diff options
Diffstat (limited to 'lib/pure/complex.nim')
| -rw-r--r-- | lib/pure/complex.nim | 161 |
1 files changed, 129 insertions, 32 deletions
diff --git a/lib/pure/complex.nim b/lib/pure/complex.nim index a8709e098..8577bf7a1 100644 --- a/lib/pure/complex.nim +++ b/lib/pure/complex.nim @@ -19,10 +19,7 @@ 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?) - + EPS = 1.0e-7 ## Epsilon used for float comparisons. type Complex* = tuple[re, im: float] @@ -30,6 +27,11 @@ type {.deprecated: [TComplex: Complex].} +proc toComplex*(x: SomeInteger): Complex = + ## Convert some integer ``x`` to a complex number. + result.re = x + result.im = 0 + proc `==` *(x, y: Complex): bool = ## Compare two complex numbers `x` and `y` for equality. result = x.re == y.re and x.im == y.im @@ -174,6 +176,12 @@ proc abs*(z: Complex): float = result = y * sqrt(1.0 + temp * temp) +proc conjugate*(z: Complex): Complex = + ## Conjugate of complex number `z`. + result.re = z.re + result.im = -z.im + + proc sqrt*(z: Complex): Complex = ## Square root for a complex number `z`. var x, y, w, r: float @@ -266,27 +274,101 @@ proc tan*(z: Complex): Complex = ## Returns the tangent of `z`. result = sin(z)/cos(z) +proc arctan*(z: Complex): Complex = + ## Returns the inverse tangent of `z`. + var i: Complex = (0.0,1.0) + result = 0.5*i*(ln(1-i*z)-ln(1+i*z)) + proc cot*(z: Complex): Complex = ## Returns the cotangent of `z`. result = cos(z)/sin(z) +proc arccot*(z: Complex): Complex = + ## Returns the inverse cotangent of `z`. + var i: Complex = (0.0,1.0) + result = 0.5*i*(ln(1-i/z)-ln(1+i/z)) + proc sec*(z: Complex): Complex = ## Returns the secant of `z`. result = 1.0/cos(z) +proc arcsec*(z: Complex): Complex = + ## Returns the inverse secant of `z`. + var i: Complex = (0.0,1.0) + result = -i*ln(i*sqrt(1-1/(z*z))+1/z) + proc csc*(z: Complex): Complex = ## Returns the cosecant of `z`. result = 1.0/sin(z) +proc arccsc*(z: Complex): Complex = + ## Returns the inverse cosecant of `z`. + var i: Complex = (0.0,1.0) + result = -i*ln(sqrt(1-1/(z*z))+i/z) + proc sinh*(z: Complex): Complex = ## Returns the hyperbolic sine of `z`. result = 0.5*(exp(z)-exp(-z)) +proc arcsinh*(z: Complex): Complex = + ## Returns the inverse hyperbolic sine of `z`. + result = ln(z+sqrt(z*z+1)) + proc cosh*(z: Complex): Complex = ## Returns the hyperbolic cosine of `z`. result = 0.5*(exp(z)+exp(-z)) +proc arccosh*(z: Complex): Complex = + ## Returns the inverse hyperbolic cosine of `z`. + result = ln(z+sqrt(z*z-1)) + +proc tanh*(z: Complex): Complex = + ## Returns the hyperbolic tangent of `z`. + result = sinh(z)/cosh(z) + +proc arctanh*(z: Complex): Complex = + ## Returns the inverse hyperbolic tangent of `z`. + result = 0.5*(ln((1+z)/(1-z))) + +proc sech*(z: Complex): Complex = + ## Returns the hyperbolic secant of `z`. + result = 2/(exp(z)+exp(-z)) + +proc arcsech*(z: Complex): Complex = + ## Returns the inverse hyperbolic secant of `z`. + result = ln(1/z+sqrt(1/z+1)*sqrt(1/z-1)) + +proc csch*(z: Complex): Complex = + ## Returns the hyperbolic cosecant of `z`. + result = 2/(exp(z)-exp(-z)) + +proc arccsch*(z: Complex): Complex = + ## Returns the inverse hyperbolic cosecant of `z`. + result = ln(1/z+sqrt(1/(z*z)+1)) + +proc coth*(z: Complex): Complex = + ## Returns the hyperbolic cotangent of `z`. + result = cosh(z)/sinh(z) + +proc arccoth*(z: Complex): Complex = + ## Returns the inverse hyperbolic cotangent of `z`. + result = 0.5*(ln(1+1/z)-ln(1-1/z)) + +proc phase*(z: Complex): float = + ## Returns the phase of `z`. + arctan2(z.im, z.re) + +proc polar*(z: Complex): tuple[r, phi: float] = + ## Returns `z` in polar coordinates. + result.r = abs(z) + result.phi = phase(z) + +proc rect*(r: float, phi: float): Complex = + ## Returns the complex number with polar coordinates `r` and `phi`. + result.re = r * cos(phi) + result.im = r * sin(phi) + proc `$`*(z: Complex): string = ## Returns `z`'s string representation as ``"(re, im)"``. @@ -306,41 +388,56 @@ when isMainModule: 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( 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( abs(oo) == sqrt(2.0) ) + assert( conjugate(a) == (1.0, -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,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( pow(a,m1) =~ (0.2, -0.4) ) - assert( ln(a) =~ (0.804718956217050, 1.107148717794090) ) + 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( 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( arctan(a) =~ (1.338972522294494, 0.402359478108525) ) - assert( cosh(a) =~ (-0.642148124715520, 1.068607421382778) ) + assert( cosh(a) =~ (-0.642148124715520, 1.068607421382778) ) assert( sinh(a) =~ (-0.489056259041294, 1.403119250622040) ) - - + assert( tanh(a) =~ (1.1667362572409199,-0.243458201185725) ) + assert( sech(a) =~ 1/cosh(a) ) + assert( csch(a) =~ 1/sinh(a) ) + assert( coth(a) =~ 1/tanh(a) ) + assert( arccosh(a) =~ (1.528570919480998, 1.14371774040242) ) + assert( arcsinh(a) =~ (1.469351744368185, 1.06344002357775) ) + assert( arctanh(a) =~ (0.173286795139986, 1.17809724509617) ) + assert( arcsech(a) =~ arccosh(1/a) ) + assert( arccsch(a) =~ arcsinh(1/a) ) + assert( arccoth(a) =~ arctanh(1/a) ) + + assert( phase(a) == 1.1071487177940904 ) + var t = polar(a) + assert( rect(t.r, t.phi) =~ a ) + assert( rect(1.0, 2.0) =~ (-0.4161468365471424, 0.9092974268256817) ) |