feat: methods on Complex (#24)

Adds all of the pocomath functions on Complex that do not depend on any unimplemented types or config properties, except quotient and roundquotient, where the design is murky. To get this working, adds some additional features:
  * Allows conversions to generic types, with the matched type
    determined from the return value of the built convertor
  * Adds predicate-based type patterns
  * Adds conversion from any non-complex type T to Complex(T)
  * Adds check for recursive loops in resolve (a key/signature depending on itself)

Reviewed-on: #24
Co-authored-by: Glen Whitney <glen@studioinfinity.org>
Co-committed-by: Glen Whitney <glen@studioinfinity.org>

src/complex/arithmetic.js

@@ -0,0 +1,69 @@
+import {Complex} from './Complex.js'
+import {promoteBinary, promoteUnary} from './helpers.js'
+import {ResolutionError} from '#core/helpers.js'
+import {match} from '#core/TypePatterns.js'
+import {ReturnsAs} from '#generic/helpers.js'
+
+export const absquare = match(Complex, (math, C) => {
+   const compAbsq = math.absquare.resolve([C.Component])
+   const R = compAbsq.returns
+   const add = math.add.resolve([R,R])
+   return ReturnsAs(add, z => add(compAbsq(z.re), compAbsq(z.im)))
+})
+
+export const add = promoteBinary('add')
+
+export const conj = match(Complex, (math, C) => {
+   const neg = math.negate.resolve(C.Component)
+   const compConj = math.conj.resolve(C.Component)
+   const cplx = math.complex.resolve([compConj.returns, neg.returns])
+   return ReturnsAs(cplx, z => cplx(compConj(z.re), neg(z.im)))
+})
+
+export const divide = [
+   match([Complex, T => !T.complex], (math, [C, R]) => {
+      const div = math.divide.resolve([C.Component, R])
+      const cplx = math.complex.resolve([div.returns, div.returns])
+      return ReturnsAs(cplx, (z, r) => cplx(div(z.re, r), div(z.im, r)))
+   }),
+   match([Complex, Complex], (math, [W, Z]) => {
+      const inv = math.invert.resolve(Z)
+      const mult = math.multiply.resolve([W, inv.returns])
+      return ReturnsAs(mult, (w, z) => mult(w, inv(z)))
+   })
+]
+
+export const invert = match(Complex, (math, C) => {
+   const conj = math.conj.resolve(C)
+   const norm = math.absquare.resolve(C)
+   const div = math.divide.resolve([C.Component, norm.returns])
+   const cplx = math.complex.resolve([div.returns, div.returns])
+   return ReturnsAs(cplx, z => {
+      const c = conj(z)
+      const d = norm(z)
+      return cplx(div(c.re, d), div(c.im, d))
+   })
+})
+
+// We want this to work for complex numbers, quaternions, octonions, etc
+// See https://math.ucr.edu/home/baez/octonions/node5.html
+export const multiply = match([Complex, Complex], (math, [W, Z]) => {
+   const conj = math.conj.resolve(W.Component)
+   if (conj.returns !== W.Component) {
+      throw new ResolutionError(
+         `conjugation on ${W.Component} returns other type (${conj.returns})`)
+   }
+   const mWZ = math.multiply.resolve([W.Component, Z.Component])
+   const mZW = math.multiply.resolve([Z.Component, W.Component])
+   const sub = math.subtract.resolve([mWZ.returns, mZW.returns])
+   const add = math.add.resolve([mWZ.returns, mZW.returns])
+   const cplx = math.complex.resolve([sub.returns, add.returns])
+   return ReturnsAs(cplx, (w, z) => {
+      const real = sub(mWZ(     w.re,  z.re), mZW(z.im, conj(w.im)))
+      const imag = add(mWZ(conj(w.re), z.im), mZW(z.re,      w.im))
+      return cplx(real, imag)
+   })
+})
+
+export const negate = promoteUnary('negate')
+export const subtract = promoteBinary('subtract')