G.1.1 Complex Types
Static Semantics
The generic library
package Numerics.Generic_Complex_Types has the following declaration:
generic
type Real
is digits <>;
package Ada.Numerics.Generic_Complex_Types
with Pure, Nonblocking
is
type Complex
is
record
Re, Im : Real'Base;
end record;
type Imaginary
is private
with Preelaborable_Initialization;
i :
constant Imaginary;
j :
constant Imaginary;
function Re (X : Complex)
return Real'Base;
function Im (X : Complex)
return Real'Base;
function Im (X : Imaginary)
return Real'Base;
procedure Set_Re (X :
in out Complex;
Re :
in Real'Base);
procedure Set_Im (X :
in out Complex;
Im :
in Real'Base);
procedure Set_Im (X :
out Imaginary;
Im :
in Real'Base);
function Compose_From_Cartesian (Re, Im : Real'Base)
return Complex;
function Compose_From_Cartesian (Re : Real'Base)
return Complex;
function Compose_From_Cartesian (Im : Imaginary)
return Complex;
function Modulus (X : Complex)
return Real'Base;
function "
abs" (Right : Complex)
return Real'Base
renames Modulus;
function Argument (X : Complex)
return Real'Base;
function Argument (X : Complex;
Cycle : Real'Base)
return Real'Base;
function Compose_From_Polar (Modulus, Argument : Real'Base)
return Complex;
function Compose_From_Polar (Modulus, Argument, Cycle : Real'Base)
return Complex;
function "+" (Right : Complex)
return Complex;
function "-" (Right : Complex)
return Complex;
function Conjugate (X : Complex)
return Complex;
function "+" (Left, Right : Complex) return Complex;
function "-" (Left, Right : Complex) return Complex;
function "*" (Left, Right : Complex) return Complex;
function "/" (Left, Right : Complex) return Complex;
function "**" (Left : Complex; Right : Integer) return Complex;
function "+" (Right : Imaginary)
return Imaginary;
function "-" (Right : Imaginary)
return Imaginary;
function Conjugate (X : Imaginary)
return Imaginary
renames "-";
function "
abs" (Right : Imaginary)
return Real'Base;
function "+" (Left, Right : Imaginary) return Imaginary;
function "-" (Left, Right : Imaginary) return Imaginary;
function "*" (Left, Right : Imaginary) return Real'Base;
function "/" (Left, Right : Imaginary) return Real'Base;
function "**" (Left : Imaginary; Right : Integer) return Complex;
function "<" (Left, Right : Imaginary) return Boolean;
function "<=" (Left, Right : Imaginary) return Boolean;
function ">" (Left, Right : Imaginary) return Boolean;
function ">=" (Left, Right : Imaginary) return Boolean;
function "+" (Left : Complex; Right : Real'Base) return Complex;
function "+" (Left : Real'Base; Right : Complex) return Complex;
function "-" (Left : Complex; Right : Real'Base) return Complex;
function "-" (Left : Real'Base; Right : Complex) return Complex;
function "*" (Left : Complex; Right : Real'Base) return Complex;
function "*" (Left : Real'Base; Right : Complex) return Complex;
function "/" (Left : Complex; Right : Real'Base) return Complex;
function "/" (Left : Real'Base; Right : Complex) return Complex;
function "+" (Left : Complex; Right : Imaginary) return Complex;
function "+" (Left : Imaginary; Right : Complex) return Complex;
function "-" (Left : Complex; Right : Imaginary) return Complex;
function "-" (Left : Imaginary; Right : Complex) return Complex;
function "*" (Left : Complex; Right : Imaginary) return Complex;
function "*" (Left : Imaginary; Right : Complex) return Complex;
function "/" (Left : Complex; Right : Imaginary) return Complex;
function "/" (Left : Imaginary; Right : Complex) return Complex;
function "+" (Left : Imaginary; Right : Real'Base) return Complex;
function "+" (Left : Real'Base; Right : Imaginary) return Complex;
function "-" (Left : Imaginary; Right : Real'Base) return Complex;
function "-" (Left : Real'Base; Right : Imaginary) return Complex;
function "*" (Left : Imaginary; Right : Real'Base) return Imaginary;
function "*" (Left : Real'Base; Right : Imaginary) return Imaginary;
function "/" (Left : Imaginary; Right : Real'Base) return Imaginary;
function "/" (Left : Real'Base; Right : Imaginary) return Imaginary;
private
type Imaginary is new Real'Base;
i : constant Imaginary := 1.0;
j : constant Imaginary := 1.0;
end Ada.Numerics.Generic_Complex_Types;
The library package Numerics.Complex_Types
is declared pure and defines the same types, constants, and subprograms
as Numerics.Generic_Complex_Types, except that the predefined type Float
is systematically substituted for Real'Base throughout. Nongeneric equivalents
of Numerics.Generic_Complex_Types for each of the other predefined floating
point types are defined similarly, with the names Numerics.Short_Complex_Types,
Numerics.Long_Complex_Types, etc.
Complex is a visible type with Cartesian components.
Imaginary is a private type; its full type is derived
from Real'Base.
The arithmetic operations
and the Re, Im, Modulus, Argument, and Conjugate functions have their
usual mathematical meanings. When applied to a parameter of pure-imaginary
type, the “imaginary-part” function Im yields the value of
its parameter, as the corresponding real value. The remaining subprograms
have the following meanings:
The Set_Re and Set_Im procedures replace the designated
component of a complex parameter with the given real value; applied to
a parameter of pure-imaginary type, the Set_Im procedure replaces the
value of that parameter with the imaginary value corresponding to the
given real value.
The Compose_From_Cartesian function constructs
a complex value from the given real and imaginary components. If only
one component is given, the other component is implicitly zero.
The Compose_From_Polar function constructs a complex
value from the given modulus (radius) and argument (angle). When the
value of the parameter Modulus is positive (resp., negative), the result
is the complex value represented by the point in the complex plane lying
at a distance from the origin given by the absolute value of Modulus
and forming an angle measured counterclockwise from the positive (resp.,
negative) real axis given by the value of the parameter Argument.
When the Cycle parameter is specified, the result
of the Argument function and the parameter Argument of the Compose_From_Polar
function are measured in units such that a full cycle of revolution has
the given value; otherwise, they are measured in radians.
The computed results
of the mathematically multivalued functions are rendered single-valued
by the following conventions, which are meant to imply the principal
branch:
The result of the Modulus function is nonnegative.
The result of the Argument function is in the quadrant
containing the point in the complex plane represented by the parameter
X. This may be any quadrant (I through IV); thus, the range of the Argument
function is approximately –π to π (–Cycle/2.0
to Cycle/2.0, if the parameter Cycle is specified).
When the point represented by the parameter X lies on the negative real
axis, the result approximates
π (resp., –π) when the
sign of the imaginary component of X is positive (resp., negative), if
Real'Signed_Zeros is True;
π, if Real'Signed_Zeros is False.
Because a result lying on or near one of the axes
may not be exactly representable, the approximation inherent in computing
the result may place it in an adjacent quadrant, close to but on the
wrong side of the axis.
Dynamic Semantics
The exception Numerics.Argument_Error is raised by
the Argument and Compose_From_Polar functions with specified cycle, signaling
a parameter value outside the domain of the corresponding mathematical
function, when the value of the parameter Cycle is zero or negative.
The
exception Constraint_Error is raised by the division operator when the
value of the right operand is zero, and by the exponentiation operator
when the value of the left operand is zero and the value of the exponent
is negative, provided that Real'Machine_Overflows is True; when Real'Machine_Overflows
is False, the result is unspecified.
Constraint_Error
can also be raised when a finite result overflows (see
G.2.6).
Implementation Requirements
In the implementation of Numerics.Generic_Complex_Types,
the range of intermediate values allowed during the calculation of a
final result shall not be affected by any range constraint of the subtype
Real.
In
the following cases, evaluation of a complex arithmetic operation shall
yield the
prescribed result, provided that the preceding rules
do not call for an exception to be raised:
The results of the Re, Im, and Compose_From_Cartesian
functions are exact.
The real (resp., imaginary) component of the result
of a binary addition operator that yields a result of complex type is
exact when either of its operands is of pure-imaginary (resp., real)
type.
The real (resp., imaginary) component of the result
of a binary subtraction operator that yields a result of complex type
is exact when its right operand is of pure-imaginary (resp., real) type.
The real component of the result of the Conjugate
function for the complex type is exact.
When the point in the complex plane represented
by the parameter X lies on the nonnegative real axis, the Argument function
yields a result of zero.
When the value of the parameter Modulus is zero,
the Compose_From_Polar function yields a result of zero.
When the value of the parameter Argument is equal
to a multiple of the quarter cycle, the result of the Compose_From_Polar
function with specified cycle lies on one of the axes. In this case,
one of its components is zero, and the other has the magnitude of the
parameter Modulus.
Exponentiation by a zero exponent yields the value
one. Exponentiation by a unit exponent yields the value of the left operand.
Exponentiation of the value one yields the value one. Exponentiation
of the value zero yields the value zero, provided that the exponent is
nonzero. When the left operand is of pure-imaginary type, one component
of the result of the exponentiation operator is zero.
When the result, or a result component, of any operator
of Numerics.Generic_Complex_Types has a mathematical definition in terms
of a single arithmetic or relational operation, that result or result
component exhibits the accuracy of the corresponding operation of the
type Real.
Other accuracy requirements for the Modulus, Argument,
and Compose_From_Polar functions, and accuracy requirements for the multiplication
of a pair of complex operands or for division by a complex operand, all
of which apply only in the strict mode, are given in
G.2.6.
The sign of a zero result or zero result component
yielded by a complex arithmetic operation or function is implementation
defined when Real'Signed_Zeros is True.
Implementation Permissions
The nongeneric equivalent packages can be actual
instantiations of the generic package for the appropriate predefined
type, though that is not required.
Implementations may obtain the result of exponentiation
of a complex or pure-imaginary operand by repeated complex multiplication,
with arbitrary association of the factors and with a possible final complex
reciprocation (when the exponent is negative). Implementations are also
permitted to obtain the result of exponentiation of a complex operand,
but not of a pure-imaginary operand, by converting the left operand to
a polar representation; exponentiating the modulus by the given exponent;
multiplying the argument by the given exponent; and reconverting to a
Cartesian representation. Because of this implementation freedom, no
accuracy requirement is imposed on complex exponentiation (except for
the prescribed results given above, which apply regardless of the implementation
method chosen).
Implementation Advice
Because the usual mathematical meaning of multiplication
of a complex operand and a real operand is that of the scaling of both
components of the former by the latter, an implementation should not
perform this operation by first promoting the real operand to complex
type and then performing a full complex multiplication. In systems that,
in the future, support an Ada binding to ISO/IEC 60559:2020, the latter
technique will not generate the required result when one of the components
of the complex operand is infinite. (Explicit multiplication of the infinite
component by the zero component obtained during promotion yields a NaN
that propagates into the final result.) Analogous advice applies in the
case of multiplication of a complex operand and a pure-imaginary operand,
and in the case of division of a complex operand by a real or pure-imaginary
operand.
Likewise, because the usual mathematical meaning
of addition of a complex operand and a real operand is that the imaginary
operand remains unchanged, an implementation should not perform this
operation by first promoting the real operand to complex type and then
performing a full complex addition. In implementations in which the Signed_Zeros
attribute of the component type is True (and which therefore conform
to ISO/IEC 60559:2020 in regard to the handling of the sign of zero in
predefined arithmetic operations), the latter technique will not generate
the required result when the imaginary component of the complex operand
is a negatively signed zero. (Explicit addition of the negative zero
to the zero obtained during promotion yields a positive zero.) Analogous
advice applies in the case of addition of a complex operand and a pure-imaginary
operand, and in the case of subtraction of a complex operand and a real
or pure-imaginary operand.
Implementations in which Real'Signed_Zeros is True
should attempt to provide a rational treatment of the signs of zero results
and result components. As one example, the result of the Argument function
should have the sign of the imaginary component of the parameter X when
the point represented by that parameter lies on the positive real axis;
as another, the sign of the imaginary component of the Compose_From_Polar
function should be the same as (resp., the opposite of) that of the Argument
parameter when that parameter has a value of zero and the Modulus parameter
has a nonnegative (resp., negative) value.
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