Add quadratic-complex-solutions

- Also switch to potentially more numerically stable quadratic formulas
- Closes racket#21

math-doc/math/scribblings/math-number-theory.scrbl

@@ -793,6 +793,14 @@ lowest term is the number @math-style{(p+r)/(q+s)}.
 @; ----------------------------------------
 @section[#:tag "quadratics"]{The Quadratic Equation}
 
+@defproc[(quadratic-complex-solutions [a Real] [b Real] [c Real]) (Listof Complex)]{
+Returns a list of all complex solutions to the equation @math-style{a x^2 + b x +c = 0}.
+  @interaction[#:eval untyped-eval
+                      (quadratic-solutions 1 0 -1)
+                      (quadratic-solutions 1 2 1)
+                      (quadratic-solutions 1 0 1)]
+}
+
 @defproc[(quadratic-solutions [a Real] [b Real] [c Real]) (Listof Real)]{
 Returns a list of all real solutions to the equation @math-style{a x^2 + b x +c = 0}.
   @interaction[#:eval untyped-eval

math-lib/math/private/number-theory/quadratic.rkt

@@ -3,21 +3,22 @@
 (require racket/list
          "types.rkt")
 
-(provide quadratic-solutions
+(provide quadratic-complex-solutions
+         quadratic-solutions
          quadratic-integer-solutions
          quadratic-natural-solutions)
 
+(: quadratic-complex-solutions  : Real Real Real -> (Listof Complex))
+(define (quadratic-complex-solutions a b c)
+  ; return list of solutions to a a x^2 + b x + c = 0
+  (let* ([d (- (* b b) (* 4 a c))]
+         [sign (if (>= 0 (real-part (* (conjugate b) (sqrt d)))) 1 -1)]
+         [q (/ (+ b (* sign d)) -2)])
+    (list (/ q a) (/ c q))))
+
 (: quadratic-solutions : Real Real Real -> (Listof Real))
 (define (quadratic-solutions a b c)
-  ; return list of solutions to a a x^2 + b x + c = 0
-  (let ([d (- (* b b) (* 4 a c))])
-    (cond
-      [(< d 0) '()]
-      [(= d 0) (list (/ b (* -2 a)))]
-      [else
-       (let ([sqrt-d (sqrt d)])
-         (list (/ (- (- b) sqrt-d) (* 2 a))
-               (/ (+ (- b) sqrt-d) (* 2 a))))])))
+  (filter real? (quadratic-solutions a b c)))
 
 (: quadratic-integer-solutions : Real Real Real -> (Listof Integer))
 (define (quadratic-integer-solutions a b c)

math-test/math/tests/number-theory-tests.rkt

@@ -3,6 +3,9 @@
 (require typed/rackunit)
 
 ; "quadratic.rkt"
+(check-equal? (quadratic-complex-solutions 1 0 -4) '(-2 2))
+(check-equal? (quadratic-complex-solutions 1 0 +4) '(0-2i 0+2i))
+(check-equal? (quadratic-complex-solutions 1 0 0)  '(0))
 (check-equal? (quadratic-solutions 1 0 -4) '(-2 2))
 (check-equal? (quadratic-solutions 1 0 +4) '())
 (check-equal? (quadratic-solutions 1 0 0)  '(0))