
| fnDeriv := quat.Exp(d.Real) |
| return Number{ |
| Real: fnDeriv, |
| Dual: quat.Mul(fnDeriv, d.Dual), |
| } |
| } |
| |
| // 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 Log(d Number) Number { |
| switch { |
| case d.Real == zeroQuat: |
| return Number{ |
| Real: quat.Log(d.Real), |
| Dual: quat.Inf(), |
| } |
| case quat.IsInf(d.Real): |
| return Number{ |
| Real: quat.Log(d.Real), |
| Dual: zeroQuat, |
| } |
| } |
| return Number{ |
| Real: quat.Log(d.Real), |
| Dual: quat.Mul(d.Dual, quat.Inv(d.Real)), |
| } |
| } |
| |
| // Sin returns the sine of d. |
| // |
| // Special cases are: |
| // Sin(±0) = (±0+Nϵ) |
| // Sin(±Inf) = NaN |
| // Sin(NaN) = NaN |
| func Sin(d Number) Number { |
| if d.Real == zeroQuat { |
| return d |
| } |
| fn := quat.Sin(d.Real) |
| deriv := quat.Cos(d.Real) |
| return Number{ |
| Real: fn, |
| Dual: quat.Mul(deriv, d.Dual), |
| } |
| } |
| |
| // Cos returns the cosine of d. |
| // |
| // Special cases are: |
| // Cos(±Inf) = NaN |
| // Cos(NaN) = NaN |
| func Cos(d Number) Number { |
| fn := quat.Cos(d.Real) |
| deriv := quat.Scale(-1, quat.Sin(d.Real)) |
| return Number{ |
| Real: fn, |
| Dual: quat.Mul(deriv, d.Dual), |
| } |
| } |
| |
| // Tan returns the tangent of d. |
| // |
| // Special cases are: |
| // Tan(±0) = (±0+Nϵ) |
| // Tan(±Inf) = NaN |
| // Tan(NaN) = NaN |
| func Tan(d Number) Number { |
| if d.Real == zeroQuat { |
| return d |
| } |
| fn := quat.Tan(d.Real) |
| deriv := addRealQuat(1, quat.Mul(fn, fn)) |
| return Number{ |
| Real: fn, |
| Dual: quat.Mul(deriv, d.Dual), |
| } |
| } |
| |
| // 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 Asin(d Number) Number { |
| if d.Real == zeroQuat { |
| return d |
| } else if m := quat.Abs(d.Real); m >= 1 { |
| if m == 1 { |
| return Number{ |
| Real: quat.Asin(d.Real), |
| Dual: quat.Inf(), |
| } |
| } |
| return Number{ |
| Real: quat.NaN(), |
| Dual: quat.NaN(), |
| } |
| } |
| fn := quat.Asin(d.Real) |
| deriv := quat.Inv(quat.Sqrt(subRealQuat(1, quat.Mul(d.Real, d.Real)))) |
| return Number{ |
| Real: fn, |
| Dual: quat.Mul(deriv, d.Dual), |
| } |
| } |
| |
| // 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 Acos(d Number) Number { |
| if m := quat.Abs(d.Real); m >= 1 { |
| if m == 1 { |
| return Number{ |
| Real: quat.Acos(d.Real), |
| Dual: quat.Inf(), |
| } |
| } |
| return Number{ |
| Real: quat.NaN(), |
| Dual: quat.NaN(), |
| } |
| } |
| fn := quat.Acos(d.Real) |
| deriv := quat.Scale(-1, quat.Inv(quat.Sqrt(subRealQuat(1, quat.Mul(d.Real, d.Real))))) |
| return Number{ |
| Real: fn, |
| Dual: quat.Mul(deriv, d.Dual), |
| } |
| } |
| |
| // Atan returns the inverse tangent of d. |
| // |
| // Special cases are: |
| // Atan(±0) = (±0+Nϵ) |
| // Atan(±Inf) = (±Pi/2+0ϵ) |
| func Atan(d Number) Number { |
| if d.Real == zeroQuat { |
| return d |
| } |
| fn := quat.Atan(d.Real) |
| deriv := quat.Inv(addRealQuat(1, quat.Mul(d.Real, d.Real))) |
| return Number{ |
| Real: fn, |
| Dual: quat.Mul(deriv, d.Dual), |
| } |
| } |