additions to complex module

1 files changed, 205 insertions, 6 deletions

diff --git a/lib/pure/complex.nim b/lib/pure/complex.nim
index c06451ca8..df08ace72 100755
--- a/lib/pure/complex.nim
+++ b/lib/pure/complex.nim
@@ -10,37 +10,67 @@
 
 
 ## This module implements complex numbers.
-
 {.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
+ 
+const
+  EPS = 5.0e-6 ## Epsilon used for float comparisons (should be smaller
+               ## if float is really float64, but w/ the current version
+               ## it seems to be float32?)
+
 
 type
-  TComplex* = tuple[re, im: float] 
+  TComplex* = tuple[re, im: float]
     ## a complex number, consisting of a real and an imaginary part
 
 proc `==` *(x, y: TComplex): bool =
   ## Compare two complex numbers `x` and `y` for equality.
   result = x.re == y.re and x.im == y.im
 
+proc `=~` *(x, y: TComplex): bool =
+  ## Compare two complex numbers `x` and `y` approximately.
+  result = abs(x.re-y.re)<EPS and abs(x.im-y.im)<EPS
+
 proc `+` *(x, y: TComplex): TComplex =
   ## Add two complex numbers.
   result.re = x.re + y.re
   result.im = x.im + y.im
 
-proc `-` *(x, y: TComplex): TComplex =
-  ## Subtract two complex numbers.
-  result.re = x.re - y.re
-  result.im = x.im - y.im
+proc `+` *(x: TComplex, y: float): TComplex =
+  ## Add complex `x` to float `y`.
+  result.re = x.re + y
+  result.im = x.im
+
+proc `+` *(x: float, y: TComplex): TComplex =
+  ## Add float `x` to complex `y`.
+  result.re = x + y.re
+  result.im = y.im
+
 
 proc `-` *(z: TComplex): TComplex =
   ## Unary minus for complex numbers.
   result.re = -z.re
   result.im = -z.im
 
+proc `-` *(x, y: TComplex): TComplex =
+  ## Subtract two complex numbers.
+  result.re = x.re - y.re
+  result.im = x.im - y.im
+
+proc `-` *(x: TComplex, y: float): TComplex =
+  ## Subtracts float `y` from complex `x`.
+  result = x + (-y)
+
+proc `-` *(x: float, y: TComplex): TComplex =
+  ## Subtracts complex `y` from float `x`.
+  result = x + (-y)
+
+
 proc `/` *(x, y: TComplex): TComplex =
   ## Divide `x` by `y`.
   var
@@ -56,11 +86,33 @@ proc `/` *(x, y: TComplex): TComplex =
     result.re = (x.re + r * x.im) / den
     result.im = (x.im - r * x.re) / den
 
+proc `/` *(x : TComplex, y: float ): TComplex =
+  ## Divide complex `x` by float `y`.
+  result.re = x.re/y
+  result.im = x.im/y
+
+proc `/` *(x : float, y: TComplex ): TComplex =
+  ## Divide float `x` by complex `y`.
+  var num : TComplex = (x, 0.0)
+  result = num/y
+
+
 proc `*` *(x, y: TComplex): TComplex =
   ## Multiply `x` with `y`.
   result.re = x.re * y.re - x.im * y.im
   result.im = x.im * y.re + x.re * y.im
 
+proc `*` *(x: float, y: TComplex): TComplex =
+  ## Multiply float `x` with complex `y`.
+  result.re = x * y.re
+  result.im = x * y.im
+
+proc `*` *(x: TComplex, y: float): TComplex =
+  ## Multiply complex `x` with float `y`.
+  result.re = x.re * y
+  result.im = x.im * y
+
+
 proc abs*(z: TComplex): float =
   ## Return the distance from (0,0) to `z`.
 
@@ -80,6 +132,7 @@ proc abs*(z: TComplex): float =
     temp = x / y
     result = y * sqrt(1.0 + temp * temp)
 
+
 proc sqrt*(z: TComplex): TComplex =
   ## Square root for a complex number `z`.
   var x, y, w, r: float
@@ -103,4 +156,150 @@ proc sqrt*(z: TComplex): TComplex =
       else:           result.im = -w
       result.re = z.im / (result.im + result.im)
 
+
+proc exp*(z: TComplex): TComplex =
+  ## e raised to the power `z`.
+  var rho   = exp(z.re)
+  var theta = z.im
+  result.re = rho*cos(theta)
+  result.im = rho*sin(theta)
+
+
+proc ln*(z: TComplex): TComplex =
+  ## Returns the natural log of `z`.
+  result.re = ln(abs(z))
+  result.im = arctan2(z.im,z.re)
+
+proc log10*(z: TComplex): TComplex =
+  ## Returns the log base 10 of `z`.
+  result = ln(z)/ln(10.0)
+
+proc log2*(z: TComplex): TComplex =
+  ## Returns the log base 2 of `z`.
+  result = ln(z)/ln(2.0)
+
+
+proc pow*(x, y: TComplex): TComplex =
+  ## `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 = 1.0/x
+  else:
+    var rho   = sqrt(x.re*x.re + x.im*x.im)
+    var theta = arctan2(x.im,x.re)
+    var s     = pow(rho,y.re) * exp(-y.im*theta)
+    var r     = y.re*theta + y.im*ln(rho)
+    result.re = s*cos(r)
+    result.im = s*sin(r)
+           
+
+proc sin*(z: TComplex): TComplex =
+  ## Returns the sine of `z`.
+  result.re = sin(z.re)*cosh(z.im)
+  result.im = cos(z.re)*sinh(z.im)
+
+proc arcsin*(z: TComplex): TComplex =
+  ## Returns the inverse sine of `z`.
+  var i: TComplex = (0.0,1.0)
+  result = -i*ln(i*z + sqrt(1.0-z*z))
+
+proc cos*(z: TComplex): TComplex =
+  ## Returns the cosine of `z`.
+  result.re = cos(z.re)*cosh(z.im)
+  result.im = -sin(z.re)*sinh(z.im)
+
+proc arccos*(z: TComplex): TComplex =
+  ## Returns the inverse cosine of `z`.
+  var i: TComplex = (0.0,1.0)
+  result = -i*ln(z + sqrt(z*z-1.0))
+
+proc tan*(z: TComplex): TComplex =
+  ## Returns the tangent of `z`.
+  result = sin(z)/cos(z)
+
+proc cot*(z: TComplex): TComplex =
+  ## Returns the cotangent of `z`.
+  result = cos(z)/sin(z)
+
+proc sec*(z: TComplex): TComplex =
+  ## Returns the secant of `z`.
+  result = 1.0/cos(z)
+
+proc csc*(z: TComplex): TComplex =
+  ## Returns the cosecant of `z`.
+  result = 1.0/sin(z)
+
+
+proc sinh*(z: TComplex): TComplex =
+  ## Returns the hyperbolic sine of `z`.
+  result = 0.5*(exp(z)-exp(-z))
+
+proc cosh*(z: TComplex): TComplex =
+  ## Returns the hyperbolic cosine of `z`.
+  result = 0.5*(exp(z)+exp(-z))
+
+
+proc `$`*(z: TComplex): string =
+  ## Returns `z`'s string representation as ``"(re, im)"``.
+  result = "(" & $z.re & ", " & $z.im & ")"
+
 {.pop.}
+
+
+when isMainModule:
+  var z = (0.0, 0.0)
+  var oo = (1.0,1.0)
+  var a = (1.0, 2.0)
+  var b = (-1.0, -2.0)
+  var m1 = (-1.0, 0.0)
+  var i = (0.0,1.0)
+  var one = (1.0,0.0)
+  var tt = (10.0, 20.0)
+  var ipi = (0.0, -PI)
+ 
+  assert( a         == a )
+  assert( (a-a)     == z )
+  assert( (a+b)     == z )
+  assert( (a/b)     == m1 )
+  assert( (1.0/a)   == (0.2, -0.4) )
+  assert( (a*b)     == (3.0, -4.0) )
+  assert( 10.0*a    == tt )
+  assert( a*10.0    == tt )
+  assert( tt/10.0   == a )
+  assert( oo+(-1.0) == i )
+  assert( (-1.0)+oo == i )
+  assert( abs(oo)   == sqrt(2.0) )
+  assert( sqrt(m1)  == i )
+  assert( exp(ipi)  =~ m1 )
+ 
+  assert( pow(a,b)   =~ (-3.72999124927876, -1.68815826725068) )
+  assert( pow(z,a)   =~ (0.0, 0.0) )
+  assert( pow(z,z)   =~ (1.0, 0.0) )
+  assert( pow(a,one) =~ a )
+  assert( pow(a,m1)  =~ (0.2, -0.4) )
+
+  assert( ln(a)    =~ (0.804718956217050, 1.107148717794090) )
+  assert( log10(a) =~ (0.349485002168009, 0.480828578784234) )
+  assert( log2(a)  =~ (1.16096404744368, 1.59727796468811) )
+
+  assert( sin(a)    =~ (3.16577851321617, 1.95960104142161) )
+  assert( cos(a)    =~ (2.03272300701967, -3.05189779915180) )
+  assert( tan(a)    =~ (0.0338128260798967, 1.0147936161466335) )
+  assert( cot(a)    =~ 1.0/tan(a) )
+  assert( sec(a)    =~ 1.0/cos(a) )
+  assert( csc(a)    =~ 1.0/sin(a) )
+  assert( arcsin(a) =~ (0.427078586392476, 1.528570919480998) )
+  assert( arccos(a) =~ (1.14371774040242, -1.52857091948100) )
+
+  assert( cosh(a) =~ (-0.642148124715520, 1.068607421382778) ) 
+  assert( sinh(a) =~ (-0.489056259041294, 1.403119250622040) )
+
+