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|
#
#
# Nim's Runtime Library
# (c) Copyright 2010 Andreas Rumpf
#
# See the file "copying.txt", included in this
# distribution, for details about the copyright.
#
## This module implements complex numbers.
## Complex numbers are currently implemented as generic on a 64-bit or 32-bit float.
{.push checks: off, line_dir: off, stack_trace: off, debugger: off.}
# the user does not want to trace a part of the standard library!
import math
type
Complex*[T: SomeFloat] = object
re*, im*: T
## A complex number, consisting of a real and an imaginary part.
Complex64* = Complex[float64]
## Alias for a pair of 64-bit floats.
Complex32* = Complex[float32]
## Alias for a pair of 32-bit floats.
func complex*[T: SomeFloat](re: T; im: T = 0.0): Complex[T] =
result.re = re
result.im = im
func complex32*(re: float32; im: float32 = 0.0): Complex[float32] =
result.re = re
result.im = im
func complex64*(re: float64; im: float64 = 0.0): Complex[float64] =
result.re = re
result.im = im
template im*(arg: typedesc[float32]): Complex32 = complex[float32](0, 1)
template im*(arg: typedesc[float64]): Complex64 = complex[float64](0, 1)
template im*(arg: float32): Complex32 = complex[float32](0, arg)
template im*(arg: float64): Complex64 = complex[float64](0, arg)
func abs*[T](z: Complex[T]): T =
## Returns the distance from (0,0) to ``z``.
result = hypot(z.re, z.im)
func abs2*[T](z: Complex[T]): T =
## Returns the squared distance from (0,0) to ``z``.
result = z.re*z.re + z.im*z.im
func conjugate*[T](z: Complex[T]): Complex[T] =
## Conjugates of complex number ``z``.
result.re = z.re
result.im = -z.im
func inv*[T](z: Complex[T]): Complex[T] =
## Multiplicatives inverse of complex number ``z``.
conjugate(z) / abs2(z)
func `==` *[T](x, y: Complex[T]): bool =
## Compares two complex numbers ``x`` and ``y`` for equality.
result = x.re == y.re and x.im == y.im
func `+` *[T](x: T; y: Complex[T]): Complex[T] =
## Adds a real number to a complex number.
result.re = x + y.re
result.im = y.im
func `+` *[T](x: Complex[T]; y: T): Complex[T] =
## Adds a complex number to a real number.
result.re = x.re + y
result.im = x.im
func `+` *[T](x, y: Complex[T]): Complex[T] =
## Adds two complex numbers.
result.re = x.re + y.re
result.im = x.im + y.im
func `-` *[T](z: Complex[T]): Complex[T] =
## Unary minus for complex numbers.
result.re = -z.re
result.im = -z.im
func `-` *[T](x: T; y: Complex[T]): Complex[T] =
## Subtracts a complex number from a real number.
x + (-y)
func `-` *[T](x: Complex[T]; y: T): Complex[T] =
## Subtracts a real number from a complex number.
result.re = x.re - y
result.im = x.im
func `-` *[T](x, y: Complex[T]): Complex[T] =
## Subtracts two complex numbers.
result.re = x.re - y.re
result.im = x.im - y.im
func `/` *[T](x: Complex[T]; y: T): Complex[T] =
## Divides complex number ``x`` by real number ``y``.
result.re = x.re / y
result.im = x.im / y
func `/` *[T](x: T; y: Complex[T]): Complex[T] =
## Divides real number ``x`` by complex number ``y``.
result = x * inv(y)
func `/` *[T](x, y: Complex[T]): Complex[T] =
## Divides ``x`` by ``y``.
var r, den: T
if abs(y.re) < abs(y.im):
r = y.re / y.im
den = y.im + r * y.re
result.re = (x.re * r + x.im) / den
result.im = (x.im * r - x.re) / den
else:
r = y.im / y.re
den = y.re + r * y.im
result.re = (x.re + r * x.im) / den
result.im = (x.im - r * x.re) / den
func `*` *[T](x: T; y: Complex[T]): Complex[T] =
## Multiplies a real number and a complex number.
result.re = x * y.re
result.im = x * y.im
func `*` *[T](x: Complex[T]; y: T): Complex[T] =
## Multiplies a complex number with a real number.
result.re = x.re * y
result.im = x.im * y
func `*` *[T](x, y: Complex[T]): Complex[T] =
## Multiplies ``x`` with ``y``.
result.re = x.re * y.re - x.im * y.im
result.im = x.im * y.re + x.re * y.im
func `+=` *[T](x: var Complex[T]; y: Complex[T]) =
## Adds ``y`` to ``x``.
x.re += y.re
x.im += y.im
func `-=` *[T](x: var Complex[T]; y: Complex[T]) =
## Subtracts ``y`` from ``x``.
x.re -= y.re
x.im -= y.im
func `*=` *[T](x: var Complex[T]; y: Complex[T]) =
## Multiplies ``y`` to ``x``.
let im = x.im * y.re + x.re * y.im
x.re = x.re * y.re - x.im * y.im
x.im = im
func `/=` *[T](x: var Complex[T]; y: Complex[T]) =
## Divides ``x`` by ``y`` in place.
x = x / y
func sqrt*[T](z: Complex[T]): Complex[T] =
## Square root for a complex number ``z``.
var x, y, w, r: T
if z.re == 0.0 and z.im == 0.0:
result = z
else:
x = abs(z.re)
y = abs(z.im)
if x >= y:
r = y / x
w = sqrt(x) * sqrt(0.5 * (1.0 + sqrt(1.0 + r * r)))
else:
r = x / y
w = sqrt(y) * sqrt(0.5 * (r + sqrt(1.0 + r * r)))
if z.re >= 0.0:
result.re = w
result.im = z.im / (w * 2.0)
else:
result.im = if z.im >= 0.0: w else: -w
result.re = z.im / (result.im + result.im)
func exp*[T](z: Complex[T]): Complex[T] =
## ``e`` raised to the power ``z``.
var
rho = exp(z.re)
theta = z.im
result.re = rho * cos(theta)
result.im = rho * sin(theta)
func ln*[T](z: Complex[T]): Complex[T] =
## Returns the natural log of ``z``.
result.re = ln(abs(z))
result.im = arctan2(z.im, z.re)
func log10*[T](z: Complex[T]): Complex[T] =
## Returns the log base 10 of ``z``.
result = ln(z) / ln(10.0)
func log2*[T](z: Complex[T]): Complex[T] =
## Returns the log base 2 of ``z``.
result = ln(z) / ln(2.0)
func pow*[T](x, y: Complex[T]): Complex[T] =
## ``x`` raised to the power ``y``.
if x.re == 0.0 and x.im == 0.0:
if y.re == 0.0 and y.im == 0.0:
result.re = 1.0
result.im = 0.0
else:
result.re = 0.0
result.im = 0.0
elif y.re == 1.0 and y.im == 0.0:
result = x
elif y.re == -1.0 and y.im == 0.0:
result = T(1.0) / x
else:
var
rho = abs(x)
theta = arctan2(x.im, x.re)
s = pow(rho, y.re) * exp(-y.im * theta)
r = y.re * theta + y.im * ln(rho)
result.re = s * cos(r)
result.im = s * sin(r)
func pow*[T](x: Complex[T]; y: T): Complex[T] =
## Complex number ``x`` raised to the power ``y``.
pow(x, complex[T](y))
func sin*[T](z: Complex[T]): Complex[T] =
## Returns the sine of ``z``.
result.re = sin(z.re) * cosh(z.im)
result.im = cos(z.re) * sinh(z.im)
func arcsin*[T](z: Complex[T]): Complex[T] =
## Returns the inverse sine of ``z``.
result = -im(T) * ln(im(T) * z + sqrt(T(1.0) - z*z))
func cos*[T](z: Complex[T]): Complex[T] =
## Returns the cosine of ``z``.
result.re = cos(z.re) * cosh(z.im)
result.im = -sin(z.re) * sinh(z.im)
func arccos*[T](z: Complex[T]): Complex[T] =
## Returns the inverse cosine of ``z``.
result = -im(T) * ln(z + sqrt(z*z - T(1.0)))
func tan*[T](z: Complex[T]): Complex[T] =
## Returns the tangent of ``z``.
result = sin(z) / cos(z)
func arctan*[T](z: Complex[T]): Complex[T] =
## Returns the inverse tangent of ``z``.
result = T(0.5)*im(T) * (ln(T(1.0) - im(T)*z) - ln(T(1.0) + im(T)*z))
func cot*[T](z: Complex[T]): Complex[T] =
## Returns the cotangent of ``z``.
result = cos(z)/sin(z)
func arccot*[T](z: Complex[T]): Complex[T] =
## Returns the inverse cotangent of ``z``.
result = T(0.5)*im(T) * (ln(T(1.0) - im(T)/z) - ln(T(1.0) + im(T)/z))
func sec*[T](z: Complex[T]): Complex[T] =
## Returns the secant of ``z``.
result = T(1.0) / cos(z)
func arcsec*[T](z: Complex[T]): Complex[T] =
## Returns the inverse secant of ``z``.
result = -im(T) * ln(im(T) * sqrt(1.0 - 1.0/(z*z)) + T(1.0)/z)
func csc*[T](z: Complex[T]): Complex[T] =
## Returns the cosecant of ``z``.
result = T(1.0) / sin(z)
func arccsc*[T](z: Complex[T]): Complex[T] =
## Returns the inverse cosecant of ``z``.
result = -im(T) * ln(sqrt(T(1.0) - T(1.0)/(z*z)) + im(T)/z)
func sinh*[T](z: Complex[T]): Complex[T] =
## Returns the hyperbolic sine of ``z``.
result = T(0.5) * (exp(z) - exp(-z))
func arcsinh*[T](z: Complex[T]): Complex[T] =
## Returns the inverse hyperbolic sine of ``z``.
result = ln(z + sqrt(z*z + 1.0))
func cosh*[T](z: Complex[T]): Complex[T] =
## Returns the hyperbolic cosine of ``z``.
result = T(0.5) * (exp(z) + exp(-z))
func arccosh*[T](z: Complex[T]): Complex[T] =
## Returns the inverse hyperbolic cosine of ``z``.
result = ln(z + sqrt(z*z - T(1.0)))
func tanh*[T](z: Complex[T]): Complex[T] =
## Returns the hyperbolic tangent of ``z``.
result = sinh(z) / cosh(z)
func arctanh*[T](z: Complex[T]): Complex[T] =
## Returns the inverse hyperbolic tangent of ``z``.
result = T(0.5) * (ln((T(1.0)+z) / (T(1.0)-z)))
func sech*[T](z: Complex[T]): Complex[T] =
## Returns the hyperbolic secant of ``z``.
result = T(2.0) / (exp(z) + exp(-z))
func arcsech*[T](z: Complex[T]): Complex[T] =
## Returns the inverse hyperbolic secant of ``z``.
result = ln(1.0/z + sqrt(T(1.0)/z+T(1.0)) * sqrt(T(1.0)/z-T(1.0)))
func csch*[T](z: Complex[T]): Complex[T] =
## Returns the hyperbolic cosecant of ``z``.
result = T(2.0) / (exp(z) - exp(-z))
func arccsch*[T](z: Complex[T]): Complex[T] =
## Returns the inverse hyperbolic cosecant of ``z``.
result = ln(T(1.0)/z + sqrt(T(1.0)/(z*z) + T(1.0)))
func coth*[T](z: Complex[T]): Complex[T] =
## Returns the hyperbolic cotangent of ``z``.
result = cosh(z) / sinh(z)
func arccoth*[T](z: Complex[T]): Complex[T] =
## Returns the inverse hyperbolic cotangent of ``z``.
result = T(0.5) * (ln(T(1.0) + T(1.0)/z) - ln(T(1.0) - T(1.0)/z))
func phase*[T](z: Complex[T]): T =
## Returns the phase of ``z``.
arctan2(z.im, z.re)
func polar*[T](z: Complex[T]): tuple[r, phi: T] =
## Returns ``z`` in polar coordinates.
(r: abs(z), phi: phase(z))
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)``
complex(r * cos(phi), r * sin(phi))
func `$`*(z: Complex): string =
## Returns ``z``'s string representation as ``"(re, im)"``.
result = "(" & $z.re & ", " & $z.im & ")"
{.pop.}
|