ComplexAlgebra

ECMAScript modules for exactly manipulating complex numbers of which real and imaginary parts are numbers of the form (p / q)sqrt(b), where p is an integer, q is a positive (non-zero) integer, and b is a positive, square-free integer.

Installation

npm install @kkitahara/complex-algebra @kkitahara/real-algebra

Examples

import { ExactRealAlgebra as RealAlgebra } from '@kkitahara/real-algebra'
import { ComplexAlgebra } from '@kkitahara/complex-algebra'
let ralg = new RealAlgebra()
let calg = new ComplexAlgebra(ralg)
let z, w, v

Generate a new real number

z = calg.num(1, 2, 5)
z.toString() // '(1 / 2)sqrt(5)'

z = calg.num(1, 2)
z.toString() // '1 / 2'

z = calg.num(3)
z.toString() // '3'

Generate a new imaginary number

z = calg.inum(1, 2, 5)
z.toString() // 'i((1 / 2)sqrt(5))'

z = calg.inum(1, 2)
z.toString() // 'i(1 / 2)'

z = calg.inum(3)
z.toString() // 'i(3)'

:warning: num and inum methods do not check if (the absolute value of) the 3rd parameter is a square-free integer or not (must be square-free!).

Real and imaginary parts

z = calg.num(1, 2, 5)
z.re.toString() // '(1 / 2)sqrt(5)'
z.im.toString() // '0'

z = calg.inum(1, 2, 5)
z.re.toString() // '0'
z.im.toString() // '(1 / 2)sqrt(5)'

Generate from two real numbers (since v1.2.0)

let a = ralg.$(1, 2, 3)
let b = ralg.$(1, 2, 5)
z = calg.$(a, b)
z.toString() // '(1 / 2)sqrt(3) + i((1 / 2)sqrt(5))'

Copy (create a new object)

z = calg.inum(1, 2, 5)
w = calg.copy(z)
w.toString() // 'i((1 / 2)sqrt(5))'

Equality

z = calg.num(1, 2, 5)
w = calg.inum(1, 2, 5)
calg.eq(z, w) // false

w = calg.num(1, 2, 5)
calg.eq(z, w) // true

Inequality

z = calg.num(1, 2, 5)
w = calg.inum(1, 2, 5)
calg.ne(z, w) // true

w = calg.num(1, 2, 5)
calg.ne(z, w) // false

isZero

calg.isZero(calg.num(0)) // true
calg.isZero(calg.inum(0)) // true
calg.isZero(calg.num(1, 2, 5)) // false
calg.isZero(calg.inum(1, 2, 5)) // false
calg.isZero(calg.num(-1, 2, 5)) // false
calg.isZero(calg.inum(-1, 2, 5)) // false

isInteger (since v1.1.0)

calg.isInteger(calg.num(0)) // true
calg.isInteger(calg.inum(0)) // true
calg.isInteger(calg.num(1, 2)) // false
calg.isInteger(calg.inum(1, 2)) // false
calg.isInteger(calg.num(6, 3)) // true
calg.isInteger(calg.inum(6, 3)) // true
calg.isInteger(calg.num(6, 3, 2)) // false
calg.isInteger(calg.inum(6, 3, 2)) // false

Addition

z = calg.num(1, 2, 5)
w = calg.inum(1, 2)
// new object is generated
v = calg.add(z, w)
v.toString() // '(1 / 2)sqrt(5) + i(1 / 2)'

In-place addition

z = calg.num(1, 2, 5)
w = calg.inum(1, 2)
// new object is not generated
z = calg.iadd(z, w)
z.toString() // '(1 / 2)sqrt(5) + i(1 / 2)'

Subtraction

z = calg.num(1, 2, 5)
w = calg.inum(1, 2)
// new object is generated
v = calg.sub(z, w)
v.toString() // '(1 / 2)sqrt(5) + i(-1 / 2)'

In-place subtraction

z = calg.num(1, 2, 5)
w = calg.inum(1, 2)
// new object is not generated
z = calg.isub(z, w)
z.toString() // '(1 / 2)sqrt(5) + i(-1 / 2)'

Multiplication

z = calg.inum(1, 2, 5)
w = calg.inum(1, 2)
// new object is generated
v = calg.mul(z, w)
v.toString() // '-(1 / 4)sqrt(5)'

In-place multiplication

z = calg.inum(1, 2, 5)
w = calg.inum(1, 2)
// new object is not generated
z = calg.imul(z, w)
z.toString() // '-(1 / 4)sqrt(5)'

Division

z = calg.inum(1, 2, 5)
w = calg.inum(1, 2)
// new object is generated
v = calg.div(z, w)
v.toString() // 'sqrt(5)'

In-place division

z = calg.inum(1, 2, 5)
w = calg.inum(1, 2)
// new object is not generated
z = calg.idiv(z, w)
z.toString() // 'sqrt(5)'

Multiplication by -1

z = calg.inum(1, 2, 5)
// new object is generated
w = calg.neg(z)
w.toString() // 'i(-(1 / 2)sqrt(5))'

In-place multiplication by -1

z = calg.inum(1, 2, 5)
// new object is not generated
z = calg.ineg(z)
z.toString() // 'i(-(1 / 2)sqrt(5))'

Complex conjugate

z = calg.num(1, 2, 5)
w = calg.inum(1, 2, 5)
// new object is generated
v = calg.cjg(z)
v.toString() // '(1 / 2)sqrt(5)'
v = calg.cjg(w)
v.toString() // 'i(-(1 / 2)sqrt(5))'

In-place evaluation of the complex conjugate

z = calg.num(1, 2, 5)
w = calg.inum(1, 2, 5)
// new object is not generated
z = calg.icjg(z)
z.toString() // '(1 / 2)sqrt(5)'
w = calg.icjg(w)
w.toString() // 'i(-(1 / 2)sqrt(5))'

Square of the absolute value

z = calg.iadd(calg.num(3), calg.inum(4))
let a = calg.abs2(z)
a.toString() // '25'
// return value is not a complex number (but a real number)
a.re // undefined
a.im // undefined

JSON (stringify and parse)

z = calg.iadd(calg.num(1, 2, 5), calg.inum(-1, 2, 7))
let str = JSON.stringify(z)
w = JSON.parse(str, calg.reviver)
calg.eq(z, w) // true

Numerical algebra

The above codes work with built-in numbers if you use

import { RealAlgebra } from '@kkitahara/real-algebra'
import { ComplexAlgebra } from '@kkitahara/complex-algebra'
let ralg = new RealAlgebra()
let calg = new ComplexAlgebra(ralg)

instead of ExactRealAlgebra. See the documents of @kkitahara/real-algebra for more details.

ESDoc documents

For more examples, see ESDoc documents:

cd node_modules/@kkitahara/complex-algebra
npm install --only=dev
npm run doc

and open doc/index.html in your browser.

LICENSE

© 2019 Koichi Kitahara
Apache 2.0