Gaussian Integers, Part 1

November 4, 2014

We represent Gaussian integers as pairs, so a + b i is represented as (a . b):

(define (gauss a b)
  (when (not (integer? a))
    (error 'gauss "must be integer"))
  (when (not (integer? b))
    (error 'gauss "must be integer"))
  (gs a b))

The internal function gs does not enforce integer types:

(define gs cons)

The pieces of a Gaussian integer can be picked apart with the re and im functions:

(define (re x) (car x))
(define (im x) (cdr x))

Scheme offers a complex number type, but doesn’t limit it to integers. Here are conversions to and from Scheme complex numbers:

(define (gauss-from-complex x)
  (gauss (real-part x) (imag-part x)))

(define (gauss-to-complex x)
  (make-rectangular (re x) (im x)))

Zero in Gaussian integers is (0 . 0). The role of the unit in regular integers is played by four different values in Gaussian integers, 1, −1, i and −i. There is no ordering concept in Gaussian integers, but two Gaussian integers can be compared for equality:

(define (gauss-zero? x)
  (and (zero? (re x)) (zero? (im x))))

(define (gauss-unit? x)
  (or (and (= (abs (re x)) 1) (zero? (im x)))
      (and (zero? (re x)) (= (abs (im x)) 1))))

(define (gauss-conjugate x)
  (gs (re x) (- (im x))))

(define (gauss-eql? x y)
  (and (= (re x) (re y))
       (= (im x) (im y))))

The norm of Gaussian integers corresponds to the absolute value in normal integers:

(define (gauss-norm x)
  (define (square x) (* x x))
  (+ (square (re x)) (square (im x))))

The arithmetic operators follow the descriptions given above:

(define (gauss-sub . xs)
  (define (sub x y)
    (gs (- (re x) (re y)) (- (im x) (im y))))
  (cond ((null? xs) (error 'gauss-sub "no operands"))
        ((null? (cdr xs)) (gauss-negate (car xs)))
        (else (let loop ((xs (cdr xs)) (zs (car xs)))
              (if (null? xs) zs
                (loop (cdr xs) (sub zs (car xs))))))))

(define (gauss-mul . xs)
  (define (mul x y)
    (gs (- (* (re x) (re y))
           (* (im x) (im y)))
        (+ (* (re x) (im y))
           (* (im x) (re y)))))
  (let loop ((xs xs) (zs (gs 1 0)))
    (if (null? xs) zs
      (loop (cdr xs) (mul (car xs) zs)))))

(define (gauss-quotient num den)
  (let ((n (gauss-norm den))
        (r (+ (* (re num) (re den))
              (* (im num) (im den))))
        (i (- (* (re den) (im num))
              (* (re num) (im den)))))
    (gs (round (/ r n)) (round (/ i n)))))

(define (gauss-remainder num den quo)
  (gauss-sub num (gauss-mul den quo)))

You can see these functions in action at http://programmingpraxis.codepad.org/ql2rU4dd.

We’ll discuss Gaussian integers again in the next exercise. In the meantime, you might want to read this description of Gaussian integers by Keith Conrad.

Pages: 1 2

4 Responses to “Gaussian Integers, Part 1”

  1. Andras said

    Question1: is the quotient not like this (ax+by) +(ay-bx)i ?
    Question2: is it ok to use floor instead of round?

    Then in Scala:
    class GaussianInteger(val a: Int, val b: Int) {
    def addition(that: GaussianInteger): GaussianInteger = new GaussianInteger(a + that.a, b + that.b)
    def subtraction(that: GaussianInteger): GaussianInteger = new GaussianInteger(a – that.a, b – that.b)
    def crossMultiply(that: GaussianInteger): GaussianInteger = new GaussianInteger(a * that.a – b * that.b, a * that.b + b * that.a)
    def quotient(that: GaussianInteger): GaussianInteger = {
    val n = that.a * that.a + that.b * that.b
    new GaussianInteger((a * that.a – b * that.b) / n, (b * that.a – a * that.b) / n)
    }
    def remainder(that: GaussianInteger): GaussianInteger = subtraction(quotient(that))
    override def toString: String = a + ” + ” + b + “i”
    }

  2. programmingpraxis said

    The quotient is ((ax+by) + (bx-ay)i) / n; the code was right, I fixed the description. Using floor instead of round doesn’t work, because the norm of the result must be less than half the norm of the divisor, by convention; see the discussion of the Division Theorem in the paper linked from the solution page.

  3. Jussi Piitulainen said

    Using Scheme’s complex numbers, I get addition and multiplication for free. I find that Gambit-C does Gaussian rationals exactly, and for it the norm is automatically real. In Guile I had to take the real part explicitly. (I needed the norm to be real in order to test that the remainder is smaller than the divisor.)

    (define (zi-conjugate a) (make-rectangular (real-part a) (- (imag-part a))))
    (define (zi-norm a) (real-part (* a (zi-conjugate a)))) ;imag-part is zero

    (define (zi-quotient a b)
      (let ((q (/ (* a (zi-conjugate b)) (zi-norm b))))
        (make-rectangular (round (real-part q)) (round (imag-part q)))))

    (define (zi-remainder a b) (- a (* b (zi-quotient a b))))

    Printing the quotient and remainder from the paper:

    (write (cons (zi-quotient 27-23i 8+i) (zi-remainder 27-23i 8+i))) (newline)

    The Gambit-C interpreter (gsi) prints this, where the absence of decimal points indicates that the components are exact integers (with a zero real component omitted):

    (3-3i . -2i)

    Guile prints this, where the presence of the decimal points indicates that the computation may have used inexact methods (floating point, I’m sure):

    (3.0-3.0i . 0.0-2.0i)

    Floating point might fail (rounding, overflowing), but my limited tests, with all components small, were fine even in Guile.

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