std/complex

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This module implements complex numbers and basic mathematical operations on them.

Complex numbers are currently generic over 64-bit or 32-bit floats.

Example:

import std/complex
from std/math import almostEqual, sqrt

let
  z1 = complex(1.0, 2.0)
  z2 = complex(3.0, -4.0)

assert almostEqual(z1 + z2, complex(4.0, -2.0))
assert almostEqual(z1 - z2, complex(-2.0, 6.0))
assert almostEqual(z1 * z2, complex(11.0, 2.0))
assert almostEqual(z1 / z2, complex(-0.2, 0.4))

assert almostEqual(abs(z1), sqrt(5.0))
assert almostEqual(conjugate(z1), complex(1.0, -2.0))

let (r, phi) = z1.polar
assert almostEqual(rect(r, phi), z1)

Types

Complex[T] = object
  re*, im*: T
A complex number, consisting of a real and an imaginary part. Source   Edit   
Complex32 = Complex[float32]
Alias for a complex number using 32-bit floats. Source   Edit   
Complex64 = Complex[float64]
Alias for a complex number using 64-bit floats. Source   Edit  

Procs

func `$`(z: Complex): string
Returns z's string representation as "(re, im)".

Example:

doAssert $complex(1.0, 2.0) == "(1.0, 2.0)"
Source   Edit   
func `*`[T](x, y: Complex[T]): Complex[T]
Multiplies two complex numbers. Source   Edit   
func `*`[T](x: Complex[T]; y: T): Complex[T]
Multiplies a complex number with a real number. Source   Edit   
func `*`[T](x: T; y: Complex[T]): Complex[T]
Multiplies a real number with a complex number. Source   Edit   
func `*=`[T](x: var Complex[T]; y: Complex[T])
Multiplies x by y. Source   Edit   
func `+`[T](x, y: Complex[T]): Complex[T]
Adds two complex numbers. Source   Edit   
func `+`[T](x: Complex[T]; y: T): Complex[T]
Adds a complex number to a real number. Source   Edit   
func `+`[T](x: T; y: Complex[T]): Complex[T]
Adds a real number to a complex number. Source   Edit   
func `+=`[T](x: var Complex[T]; y: Complex[T])
Adds y to x. Source   Edit   
func `-`[T](x, y: Complex[T]): Complex[T]
Subtracts two complex numbers. Source   Edit   
func `-`[T](x: Complex[T]; y: T): Complex[T]
Subtracts a real number from a complex number. Source   Edit   
func `-`[T](x: T; y: Complex[T]): Complex[T]
Subtracts a complex number from a real number. Source   Edit   
func `-`[T](z: Complex[T]): Complex[T]
Unary minus for complex numbers. Source   Edit   
func `-=`[T](x: var Complex[T]; y: Complex[T])
Subtracts y from x. Source   Edit   
func `/`[T](x, y: Complex[T]): Complex[T]
Divides two complex numbers. Source   Edit   
func `/`[T](x: Complex[T]; y: T): Complex[T]
Divides a complex number by a real number. Source   Edit   
func `/`[T](x: T; y: Complex[T]): Complex[T]
Divides a real number by a complex number. Source   Edit   
func `/=`[T](x: var Complex[T]; y: Complex[T])
Divides x by y in place. Source   Edit   
func `==`[T](x, y: Complex[T]): bool
Compares two complex numbers for equality. Source   Edit   
func abs[T](z: Complex[T]): T
Returns the absolute value of z, that is the distance from (0, 0) to z. Source   Edit   
func abs2[T](z: Complex[T]): T
Returns the squared absolute value of z, that is the squared distance from (0, 0) to z. This is more efficient than abs(z) ^ 2. Source   Edit   
func almostEqual[T: SomeFloat](x, y: Complex[T]; unitsInLastPlace: Natural = 4): bool

Checks if two complex values are almost equal, using the machine epsilon.

Two complex values are considered almost equal if their real and imaginary components are almost equal.

unitsInLastPlace is the max number of units in the last place difference tolerated when comparing two numbers. The larger the value, the more error is allowed. A 0 value means that two numbers must be exactly the same to be considered equal.

The machine epsilon has to be scaled to the magnitude of the values used and multiplied by the desired precision in ULPs unless the difference is subnormal.

Source   Edit   
func arccos[T](z: Complex[T]): Complex[T]
Returns the inverse cosine of z. Source   Edit   
func arccosh[T](z: Complex[T]): Complex[T]
Returns the inverse hyperbolic cosine of z. Source   Edit   
func arccot[T](z: Complex[T]): Complex[T]
Returns the inverse cotangent of z. Source   Edit   
func arccoth[T](z: Complex[T]): Complex[T]
Returns the inverse hyperbolic cotangent of z. Source   Edit   
func arccsc[T](z: Complex[T]): Complex[T]
Returns the inverse cosecant of z. Source   Edit   
func arccsch[T](z: Complex[T]): Complex[T]
Returns the inverse hyperbolic cosecant of z. Source   Edit   
func arcsec[T](z: Complex[T]): Complex[T]
Returns the inverse secant of z. Source   Edit   
func arcsech[T](z: Complex[T]): Complex[T]
Returns the inverse hyperbolic secant of z. Source   Edit   
func arcsin[T](z: Complex[T]): Complex[T]
Returns the inverse sine of z. Source   Edit   
func arcsinh[T](z: Complex[T]): Complex[T]
Returns the inverse hyperbolic sine of z. Source   Edit   
func arctan[T](z: Complex[T]): Complex[T]
Returns the inverse tangent of z. Source   Edit   
func arctanh[T](z: Complex[T]): Complex[T]
Returns the inverse hyperbolic tangent of z. Source   Edit   
func complex[T: SomeFloat](re: T; im: T = 0.0): Complex[T]
Returns a Complex[T] with real part re and imaginary part im. Source   Edit   
func complex32(re: float32; im: float32 = 0.0): Complex32 {....raises: [],
    tags: [], forbids: [].}
Returns a Complex32 with real part re and imaginary part im. Source   Edit   
func complex64(re: float64; im: float64 = 0.0): Complex64 {....raises: [],
    tags: [], forbids: [].}
Returns a Complex64 with real part re and imaginary part im. Source   Edit   
func conjugate[T](z: Complex[T]): Complex[T]
Returns the complex conjugate of z (complex(z.re, -z.im)). Source   Edit   
func cos[T](z: Complex[T]): Complex[T]
Returns the cosine of z. Source   Edit   
func cosh[T](z: Complex[T]): Complex[T]
Returns the hyperbolic cosine of z. Source   Edit   
func cot[T](z: Complex[T]): Complex[T]
Returns the cotangent of z. Source   Edit   
func coth[T](z: Complex[T]): Complex[T]
Returns the hyperbolic cotangent of z. Source   Edit   
func csc[T](z: Complex[T]): Complex[T]
Returns the cosecant of z. Source   Edit   
func csch[T](z: Complex[T]): Complex[T]
Returns the hyperbolic cosecant of z. Source   Edit   
func exp[T](z: Complex[T]): Complex[T]
Computes the exponential function (e^z). Source   Edit   
proc formatValue(result: var string; value: Complex; specifier: string)
Standard format implementation for Complex. It makes little sense to call this directly, but it is required to exist by the & macro. For complex numbers, we add a specific 'j' specifier, which formats the value as (A+Bj) like in mathematics. Source   Edit   
func inv[T](z: Complex[T]): Complex[T]
Returns the multiplicative inverse of z (1/z). Source   Edit   
func ln[T](z: Complex[T]): Complex[T]
Returns the (principal value of the) natural logarithm of z. Source   Edit   
func log2[T](z: Complex[T]): Complex[T]

Returns the logarithm base 2 of z.

See also:

Source   Edit   
func log10[T](z: Complex[T]): Complex[T]

Returns the logarithm base 10 of z.

See also:

Source   Edit   
func phase[T](z: Complex[T]): T
Returns the phase (or argument) of z, that is the angle in polar representation.

result = arctan2(z.im, z.re)

Source   Edit   
func polar[T](z: Complex[T]): tuple[r, phi: T]
Returns z in polar coordinates.

result.r = abs(z)
result.phi = phase(z)

See also:

Source   Edit   
func pow[T](x, y: Complex[T]): Complex[T]
x raised to the power of y. Source   Edit   
func pow[T](x: Complex[T]; y: T): Complex[T]
The complex number x raised to the power of the real number y. Source   Edit   
func rect[T](r, phi: T): Complex[T]
Returns the complex number with polar coordinates r and phi.

result.re = r * cos(phi)
result.im = r * sin(phi)

See also:

Source   Edit   
func sec[T](z: Complex[T]): Complex[T]
Returns the secant of z. Source   Edit   
func sech[T](z: Complex[T]): Complex[T]
Returns the hyperbolic secant of z. Source   Edit   
func sgn[T](z: Complex[T]): Complex[T]
Returns the phase of z as a unit complex number, or 0 if z is 0. Source   Edit   
func sin[T](z: Complex[T]): Complex[T]
Returns the sine of z. Source   Edit   
func sinh[T](z: Complex[T]): Complex[T]
Returns the hyperbolic sine of z. Source   Edit   
func sqrt[T](z: Complex[T]): Complex[T]
Computes the (principal) square root of a complex number z. Source   Edit   
func tan[T](z: Complex[T]): Complex[T]
Returns the tangent of z. Source   Edit   
func tanh[T](z: Complex[T]): Complex[T]
Returns the hyperbolic tangent of z. Source   Edit  

Templates

template im(arg: float32): Complex32
Returns arg as an imaginary number (complex32(0, arg)). Source   Edit   
template im(arg: float64): Complex64
Returns arg as an imaginary number (complex64(0, arg)). Source   Edit   
template im(arg: typedesc[float32]): Complex32
Returns the imaginary unit (complex32(0, 1)). Source   Edit   
template im(arg: typedesc[float64]): Complex64
Returns the imaginary unit (complex64(0, 1)). Source   Edit  

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