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-;; Complex numbers
-
-;; by Konrad Hinsen
-;; last updated May 4, 2009
-
-;; Copyright (c) Konrad Hinsen, 2009. All rights reserved. The use
-;; and distribution terms for this software are covered by the Eclipse
-;; Public License 1.0 (http://opensource.org/licenses/eclipse-1.0.php)
-;; which can be found in the file epl-v10.html at the root of this
-;; distribution. By using this software in any fashion, you are
-;; agreeing to be bound by the terms of this license. You must not
-;; remove this notice, or any other, from this software.
-
-(ns
- #^{:author "Konrad Hinsen"
- :doc "Complex numbers
- NOTE: This library is in evolution. Most math functions are
- not implemented yet."}
- clojure.contrib.complex-numbers
- (:refer-clojure :exclude (deftype))
- (:use [clojure.contrib.types :only (deftype)]
- [clojure.contrib.generic :only (root-type)])
- (:require [clojure.contrib.generic.arithmetic :as ga]
- [clojure.contrib.generic.comparison :as gc]
- [clojure.contrib.generic.math-functions :as gm]))
-
-;
-; Complex numbers are represented as struct maps. The real and imaginary
-; parts can be of any type for which arithmetic and maths functions
-; are defined.
-;
-(defstruct complex-struct :real :imag)
-
-;
-; The general complex number type
-;
-(deftype ::complex complex
- (fn [real imag] (struct complex-struct real imag))
- (fn [c] (vals c)))
-
-(derive ::complex root-type)
-
-;
-; A specialized subtype for pure imaginary numbers. Introducing this type
-; reduces the number of operations by eliminating additions with and
-; multiplications by zero.
-;
-(deftype ::pure-imaginary imaginary
- (fn [imag] (struct complex-struct 0 imag))
- (fn [c] (list (:imag c))))
-
-(derive ::pure-imaginary ::complex)
-
-;
-; Extraction of real and imaginary parts
-;
-(def real (accessor complex-struct :real))
-(def imag (accessor complex-struct :imag))
-
-;
-; Equality and zero test
-;
-(defmethod gc/zero? ::complex
- [x]
- (let [[rx ix] (vals x)]
- (and (zero? rx) (zero? ix))))
-
-(defmethod gc/= [::complex ::complex]
- [x y]
- (let [[rx ix] (vals x)
- [ry iy] (vals y)]
- (and (gc/= rx ry) (gc/= ix iy))))
-
-(defmethod gc/= [::pure-imaginary ::pure-imaginary]
- [x y]
- (gc/= (imag x) (imag y)))
-
-(defmethod gc/= [::complex ::pure-imaginary]
- [x y]
- (let [[rx ix] (vals x)]
- (and (gc/zero? rx) (gc/= ix (imag y)))))
-
-(defmethod gc/= [::pure-imaginary ::complex]
- [x y]
- (let [[ry iy] (vals y)]
- (and (gc/zero? ry) (gc/= (imag x) iy))))
-
-(defmethod gc/= [::complex root-type]
- [x y]
- (let [[rx ix] (vals x)]
- (and (gc/zero? ix) (gc/= rx y))))
-
-(defmethod gc/= [root-type ::complex]
- [x y]
- (let [[ry iy] (vals y)]
- (and (gc/zero? iy) (gc/= x ry))))
-
-(defmethod gc/= [::pure-imaginary root-type]
- [x y]
- (and (gc/zero? (imag x)) (gc/zero? y)))
-
-(defmethod gc/= [root-type ::pure-imaginary]
- [x y]
- (and (gc/zero? x) (gc/zero? (imag y))))
-
-;
-; Addition
-;
-(defmethod ga/+ [::complex ::complex]
- [x y]
- (let [[rx ix] (vals x)
- [ry iy] (vals y)]
- (complex (ga/+ rx ry) (ga/+ ix iy))))
-
-(defmethod ga/+ [::pure-imaginary ::pure-imaginary]
- [x y]
- (imaginary (ga/+ (imag x) (imag y))))
-
-(defmethod ga/+ [::complex ::pure-imaginary]
- [x y]
- (let [[rx ix] (vals x)]
- (complex rx (ga/+ ix (imag y)))))
-
-(defmethod ga/+ [::pure-imaginary ::complex]
- [x y]
- (let [[ry iy] (vals y)]
- (complex ry (ga/+ (imag x) iy))))
-
-(defmethod ga/+ [::complex root-type]
- [x y]
- (let [[rx ix] (vals x)]
- (complex (ga/+ rx y) ix)))
-
-(defmethod ga/+ [root-type ::complex]
- [x y]
- (let [[ry iy] (vals y)]
- (complex (ga/+ x ry) iy)))
-
-(defmethod ga/+ [::pure-imaginary root-type]
- [x y]
- (complex y (imag x)))
-
-(defmethod ga/+ [root-type ::pure-imaginary]
- [x y]
- (complex x (imag y)))
-
-;
-; Negation
-;
-(defmethod ga/- ::complex
- [x]
- (let [[rx ix] (vals x)]
- (complex (ga/- rx) (ga/- ix))))
-
-(defmethod ga/- ::pure-imaginary
- [x]
- (imaginary (ga/- (imag x))))
-
-;
-; Subtraction is automatically supplied by ga/-, optimized implementations
-; can be added later...
-;
-
-;
-; Multiplication
-;
-(defmethod ga/* [::complex ::complex]
- [x y]
- (let [[rx ix] (vals x)
- [ry iy] (vals y)]
- (complex (ga/- (ga/* rx ry) (ga/* ix iy))
- (ga/+ (ga/* rx iy) (ga/* ix ry)))))
-
-(defmethod ga/* [::pure-imaginary ::pure-imaginary]
- [x y]
- (ga/- (ga/* (imag x) (imag y))))
-
-(defmethod ga/* [::complex ::pure-imaginary]
- [x y]
- (let [[rx ix] (vals x)
- iy (imag y)]
- (complex (ga/- (ga/* ix iy))
- (ga/* rx iy))))
-
-(defmethod ga/* [::pure-imaginary ::complex]
- [x y]
- (let [ix (imag x)
- [ry iy] (vals y)]
- (complex (ga/- (ga/* ix iy))
- (ga/* ix ry))))
-
-(defmethod ga/* [::complex root-type]
- [x y]
- (let [[rx ix] (vals x)]
- (complex (ga/* rx y) (ga/* ix y))))
-
-(defmethod ga/* [root-type ::complex]
- [x y]
- (let [[ry iy] (vals y)]
- (complex (ga/* x ry) (ga/* x iy))))
-
-(defmethod ga/* [::pure-imaginary root-type]
- [x y]
- (imaginary (ga/* (imag x) y)))
-
-(defmethod ga/* [root-type ::pure-imaginary]
- [x y]
- (imaginary (ga/* x (imag y))))
-
-;
-; Inversion
-;
-(ga/defmethod* ga / ::complex
- [x]
- (let [[rx ix] (vals x)
- den ((ga/qsym ga /) (ga/+ (ga/* rx rx) (ga/* ix ix)))]
- (complex (ga/* rx den) (ga/- (ga/* ix den)))))
-
-(ga/defmethod* ga / ::pure-imaginary
- [x]
- (imaginary (ga/- ((ga/qsym ga /) (imag x)))))
-
-;
-; Division is automatically supplied by ga//, optimized implementations
-; can be added later...
-;
-
-;
-; Conjugation
-;
-(defmethod gm/conjugate ::complex
- [x]
- (let [[r i] (vals x)]
- (complex r (ga/- i))))
-
-(defmethod gm/conjugate ::pure-imaginary
- [x]
- (imaginary (ga/- (imag x))))
-
-;
-; Absolute value
-;
-(defmethod gm/abs ::complex
- [x]
- (let [[r i] (vals x)]
- (gm/sqrt (ga/+ (ga/* r r) (ga/* i i)))))
-
-(defmethod gm/abs ::pure-imaginary
- [x]
- (gm/abs (imag x)))
-
-;
-; Square root
-;
-(let [one-half (/ 1 2)
- one-eighth (/ 1 8)]
- (defmethod gm/sqrt ::complex
- [x]
- (let [[r i] (vals x)]
- (if (and (gc/zero? r) (gc/zero? i))
- 0
- (let [; The basic formula would say
- ; abs (gm/sqrt (ga/+ (ga/* r r) (ga/* i i)))
- ; p (gm/sqrt (ga/* one-half (ga/+ abs r)))
- ; but the slightly more complicated one below
- ; avoids overflow for large r or i.
- ar (gm/abs r)
- ai (gm/abs i)
- r8 (ga/* one-eighth ar)
- i8 (ga/* one-eighth ai)
- abs (gm/sqrt (ga/+ (ga/* r8 r8) (ga/* i8 i8)))
- p (ga/* 2 (gm/sqrt (ga/+ abs r8)))
- q ((ga/qsym ga /) ai (ga/* 2 p))
- s (gm/sgn i)]
- (if (gc/< r 0)
- (complex q (ga/* s p))
- (complex p (ga/* s q))))))))
-
-;
-; Exponential function
-;
-(defmethod gm/exp ::complex
- [x]
- (let [[r i] (vals x)
- exp-r (gm/exp r)
- cos-i (gm/cos i)
- sin-i (gm/sin i)]
- (complex (ga/* exp-r cos-i) (ga/* exp-r sin-i))))
-
-(defmethod gm/exp ::pure-imaginary
- [x]
- (let [i (imag x)]
- (complex (gm/cos i) (gm/sin i))))