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814f644c73
extras/complex_numbers/complex_numbers.complex.m:
Fix incorrect predicate name in documentation of complex_to_polar/3.
190 lines
5.2 KiB
Mathematica
190 lines
5.2 KiB
Mathematica
%---------------------------------------------------------------------------%
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% vim: ft=mercury ts=4 sw=4 et
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%---------------------------------------------------------------------------%
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% Copyright (C) 1997-1998, 2001, 2005-2006 The University of Melbourne.
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% Copyright (C) 2015, 2018, 2025-2026 The Mercury team.
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% This file is distributed under the terms specified in COPYING.LIB.
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%---------------------------------------------------------------------------%
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%
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% File: complex.m.
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% Main author: fjh.
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% Stability: medium.
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%
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% Complex numbers.
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%
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% Note that the overloaded versions of the binary operators that
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% provide mixed-type arithmetic are defined in other modules.
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%
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% See also:
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% complex_float.m, float_complex.m
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% imag.m, complex_imag.m, imag_complex.m
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%
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%---------------------------------------------------------------------------%
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:- module complex_numbers.complex.
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:- interface.
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%---------------------------------------------------------------------------%
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% The constructor cmplx/2 is made public, but generally it is most convenient
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% to use the syntax `X + Y*i' for complex numbers, where `i' is declared in
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% module `imag'. Due to the wonders of logic programming, this works fine for
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% both constructing and pattern matching; with intermodule optimization
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% enabled, the compiler should generate equally good code for it.
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:- type complex
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---> cmplx(
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float, % real part
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float % imag part
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).
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%---------------------------------------------------------------------------%
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% Convert float to complex.
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%
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:- func complex(float) = complex.
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% Extract real part.
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%
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:- func real(complex) = float.
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% Extract imaginary part.
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%
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:- func imag(complex) = float.
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% Square of absolute value.
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%
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:- func abs2(complex) = float.
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% Absolute value (a.k.a. modulus).
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%
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:- func abs(complex) = float.
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% Argument (a.k.a. phase, or amplitude, or angle).
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% This function returns the principle value:
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%
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% for all Z, -pi < arg(Z) and arg(Z) =< pi.
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%
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:- func arg(complex) = float.
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% Complex conjugate.
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%
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:- func conj(complex) = complex.
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% Addition.
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%
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:- func complex + complex = complex.
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:- mode in + in = uo is det.
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% Subtraction.
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%
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:- func complex - complex = complex.
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:- mode in - in = uo is det.
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% Multiplication.
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%
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:- func complex * complex = complex.
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:- mode in * in = uo is det.
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% Division.
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%
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:- func complex / complex = complex.
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:- mode in / in = uo is det.
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% Unary plus.
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%
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:- func + complex = complex.
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:- mode + in = uo is det.
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% Unary minus.
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%
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:- func - complex = complex.
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:- mode - in = uo is det.
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% sqr(X) = X * X.
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%
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:- func sqr(complex) = complex.
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% Square root.
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%
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:- func sqrt(complex) = complex.
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% cis(Theta) = cos(Theta) + i * sin(Theta)
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%
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:- func cis(float) = complex.
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% polar_to_complex(R, Theta).
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% Conversion from polar coordinates.
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%
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:- func polar_to_complex(float, float) = complex.
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% complex_to_polar(Z, R, Theta).
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% Conversion to polar coordinates.
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%
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:- pred complex_to_polar(complex::in, float::out, float::out) is det.
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%---------------------------------------------------------------------------%
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%---------------------------------------------------------------------------%
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:- implementation.
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:- import_module float.
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:- import_module math.
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%---------------------------------------------------------------------------%
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complex(Real) = cmplx(Real, 0.0).
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real(cmplx(Real, _Imag)) = Real.
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imag(cmplx(_Real, Imag)) = Imag.
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cmplx(Xr, Xi) + cmplx(Yr, Yi) = cmplx(Xr + Yr, Xi + Yi).
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cmplx(Xr, Xi) - cmplx(Yr, Yi) = cmplx(Xr - Yr, Xi - Yi).
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cmplx(Xr, Xi) * cmplx(Yr, Yi) =
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cmplx(Xr * Yr - Xi * Yi, Xr * Yi + Xi * Yr).
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cmplx(Xr, Xi) / cmplx(Yr, Yi) =
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cmplx((Xr * Yr + Xi * Yi) / Div, (Xi * Yr - Xr * Yi) / Div) :-
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Div = (Yr * Yr + Yi * Yi).
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% Here's the derivation of the formula for complex division:
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% cmplx(Xr, Xi) / cmplx(Yr, Yi) =
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% (cmplx(Xr, Xi) / cmplx(Yr, Yi)) * 1.0 =
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% (cmplx(Xr, Xi) / cmplx(Yr, Yi)) * (cmplx(Yr, -Yi) / cmplx(Yr, -Yi)) =
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% (cmplx(Xr, Xi) * (cmplx(Yr, -Yi)) / (cmplx(Yr, Yi) * cmplx(Yr, -Yi)) =
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% (cmplx(Xr, Xi) * (cmplx(Yr, -Yi)) / (Yr * Yr + Yi * Yi) =
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% cmplx(Xr * Yr + Xi * Yi, Xi * Yr - Xr * Yi) / (Yr * Yr + Yi * Yi) =
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% cmplx((Xr * Yr + Xi * Yi) / Div, (Xi * Yr - Xr * Yi) / Div) :-
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% Div = (Yr * Yr + Yi * Yi).
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+ cmplx(R, I) = cmplx(+ R, + I).
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- cmplx(R, I) = cmplx(- R, - I).
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abs2(cmplx(R, I)) = R*R + I*I.
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abs(Z) = sqrt(abs2(Z)).
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arg(cmplx(R, I)) = atan2(I, R).
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conj(cmplx(R, I)) = cmplx(R, -I).
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sqr(cmplx(Re0, Im0)) = cmplx(Re, Im) :-
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Re = Re0 * Re0 - Im0 * Im0,
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Im = 2.0 * Re0 * Im0.
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sqrt(Z0) = Z :-
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complex_to_polar(Z0, Magnitude0, Theta0),
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Magnitude = sqrt(Magnitude0),
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Theta = Theta0 / 2.0,
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Z = polar_to_complex(Magnitude, Theta).
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complex_to_polar(Z, abs(Z), arg(Z)).
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polar_to_complex(Magnitude, Theta) = cmplx(Real, Imag) :-
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Real = Magnitude * cos(Theta),
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Imag = Magnitude * sin(Theta).
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cis(Theta) = cmplx(cos(Theta), sin(Theta)).
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%---------------------------------------------------------------------------%
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:- end_module complex_numbers.complex.
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%---------------------------------------------------------------------------%
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