G.1.2 Complex Elementary Functions
Static Semantics
The generic library
package Numerics.Generic_Complex_Elementary_Functions has the following
declaration:
{
AI95-00434-01}
with Ada.Numerics.Generic_Complex_Types;
generic
with package Complex_Types
is
new Ada.Numerics.Generic_Complex_Types (<>);
use Complex_Types;
package Ada.Numerics.Generic_Complex_Elementary_Functions
is
pragma Pure(Generic_Complex_Elementary_Functions);
function Sqrt (X : Complex)
return Complex;
function Log (X : Complex)
return Complex;
function Exp (X : Complex)
return Complex;
function Exp (X : Imaginary)
return Complex;
function "**" (Left : Complex; Right : Complex)
return Complex;
function "**" (Left : Complex; Right : Real'Base)
return Complex;
function "**" (Left : Real'Base; Right : Complex)
return Complex;
function Sin (X : Complex)
return Complex;
function Cos (X : Complex)
return Complex;
function Tan (X : Complex)
return Complex;
function Cot (X : Complex)
return Complex;
function Arcsin (X : Complex)
return Complex;
function Arccos (X : Complex)
return Complex;
function Arctan (X : Complex)
return Complex;
function Arccot (X : Complex)
return Complex;
function Sinh (X : Complex)
return Complex;
function Cosh (X : Complex)
return Complex;
function Tanh (X : Complex)
return Complex;
function Coth (X : Complex)
return Complex;
function Arcsinh (X : Complex)
return Complex;
function Arccosh (X : Complex)
return Complex;
function Arctanh (X : Complex)
return Complex;
function Arccoth (X : Complex)
return Complex;
end Ada.Numerics.Generic_Complex_Elementary_Functions;
{
8652/0020}
{
AI95-00126-01}
The library package Numerics.Complex_Elementary_Functions
is declared pure and defines the same subprograms as Numerics.Generic_Complex_Elementary_Functions,
except that the predefined type Float is systematically substituted for
Real'Base, and the Complex and Imaginary types exported by Numerics.Complex_Types
are systematically substituted for Complex and Imaginary, throughout.
Nongeneric equivalents of Numerics.Generic_Complex_Elementary_Functions
corresponding to each of the other predefined floating point types are
defined similarly, with the names Numerics.Short_Complex_Elementary_Functions,
Numerics.Long_Complex_Elementary_Functions, etc.
Reason: The nongeneric equivalents are
provided to allow the programmer to construct simple mathematical applications
without being required to understand and use generics.
The overloading of the Exp function for the pure-imaginary
type is provided to give the user an alternate way to compose a complex
value from a given modulus and argument. In addition to Compose_From_Polar(Rho,
Theta) (see
G.1.1), the programmer may write
Rho * Exp(i * Theta).
The imaginary (resp., real) component of the parameter
X of the forward hyperbolic (resp., trigonometric) functions and of the
Exp function (and the parameter X, itself, in the case of the overloading
of the Exp function for the pure-imaginary type) represents an angle
measured in radians, as does the imaginary (resp., real) component of
the result of the Log and inverse hyperbolic (resp., trigonometric) functions.
The functions have
their usual mathematical meanings. However, the arbitrariness inherent
in the placement of branch cuts, across which some of the complex elementary
functions exhibit discontinuities, is eliminated by the following conventions:
The imaginary component of the result of the Sqrt
and Log functions is discontinuous as the parameter X crosses the negative
real axis.
The result of the exponentiation operator when
the left operand is of complex type is discontinuous as that operand
crosses the negative real axis.
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The imaginary component of the result of the Arcsin, Arccos, and Arctanh
functions is discontinuous as the parameter X crosses the real axis to
the left of –1.0 or the right of 1.0.
{
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The real component of the result of the Arctan and Arcsinh functions
is discontinuous as the parameter X crosses the imaginary axis below
–
i or above
i.
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The real component of the result of the Arccot function is discontinuous
as the parameter X crosses the imaginary axis below –
i
or above
i.
The imaginary component of the Arccosh function
is discontinuous as the parameter X crosses the real axis to the left
of 1.0.
The imaginary component of the result of the Arccoth
function is discontinuous as the parameter X crosses the real axis between
–1.0 and 1.0.
Discussion:
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The branch cuts come from the fact that the functions in question are
really multi-valued in the complex domain, and that we have to pick one
principal value to be the result of the function. Evidently we
have much freedom in choosing where the branch cuts lie. However, we
are adhering to the following principles which seem to lead to the more
natural definitions:
A branch cut should not intersect the real
axis at a place where the corresponding real function is well-defined
(in other words, the complex function should be an extension of the corresponding
real function).
Because all the functions in question are
analytic, to ensure power series validity for the principal value, the
branch cuts should be invariant by complex conjugation.
For odd functions, to ensure that the principal
value remains an odd function, the branch cuts should be invariant by
reflection in the origin.
{
AI95-00185-01}
The computed results of the mathematically multivalued functions are
rendered single-valued by the following conventions, which are meant
to imply that the principal branch is an analytic continuation of the
corresponding real-valued function in Numerics.Generic_Elementary_Functions.
(For Arctan and Arccot, the single-argument function in question is that
obtained from the two-argument version by fixing the second argument
to be its default value.)
The real component of the result of the Sqrt and
Arccosh functions is nonnegative.
The same convention applies to the imaginary component
of the result of the Log function as applies to the result of the natural-cycle
version of the Argument function of Numerics.Generic_Complex_Types (see
G.1.1).
The range of the real (resp., imaginary) component
of the result of the Arcsin and Arctan (resp., Arcsinh and Arctanh) functions
is approximately –π/2.0 to π/2.0.
The real (resp., imaginary) component of the result
of the Arccos and Arccot (resp., Arccoth) functions ranges from 0.0 to
approximately π.
The range of the imaginary component of the result
of the Arccosh function is approximately –π to π.
In addition, the exponentiation operator inherits
the single-valuedness of the Log function.
Dynamic Semantics
The exception Numerics.Argument_Error is raised by
the exponentiation operator, signaling a parameter value outside the
domain of the corresponding mathematical function, when the value of
the left operand is zero and the real component of the exponent (or the
exponent itself, when it is of real type) is zero.
{Division_Check
[partial]} {check,
language-defined (Division_Check)} {Constraint_Error
(raised by failure of run-time check)} The
exception Constraint_Error is raised, signaling a pole of the mathematical
function (analogous to dividing by zero), in the following cases, provided
that Complex_Types.Real'Machine_Overflows is True:
by the Log, Cot, and Coth functions, when the value
of the parameter X is zero;
by the exponentiation operator, when the value
of the left operand is zero and the real component of the exponent (or
the exponent itself, when it is of real type) is negative;
by the Arctan and Arccot functions, when the value
of the parameter X is ± i;
by the Arctanh and Arccoth functions, when the
value of the parameter X is ± 1.0.
[Constraint_Error can also be raised when a finite
result overflows (see
G.2.6); this may occur
for parameter values sufficiently
near poles, and, in the case
of some of the functions, for parameter values having components of sufficiently
large magnitude.]
{unspecified [partial]}
When Complex_Types.Real'Machine_Overflows is False,
the result at poles is unspecified.
Reason: The purpose of raising Constraint_Error
(rather than Numerics.Argument_Error) at the poles of a function, when
Float_Type'Machine_Overflows is True, is to provide continuous behavior
as the actual parameters of the function approach the pole and finally
reach it.
Discussion: It is anticipated that an
Ada binding to IEC 559:1989 will be developed in the future. As part
of such a binding, the Machine_Overflows attribute of a conformant floating
point type will be specified to yield False, which will permit implementations
of the complex elementary functions to deliver results with an infinite
component (and set the overflow flag defined by the binding) instead
of raising Constraint_Error in overflow situations, when traps are disabled.
Similarly, it is appropriate for the complex elementary functions to
deliver results with an infinite component (and set the zero-divide flag
defined by the binding) instead of raising Constraint_Error at poles,
when traps are disabled. Finally, such a binding should also specify
the behavior of the complex elementary functions, when sensible, given
parameters with infinite components.
Implementation Requirements
In the implementation of Numerics.Generic_Complex_Elementary_Functions,
the range of intermediate values allowed during the calculation of a
final result shall not be affected by any range constraint of the subtype
Complex_Types.Real.
Implementation Note: Implementations
of Numerics.Generic_Complex_Elementary_Functions written in Ada should
therefore avoid declaring local variables of subtype Complex_Types.Real;
the subtype Complex_Types.Real'Base should be used instead.
{prescribed
result (for the evaluation of a complex elementary function)}
In the following cases, evaluation of a complex elementary
function shall yield the
prescribed result (or a result having
the prescribed component), provided that the preceding rules do not call
for an exception to be raised:
When the parameter X has the value zero, the Sqrt,
Sin, Arcsin, Tan, Arctan, Sinh, Arcsinh, Tanh, and Arctanh functions
yield a result of zero; the Exp, Cos, and Cosh functions yield a result
of one; the Arccos and Arccot functions yield a real result; and the
Arccoth function yields an imaginary result.
When the parameter X has the value one, the Sqrt
function yields a result of one; the Log, Arccos, and Arccosh functions
yield a result of zero; and the Arcsin function yields a real result.
When the parameter X has the value –1.0,
the Sqrt function yields the result
i
(resp., –i), when the sign of
the imaginary component of X is positive (resp., negative), if Complex_Types.Real'Signed_Zeros
is True;
i,
if Complex_Types.Real'Signed_Zeros is False;
{
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When the parameter X has the value –1.0, the Log function yields
an imaginary result; and the Arcsin and Arccos functions yield a real
result.
When the parameter X has the value ± i,
the Log function yields an imaginary result.
Exponentiation by a zero exponent yields the value
one. Exponentiation by a unit exponent yields the value of the left operand
(as a complex value). Exponentiation of the value one yields the value
one. Exponentiation of the value zero yields the value zero.
Discussion: It is possible to give many
other prescribed results restricting the result to the real or imaginary
axis when the parameter X is appropriately restricted to easily testable
portions of the domain. We follow the proposed ISO/IEC standard for Generic_Complex_Elementary_Functions
(for Ada 83), CD 13813, in not doing so, however.
Other accuracy requirements for the complex elementary
functions, 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 elementary function is implementation defined when
Complex_Types.Real'Signed_Zeros is True.
Implementation defined: The sign of a
zero result (or a component thereof) from any operator or function in
Numerics.Generic_Complex_Elementary_Functions, when Complex_Types.Real'Signed_Zeros
is True.
Implementation Permissions
The nongeneric equivalent packages may, but need
not, be actual instantiations of the generic package with the appropriate
predefined nongeneric equivalent of Numerics.Generic_Complex_Types; if
they are, then the latter shall have been obtained by actual instantiation
of Numerics.Generic_Complex_Types.
The exponentiation operator may be implemented in
terms of the Exp and Log functions. Because this implementation yields
poor accuracy in some parts of the domain, no accuracy requirement is
imposed on complex exponentiation.
{unspecified
[partial]} The implementation of the Exp function
of a complex parameter X is allowed to raise the exception Constraint_Error,
signaling overflow, when the real component of X exceeds an unspecified
threshold that is approximately log(
Complex_Types.Real'Safe_Last).
This permission recognizes the impracticality of avoiding overflow in
the marginal case that the exponential of the real component of X exceeds
the safe range of Complex_Types.Real but both components of the final
result do not. Similarly, the Sin and Cos (resp., Sinh and Cosh) functions
are allowed to raise the exception Constraint_Error, signaling overflow,
when the absolute value of the imaginary (resp., real) component of the
parameter X exceeds an unspecified threshold that is approximately log(
Complex_Types.Real'Safe_Last)
+ log(2.0).
{unspecified [partial]}
This permission recognizes the impracticality of
avoiding overflow in the marginal case that the hyperbolic sine or cosine
of the imaginary (resp., real) component of X exceeds the safe range
of Complex_Types.Real but both components of the final result do not.
Implementation Advice
Implementations in which Complex_Types.Real'Signed_Zeros
is True should attempt to provide a rational treatment of the signs of
zero results and result components. For example, many of the complex
elementary functions have components that are odd functions of one of
the parameter components; in these cases, the result component should
have the sign of the parameter component at the origin. Other complex
elementary functions have zero components whose sign is opposite that
of a parameter component at the origin, or is always positive or always
negative.
Implementation Advice: If Complex_Types.Real'Signed_Zeros
is true for Numerics.Generic_Complex_Elementary_Functions, a rational
treatment of the signs of zero results and result components should be
provided.
Wording Changes from Ada 83
The semantics
of Numerics.Generic_Complex_Elementary_Functions differs from Generic_Complex_Elementary_Functions
as defined in ISO/IEC CD 13814 (for Ada 83) in the following ways:
The generic package is a child unit of the
package defining the Argument_Error exception.
The proposed Generic_Complex_Elementary_Functions
standard (for Ada 83) specified names for the nongeneric equivalents,
if provided. Here, those nongeneric equivalents are required.
The generic package imports an instance of
Numerics.Generic_Complex_Types rather than a long list of individual
types and operations exported by such an instance.
The dependence of the imaginary component
of the Sqrt and Log functions on the sign of a zero parameter component
is tied to the value of Complex_Types.Real'Signed_Zeros.
Conformance to accuracy requirements is conditional.
Wording Changes from Ada 95
{
8652/0020}
{
AI95-00126-01}
Corrigendum: Explicitly stated that the nongeneric equivalents
of Generic_Complex_Elementary_Functions are pure.
{
AI95-00185-01}
Corrected various inconsistencies in the definition of the branch cuts.
Sponsored by Ada-Europe