Annex G
(normative)
Numerics
1 The Numerics Annex specifies
2 * features for complex arithmetic, including complex I/O;
3 * a mode ("strict mode"), in which the predefined arithmetic operations
of floating point and fixed point types and the functions and
operations of various predefined packages have to provide guaranteed
accuracy or conform to other numeric performance requirements, which
the Numerics Annex also specifies;
4 * a mode ("relaxed mode"), in which no accuracy or other numeric
performance requirements need be satisfied, as for implementations not
conforming to the Numerics Annex;
5/2 * {AI95-00296-01} models of floating point and fixed point arithmetic on
which the accuracy requirements of strict mode are based;
6/2 * {AI95-00296-01} the definitions of the model-oriented attributes of
floating point types that apply in the strict mode; and
6.1/2 * {AI95-00296-01} features for the manipulation of real and complex
vectors and matrices.
Implementation Advice
7/3 {AI05-0229-1} If Fortran (respectively, C) is widely supported in the
target environment, implementations supporting the Numerics Annex should
provide the child package Interfaces.Fortran (respectively, Interfaces.C)
specified in Annex B and should support a convention_identifier of Fortran
(respectively, C) for the Convention aspect (see Annex B), thus allowing Ada
programs to interface with programs written in that language.
7.a.1/2 Implementation Advice: If Fortran (respectively, C) is supported
in the target environment, then interfacing to Fortran
(respectively, C) should be supported as specified in Annex B.
Extensions to Ada 83
7.a This Annex is new to Ada 95.
G.1 Complex Arithmetic
1 Types and arithmetic operations for complex arithmetic are provided in
Generic_Complex_Types, which is defined in G.1.1. Implementation-defined
approximations to the complex analogs of the mathematical functions known as
the "elementary functions" are provided by the subprograms in Generic_Complex_-
Elementary_Functions, which is defined in G.1.2. Both of these library units
are generic children of the predefined package Numerics (see A.5). Nongeneric
equivalents of these generic packages for each of the predefined floating
point types are also provided as children of Numerics.
1.a Implementation defined: The accuracy actually achieved by the
complex elementary functions and by other complex arithmetic
operations.
1.b Discussion: Complex arithmetic is defined in the Numerics Annex,
rather than in the core, because it is considered to be a
specialized need of (some) numeric applications.
G.1.1 Complex Types
Static Semantics
1 The generic library package Numerics.Generic_Complex_Types has the
following declaration:
2/1 {8652/0020} {AI95-00126-01} generic
type Real is digits <>;
package Ada.Numerics.Generic_Complex_Types is
pragma Pure(Generic_Complex_Types);
3 type Complex is
record
Re, Im : Real'Base;
end record;
4/2 {AI95-00161-01} type Imaginary is private;
pragma Preelaborable_Initialization(Imaginary);
5 i : constant Imaginary;
j : constant Imaginary;
6 function Re (X : Complex) return Real'Base;
function Im (X : Complex) return Real'Base;
function Im (X : Imaginary) return Real'Base;
7 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);
8 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;
9 function Modulus (X : Complex) return Real'Base;
function "abs" (Right : Complex) return Real'Base renames Modulus;
10 function Argument (X : Complex) return Real'Base;
function Argument (X : Complex;
Cycle : Real'Base) return Real'Base;
11 function Compose_From_Polar (Modulus, Argument : Real'Base)
return Complex;
function Compose_From_Polar (Modulus, Argument, Cycle : Real'Base)
return Complex;
12 function "+" (Right : Complex) return Complex;
function "-" (Right : Complex) return Complex;
function Conjugate (X : Complex) return Complex;
13 function "+" (Left, Right : Complex) return Complex;
function "-" (Left, Right : Complex) return Complex;
function "*" (Left, Right : Complex) return Complex;
function "/" (Left, Right : Complex) return Complex;
14 function "**" (Left : Complex; Right : Integer) return Complex;
15 function "+" (Right : Imaginary) return Imaginary;
function "-" (Right : Imaginary) return Imaginary;
function Conjugate
(X : Imaginary) return Imaginary renames "-";
function "abs" (Right : Imaginary) return Real'Base;
16 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;
17 function "**" (Left : Imaginary; Right : Integer) return Complex;
18 function "<" (Left, Right : Imaginary) return Boolean;
function "<=" (Left, Right : Imaginary) return Boolean;
function ">" (Left, Right : Imaginary) return Boolean;
function ">=" (Left, Right : Imaginary) return Boolean;
19 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;
20 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;
21 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;
22 private
23 type Imaginary is new Real'Base;
i : constant Imaginary := 1.0;
j : constant Imaginary := 1.0;
24 end Ada.Numerics.Generic_Complex_Types;
25/1 {8652/0020} {AI95-00126-01} 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.
25.a Reason: The nongeneric equivalents are provided to allow the
programmer to construct simple mathematical applications without
being required to understand and use generics.
25.b Reason: The nongeneric equivalents all export the types Complex
and Imaginary and the constants i and j (rather than uniquely
named types and constants, such as Short_Complex, Long_Complex,
etc.) to preserve their equivalence to actual instantiations of
the generic package and to allow the programmer to change the
precision of an application globally by changing a single context
clause.
26/2 {AI95-00434-01} [Complex is a visible type with Cartesian components.]
26.a Reason: The Cartesian representation is far more common than the
polar representation, in practice. The accuracy of the results of
the complex arithmetic operations and of the complex elementary
functions is dependent on the representation; thus, implementers
need to know that representation. The type is visible so that
complex "literals" can be written in aggregate notation, if
desired.
27 [Imaginary is a private type; its full type is derived from Real'Base.]
27.a Reason: The Imaginary type and the constants i and j are provided
for two reasons:
27.b * They allow complex "literals" to be written in the alternate
form of a + b*i (or a + b*j), if desired. Of course, in some
contexts the sum will need to be parenthesized.
27.c * When an Ada binding to IEC 559:1989 that provides (signed)
infinities as the result of operations that overflow becomes
available, it will be important to allow arithmetic between
pure-imaginary and complex operands without requiring the
former to be represented as (or promoted to) complex values
with a real component of zero. For example, the multiplication
of a + b*i by d*i should yield -b· d + a· d*i, but if one
cannot avoid representing the pure-imaginary value d*i as the
complex value 0.0 + d*i, then a NaN ("Not-a-Number") could be
produced as the result of multiplying a by 0.0 (e.g., when a
is infinite); the NaN could later trigger an exception.
Providing the Imaginary type and overloadings of the
arithmetic operators for mixtures of Imaginary and Complex
operands gives the programmer the same control over avoiding
premature coercion of pure-imaginary values to complex as is
already provided for pure-real values.
27.d Reason: The Imaginary type is private, rather than being visibly
derived from Real'Base, for two reasons:
27.e * to preclude implicit conversions of real literals to the
Imaginary type (such implicit conversions would make many
common arithmetic expressions ambiguous); and
27.f * to suppress the implicit derivation of the multiplication,
division, and absolute value operators with Imaginary operands
and an Imaginary result (the result type would be incorrect).
27.g Reason: The base subtype Real'Base is used for the component type
of Complex, the parent type of Imaginary, and the parameter and
result types of some of the subprograms to maximize the chances of
being able to pass meaningful values into the subprograms and
receive meaningful results back. The generic formal parameter Real
therefore plays only one role, that of providing the precision to
be maintained in complex arithmetic calculations. Thus, the
subprograms in Numerics.Generic_Complex_Types share with those in
Numerics.Generic_Elementary_Functions, and indeed even with the
predefined arithmetic operations (see 4.5), the property of being
free of range checks on input and output, i.e., of being able to
exploit the base range of the relevant floating point type fully.
As a result, the user loses the ability to impose
application-oriented bounds on the range of values that the
components of a complex variable can acquire; however, it can be
argued that few, if any, applications have a naturally square
domain (as opposed to a circular domain) anyway.
28 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:
28.a Reason: The middle case can be understood by considering the
parameter of pure-imaginary type to represent a complex value with
a zero real part.
29 * 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.
30 * 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.
31 * 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.
32 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.
33 The computed results of the mathematically multivalued functions are
rendered single-valued by the following conventions, which are meant to imply
the principal branch:
34 * The result of the Modulus function is nonnegative.
35 * 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 -PI to PI (-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
36 * PI (resp., -PI) when the sign of the imaginary component of X is
positive (resp., negative), if Real'Signed_Zeros is True;
37 * PI, if Real'Signed_Zeros is False.
38 * 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
39 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.
40 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).]
40.a 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 arithmetic operations 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 arithmetic operations to deliver
results with infinite components (and set the zero-divide flag
defined by the binding) instead of raising Constraint_Error in the
situations defined above, when traps are disabled. Finally, such a
binding should also specify the behavior of the complex arithmetic
operations, when sensible, given operands with infinite
components.
Implementation Requirements
41 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.
41.a Implementation Note: Implementations of
Numerics.Generic_Complex_Types written in Ada should therefore
avoid declaring local variables of subtype Real; the subtype
Real'Base should be used instead.
42 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:
43 * The results of the Re, Im, and Compose_From_Cartesian functions are
exact.
44 * 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.
44.a Ramification: The result of the addition operator is exact when
one of its operands is of real type and the other is of
pure-imaginary type. In this particular case, the operator is
analogous to the Compose_From_Cartesian function; it performs no
arithmetic.
45 * 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.
46 * The real component of the result of the Conjugate function for the
complex type is exact.
47 * 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.
47.a Discussion: Argument(X + i*Y) is analogous to EF.Arctan(Y, X),
where EF is an appropriate instance of
Numerics.Generic_Elementary_Functions, except when X and Y are
both zero, in which case the former yields the value zero while
the latter raises Numerics.Argument_Error.
48 * When the value of the parameter Modulus is zero, the
Compose_From_Polar function yields a result of zero.
49 * 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.
50 * 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.
51 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.
52 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.
53 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.
53.a Implementation defined: The sign of a zero result (or a component
thereof) from any operator or function in
Numerics.Generic_Complex_Types, when Real'Signed_Zeros is True.
Implementation Permissions
54 The nongeneric equivalent packages may, but need not, be actual
instantiations of the generic package for the appropriate predefined type.
55/2 {8652/0091} {AI95-00434-01} 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
56 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 IEC 559:1989, 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.
56.a/2 Implementation Advice: Mixed real and complex operations (as well
as pure-imaginary and complex operations) should not be performed
by converting the real (resp. pure-imaginary) operand to complex.
57 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 IEC 559:1989 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.
58 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.
58.a.1/3 Implementation Advice: If Real'Signed_Zeros is True for Numerics.-
Generic_Complex_Types, a rational treatment of the signs of zero
results and result components should be provided.
Wording Changes from Ada 83
58.a The semantics of Numerics.Generic_Complex_Types differs from
Generic_Complex_Types as defined in ISO/IEC CD 13813 (for Ada 83)
in the following ways:
58.b * The generic package is a child of the package defining the
Argument_Error exception.
58.c * The nongeneric equivalents export types and constants with the
same names as those exported by the generic package, rather
than with names unique to the package.
58.d * Implementations are not allowed to impose an optional
restriction that the generic actual parameter associated with
Real be unconstrained. (In view of the ability to declare
variables of subtype Real'Base in implementations of
Numerics.Generic_Complex_Types, this flexibility is no longer
needed.)
58.e * The dependence of the Argument function on the sign of a zero
parameter component is tied to the value of Real'Signed_Zeros.
58.f * Conformance to accuracy requirements is conditional.
Extensions to Ada 95
58.g/2 {AI95-00161-01} Amendment Correction: Added a pragma
Preelaborable_Initialization to type Imaginary, so that it can be
used in preelaborated units.
Wording Changes from Ada 95
58.h/2 {8652/0020} {AI95-00126-01} Corrigendum: Explicitly stated that
the nongeneric equivalents of Generic_Complex_Types are pure.
G.1.2 Complex Elementary Functions
Static Semantics
1 The generic library package Numerics.Generic_Complex_Elementary_Functions
has the following declaration:
2/2 {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);
3 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;
4 function Sin (X : Complex) return Complex;
function Cos (X : Complex) return Complex;
function Tan (X : Complex) return Complex;
function Cot (X : Complex) return Complex;
5 function Arcsin (X : Complex) return Complex;
function Arccos (X : Complex) return Complex;
function Arctan (X : Complex) return Complex;
function Arccot (X : Complex) return Complex;
6 function Sinh (X : Complex) return Complex;
function Cosh (X : Complex) return Complex;
function Tanh (X : Complex) return Complex;
function Coth (X : Complex) return Complex;
7 function Arcsinh (X : Complex) return Complex;
function Arccosh (X : Complex) return Complex;
function Arctanh (X : Complex) return Complex;
function Arccoth (X : Complex) return Complex;
8 end Ada.Numerics.Generic_Complex_Elementary_Functions;
9/1 {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.
9.a Reason: The nongeneric equivalents are provided to allow the
programmer to construct simple mathematical applications without
being required to understand and use generics.
10 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).
11 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.
12 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:
13 * The imaginary component of the result of the Sqrt and Log functions is
discontinuous as the parameter X crosses the negative real axis.
14 * The result of the exponentiation operator when the left operand is of
complex type is discontinuous as that operand crosses the negative
real axis.
15/2 * {AI95-00185-01} 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.
16/2 * {AI95-00185-01} 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.
17/2 * {AI95-00185-01} 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.
18 * The imaginary component of the Arccosh function is discontinuous as
the parameter X crosses the real axis to the left of 1.0.
19 * 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.
19.a/2 Discussion: {AI95-00185-01} 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:
19.b/2 * 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).
19.c/2 * 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.
19.d/2 * For odd functions, to ensure that the principal value remains
an odd function, the branch cuts should be invariant by
reflection in the origin.
20/2 {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.)
21 * The real component of the result of the Sqrt and Arccosh functions is
nonnegative.
22 * 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).
23 * The range of the real (resp., imaginary) component of the result of
the Arcsin and Arctan (resp., Arcsinh and Arctanh) functions is
approximately -PI/2.0 to PI/2.0.
24 * The real (resp., imaginary) component of the result of the Arccos and
Arccot (resp., Arccoth) functions ranges from 0.0 to approximately
PI.
25 * The range of the imaginary component of the result of the Arccosh
function is approximately -PI to PI.
26 In addition, the exponentiation operator inherits the single-valuedness of
the Log function.
Dynamic Semantics
27 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.
28 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:
29 * by the Log, Cot, and Coth functions, when the value of the parameter X
is zero;
30 * 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;
31 * by the Arctan and Arccot functions, when the value of the parameter X
is ± i;
32 * by the Arctanh and Arccoth functions, when the value of the parameter
X is ± 1.0.
33 [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.] When Complex_Types.Real'Machine_Overflows is
False, the result at poles is unspecified.
33.a 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.
33.b 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
34 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.
34.a 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.
35 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:
36 * 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.
37 * 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.
38 * When the parameter X has the value -1.0, the Sqrt function yields the
result
39 * i (resp., -i), when the sign of the imaginary component of X is
positive (resp., negative), if Complex_Types.Real'Signed_Zeros is
True;
40 * i, if Complex_Types.Real'Signed_Zeros is False;
41/2 * {AI95-00434-01} 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.
42 * When the parameter X has the value ± i, the Log function yields an
imaginary result.
43 * 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.
43.a 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.
44 Other accuracy requirements for the complex elementary functions, which
apply only in the strict mode, are given in G.2.6.
45 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.
45.a 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
46 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.
47 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.
48 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). 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
49 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.
49.a.1/3 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
49.a 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:
49.b * The generic package is a child unit of the package defining
the Argument_Error exception.
49.c * The proposed Generic_Complex_Elementary_Functions standard
(for Ada 83) specified names for the nongeneric equivalents,
if provided. Here, those nongeneric equivalents are required.
49.d * 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.
49.e * 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.
49.f * Conformance to accuracy requirements is conditional.
Wording Changes from Ada 95
49.g/2 {8652/0020} {AI95-00126-01} Corrigendum: Explicitly stated that
the nongeneric equivalents of Generic_Complex_Elementary_Functions
are pure.
49.h/2 {AI95-00185-01} Corrected various inconsistencies in the
definition of the branch cuts.
G.1.3 Complex Input-Output
1 The generic package Text_IO.Complex_IO defines procedures for the
formatted input and output of complex values. The generic actual parameter in
an instantiation of Text_IO.Complex_IO is an instance of
Numerics.Generic_Complex_Types for some floating point subtype. Exceptional
conditions are reported by raising the appropriate exception defined in
Text_IO.
1.a Implementation Note: An implementation of Text_IO.Complex_IO can
be built around an instance of Text_IO.Float_IO for the base
subtype of Complex_Types.Real, where Complex_Types is the generic
formal package parameter of Text_IO.Complex_IO. There is no need
for an implementation of Text_IO.Complex_IO to parse real values.
Static Semantics
2 The generic library package Text_IO.Complex_IO has the following
declaration:
2.a Ramification: Because this is a child of Text_IO, the declarations
of the visible part of Text_IO are directly visible within it.
3 with Ada.Numerics.Generic_Complex_Types;
generic
with package Complex_Types is
new Ada.Numerics.Generic_Complex_Types (<>);
package Ada.Text_IO.Complex_IO is
4 use Complex_Types;
5 Default_Fore : Field := 2;
Default_Aft : Field := Real'Digits - 1;
Default_Exp : Field := 3;
6 procedure Get (File : in File_Type;
Item : out Complex;
Width : in Field := 0);
procedure Get (Item : out Complex;
Width : in Field := 0);
7 procedure Put (File : in File_Type;
Item : in Complex;
Fore : in Field := Default_Fore;
Aft : in Field := Default_Aft;
Exp : in Field := Default_Exp);
procedure Put (Item : in Complex;
Fore : in Field := Default_Fore;
Aft : in Field := Default_Aft;
Exp : in Field := Default_Exp);
8 procedure Get (From : in String;
Item : out Complex;
Last : out Positive);
procedure Put (To : out String;
Item : in Complex;
Aft : in Field := Default_Aft;
Exp : in Field := Default_Exp);
9 end Ada.Text_IO.Complex_IO;
9.1/2 {AI95-00328-01} The library package Complex_Text_IO defines the same
subprograms as Text_IO.Complex_IO, except that the predefined type Float is
systematically substituted for Real, and the type
Numerics.Complex_Types.Complex is systematically substituted for Complex
throughout. Nongeneric equivalents of Text_IO.Complex_IO corresponding to each
of the other predefined floating point types are defined similarly, with the
names Short_Complex_Text_IO, Long_Complex_Text_IO, etc.
9.a/2 Reason: The nongeneric equivalents are provided to allow the
programmer to construct simple mathematical applications without
being required to understand and use generics.
10 The semantics of the Get and Put procedures are as follows:
11 procedure Get (File : in File_Type;
Item : out Complex;
Width : in Field := 0);
procedure Get (Item : out Complex;
Width : in Field := 0);
12/1 {8652/0092} {AI95-00029-01} The input sequence is a pair of
optionally signed real literals representing the real and
imaginary components of a complex value. These components have the
format defined for the corresponding Get procedure of an instance
of Text_IO.Float_IO (see A.10.9) for the base subtype of
Complex_Types.Real. The pair of components may be separated by a
comma or surrounded by a pair of parentheses or both. Blanks are
freely allowed before each of the components and before the
parentheses and comma, if either is used. If the value of the
parameter Width is zero, then
13 * line and page terminators are also allowed in these places;
14 * the components shall be separated by at least one blank or
line terminator if the comma is omitted; and
15 * reading stops when the right parenthesis has been read, if the
input sequence includes a left parenthesis, or when the
imaginary component has been read, otherwise.
15.1 If a nonzero value of Width is supplied, then
16 * the components shall be separated by at least one blank if the
comma is omitted; and
17 * exactly Width characters are read, or the characters (possibly
none) up to a line terminator, whichever comes first (blanks
are included in the count).
17.a Reason: The parenthesized and comma-separated form is the form
produced by Put on output (see below), and also by list-directed
output in Fortran. The other allowed forms match several common
styles of edit-directed output in Fortran, allowing most
preexisting Fortran data files containing complex data to be read
easily. When such files contain complex values with no separation
between the real and imaginary components, the user will have to
read those components separately, using an instance of
Text_IO.Float_IO.
18 Returns, in the parameter Item, the value of type Complex that
corresponds to the input sequence.
19 The exception Text_IO.Data_Error is raised if the input sequence
does not have the required syntax or if the components of the
complex value obtained are not of the base subtype of
Complex_Types.Real.
20 procedure Put (File : in File_Type;
Item : in Complex;
Fore : in Field := Default_Fore;
Aft : in Field := Default_Aft;
Exp : in Field := Default_Exp);
procedure Put (Item : in Complex;
Fore : in Field := Default_Fore;
Aft : in Field := Default_Aft;
Exp : in Field := Default_Exp);
21 Outputs the value of the parameter Item as a pair of decimal
literals representing the real and imaginary components of the
complex value, using the syntax of an aggregate. More
specifically,
22 * outputs a left parenthesis;
23 * outputs the value of the real component of the parameter Item
with the format defined by the corresponding Put procedure of
an instance of Text_IO.Float_IO for the base subtype of
Complex_Types.Real, using the given values of Fore, Aft, and
Exp;
24 * outputs a comma;
25 * outputs the value of the imaginary component of the parameter
Item with the format defined by the corresponding Put
procedure of an instance of Text_IO.Float_IO for the base
subtype of Complex_Types.Real, using the given values of Fore,
Aft, and Exp;
26 * outputs a right parenthesis.
26.a Discussion: If the file has a bounded line length, a line
terminator may be output implicitly before any element of the
sequence itemized above.
26.b Discussion: The option of outputting the complex value as a pair
of reals without additional punctuation is not provided, since it
can be accomplished by outputting the real and imaginary
components of the complex value separately.
27 procedure Get (From : in String;
Item : out Complex;
Last : out Positive);
28/2 {AI95-00434-01} Reads a complex value from the beginning of the
given string, following the same rule as the Get procedure that
reads a complex value from a file, but treating the end of the
string as a file terminator. Returns, in the parameter Item, the
value of type Complex that corresponds to the input sequence.
Returns in Last the index value such that From(Last) is the last
character read.
29 The exception Text_IO.Data_Error is raised if the input sequence
does not have the required syntax or if the components of the
complex value obtained are not of the base subtype of
Complex_Types.Real.
30 procedure Put (To : out String;
Item : in Complex;
Aft : in Field := Default_Aft;
Exp : in Field := Default_Exp);
31 Outputs the value of the parameter Item to the given string as a
pair of decimal literals representing the real and imaginary
components of the complex value, using the syntax of an aggregate.
More specifically,
32 * a left parenthesis, the real component, and a comma are left
justified in the given string, with the real component having
the format defined by the Put procedure (for output to a file)
of an instance of Text_IO.Float_IO for the base subtype of
Complex_Types.Real, using a value of zero for Fore and the
given values of Aft and Exp;
33 * the imaginary component and a right parenthesis are right
justified in the given string, with the imaginary component
having the format defined by the Put procedure (for output to
a file) of an instance of Text_IO.Float_IO for the base
subtype of Complex_Types.Real, using a value for Fore that
completely fills the remainder of the string, together with
the given values of Aft and Exp.
33.a Reason: This rule is the one proposed in LSN-1051. Other rules
were considered, including one that would have read "Outputs the
value of the parameter Item to the given string, following the
same rule as for output to a file, using a value for Fore such
that the sequence of characters output exactly fills, or comes
closest to filling, the string; in the latter case, the string is
filled by inserting one extra blank immediately after the
comma." While this latter rule might be considered the closest analogue
to the rule for output to a string in Text_IO.Float_IO, it
requires a more difficult and inefficient implementation involving
special cases when the integer part of one component is
substantially longer than that of the other and the string is too
short to allow both to be preceded by blanks. Unless such a
special case applies, the latter rule might produce better
columnar output if several such strings are ultimately output to a
file, but very nearly the same output can be produced by
outputting to the file directly, with the appropriate value of
Fore; in any case, it might validly be assumed that output to a
string is intended for further computation rather than for
display, so that the precise formatting of the string to achieve a
particular appearance is not the major concern.
34 The exception Text_IO.Layout_Error is raised if the given string
is too short to hold the formatted output.
Implementation Permissions
35 Other exceptions declared (by renaming) in Text_IO may be raised by the
preceding procedures in the appropriate circumstances, as for the
corresponding procedures of Text_IO.Float_IO.
Extensions to Ada 95
35.a/2 {AI95-00328-01} Nongeneric equivalents for Text_IO.Complex_IO are
added, to be consistent with all other language-defined Numerics
generic packages.
Wording Changes from Ada 95
35.b/2 {8652/0092} {AI95-00029-01} Corrigendum: Clarified that the syntax
of values read by Complex_IO is the same as that read by
Text_IO.Float_IO.
G.1.4 The Package Wide_Text_IO.Complex_IO
Static Semantics
1 Implementations shall also provide the generic library package
Wide_Text_IO.Complex_IO. Its declaration is obtained from that of
Text_IO.Complex_IO by systematically replacing Text_IO by Wide_Text_IO and
String by Wide_String; the description of its behavior is obtained by
additionally replacing references to particular characters (commas,
parentheses, etc.) by those for the corresponding wide characters.
G.1.5 The Package Wide_Wide_Text_IO.Complex_IO
Static Semantics
1/2 {AI95-00285-01} Implementations shall also provide the generic library
package Wide_Wide_Text_IO.Complex_IO. Its declaration is obtained from that of
Text_IO.Complex_IO by systematically replacing Text_IO by Wide_Wide_Text_IO
and String by Wide_Wide_String; the description of its behavior is obtained by
additionally replacing references to particular characters (commas,
parentheses, etc.) by those for the corresponding wide wide characters.
Extensions to Ada 95
1.a/2 {AI95-00285-01} Package Wide_Wide_Text_IO.Complex_IO is new. (At
least it wasn't called Incredibly_Wide_Text_IO.Complex_IO; maybe
next time.)
G.2 Numeric Performance Requirements
Implementation Requirements
1 Implementations shall provide a user-selectable mode in which the accuracy
and other numeric performance requirements detailed in the following
subclauses are observed. This mode, referred to as the strict mode, may or may
not be the default mode; it directly affects the results of the predefined
arithmetic operations of real types and the results of the subprograms in
children of the Numerics package, and indirectly affects the operations in
other language defined packages. Implementations shall also provide the
opposing mode, which is known as the relaxed mode.
1.a Reason: On the assumption that the users of an implementation that
does not support the Numerics Annex have no particular need for
numerical performance, such an implementation has no obligation to
meet any particular requirements in this area. On the other hand,
users of an implementation that does support the Numerics Annex
are provided with a way of ensuring that their programs achieve a
known level of numerical performance and that the performance is
portable to other such implementations. The relaxed mode is
provided to allow implementers to offer an efficient but not fully
accurate alternative in the case that the strict mode entails a
time overhead that some users may find excessive. In some of its
areas of impact, the relaxed mode may be fully equivalent to the
strict mode.
1.b Implementation Note: The relaxed mode may, for example, be used to
exploit the implementation of (some of) the elementary functions
in hardware, when available. Such implementations often do not
meet the accuracy requirements of the strict mode, or do not meet
them over the specified range of parameter values, but compensate
in other ways that may be important to the user, such as their
extreme speed.
1.c Ramification: For implementations supporting the Numerics Annex,
the choice of mode has no effect on the selection of a
representation for a real type or on the values of attributes of a
real type.
Implementation Permissions
2 Either mode may be the default mode.
2.a Implementation defined: Whether the strict mode or the relaxed
mode is the default.
3 The two modes need not actually be different.
Extensions to Ada 83
3.a The choice between strict and relaxed numeric performance was not
available in Ada 83.
G.2.1 Model of Floating Point Arithmetic
1 In the strict mode, the predefined operations of a floating point type
shall satisfy the accuracy requirements specified here and shall avoid or
signal overflow in the situations described. This behavior is presented in
terms of a model of floating point arithmetic that builds on the concept of
the canonical form (see A.5.3).
Static Semantics
2 Associated with each floating point type is an infinite set of model
numbers. The model numbers of a type are used to define the accuracy
requirements that have to be satisfied by certain predefined operations of the
type; through certain attributes of the model numbers, they are also used to
explain the meaning of a user-declared floating point type declaration. The
model numbers of a derived type are those of the parent type; the model
numbers of a subtype are those of its type.
3 The model numbers of a floating point type T are zero and all the values
expressible in the canonical form (for the type T), in which mantissa has
T'Model_Mantissa digits and exponent has a value greater than or equal to
T'Model_Emin. (These attributes are defined in G.2.2.)
3.a Discussion: The model is capable of describing the behavior of
most existing hardware that has a mantissa-exponent
representation. As applied to a type T, it is parameterized by the
values of T'Machine_Radix, T'Model_Mantissa, T'Model_Emin,
T'Safe_First, and T'Safe_Last. The values of these attributes are
determined by how, and how well, the hardware behaves. They in
turn determine the set of model numbers and the safe range of the
type, which figure in the accuracy and range (overflow avoidance)
requirements.
3.b In hardware that is free of arithmetic anomalies,
T'Model_Mantissa, T'Model_Emin, T'Safe_First, and T'Safe_Last will
yield the same values as T'Machine_Mantissa, T'Machine_Emin,
T'Base'First, and T'Base'Last, respectively, and the model numbers
in the safe range of the type T will coincide with the machine
numbers of the type T. In less perfect hardware, it is not
possible for the model-oriented attributes to have these optimal
values, since the hardware, by definition, and therefore the
implementation, cannot conform to the stringencies of the
resulting model; in this case, the values yielded by the
model-oriented parameters have to be made more conservative (i.e.,
have to be penalized), with the result that the model numbers are
more widely separated than the machine numbers, and the safe range
is a subrange of the base range. The implementation will then be
able to conform to the requirements of the weaker model defined by
the sparser set of model numbers and the smaller safe range.
4 A model interval of a floating point type is any interval whose bounds are
model numbers of the type. The model interval of a type T associated with a
value v is the smallest model interval of T that includes v. (The model
interval associated with a model number of a type consists of that number
only.)
Implementation Requirements
5 The accuracy requirements for the evaluation of certain predefined
operations of floating point types are as follows.
5.a Discussion: This subclause does not cover the accuracy of an
operation of a static expression; such operations have to be
evaluated exactly (see 4.9). It also does not cover the accuracy
of the predefined attributes of a floating point subtype that
yield a value of the type; such operations also yield exact
results (see 3.5.8 and A.5.3).
6 An operand interval is the model interval, of the type specified for the
operand of an operation, associated with the value of the operand.
7 For any predefined arithmetic operation that yields a result of a floating
point type T, the required bounds on the result are given by a model interval
of T (called the result interval) defined in terms of the operand values as
follows:
8 * The result interval is the smallest model interval of T that includes
the minimum and the maximum of all the values obtained by applying the
(exact) mathematical operation to values arbitrarily selected from the
respective operand intervals.
9 The result interval of an exponentiation is obtained by applying the above
rule to the sequence of multiplications defined by the exponent, assuming
arbitrary association of the factors, and to the final division in the case of
a negative exponent.
10 The result interval of a conversion of a numeric value to a floating point
type T is the model interval of T associated with the operand value, except
when the source expression is of a fixed point type with a small that is not a
power of T'Machine_Radix or is a fixed point multiplication or division either
of whose operands has a small that is not a power of T'Machine_Radix; in these
cases, the result interval is implementation defined.
10.a Implementation defined: The result interval in certain cases of
fixed-to-float conversion.
11 For any of the foregoing operations, the implementation shall deliver a
value that belongs to the result interval when both bounds of the result
interval are in the safe range of the result type T, as determined by the
values of T'Safe_First and T'Safe_Last; otherwise,
12 * if T'Machine_Overflows is True, the implementation shall either
deliver a value that belongs to the result interval or raise
Constraint_Error;
13 * if T'Machine_Overflows is False, the result is implementation defined.
13.a Implementation defined: The result of a floating point arithmetic
operation in overflow situations, when the Machine_Overflows
attribute of the result type is False.
14 For any predefined relation on operands of a floating point type T, the
implementation may deliver any value (i.e., either True or False) obtained by
applying the (exact) mathematical comparison to values arbitrarily chosen from
the respective operand intervals.
15 The result of a membership test is defined in terms of comparisons of the
operand value with the lower and upper bounds of the given range or type mark
(the usual rules apply to these comparisons).
Implementation Permissions
16 If the underlying floating point hardware implements division as
multiplication by a reciprocal, the result interval for division (and
exponentiation by a negative exponent) is implementation defined.
16.a Implementation defined: The result interval for division (or
exponentiation by a negative exponent), when the floating point
hardware implements division as multiplication by a reciprocal.
Wording Changes from Ada 83
16.b The Ada 95 model numbers of a floating point type that are in the
safe range of the type are comparable to the Ada 83 safe numbers
of the type. There is no analog of the Ada 83 model numbers. The
Ada 95 model numbers, when not restricted to the safe range, are
an infinite set.
Inconsistencies With Ada 83
16.c Giving the model numbers the hardware radix, instead of always a
radix of two, allows (in conjunction with other changes) some
borderline declared types to be represented with less precision
than in Ada 83 (i.e., with single precision, whereas Ada 83 would
have used double precision). Because the lower precision satisfies
the requirements of the model (and did so in Ada 83 as well), this
change is viewed as a desirable correction of an anomaly, rather
than a worrisome inconsistency. (Of course, the wider
representation chosen in Ada 83 also remains eligible for
selection in Ada 95.)
16.d As an example of this phenomenon, assume that Float is represented
in single precision and that a double precision type is also
available. Also assume hexadecimal hardware with clean properties,
for example certain IBM hardware. Then,
16.e type T is digits Float'Digits range -Float'Last .. Float'Last;
16.f results in T being represented in double precision in Ada 83 and
in single precision in Ada 95. The latter is intuitively correct;
the former is counterintuitive. The reason why the double
precision type is used in Ada 83 is that Float has model and safe
numbers (in Ada 83) with 21 binary digits in their mantissas, as
is required to model the hypothesized hexadecimal hardware using a
binary radix; thus Float'Last, which is not a model number, is
slightly outside the range of safe numbers of the single precision
type, making that type ineligible for selection as the
representation of T even though it provides adequate precision. In
Ada 95, Float'Last (the same value as before) is a model number
and is in the safe range of Float on the hypothesized hardware,
making Float eligible for the representation of T.
Extensions to Ada 83
16.g Giving the model numbers the hardware radix allows for practical
implementations on decimal hardware.
Wording Changes from Ada 83
16.h The wording of the model of floating point arithmetic has been
simplified to a large extent.
G.2.2 Model-Oriented Attributes of Floating Point Types
1 In implementations that support the Numerics Annex, the model-oriented
attributes of floating point types shall yield the values defined here, in
both the strict and the relaxed modes. These definitions add conditions to
those in A.5.3.
Static Semantics
2 For every subtype S of a floating point type T:
3/2 {AI95-00256-01} S'Model_Mantissa
Yields the number of digits in the mantissa of the canonical
form of the model numbers of T (see A.5.3). The value of this
attribute shall be greater than or equal to
3.1/2 Ceiling(d · log(10) / log(T'Machine_Radix)) + g
3.2/2 where d is the requested decimal precision of T, and g is 0 if
T'Machine_Radix is a positive power of 10 and 1 otherwise. In
addition, T'Model_Mantissa shall be less than or equal to the
value of T'Machine_Mantissa. This attribute yields a value of
the type universal_integer.
3.a Ramification: S'Model_Epsilon, which is defined in terms of
S'Model_Mantissa (see A.5.3), yields the absolute value of the
difference between one and the next model number of the type T
above one. It is equal to or larger than the absolute value of the
difference between one and the next machine number of the type T
above one.
4 S'Model_Emin
Yields the minimum exponent of the canonical form of the model
numbers of T (see A.5.3). The value of this attribute shall be
greater than or equal to the value of T'Machine_Emin. This
attribute yields a value of the type universal_integer.
4.a Ramification: S'Model_Small, which is defined in terms of
S'Model_Emin (see A.5.3), yields the smallest positive (nonzero)
model number of the type T.
5 S'Safe_First
Yields the lower bound of the safe range of T. The value of
this attribute shall be a model number of T and greater than
or equal to the lower bound of the base range of T. In
addition, if T is declared by a floating_point_definition or
is derived from such a type, and the
floating_point_definition includes a
real_range_specification specifying a lower bound of lb, then
the value of this attribute shall be less than or equal to lb;
otherwise, it shall be less than or equal to -10.0 (4 · d),
where d is the requested decimal precision of T. This
attribute yields a value of the type universal_real.
6 S'Safe_Last Yields the upper bound of the safe range of T. The value of
this attribute shall be a model number of T and less than or
equal to the upper bound of the base range of T. In addition,
if T is declared by a floating_point_definition or is derived
from such a type, and the floating_point_definition includes a
real_range_specification specifying an upper bound of ub, then
the value of this attribute shall be greater than or equal to
ub; otherwise, it shall be greater than or equal to 10.0 (4 ·
d), where d is the requested decimal precision of T. This
attribute yields a value of the type universal_real.
7 S'Model Denotes a function (of a parameter X) whose specification is
given in A.5.3. If X is a model number of T, the function
yields X; otherwise, it yields the value obtained by rounding
or truncating X to either one of the adjacent model numbers of
T. Constraint_Error is raised if the resulting model number is
outside the safe range of S. A zero result has the sign of X
when S'Signed_Zeros is True.
8 Subject to the constraints given above, the values of S'Model_Mantissa and
S'Safe_Last are to be maximized, and the values of S'Model_Emin and
S'Safe_First minimized, by the implementation as follows:
9 * First, S'Model_Mantissa is set to the largest value for which values
of S'Model_Emin, S'Safe_First, and S'Safe_Last can be chosen so that
the implementation satisfies the strict-mode requirements of G.2.1 in
terms of the model numbers and safe range induced by these attributes.
10 * Next, S'Model_Emin is set to the smallest value for which values of
S'Safe_First and S'Safe_Last can be chosen so that the implementation
satisfies the strict-mode requirements of G.2.1 in terms of the model
numbers and safe range induced by these attributes and the previously
determined value of S'Model_Mantissa.
11/3 * {AI05-0092-1} Finally, S'Safe_First and S'Safe_Last are set (in
either order) to the smallest and largest values, respectively, for
which the implementation satisfies the strict-mode requirements of
G.2.1 in terms of the model numbers and safe range induced by these
attributes and the previously determined values of S'Model_Mantissa
and S'Model_Emin.
11.a Ramification: The following table shows appropriate attribute
values for IEEE basic single and double precision types (ANSI/IEEE
Std 754-1985, IEC 559:1989). Here, we use the names IEEE_Float_32
and IEEE_Float_64, the names that would typically be declared in
package Interfaces, in an implementation that supports IEEE
arithmetic. In such an implementation, the attributes would
typically be the same for Standard.Float and Long_Float,
respectively.
11.b Attribute IEEE_Float_32 IEEE_Float_64
11.c 'Machine_Radix 2 2
'Machine_Mantissa 24 53
'Machine_Emin -125 -1021
'Machine_Emax 128 1024
'Denorm True True
'Machine_Rounds True True
'Machine_Overflows True/False True/False
'Signed_Zeros should be True should be True
11.d 'Model_Mantissa (same as 'Machine_Mantissa) (same as 'Machine_Mantissa)
'Model_Emin (same as 'Machine_Emin) (same as 'Machine_Emin)
'Model_Epsilon 2.0**(-23) 2.0**(-52)
'Model_Small 2.0**(-126) 2.0**(-1022)
'Safe_First -2.0**128*(1.0-2.0**(-24)) -2.0**1024*(1.0-2.0**(-53))
'Safe_Last 2.0**128*(1.0-2.0**(-24)) 2.0**1024*(1.0-2.0**(-53))
11.e 'Digits 6 15
'Base'Digits (same as 'Digits) (same as 'Digits)
11.f 'First (same as 'Safe_First) (same as 'Safe_First)
'Last (same as 'Safe_Last) (same as 'Safe_Last)
'Size 32 64
11.g Note: 'Machine_Overflows can be True or False, depending on
whether the Ada implementation raises Constraint_Error or delivers
a signed infinity in overflow and zerodivide situations (and at
poles of the elementary functions).
Wording Changes from Ada 95
11.h/2 {AI95-00256-01} Corrected the definition of Model_Mantissa to
match that given in 3.5.8.
G.2.3 Model of Fixed Point Arithmetic
1 In the strict mode, the predefined arithmetic operations of a fixed point
type shall satisfy the accuracy requirements specified here and shall avoid or
signal overflow in the situations described.
Implementation Requirements
2 The accuracy requirements for the predefined fixed point arithmetic
operations and conversions, and the results of relations on fixed point
operands, are given below.
2.a Discussion: This subclause does not cover the accuracy of an
operation of a static expression; such operations have to be
evaluated exactly (see 4.9).
3 The operands of the fixed point adding operators, absolute value, and
comparisons have the same type. These operations are required to yield exact
results, unless they overflow.
4 Multiplications and divisions are allowed between operands of any two
fixed point types; the result has to be (implicitly or explicitly) converted
to some other numeric type. For purposes of defining the accuracy rules, the
multiplication or division and the conversion are treated as a single
operation whose accuracy depends on three types (those of the operands and the
result). For decimal fixed point types, the attribute T'Round may be used to
imply explicit conversion with rounding (see 3.5.10).
5 When the result type is a floating point type, the accuracy is as given in
G.2.1. For some combinations of the operand and result types in the remaining
cases, the result is required to belong to a small set of values called the
perfect result set; for other combinations, it is required merely to belong to
a generally larger and implementation-defined set of values called the close
result set. When the result type is a decimal fixed point type, the perfect
result set contains a single value; thus, operations on decimal types are
always fully specified.
5.a Implementation defined: The definition of close result set, which
determines the accuracy of certain fixed point multiplications and
divisions.
6 When one operand of a fixed-fixed multiplication or division is of type
universal_real, that operand is not implicitly converted in the usual sense,
since the context does not determine a unique target type, but the accuracy of
the result of the multiplication or division (i.e., whether the result has to
belong to the perfect result set or merely the close result set) depends on
the value of the operand of type universal_real and on the types of the other
operand and of the result.
6.a Discussion: We need not consider here the multiplication or
division of two such operands, since in that case either the
operation is evaluated exactly (i.e., it is an operation of a
static expression all of whose operators are of a root numeric
type) or it is considered to be an operation of a floating point
type.
7 For a fixed point multiplication or division whose (exact) mathematical
result is v, and for the conversion of a value v to a fixed point type, the
perfect result set and close result set are defined as follows:
8 * If the result type is an ordinary fixed point type with a small of s,
9 * if v is an integer multiple of s, then the perfect result set
contains only the value v;
10 * otherwise, it contains the integer multiple of s just below v and
the integer multiple of s just above v.
11 The close result set is an implementation-defined set of consecutive
integer multiples of s containing the perfect result set as a subset.
12 * If the result type is a decimal type with a small of s,
13 * if v is an integer multiple of s, then the perfect result set
contains only the value v;
14/3 * {AI05-0264-1} otherwise, if truncation applies, then it contains
only the integer multiple of s in the direction toward zero,
whereas if rounding applies, then it contains only the nearest
integer multiple of s (with ties broken by rounding away from
zero).
15 The close result set is an implementation-defined set of consecutive
integer multiples of s containing the perfect result set as a subset.
15.a Ramification: As a consequence of subsequent rules, this case does
not arise when the operand types are also decimal types.
16 * If the result type is an integer type,
17 * if v is an integer, then the perfect result set contains only the
value v;
18 * otherwise, it contains the integer nearest to the value v (if v
lies equally distant from two consecutive integers, the perfect
result set contains the one that is further from zero).
19 The close result set is an implementation-defined set of consecutive
integers containing the perfect result set as a subset.
20 The result of a fixed point multiplication or division shall belong either
to the perfect result set or to the close result set, as described below, if
overflow does not occur. In the following cases, if the result type is a fixed
point type, let s be its small; otherwise, i.e. when the result type is an
integer type, let s be 1.0.
21 * For a multiplication or division neither of whose operands is of type
universal_real, let l and r be the smalls of the left and right
operands. For a multiplication, if (l · r) / s is an integer or the
reciprocal of an integer (the smalls are said to be "compatible" in
this case), the result shall belong to the perfect result set;
otherwise, it belongs to the close result set. For a division, if l /
(r · s) is an integer or the reciprocal of an integer (i.e., the
smalls are compatible), the result shall belong to the perfect result
set; otherwise, it belongs to the close result set.
21.a Ramification: When the operand and result types are all decimal
types, their smalls are necessarily compatible; the same is true
when they are all ordinary fixed point types with binary smalls.
22 * For a multiplication or division having one universal_real operand
with a value of v, note that it is always possible to factor v as an
integer multiple of a "compatible" small, but the integer multiple may
be "too big." If there exists a factorization in which that multiple
is less than some implementation-defined limit, the result shall
belong to the perfect result set; otherwise, it belongs to the close
result set.
22.a Implementation defined: Conditions on a universal_real operand of
a fixed point multiplication or division for which the result
shall be in the perfect result set.
23 A multiplication P * Q of an operand of a fixed point type F by an operand
of an integer type I, or vice-versa, and a division P / Q of an operand of a
fixed point type F by an operand of an integer type I, are also allowed. In
these cases, the result has a type of F; explicit conversion of the result is
never required. The accuracy required in these cases is the same as that
required for a multiplication F(P * Q) or a division F(P / Q) obtained by
interpreting the operand of the integer type to have a fixed point type with a
small of 1.0.
24 The accuracy of the result of a conversion from an integer or fixed point
type to a fixed point type, or from a fixed point type to an integer type, is
the same as that of a fixed point multiplication of the source value by a
fixed point operand having a small of 1.0 and a value of 1.0, as given by the
foregoing rules. The result of a conversion from a floating point type to a
fixed point type shall belong to the close result set. The result of a
conversion of a universal_real operand to a fixed point type shall belong to
the perfect result set.
25 The possibility of overflow in the result of a predefined arithmetic
operation or conversion yielding a result of a fixed point type T is analogous
to that for floating point types, except for being related to the base range
instead of the safe range. If all of the permitted results belong to the base
range of T, then the implementation shall deliver one of the permitted
results; otherwise,
26 * if T'Machine_Overflows is True, the implementation shall either
deliver one of the permitted results or raise Constraint_Error;
27 * if T'Machine_Overflows is False, the result is implementation defined.
27.a Implementation defined: The result of a fixed point arithmetic
operation in overflow situations, when the Machine_Overflows
attribute of the result type is False.
Inconsistencies With Ada 83
27.b Since the values of a fixed point type are now just the integer
multiples of its small, the possibility of using extra bits
available in the chosen representation for extra accuracy rather
than for increasing the base range would appear to be removed,
raising the possibility that some fixed point expressions will
yield less accurate results than in Ada 83. However, this is
partially offset by the ability of an implementation to choose a
smaller default small than before. Of course, if it does so for a
type T then T'Small will have a different value than it previously
had.
27.c The accuracy requirements in the case of incompatible smalls are
relaxed to foster wider support for nonbinary smalls. If this
relaxation is exploited for a type that was previously supported,
lower accuracy could result; however, there is no particular
incentive to exploit the relaxation in such a case.
Wording Changes from Ada 83
27.d The fixed point accuracy requirements are now expressed without
reference to model or safe numbers, largely because the full
generality of the former model was never exploited in the case of
fixed point types (particularly in regard to operand
perturbation). Although the new formulation in terms of perfect
result sets and close result sets is still verbose, it can be seen
to distill down to two cases:
27.e * a case where the result must be the exact result, if the exact
result is representable, or, if not, then either one of the
adjacent values of the type (in some subcases only one of
those adjacent values is allowed);
27.f * a case where the accuracy is not specified by the language.
G.2.4 Accuracy Requirements for the Elementary Functions
1 In the strict mode, the performance of
Numerics.Generic_Elementary_Functions shall be as specified here.
Implementation Requirements
2 When an exception is not raised, the result of evaluating a function in an
instance EF of Numerics.Generic_Elementary_Functions belongs to a result
interval, defined as the smallest model interval of EF.Float_Type that
contains all the values of the form f · (1.0 + d), where f is the exact value
of the corresponding mathematical function at the given parameter values, d is
a real number, and |d| is less than or equal to the function's maximum
relative error. The function delivers a value that belongs to the result
interval when both of its bounds belong to the safe range of EF.Float_Type;
otherwise,
3 * if EF.Float_Type'Machine_Overflows is True, the function either
delivers a value that belongs to the result interval or raises
Constraint_Error, signaling overflow;
4 * if EF.Float_Type'Machine_Overflows is False, the result is
implementation defined.
4.a Implementation defined: The result of an elementary function
reference in overflow situations, when the Machine_Overflows
attribute of the result type is False.
5 The maximum relative error exhibited by each function is as follows:
6 * 2.0 · EF.Float_Type'Model_Epsilon, in the case of the Sqrt, Sin, and
Cos functions;
7 * 4.0 · EF.Float_Type'Model_Epsilon, in the case of the Log, Exp, Tan,
Cot, and inverse trigonometric functions; and
8 * 8.0 · EF.Float_Type'Model_Epsilon, in the case of the forward and
inverse hyperbolic functions.
9 The maximum relative error exhibited by the exponentiation operator, which
depends on the values of the operands, is (4.0 + |Right · log(Left)| / 32.0) ·
EF.Float_Type'Model_Epsilon.
10 The maximum relative error given above applies throughout the domain of
the forward trigonometric functions when the Cycle parameter is specified.
When the Cycle parameter is omitted, the maximum relative error given above
applies only when the absolute value of the angle parameter X is less than or
equal to some implementation-defined angle threshold, which shall be at least
EF.Float_Type'Machine_Radix (Floor(EF.Float_Type'Machine_Mantissa/2)). Beyond
the angle threshold, the accuracy of the forward trigonometric functions is
implementation defined.
10.a Implementation defined: The value of the angle threshold, within
which certain elementary functions, complex arithmetic operations,
and complex elementary functions yield results conforming to a
maximum relative error bound.
10.b Implementation defined: The accuracy of certain elementary
functions for parameters beyond the angle threshold.
10.c Implementation Note: The angle threshold indirectly determines the
amount of precision that the implementation has to maintain during
argument reduction.
11/2 {AI95-00434-01} The prescribed results specified in A.5.1 for certain
functions at particular parameter values take precedence over the maximum
relative error bounds; effectively, they narrow to a single value the result
interval allowed by the maximum relative error bounds. Additional rules with a
similar effect are given by table G-1 for the inverse trigonometric functions,
at particular parameter values for which the mathematical result is possibly
not a model number of EF.Float_Type (or is, indeed, even transcendental). In
each table entry, the values of the parameters are such that the result lies
on the axis between two quadrants; the corresponding accuracy rule, which
takes precedence over the maximum relative error bounds, is that the result
interval is the model interval of EF.Float_Type associated with the exact
mathematical result given in the table.
12/1 This paragraph was deleted.
13 The last line of the table is meant to apply when
EF.Float_Type'Signed_Zeros is False; the two lines just above it, when
EF.Float_Type'Signed_Zeros is True and the parameter Y has a zero value with
the indicated sign.
Table G-1: Tightly Approximated Elementary Function Results
Function
Value of X
Value of Y
Exact Result
when Cycle
Specified
Exact Result
when Cycle
Omitted
Arcsin
1.0
n.a.
Cycle/4.0
PI/2.0
Arcsin
-1.0
n.a.
-Cycle/4.0
-PI/2.0
Arccos
0.0
n.a.
Cycle/4.0
PI/2.0
Arccos
-1.0
n.a.
Cycle/2.0
PI
Arctan and Arccot
0.0
positive
Cycle/4.0
PI/2.0
Arctan and Arccot
0.0
negative
-Cycle/4.0
-PI/2.0
Arctan and Arccot
negative
+0.0
Cycle/2.0
PI
Arctan and Arccot
negative
-0.0
-Cycle/2.0
-PI
Arctan and Arccot
negative
0.0
Cycle/2.0
PI
14 The amount by which the result of an inverse trigonometric function is
allowed to spill over into a quadrant adjacent to the one corresponding to the
principal branch, as given in A.5.1, is limited. The rule is that the result
belongs to the smallest model interval of EF.Float_Type that contains both
boundaries of the quadrant corresponding to the principal branch. This rule
also takes precedence over the maximum relative error bounds, effectively
narrowing the result interval allowed by them.
15 Finally, the following specifications also take precedence over the
maximum relative error bounds:
16 * The absolute value of the result of the Sin, Cos, and Tanh functions
never exceeds one.
17 * The absolute value of the result of the Coth function is never less
than one.
18 * The result of the Cosh function is never less than one.
Implementation Advice
19 The versions of the forward trigonometric functions without a Cycle
parameter should not be implemented by calling the corresponding version with
a Cycle parameter of 2.0*Numerics.Pi, since this will not provide the required
accuracy in some portions of the domain. For the same reason, the version of
Log without a Base parameter should not be implemented by calling the
corresponding version with a Base parameter of Numerics.e.
19.a.1/2 Implementation Advice: For elementary functions, the forward
trigonometric functions without a Cycle parameter should not be
implemented by calling the corresponding version with a Cycle
parameter. Log without a Base parameter should not be implemented
by calling Log with a Base parameter.
Wording Changes from Ada 83
19.a The semantics of Numerics.Generic_Elementary_Functions differs
from Generic_Elementary_Functions as defined in ISO/IEC DIS 11430
(for Ada 83) in the following ways related to the accuracy
specified for strict mode:
19.b * The maximum relative error bounds use the Model_Epsilon
attribute instead of the Base'Epsilon attribute.
19.c * The accuracy requirements are expressed in terms of result
intervals that are model intervals. On the one hand, this
facilitates the description of the required results in the
presence of underflow; on the other hand, it slightly relaxes
the requirements expressed in ISO/IEC DIS 11430.
G.2.5 Performance Requirements for Random Number Generation
1 In the strict mode, the performance of Numerics.Float_Random and
Numerics.Discrete_Random shall be as specified here.
Implementation Requirements
2 Two different calls to the time-dependent Reset procedure shall reset the
generator to different states, provided that the calls are separated in time
by at least one second and not more than fifty years.
3 The implementation's representations of generator states and its
algorithms for generating random numbers shall yield a period of at least
2(31)-2; much longer periods are desirable but not required.
4 The implementations of Numerics.Float_Random.Random and
Numerics.Discrete_Random.Random shall pass at least 85% of the individual
trials in a suite of statistical tests. For Numerics.Float_Random, the tests
are applied directly to the floating point values generated (i.e., they are
not converted to integers first), while for Numerics.Discrete_Random they are
applied to the generated values of various discrete types. Each test suite
performs 6 different tests, with each test repeated 10 times, yielding a total
of 60 individual trials. An individual trial is deemed to pass if the
chi-square value (or other statistic) calculated for the observed counts or
distribution falls within the range of values corresponding to the 2.5 and
97.5 percentage points for the relevant degrees of freedom (i.e., it shall be
neither too high nor too low). For the purpose of determining the degrees of
freedom, measurement categories are combined whenever the expected counts are
fewer than 5.
4.a Implementation Note: In the floating point random number test
suite, the generator is reset to a time-dependent state at the
beginning of the run. The test suite incorporates the following
tests, adapted from D. E. Knuth, The Art of Computer Programming,
vol. 2: Seminumerical Algorithms. In the descriptions below, the
given number of degrees of freedom is the number before reduction
due to any necessary combination of measurement categories with
small expected counts; it is one less than the number of
measurement categories.
4.b * Proportional Distribution Test (a variant of the
Equidistribution Test). The interval 0.0 .. 1.0 is partitioned
into K subintervals. K is chosen randomly between 4 and 25 for
each repetition of the test, along with the boundaries of the
subintervals (subject to the constraint that at least 2 of the
subintervals have a width of 0.001 or more). 5000 random
floating point numbers are generated. The counts of random
numbers falling into each subinterval are tallied and compared
with the expected counts, which are proportional to the widths
of the subintervals. The number of degrees of freedom for the
chi-square test is K-1.
4.c * Gap Test. The bounds of a range A .. B, with 0.0 <= A < B <=
1.0, are chosen randomly for each repetition of the test,
subject to the constraint that 0.2 <= B-A <= 0.6. Random
floating point numbers are generated until 5000 falling into
the range A .. B have been encountered. Each of these 5000 is
preceded by a "gap" (of length greater than or equal to 0) of
consecutive random numbers not falling into the range A .. B.
The counts of gaps of each length from 0 to 15, and of all
lengths greater than 15 lumped together, are tallied and
compared with the expected counts. Let P = B-A. The
probability that a gap has a length of L is (1-P) (L) · P for
L <= 15, while the probability that a gap has a length of 16
or more is (1-P) (16). The number of degrees of freedom for
the chi-square test is 16.
4.d * Permutation Test. 5000 tuples of 4 different random floating
point numbers are generated. (An entire 4-tuple is discarded
in the unlikely event that it contains any two exactly equal
components.) The counts of each of the 4! = 24 possible
relative orderings of the components of the 4-tuples are
tallied and compared with the expected counts. Each of the
possible relative orderings has an equal probability. The
number of degrees of freedom for the chi-square test is 23.
4.e * Increasing-Runs Test. Random floating point numbers are
generated until 5000 increasing runs have been observed. An
"increasing run" is a sequence of random numbers in strictly
increasing order; it is followed by a random number that is
strictly smaller than the preceding random number. (A run
under construction is entirely discarded in the unlikely event
that one random number is followed immediately by an exactly
equal random number.) The decreasing random number that
follows an increasing run is discarded and not included with
the next increasing run. The counts of increasing runs of each
length from 1 to 4, and of all lengths greater than 4 lumped
together, are tallied and compared with the expected counts.
The probability that an increasing run has a length of L is
1/L! - 1/(L+1)! for L <= 4, while the probability that an
increasing run has a length of 5 or more is 1/5!. The number
of degrees of freedom for the chi-square test is 4.
4.f * Decreasing-Runs Test. The test is similar to the Increasing
Runs Test, but with decreasing runs.
4.g * Maximum-of-t Test (with t = 5). 5000 tuples of 5 random
floating point numbers are generated. The maximum of the
components of each 5-tuple is determined and raised to the 5th
power. The uniformity of the resulting values over the range
0.0 .. 1.0 is tested as in the Proportional Distribution Test.
4.h Implementation Note: In the discrete random number test suite,
Numerics.Discrete_Random is instantiated as described below. The
generator is reset to a time-dependent state after each
instantiation. The test suite incorporates the following tests,
adapted from D. E. Knuth (op. cit.) and other sources. The given
number of degrees of freedom for the chi-square test is reduced by
any necessary combination of measurement categories with small
expected counts, as described above.
4.i * Equidistribution Test. In each repetition of the test, a
number R between 2 and 30 is chosen randomly, and
Numerics.Discrete_Random is instantiated with an integer
subtype whose range is 1 .. R. 5000 integers are generated
randomly from this range. The counts of occurrences of each
integer in the range are tallied and compared with the
expected counts, which have equal probabilities. The number of
degrees of freedom for the chi-square test is R-1.
4.j * Simplified Poker Test. Numerics.Discrete_Random is
instantiated once with an enumeration subtype representing the
13 denominations (Two through Ten, Jack, Queen, King, and Ace)
of an infinite deck of playing cards. 2000 "poker" hands
(5-tuples of values of this subtype) are generated randomly.
The counts of hands containing exactly K different
denominations (1 <= K <= 5) are tallied and compared with the
expected counts. The probability that a hand contains exactly
K different denominations is given by a formula in Knuth. The
number of degrees of freedom for the chi-square test is 4.
4.k * Coupon Collector's Test. Numerics.Discrete_Random is
instantiated in each repetition of the test with an integer
subtype whose range is 1 .. R, where R varies systematically
from 2 to 11. Integers are generated randomly from this range
until each value in the range has occurred, and the number K
of integers generated is recorded. This constitutes a "coupon
collector's segment" of length K. 2000 such segments are
generated. The counts of segments of each length from R to
R+29, and of all lengths greater than R+29 lumped together,
are tallied and compared with the expected counts. The
probability that a segment has any given length is given by
formulas in Knuth. The number of degrees of freedom for the
chi-square test is 30.
4.l * Craps Test (Lengths of Games). Numerics.Discrete_Random is
instantiated once with an integer subtype whose range is 1 ..
6 (representing the six numbers on a die). 5000 craps games
are played, and their lengths are recorded. (The length of a
craps game is the number of rolls of the pair of dice required
to produce a win or a loss. A game is won on the first roll if
the dice show 7 or 11; it is lost if they show 2, 3, or 12. If
the dice show some other sum on the first roll, it is called
the point, and the game is won if and only if the point is
rolled again before a 7 is rolled.) The counts of games of
each length from 1 to 18, and of all lengths greater than 18
lumped together, are tallied and compared with the expected
counts. For 2 <= S <= 12, let D (S) be the probability that a
roll of a pair of dice shows the sum S, and let Q (S)(L) = D
(S) · (1 - (D (S) + D (7))) (L-2) · (D (S) + D (7)). Then, the
probability that a game has a length of 1 is D (7) + D (11) +
D (2) + D (3) + D (12) and, for L > 1, the probability that a
game has a length of L is Q (4)(L) + Q (5)(L) + Q (6)(L) + Q
(8)(L) + Q (9)(L) + Q (10)(L). The number of degrees of
freedom for the chi-square test is 18.
4.m * Craps Test (Lengths of Passes). This test is similar to the
last, but enough craps games are played for 3000 losses to
occur. A string of wins followed by a loss is called a pass,
and its length is the number of wins preceding the loss. The
counts of passes of each length from 0 to 7, and of all
lengths greater than 7 lumped together, are tallied and
compared with the expected counts. For L >= 0, the probability
that a pass has a length of L is W (L) · (1-W), where W, the
probability that a game ends in a win, is 244.0/495.0. The
number of degrees of freedom for the chi-square test is 8.
4.n * Collision Test. Numerics.Discrete_Random is instantiated once
with an integer or enumeration type representing binary bits.
15 successive calls on the Random function are used to obtain
the bits of a 15-bit binary integer between 0 and 32767. 3000
such integers are generated, and the number of collisions
(integers previously generated) is counted and compared with
the expected count. A chi-square test is not used to assess
the number of collisions; rather, the limits on the number of
collisions, corresponding to the 2.5 and 97.5 percentage
points, are (from formulas in Knuth) 112 and 154. The test
passes if and only if the number of collisions is in this
range.
G.2.6 Accuracy Requirements for Complex Arithmetic
1 In the strict mode, the performance of Numerics.Generic_Complex_Types and
Numerics.Generic_Complex_Elementary_Functions shall be as specified here.
Implementation Requirements
2 When an exception is not raised, the result of evaluating a real function
of an instance CT of Numerics.Generic_Complex_Types (i.e., a function that
yields a value of subtype CT.Real'Base or CT.Imaginary) belongs to a result
interval defined as for a real elementary function (see G.2.4).
3 When an exception is not raised, each component of the result of
evaluating a complex function of such an instance, or of an instance of
Numerics.Generic_Complex_Elementary_Functions obtained by instantiating the
latter with CT (i.e., a function that yields a value of subtype CT.Complex),
also belongs to a result interval. The result intervals for the components of
the result are either defined by a maximum relative error bound or by a
maximum box error bound. When the result interval for the real (resp.,
imaginary) component is defined by maximum relative error, it is defined as
for that of a real function, relative to the exact value of the real (resp.,
imaginary) part of the result of the corresponding mathematical function. When
defined by maximum box error, the result interval for a component of the
result is the smallest model interval of CT.Real that contains all the values
of the corresponding part of f · (1.0 + d), where f is the exact complex value
of the corresponding mathematical function at the given parameter values, d is
complex, and |d| is less than or equal to the given maximum box error. The
function delivers a value that belongs to the result interval (or a value both
of whose components belong to their respective result intervals) when both
bounds of the result interval(s) belong to the safe range of CT.Real;
otherwise,
3.a Discussion: The maximum relative error could be specified
separately for each component, but we do not take advantage of
that freedom here.
3.b Discussion: Note that f · (1.0 + d) defines a small circular
region of the complex plane centered at f, and the result
intervals for the real and imaginary components of the result
define a small rectangular box containing that circle.
3.c Reason: Box error is used when the computation of the result risks
loss of significance in a component due to cancellation.
3.d Ramification: The components of a complex function that exhibits
bounded relative error in each component have to have the correct
sign. In contrast, one of the components of a complex function
that exhibits bounded box error may have the wrong sign, since the
dimensions of the box containing the result are proportional to
the modulus of the mathematical result and not to either component
of the mathematical result individually. Thus, for example, the
box containing the computed result of a complex function whose
mathematical result has a large modulus but lies very close to the
imaginary axis might well straddle that axis, allowing the real
component of the computed result to have the wrong sign. In this
case, the distance between the computed result and the
mathematical result is, nevertheless, a small fraction of the
modulus of the mathematical result.
4 * if CT.Real'Machine_Overflows is True, the function either delivers a
value that belongs to the result interval (or a value both of whose
components belong to their respective result intervals) or raises
Constraint_Error, signaling overflow;
5 * if CT.Real'Machine_Overflows is False, the result is implementation
defined.
5.a Implementation defined: The result of a complex arithmetic
operation or complex elementary function reference in overflow
situations, when the Machine_Overflows attribute of the
corresponding real type is False.
6/2 {AI95-00434-01} The error bounds for particular complex functions are
tabulated in table G-2. In the table, the error bound is given as the
coefficient of CT.Real'Model_Epsilon.
7/1 This paragraph was deleted.
Table G-2: Error Bounds for Particular Complex Functions
Function or Operator
Nature of
Result
Nature of
Bound
Error Bound
Modulus
real
max. rel. error
3.0
Argument
real
max. rel. error
4.0
Compose_From_Polar
complex
max. rel. error
3.0
"*" (both operands complex)
complex
max. box error
5.0
"/" (right operand complex)
complex
max. box error
13.0
Sqrt
complex
max. rel. error
6.0
Log
complex
max. box error
13.0
Exp (complex parameter)
complex
max. rel. error
7.0
Exp (imaginary parameter)
complex
max. rel. error
2.0
Sin, Cos, Sinh, and Cosh
complex
max. rel. error
11.0
Tan, Cot, Tanh, and Coth
complex
max. rel. error
35.0
inverse trigonometric
complex
max. rel. error
14.0
inverse hyperbolic
complex
max. rel. error
14.0
8 The maximum relative error given above applies throughout the domain of
the Compose_From_Polar function when the Cycle parameter is specified. When
the Cycle parameter is omitted, the maximum relative error applies only when
the absolute value of the parameter Argument is less than or equal to the
angle threshold (see G.2.4). For the Exp function, and for the forward
hyperbolic (resp., trigonometric) functions, the maximum relative error given
above likewise applies only when the absolute value of the imaginary (resp.,
real) component of the parameter X (or the absolute value of the parameter
itself, in the case of the Exp function with a parameter of pure-imaginary
type) is less than or equal to the angle threshold. For larger angles, the
accuracy is implementation defined.
8.a Implementation defined: The accuracy of certain complex arithmetic
operations and certain complex elementary functions for parameters
(or components thereof) beyond the angle threshold.
9 The prescribed results specified in G.1.2 for certain functions at
particular parameter values take precedence over the error bounds;
effectively, they narrow to a single value the result interval allowed by the
error bounds for a component of the result. Additional rules with a similar
effect are given below for certain inverse trigonometric and inverse
hyperbolic functions, at particular parameter values for which a component of
the mathematical result is transcendental. In each case, the accuracy rule,
which takes precedence over the error bounds, is that the result interval for
the stated result component is the model interval of CT.Real associated with
the component's exact mathematical value. The cases in question are as
follows:
10 * When the parameter X has the value zero, the real (resp., imaginary)
component of the result of the Arccot (resp., Arccoth) function is in
the model interval of CT.Real associated with the value PI/2.0.
11 * When the parameter X has the value one, the real component of the
result of the Arcsin function is in the model interval of CT.Real
associated with the value PI/2.0.
12 * When the parameter X has the value -1.0, the real component of the
result of the Arcsin (resp., Arccos) function is in the model interval
of CT.Real associated with the value -PI/2.0 (resp., PI).
12.a Discussion: It is possible to give many other prescribed results
in which a component of the parameter is restricted to a similar
model interval when the parameter X is appropriately restricted to
an easily testable portion of the domain. We follow the proposed
ISO/IEC standard for Generic_Complex_Elementary_Functions (for Ada
83) in not doing so, however.
13/2 {AI95-00434-01} The amount by which a component of the result of an
inverse trigonometric or inverse hyperbolic function is allowed to spill over
into a quadrant adjacent to the one corresponding to the principal branch, as
given in G.1.2, is limited. The rule is that the result belongs to the
smallest model interval of CT.Real that contains both boundaries of the
quadrant corresponding to the principal branch. This rule also takes
precedence over the maximum error bounds, effectively narrowing the result
interval allowed by them.
14 Finally, the results allowed by the error bounds are narrowed by one
further rule: The absolute value of each component of the result of the Exp
function, for a pure-imaginary parameter, never exceeds one.
Implementation Advice
15 The version of the Compose_From_Polar function without a Cycle parameter
should not be implemented by calling the corresponding version with a Cycle
parameter of 2.0*Numerics.Pi, since this will not provide the required
accuracy in some portions of the domain.
15.a.1/2 Implementation Advice: For complex arithmetic, the
Compose_From_Polar function without a Cycle parameter should not
be implemented by calling Compose_From_Polar with a Cycle
parameter.
Wording Changes from Ada 83
15.a The semantics of Numerics.Generic_Complex_Types and
Numerics.Generic_Complex_Elementary_Functions differs from
Generic_Complex_Types and Generic_Complex_Elementary_Functions as
defined in ISO/IEC CDs 13813 and 13814 (for Ada 83) in ways
analogous to those identified for the elementary functions in
G.2.4. In addition, we do not generally specify the signs of zero
results (or result components), although those proposed standards
do.
G.3 Vector and Matrix Manipulation
1/2 {AI95-00296-01} Types and operations for the manipulation of real vectors
and matrices are provided in Generic_Real_Arrays, which is defined in G.3.1.
Types and operations for the manipulation of complex vectors and matrices are
provided in Generic_Complex_Arrays, which is defined in G.3.2. Both of these
library units are generic children of the predefined package Numerics (see
A.5). Nongeneric equivalents of these packages for each of the predefined
floating point types are also provided as children of Numerics.
1.a/2 Discussion: Vector and matrix manipulation is defined in the
Numerics Annex, rather than in the core, because it is considered
to be a specialized need of (some) numeric applications.
1.b/2 These packages provide facilities that are similar to and replace
those found in ISO/IEC 13813:1998 Information technology -
Programming languages - Generic packages of real and complex type
declarations and basic operations for Ada (including vector and
matrix types). (The other facilities provided by that Standard
were already provided in Ada 95.) In addition to the main
facilities of that Standard, these packages also include
subprograms for the solution of linear equations, matrix
inversion, determinants, and the determination of the eigenvalues
and eigenvectors of real symmetric matrices and Hermitian
matrices.
Extensions to Ada 95
1.c/3 {AI95-00296-01} {AI05-0299-1} This subclause It just provides an
introduction to the following subclauses.
G.3.1 Real Vectors and Matrices
Static Semantics
1/2 {AI95-00296-01} {AI95-00418-01} The generic library package
Numerics.Generic_Real_Arrays has the following declaration:
2/2 generic
type Real is digits <>;
package Ada.Numerics.Generic_Real_Arrays is
pragma Pure(Generic_Real_Arrays);
3/2 -- Types
4/2 type Real_Vector is array (Integer range <>) of Real'Base;
type Real_Matrix is array (Integer range <>, Integer range <>)
of Real'Base;
5/2 -- Subprograms for Real_Vector types
6/2 -- Real_Vector arithmetic operations
7/2 function "+" (Right : Real_Vector) return Real_Vector;
function "-" (Right : Real_Vector) return Real_Vector;
function "abs" (Right : Real_Vector) return Real_Vector;
8/2 function "+" (Left, Right : Real_Vector) return Real_Vector;
function "-" (Left, Right : Real_Vector) return Real_Vector;
9/2 function "*" (Left, Right : Real_Vector) return Real'Base;
10/2 function "abs" (Right : Real_Vector) return Real'Base;
11/2 -- Real_Vector scaling operations
12/2 function "*" (Left : Real'Base; Right : Real_Vector)
return Real_Vector;
function "*" (Left : Real_Vector; Right : Real'Base)
return Real_Vector;
function "/" (Left : Real_Vector; Right : Real'Base)
return Real_Vector;
13/2 -- Other Real_Vector operations
14/2 function Unit_Vector (Index : Integer;
Order : Positive;
First : Integer := 1) return Real_Vector;
15/2 -- Subprograms for Real_Matrix types
16/2 -- Real_Matrix arithmetic operations
17/2 function "+" (Right : Real_Matrix) return Real_Matrix;
function "-" (Right : Real_Matrix) return Real_Matrix;
function "abs" (Right : Real_Matrix) return Real_Matrix;
function Transpose (X : Real_Matrix) return Real_Matrix;
18/2 function "+" (Left, Right : Real_Matrix) return Real_Matrix;
function "-" (Left, Right : Real_Matrix) return Real_Matrix;
function "*" (Left, Right : Real_Matrix) return Real_Matrix;
19/2 function "*" (Left, Right : Real_Vector) return Real_Matrix;
20/2 function "*" (Left : Real_Vector; Right : Real_Matrix)
return Real_Vector;
function "*" (Left : Real_Matrix; Right : Real_Vector)
return Real_Vector;
21/2 -- Real_Matrix scaling operations
22/2 function "*" (Left : Real'Base; Right : Real_Matrix)
return Real_Matrix;
function "*" (Left : Real_Matrix; Right : Real'Base)
return Real_Matrix;
function "/" (Left : Real_Matrix; Right : Real'Base)
return Real_Matrix;
23/2 -- Real_Matrix inversion and related operations
24/2 function Solve
(A : Real_Matrix; X : Real_Vector) return Real_Vector;
function Solve (A, X : Real_Matrix) return Real_Matrix;
function Inverse (A : Real_Matrix) return Real_Matrix;
function Determinant (A : Real_Matrix) return Real'Base;
25/2 -- Eigenvalues and vectors of a real symmetric matrix
26/2 function Eigenvalues (A : Real_Matrix) return Real_Vector;
27/2 procedure Eigensystem (A : in Real_Matrix;
Values : out Real_Vector;
Vectors : out Real_Matrix);
28/2 -- Other Real_Matrix operations
29/2 function Unit_Matrix (Order : Positive;
First_1, First_2 : Integer := 1)
return Real_Matrix;
30/2 end Ada.Numerics.Generic_Real_Arrays;
31/2 {AI95-00296-01} The library package Numerics.Real_Arrays is declared pure
and defines the same types and subprograms as Numerics.Generic_Real_Arrays,
except that the predefined type Float is systematically substituted for
Real'Base throughout. Nongeneric equivalents for each of the other predefined
floating point types are defined similarly, with the names
Numerics.Short_Real_Arrays, Numerics.Long_Real_Arrays, etc.
31.a/2 Reason: The nongeneric equivalents are provided to allow the
programmer to construct simple mathematical applications without
being required to understand and use generics, and to be
consistent with other Numerics packages.
32/2 {AI95-00296-01} Two types are defined and exported by
Numerics.Generic_Real_Arrays. The composite type Real_Vector is provided to
represent a vector with components of type Real; it is defined as an
unconstrained, one-dimensional array with an index of type Integer. The
composite type Real_Matrix is provided to represent a matrix with components
of type Real; it is defined as an unconstrained, two-dimensional array with
indices of type Integer.
33/2 {AI95-00296-01} The effect of the various subprograms is as described
below. In most cases the subprograms are described in terms of corresponding
scalar operations of the type Real; any exception raised by those operations
is propagated by the array operation. Moreover, the accuracy of the result for
each individual component is as defined for the scalar operation unless stated
otherwise.
34/2 {AI95-00296-01} In the case of those operations which are defined to
involve an inner product, Constraint_Error may be raised if an intermediate
result is outside the range of Real'Base even though the mathematical final
result would not be.
35/2 function "+" (Right : Real_Vector) return Real_Vector;
function "-" (Right : Real_Vector) return Real_Vector;
function "abs" (Right : Real_Vector) return Real_Vector;
36/2 {AI95-00296-01} Each operation returns the result of applying the
corresponding operation of the type Real to each component of
Right. The index range of the result is Right'Range.
37/2 function "+" (Left, Right : Real_Vector) return Real_Vector;
function "-" (Left, Right : Real_Vector) return Real_Vector;
38/2 {AI95-00296-01} Each operation returns the result of applying the
corresponding operation of the type Real to each component of Left
and the matching component of Right. The index range of the result
is Left'Range. Constraint_Error is raised if Left'Length is not
equal to Right'Length.
39/2 function "*" (Left, Right : Real_Vector) return Real'Base;
40/2 {AI95-00296-01} This operation returns the inner product of Left
and Right. Constraint_Error is raised if Left'Length is not equal
to Right'Length. This operation involves an inner product.
41/2 function "abs" (Right : Real_Vector) return Real'Base;
42/2 {AI95-00418-01} This operation returns the L2-norm of Right (the
square root of the inner product of the vector with itself).
42.a/2 Discussion: Normalization of vectors is a frequent enough
operation that it is useful to provide the norm as a basic
operation. Furthermore, implementing the norm is not entirely
straightforward, because the inner product might overflow while
the final norm does not. An implementation cannot merely return
Sqrt (X * X), it has to cope with a possible overflow of the inner
product.
42.b/2 Implementation Note: While the definition is given in terms of an
inner product, the norm doesn't "involve an inner product" in the
technical sense. The reason is that it has accuracy requirements
substantially different from those applicable to inner products;
and that cancellations cannot occur, because all the terms are
positive, so there is no possibility of intermediate overflow.
43/2 function "*" (Left : Real'Base; Right : Real_Vector) return Real_Vector;
44/2 {AI95-00296-01} This operation returns the result of multiplying
each component of Right by the scalar Left using the "*" operation
of the type Real. The index range of the result is Right'Range.
45/2 function "*" (Left : Real_Vector; Right : Real'Base) return Real_Vector;
function "/" (Left : Real_Vector; Right : Real'Base) return Real_Vector;
46/2 {AI95-00296-01} Each operation returns the result of applying the
corresponding operation of the type Real to each component of Left
and to the scalar Right. The index range of the result is
Left'Range.
47/2 function Unit_Vector (Index : Integer;
Order : Positive;
First : Integer := 1) return Real_Vector;
48/2 {AI95-00296-01} This function returns a unit vector with Order
components and a lower bound of First. All components are set to
0.0 except for the Index component which is set to 1.0.
Constraint_Error is raised if Index < First, Index > First + Order
- 1 or if First + Order - 1 > Integer'Last.
49/2 function "+" (Right : Real_Matrix) return Real_Matrix;
function "-" (Right : Real_Matrix) return Real_Matrix;
function "abs" (Right : Real_Matrix) return Real_Matrix;
50/2 {AI95-00296-01} Each operation returns the result of applying the
corresponding operation of the type Real to each component of
Right. The index ranges of the result are those of Right.
51/2 function Transpose (X : Real_Matrix) return Real_Matrix;
52/2 {AI95-00296-01} This function returns the transpose of a matrix X.
The first and second index ranges of the result are X'Range(2) and
X'Range(1) respectively.
53/2 function "+" (Left, Right : Real_Matrix) return Real_Matrix;
function "-" (Left, Right : Real_Matrix) return Real_Matrix;
54/2 {AI95-00296-01} Each operation returns the result of applying the
corresponding operation of the type Real to each component of Left
and the matching component of Right. The index ranges of the
result are those of Left. Constraint_Error is raised if
Left'Length(1) is not equal to Right'Length(1) or Left'Length(2)
is not equal to Right'Length(2).
55/2 function "*" (Left, Right : Real_Matrix) return Real_Matrix;
56/2 {AI95-00296-01} This operation provides the standard mathematical
operation for matrix multiplication. The first and second index
ranges of the result are Left'Range(1) and Right'Range(2)
respectively. Constraint_Error is raised if Left'Length(2) is not
equal to Right'Length(1). This operation involves inner products.
57/2 function "*" (Left, Right : Real_Vector) return Real_Matrix;
58/2 {AI95-00296-01} This operation returns the outer product of a
(column) vector Left by a (row) vector Right using the operation
"*" of the type Real for computing the individual components. The
first and second index ranges of the result are Left'Range and
Right'Range respectively.
59/2 function "*" (Left : Real_Vector; Right : Real_Matrix) return Real_Vector;
60/2 {AI95-00296-01} This operation provides the standard mathematical
operation for multiplication of a (row) vector Left by a matrix
Right. The index range of the (row) vector result is
Right'Range(2). Constraint_Error is raised if Left'Length is not
equal to Right'Length(1). This operation involves inner products.
61/2 function "*" (Left : Real_Matrix; Right : Real_Vector) return Real_Vector;
62/2 {AI95-00296-01} This operation provides the standard mathematical
operation for multiplication of a matrix Left by a (column) vector
Right. The index range of the (column) vector result is
Left'Range(1). Constraint_Error is raised if Left'Length(2) is not
equal to Right'Length. This operation involves inner products.
63/2 function "*" (Left : Real'Base; Right : Real_Matrix) return Real_Matrix;
64/2 {AI95-00296-01} This operation returns the result of multiplying
each component of Right by the scalar Left using the "*" operation
of the type Real. The index ranges of the result are those of
Right.
65/2 function "*" (Left : Real_Matrix; Right : Real'Base) return Real_Matrix;
function "/" (Left : Real_Matrix; Right : Real'Base) return Real_Matrix;
66/2 {AI95-00296-01} Each operation returns the result of applying the
corresponding operation of the type Real to each component of Left
and to the scalar Right. The index ranges of the result are those
of Left.
67/2 function Solve (A : Real_Matrix; X : Real_Vector) return Real_Vector;
68/2 {AI95-00296-01} This function returns a vector Y such that X is
(nearly) equal to A * Y. This is the standard mathematical
operation for solving a single set of linear equations. The index
range of the result is A'Range(2). Constraint_Error is raised if
A'Length(1), A'Length(2), and X'Length are not equal.
Constraint_Error is raised if the matrix A is ill-conditioned.
68.a/2 Discussion: The text says that Y is such that "X is (nearly) equal
to A * Y" rather than "X is equal to A * Y" because rounding
errors may mean that there is no value of Y such that X is exactly
equal to A * Y. On the other hand it does not mean that any old
rough value will do. The algorithm given under
Implementation Advice should be followed.
68.b/2 The requirement to raise Constraint_Error if the matrix is
ill-conditioned is really a reflection of what will happen if the
matrix is ill-conditioned. See Implementation Advice. We do not
make any attempt to define ill-conditioned formally.
68.c/2 These remarks apply to all versions of Solve and Inverse.
69/2 function Solve (A, X : Real_Matrix) return Real_Matrix;
70/2 {AI95-00296-01} This function returns a matrix Y such that X is
(nearly) equal to A * Y. This is the standard mathematical
operation for solving several sets of linear equations. The index
ranges of the result are A'Range(2) and X'Range(2).
Constraint_Error is raised if A'Length(1), A'Length(2), and
X'Length(1) are not equal. Constraint_Error is raised if the
matrix A is ill-conditioned.
71/2 function Inverse (A : Real_Matrix) return Real_Matrix;
72/2 {AI95-00296-01} This function returns a matrix B such that A * B
is (nearly) equal to the unit matrix. The index ranges of the
result are A'Range(2) and A'Range(1). Constraint_Error is raised
if A'Length(1) is not equal to A'Length(2). Constraint_Error is
raised if the matrix A is ill-conditioned.
73/2 function Determinant (A : Real_Matrix) return Real'Base;
74/2 {AI95-00296-01} This function returns the determinant of the
matrix A. Constraint_Error is raised if A'Length(1) is not equal
to A'Length(2).
75/2 function Eigenvalues(A : Real_Matrix) return Real_Vector;
76/2 {AI95-00296-01} This function returns the eigenvalues of the
symmetric matrix A as a vector sorted into order with the largest
first. Constraint_Error is raised if A'Length(1) is not equal to
A'Length(2). The index range of the result is A'Range(1).
Argument_Error is raised if the matrix A is not symmetric.
77/2 procedure Eigensystem(A : in Real_Matrix;
Values : out Real_Vector;
Vectors : out Real_Matrix);
78/3 {AI95-00296-01} {AI05-0047-1} This procedure computes both the
eigenvalues and eigenvectors of the symmetric matrix A. The out
parameter Values is the same as that obtained by calling the
function Eigenvalues. The out parameter Vectors is a matrix whose
columns are the eigenvectors of the matrix A. The order of the
columns corresponds to the order of the eigenvalues. The
eigenvectors are normalized and mutually orthogonal (they are
orthonormal), including when there are repeated eigenvalues.
Constraint_Error is raised if A'Length(1) is not equal to
A'Length(2), or if Values'Range is not equal to A'Range(1), or if
the index ranges of the parameter Vectors are not equal to those
of A. Argument_Error is raised if the matrix A is not symmetric.
Constraint_Error is also raised in implementation-defined
circumstances if the algorithm used does not converge quickly
enough.
78.a/3 Ramification: {AI05-0047-1} There is no requirement on the
absolute direction of the returned eigenvectors. Thus they might
be multiplied by -1. It is only the ratios of the components that
matter. This is standard practice.
79/2 function Unit_Matrix (Order : Positive;
First_1, First_2 : Integer := 1) return Real_Matrix;
80/2 {AI95-00296-01} This function returns a square unit matrix with
Order**2 components and lower bounds of First_1 and First_2 (for
the first and second index ranges respectively). All components
are set to 0.0 except for the main diagonal, whose components are
set to 1.0. Constraint_Error is raised if First_1 + Order - 1 >
Integer'Last or First_2 + Order - 1 > Integer'Last.
Implementation Requirements
81/2 {AI95-00296-01} Accuracy requirements for the subprograms Solve, Inverse,
Determinant, Eigenvalues and Eigensystem are implementation defined.
81.a/2 Implementation defined: The accuracy requirements for the
subprograms Solve, Inverse, Determinant, Eigenvalues and
Eigensystem for type Real_Matrix.
82/2 For operations not involving an inner product, the accuracy requirements
are those of the corresponding operations of the type Real in both the strict
mode and the relaxed mode (see G.2).
83/2 For operations involving an inner product, no requirements are specified
in the relaxed mode. In the strict mode the modulus of the absolute error of
the inner product X*Y shall not exceed g*abs(X)*abs(Y) where g is defined as
84/2 g = X'Length * Real'Machine_Radix**(1 - Real'Model_Mantissa)
85/2 {AI95-00418-01} For the L2-norm, no accuracy requirements are specified
in the relaxed mode. In the strict mode the relative error on the norm shall
not exceed g / 2.0 + 3.0 * Real'Model_Epsilon where g is defined as above.
85.a/2 Reason: This is simply the combination of the error on the inner
product with the error on Sqrt. A first order computation would
lead to 2.0 * Real'Model_Epsilon above, but we are adding an extra
Real'Model_Epsilon to account for higher order effects.
Documentation Requirements
86/2 {AI95-00296-01} Implementations shall document any techniques used to
reduce cancellation errors such as extended precision arithmetic.
86.a/2 Documentation Requirement: Any techniques used to reduce
cancellation errors in Numerics.Generic_Real_Arrays shall be
documented.
86.b/2 Implementation Note: The above accuracy requirement is met by the
canonical implementation of the inner product by multiplication
and addition using the corresponding operations of type Real'Base
and performing the cumulative addition using ascending indices.
Note however, that some hardware provides special operations for
the computation of the inner product and although these may be
fast they may not meet the accuracy requirement specified. See
Accuracy and Stability of Numerical Algorithms By N J Higham (ISBN
0-89871-355-2), Section 3.1.
86.c/3 {AI05-0047-1} Note moreover that the componentwise accuracy
requirements are not met by subcubic methods for matrix
multiplication such as that devised by Strassen. These methods,
which are typically used for the fast multiplication of very large
matrices (e.g. order more than a few thousands), have normwise
accuracy properties. If it is desired to use such methods, then
distinct subprograms should be provided (perhaps in a child
package). See Section 22.2.2 in the above reference.
Implementation Permissions
87/2 {AI95-00296-01} The nongeneric equivalent packages may, but need not, be
actual instantiations of the generic package for the appropriate predefined
type.
Implementation Advice
88/3 {AI95-00296-01} {AI05-0264-1} Implementations should implement the Solve
and Inverse functions using established techniques such as LU decomposition
with row interchanges followed by back and forward substitution.
Implementations are recommended to refine the result by performing an
iteration on the residuals; if this is done, then it should be documented.
88.a/2 Implementation Advice: Solve and Inverse for
Numerics.Generic_Real_Arrays should be implemented using
established techniques such as LU decomposition and the result
should be refined by an iteration on the residuals.
89/2 It is not the intention that any special provision should be made to
determine whether a matrix is ill-conditioned or not. The naturally occurring
overflow (including division by zero) which will result from executing these
functions with an ill-conditioned matrix and thus raise Constraint_Error is
sufficient.
89.a/2 Discussion: There isn't any advice for the implementation to
document with this paragraph.
90/2 The test that a matrix is symmetric should be performed by using the
equality operator to compare the relevant components.
90.a/2 Implementation Advice: The equality operator should be used to
test that a matrix in Numerics.Generic_Real_Arrays is symmetric.
91/3 {AI05-0047-1} An implementation should minimize the circumstances under
which the algorithm used for Eigenvalues and Eigensystem fails to converge.
91.a.1/3 Implementation Advice: An implementation should minimize the
circumstances under which the algorithm used for
Numerics.Generic_Real_Arrays.Eigenvalues and
Numerics.Generic_Real_Arrays.Eigensystem fails to converge.
91.a/3 Implementation Note: J. H. Wilkinson is the acknowledged expert in
this area. See for example Wilkinson, J. H., and Reinsch, C. ,
Linear Algebra , vol II of Handbook for Automatic Computation,
Springer-Verlag, or Wilkinson, J. H., The Algebraic Eigenvalue
Problem, Oxford University Press.
Extensions to Ada 95
91.b/2 {AI95-00296-01} The package Numerics.Generic_Real_Arrays and its
nongeneric equivalents are new.
Wording Changes from Ada 2005
91.c/3 {AI05-0047-1} Correction: Corrected various accuracy and
definition issues.
G.3.2 Complex Vectors and Matrices
Static Semantics
1/2 {AI95-00296-01} The generic library package
Numerics.Generic_Complex_Arrays has the following declaration:
2/2 with Ada.Numerics.Generic_Real_Arrays, Ada.Numerics.Generic_Complex_Types;
generic
with package Real_Arrays is new
Ada.Numerics.Generic_Real_Arrays (<>);
use Real_Arrays;
with package Complex_Types is new
Ada.Numerics.Generic_Complex_Types (Real);
use Complex_Types;
package Ada.Numerics.Generic_Complex_Arrays is
pragma Pure(Generic_Complex_Arrays);
3/2 -- Types
4/2 type Complex_Vector is array (Integer range <>) of Complex;
type Complex_Matrix is array (Integer range <>,
Integer range <>) of Complex;
5/2 -- Subprograms for Complex_Vector types
6/2 -- Complex_Vector selection, conversion and composition operations
7/2 function Re (X : Complex_Vector) return Real_Vector;
function Im (X : Complex_Vector) return Real_Vector;
8/2 procedure Set_Re (X : in out Complex_Vector;
Re : in Real_Vector);
procedure Set_Im (X : in out Complex_Vector;
Im : in Real_Vector);
9/2 function Compose_From_Cartesian (Re : Real_Vector)
return Complex_Vector;
function Compose_From_Cartesian (Re, Im : Real_Vector)
return Complex_Vector;
10/2 function Modulus (X : Complex_Vector) return Real_Vector;
function "abs" (Right : Complex_Vector) return Real_Vector
renames Modulus;
function Argument (X : Complex_Vector) return Real_Vector;
function Argument (X : Complex_Vector;
Cycle : Real'Base) return Real_Vector;
11/2 function Compose_From_Polar (Modulus, Argument : Real_Vector)
return Complex_Vector;
function Compose_From_Polar (Modulus, Argument : Real_Vector;
Cycle : Real'Base)
return Complex_Vector;
12/2 -- Complex_Vector arithmetic operations
13/2 function "+" (Right : Complex_Vector) return Complex_Vector;
function "-" (Right : Complex_Vector) return Complex_Vector;
function Conjugate (X : Complex_Vector) return Complex_Vector;
14/2 function "+" (Left, Right : Complex_Vector) return Complex_Vector;
function "-" (Left, Right : Complex_Vector) return Complex_Vector;
15/2 function "*" (Left, Right : Complex_Vector) return Complex;
16/3 {AI05-0047-1}
function "abs" (Right : Complex_Vector) return Real'Base;
17/2 -- Mixed Real_Vector and Complex_Vector arithmetic operations
18/2 function "+" (Left : Real_Vector;
Right : Complex_Vector) return Complex_Vector;
function "+" (Left : Complex_Vector;
Right : Real_Vector) return Complex_Vector;
function "-" (Left : Real_Vector;
Right : Complex_Vector) return Complex_Vector;
function "-" (Left : Complex_Vector;
Right : Real_Vector) return Complex_Vector;
19/2 function "*" (Left : Real_Vector; Right : Complex_Vector)
return Complex;
function "*" (Left : Complex_Vector; Right : Real_Vector)
return Complex;
20/2 -- Complex_Vector scaling operations
21/2 function "*" (Left : Complex;
Right : Complex_Vector) return Complex_Vector;
function "*" (Left : Complex_Vector;
Right : Complex) return Complex_Vector;
function "/" (Left : Complex_Vector;
Right : Complex) return Complex_Vector;
22/2 function "*" (Left : Real'Base;
Right : Complex_Vector) return Complex_Vector;
function "*" (Left : Complex_Vector;
Right : Real'Base) return Complex_Vector;
function "/" (Left : Complex_Vector;
Right : Real'Base) return Complex_Vector;
23/2 -- Other Complex_Vector operations
24/2 function Unit_Vector (Index : Integer;
Order : Positive;
First : Integer := 1) return Complex_Vector;
25/2 -- Subprograms for Complex_Matrix types
26/2 -- Complex_Matrix selection, conversion and composition operations
27/2 function Re (X : Complex_Matrix) return Real_Matrix;
function Im (X : Complex_Matrix) return Real_Matrix;
28/2 procedure Set_Re (X : in out Complex_Matrix;
Re : in Real_Matrix);
procedure Set_Im (X : in out Complex_Matrix;
Im : in Real_Matrix);
29/2 function Compose_From_Cartesian (Re : Real_Matrix)
return Complex_Matrix;
function Compose_From_Cartesian (Re, Im : Real_Matrix)
return Complex_Matrix;
30/2 function Modulus (X : Complex_Matrix) return Real_Matrix;
function "abs" (Right : Complex_Matrix) return Real_Matrix
renames Modulus;
31/2 function Argument (X : Complex_Matrix) return Real_Matrix;
function Argument (X : Complex_Matrix;
Cycle : Real'Base) return Real_Matrix;
32/2 function Compose_From_Polar (Modulus, Argument : Real_Matrix)
return Complex_Matrix;
function Compose_From_Polar (Modulus, Argument : Real_Matrix;
Cycle : Real'Base)
return Complex_Matrix;
33/2 -- Complex_Matrix arithmetic operations
34/2 function "+" (Right : Complex_Matrix) return Complex_Matrix;
function "-" (Right : Complex_Matrix) return Complex_Matrix;
function Conjugate (X : Complex_Matrix) return Complex_Matrix;
function Transpose (X : Complex_Matrix) return Complex_Matrix;
35/2 function "+" (Left, Right : Complex_Matrix) return Complex_Matrix;
function "-" (Left, Right : Complex_Matrix) return Complex_Matrix;
function "*" (Left, Right : Complex_Matrix) return Complex_Matrix;
36/2 function "*" (Left, Right : Complex_Vector) return Complex_Matrix;
37/2 function "*" (Left : Complex_Vector;
Right : Complex_Matrix) return Complex_Vector;
function "*" (Left : Complex_Matrix;
Right : Complex_Vector) return Complex_Vector;
38/2 -- Mixed Real_Matrix and Complex_Matrix arithmetic operations
39/2 function "+" (Left : Real_Matrix;
Right : Complex_Matrix) return Complex_Matrix;
function "+" (Left : Complex_Matrix;
Right : Real_Matrix) return Complex_Matrix;
function "-" (Left : Real_Matrix;
Right : Complex_Matrix) return Complex_Matrix;
function "-" (Left : Complex_Matrix;
Right : Real_Matrix) return Complex_Matrix;
function "*" (Left : Real_Matrix;
Right : Complex_Matrix) return Complex_Matrix;
function "*" (Left : Complex_Matrix;
Right : Real_Matrix) return Complex_Matrix;
40/2 function "*" (Left : Real_Vector;
Right : Complex_Vector) return Complex_Matrix;
function "*" (Left : Complex_Vector;
Right : Real_Vector) return Complex_Matrix;
41/2 function "*" (Left : Real_Vector;
Right : Complex_Matrix) return Complex_Vector;
function "*" (Left : Complex_Vector;
Right : Real_Matrix) return Complex_Vector;
function "*" (Left : Real_Matrix;
Right : Complex_Vector) return Complex_Vector;
function "*" (Left : Complex_Matrix;
Right : Real_Vector) return Complex_Vector;
42/2 -- Complex_Matrix scaling operations
43/2 function "*" (Left : Complex;
Right : Complex_Matrix) return Complex_Matrix;
function "*" (Left : Complex_Matrix;
Right : Complex) return Complex_Matrix;
function "/" (Left : Complex_Matrix;
Right : Complex) return Complex_Matrix;
44/2 function "*" (Left : Real'Base;
Right : Complex_Matrix) return Complex_Matrix;
function "*" (Left : Complex_Matrix;
Right : Real'Base) return Complex_Matrix;
function "/" (Left : Complex_Matrix;
Right : Real'Base) return Complex_Matrix;
45/2 -- Complex_Matrix inversion and related operations
46/2 function Solve (A : Complex_Matrix; X : Complex_Vector)
return Complex_Vector;
function Solve (A, X : Complex_Matrix) return Complex_Matrix;
function Inverse (A : Complex_Matrix) return Complex_Matrix;
function Determinant (A : Complex_Matrix) return Complex;
47/2 -- Eigenvalues and vectors of a Hermitian matrix
48/2 function Eigenvalues(A : Complex_Matrix) return Real_Vector;
49/2 procedure Eigensystem(A : in Complex_Matrix;
Values : out Real_Vector;
Vectors : out Complex_Matrix);
50/2 -- Other Complex_Matrix operations
51/2 function Unit_Matrix (Order : Positive;
First_1, First_2 : Integer := 1)
return Complex_Matrix;
52/2 end Ada.Numerics.Generic_Complex_Arrays;
53/2 {AI95-00296-01} The library package Numerics.Complex_Arrays is declared
pure and defines the same types and subprograms as
Numerics.Generic_Complex_Arrays, except that the predefined type Float is
systematically substituted for Real'Base, and the Real_Vector and Real_Matrix
types exported by Numerics.Real_Arrays are systematically substituted for
Real_Vector and Real_Matrix, and the Complex type exported by
Numerics.Complex_Types is systematically substituted for Complex, throughout.
Nongeneric equivalents for each of the other predefined floating point types
are defined similarly, with the names Numerics.Short_Complex_Arrays,
Numerics.Long_Complex_Arrays, etc.
54/2 {AI95-00296-01} Two types are defined and exported by
Numerics.Generic_Complex_Arrays. The composite type Complex_Vector is provided
to represent a vector with components of type Complex; it is defined as an
unconstrained one-dimensional array with an index of type Integer. The
composite type Complex_Matrix is provided to represent a matrix with
components of type Complex; it is defined as an unconstrained, two-dimensional
array with indices of type Integer.
55/2 {AI95-00296-01} The effect of the various subprograms is as described
below. In many cases they are described in terms of corresponding scalar
operations in Numerics.Generic_Complex_Types. Any exception raised by those
operations is propagated by the array subprogram. Moreover, any constraints on
the parameters and the accuracy of the result for each individual component
are as defined for the scalar operation.
56/2 {AI95-00296-01} In the case of those operations which are defined to
involve an inner product, Constraint_Error may be raised if an intermediate
result has a component outside the range of Real'Base even though the final
mathematical result would not.
56.a/3 Discussion: {AI05-0047-1} An inner product never involves implicit
complex conjugation. If the product of a vector with the conjugate
of another (or the same) vector is required, then this has to be
stated explicitly by writing for example X * Conjugate(Y). This
mimics the usual mathematical notation.
57/2 function Re (X : Complex_Vector) return Real_Vector;
function Im (X : Complex_Vector) return Real_Vector;
58/2 {AI95-00296-01} Each function returns a vector of the specified
Cartesian components of X. The index range of the result is
X'Range.
59/2 procedure Set_Re (X : in out Complex_Vector; Re : in Real_Vector);
procedure Set_Im (X : in out Complex_Vector; Im : in Real_Vector);
60/2 {AI95-00296-01} Each procedure replaces the specified (Cartesian)
component of each of the components of X by the value of the
matching component of Re or Im; the other (Cartesian) component of
each of the components is unchanged. Constraint_Error is raised if
X'Length is not equal to Re'Length or Im'Length.
61/2 function Compose_From_Cartesian (Re : Real_Vector)
return Complex_Vector;
function Compose_From_Cartesian (Re, Im : Real_Vector)
return Complex_Vector;
62/2 {AI95-00296-01} Each function constructs a vector of Complex
results (in Cartesian representation) formed from given vectors of
Cartesian components; when only the real components are given,
imaginary components of zero are assumed. The index range of the
result is Re'Range. Constraint_Error is raised if Re'Length is not
equal to Im'Length.
63/2 function Modulus (X : Complex_Vector) return Real_Vector;
function "abs" (Right : Complex_Vector) return Real_Vector
renames Modulus;
function Argument (X : Complex_Vector) return Real_Vector;
function Argument (X : Complex_Vector;
Cycle : Real'Base) return Real_Vector;
64/2 {AI95-00296-01} Each function calculates and returns a vector of
the specified polar components of X or Right using the
corresponding function in numerics.generic_complex_types. The
index range of the result is X'Range or Right'Range.
65/2 function Compose_From_Polar (Modulus, Argument : Real_Vector)
return Complex_Vector;
function Compose_From_Polar (Modulus, Argument : Real_Vector;
Cycle : Real'Base)
return Complex_Vector;
66/2 {AI95-00296-01} Each function constructs a vector of Complex
results (in Cartesian representation) formed from given vectors of
polar components using the corresponding function in numerics.-
generic_complex_types on matching components of Modulus and
Argument. The index range of the result is Modulus'Range.
Constraint_Error is raised if Modulus'Length is not equal to
Argument'Length.
67/2 function "+" (Right : Complex_Vector) return Complex_Vector;
function "-" (Right : Complex_Vector) return Complex_Vector;
68/2 {AI95-00296-01} Each operation returns the result of applying the
corresponding operation in numerics.generic_complex_types to each
component of Right. The index range of the result is Right'Range.
69/2 function Conjugate (X : Complex_Vector) return Complex_Vector;
70/2 {AI95-00296-01} This function returns the result of applying the
appropriate function Conjugate in numerics.generic_complex_types
to each component of X. The index range of the result is X'Range.
71/2 function "+" (Left, Right : Complex_Vector) return Complex_Vector;
function "-" (Left, Right : Complex_Vector) return Complex_Vector;
72/2 {AI95-00296-01} Each operation returns the result of applying the
corresponding operation in numerics.generic_complex_types to each
component of Left and the matching component of Right. The index
range of the result is Left'Range. Constraint_Error is raised if
Left'Length is not equal to Right'Length.
73/2 function "*" (Left, Right : Complex_Vector) return Complex;
74/2 {AI95-00296-01} This operation returns the inner product of Left
and Right. Constraint_Error is raised if Left'Length is not equal
to Right'Length. This operation involves an inner product.
75/3 {AI05-0047-1} function "abs" (Right : Complex_Vector) return Real'Base;
76/2 {AI95-00418-01} This operation returns the Hermitian L2-norm of
Right (the square root of the inner product of the vector with its
conjugate).
76.a/2 Implementation Note: While the definition is given in terms of an
inner product, the norm doesn't "involve an inner product" in the
technical sense. The reason is that it has accuracy requirements
substantially different from those applicable to inner products;
and that cancellations cannot occur, because all the terms are
positive, so there is no possibility of intermediate overflow.
77/2 function "+" (Left : Real_Vector;
Right : Complex_Vector) return Complex_Vector;
function "+" (Left : Complex_Vector;
Right : Real_Vector) return Complex_Vector;
function "-" (Left : Real_Vector;
Right : Complex_Vector) return Complex_Vector;
function "-" (Left : Complex_Vector;
Right : Real_Vector) return Complex_Vector;
78/2 {AI95-00296-01} Each operation returns the result of applying the
corresponding operation in numerics.generic_complex_types to each
component of Left and the matching component of Right. The index
range of the result is Left'Range. Constraint_Error is raised if
Left'Length is not equal to Right'Length.
79/2 function "*" (Left : Real_Vector; Right : Complex_Vector) return Complex;
function "*" (Left : Complex_Vector; Right : Real_Vector) return Complex;
80/2 {AI95-00296-01} Each operation returns the inner product of Left
and Right. Constraint_Error is raised if Left'Length is not equal
to Right'Length. These operations involve an inner product.
81/2 function "*" (Left : Complex; Right : Complex_Vector) return Complex_Vector;
82/2 {AI95-00296-01} This operation returns the result of multiplying
each component of Right by the complex number Left using the
appropriate operation "*" in numerics.generic_complex_types. The
index range of the result is Right'Range.
83/2 function "*" (Left : Complex_Vector; Right : Complex) return Complex_Vector;
function "/" (Left : Complex_Vector; Right : Complex) return Complex_Vector;
84/2 {AI95-00296-01} Each operation returns the result of applying the
corresponding operation in numerics.generic_complex_types to each
component of the vector Left and the complex number Right. The
index range of the result is Left'Range.
85/2 function "*" (Left : Real'Base;
Right : Complex_Vector) return Complex_Vector;
86/2 {AI95-00296-01} This operation returns the result of multiplying
each component of Right by the real number Left using the
appropriate operation "*" in numerics.generic_complex_types. The
index range of the result is Right'Range.
87/2 function "*" (Left : Complex_Vector;
Right : Real'Base) return Complex_Vector;
function "/" (Left : Complex_Vector;
Right : Real'Base) return Complex_Vector;
88/2 {AI95-00296-01} Each operation returns the result of applying the
corresponding operation in numerics.generic_complex_types to each
component of the vector Left and the real number Right. The index
range of the result is Left'Range.
89/2 function Unit_Vector (Index : Integer;
Order : Positive;
First : Integer := 1) return Complex_Vector;
90/2 {AI95-00296-01} This function returns a unit vector with Order
components and a lower bound of First. All components are set to
(0.0, 0.0) except for the Index component which is set to (1.0,
0.0). Constraint_Error is raised if Index < First, Index > First +
Order - 1, or if First + Order - 1 > Integer'Last.
91/2 function Re (X : Complex_Matrix) return Real_Matrix;
function Im (X : Complex_Matrix) return Real_Matrix;
92/2 {AI95-00296-01} Each function returns a matrix of the specified
Cartesian components of X. The index ranges of the result are
those of X.
93/2 procedure Set_Re (X : in out Complex_Matrix; Re : in Real_Matrix);
procedure Set_Im (X : in out Complex_Matrix; Im : in Real_Matrix);
94/2 {AI95-00296-01} Each procedure replaces the specified (Cartesian)
component of each of the components of X by the value of the
matching component of Re or Im; the other (Cartesian) component of
each of the components is unchanged. Constraint_Error is raised if
X'Length(1) is not equal to Re'Length(1) or Im'Length(1) or if
X'Length(2) is not equal to Re'Length(2) or Im'Length(2).
95/2 function Compose_From_Cartesian (Re : Real_Matrix)
return Complex_Matrix;
function Compose_From_Cartesian (Re, Im : Real_Matrix)
return Complex_Matrix;
96/2 {AI95-00296-01} Each function constructs a matrix of Complex
results (in Cartesian representation) formed from given matrices
of Cartesian components; when only the real components are given,
imaginary components of zero are assumed. The index ranges of the
result are those of Re. Constraint_Error is raised if Re'Length(1)
is not equal to Im'Length(1) or Re'Length(2) is not equal to
Im'Length(2).
97/2 function Modulus (X : Complex_Matrix) return Real_Matrix;
function "abs" (Right : Complex_Matrix) return Real_Matrix
renames Modulus;
function Argument (X : Complex_Matrix) return Real_Matrix;
function Argument (X : Complex_Matrix;
Cycle : Real'Base) return Real_Matrix;
98/2 {AI95-00296-01} Each function calculates and returns a matrix of
the specified polar components of X or Right using the
corresponding function in numerics.generic_complex_types. The
index ranges of the result are those of X or Right.
99/2 function Compose_From_Polar (Modulus, Argument : Real_Matrix)
return Complex_Matrix;
function Compose_From_Polar (Modulus, Argument : Real_Matrix;
Cycle : Real'Base)
return Complex_Matrix;
100/2 {AI95-00296-01} Each function constructs a matrix of Complex
results (in Cartesian representation) formed from given matrices
of polar components using the corresponding function in numerics.-
generic_complex_types on matching components of Modulus and
Argument. The index ranges of the result are those of Modulus.
Constraint_Error is raised if Modulus'Length(1) is not equal to
Argument'Length(1) or Modulus'Length(2) is not equal to
Argument'Length(2).
101/2 function "+" (Right : Complex_Matrix) return Complex_Matrix;
function "-" (Right : Complex_Matrix) return Complex_Matrix;
102/2 {AI95-00296-01} Each operation returns the result of applying the
corresponding operation in numerics.generic_complex_types to each
component of Right. The index ranges of the result are those of
Right.
103/2 function Conjugate (X : Complex_Matrix) return Complex_Matrix;
104/2 {AI95-00296-01} This function returns the result of applying the
appropriate function Conjugate in numerics.generic_complex_types
to each component of X. The index ranges of the result are those
of X.
105/2 function Transpose (X : Complex_Matrix) return Complex_Matrix;
106/2 {AI95-00296-01} This function returns the transpose of a matrix X.
The first and second index ranges of the result are X'Range(2) and
X'Range(1) respectively.
107/2 function "+" (Left, Right : Complex_Matrix) return Complex_Matrix;
function "-" (Left, Right : Complex_Matrix) return Complex_Matrix;
108/2 {AI95-00296-01} Each operation returns the result of applying the
corresponding operation in numerics.generic_complex_types to each
component of Left and the matching component of Right. The index
ranges of the result are those of Left. Constraint_Error is raised
if Left'Length(1) is not equal to Right'Length(1) or
Left'Length(2) is not equal to Right'Length(2).
109/2 function "*" (Left, Right : Complex_Matrix) return Complex_Matrix;
110/2 {AI95-00296-01} This operation provides the standard mathematical
operation for matrix multiplication. The first and second index
ranges of the result are Left'Range(1) and Right'Range(2)
respectively. Constraint_Error is raised if Left'Length(2) is not
equal to Right'Length(1). This operation involves inner products.
111/2 function "*" (Left, Right : Complex_Vector) return Complex_Matrix;
112/2 {AI95-00296-01} This operation returns the outer product of a
(column) vector Left by a (row) vector Right using the appropriate
operation "*" in numerics.generic_complex_types for computing the
individual components. The first and second index ranges of the
result are Left'Range and Right'Range respectively.
113/2 function "*" (Left : Complex_Vector;
Right : Complex_Matrix) return Complex_Vector;
114/2 {AI95-00296-01} This operation provides the standard mathematical
operation for multiplication of a (row) vector Left by a matrix
Right. The index range of the (row) vector result is
Right'Range(2). Constraint_Error is raised if Left'Length is not
equal to Right'Length(1). This operation involves inner products.
115/2 function "*" (Left : Complex_Matrix;
Right : Complex_Vector) return Complex_Vector;
116/2 {AI95-00296-01} This operation provides the standard mathematical
operation for multiplication of a matrix Left by a (column) vector
Right. The index range of the (column) vector result is
Left'Range(1). Constraint_Error is raised if Left'Length(2) is not
equal to Right'Length. This operation involves inner products.
117/2 function "+" (Left : Real_Matrix;
Right : Complex_Matrix) return Complex_Matrix;
function "+" (Left : Complex_Matrix;
Right : Real_Matrix) return Complex_Matrix;
function "-" (Left : Real_Matrix;
Right : Complex_Matrix) return Complex_Matrix;
function "-" (Left : Complex_Matrix;
Right : Real_Matrix) return Complex_Matrix;
118/2 {AI95-00296-01} Each operation returns the result of applying the
corresponding operation in numerics.generic_complex_types to each
component of Left and the matching component of Right. The index
ranges of the result are those of Left. Constraint_Error is raised
if Left'Length(1) is not equal to Right'Length(1) or
Left'Length(2) is not equal to Right'Length(2).
119/2 function "*" (Left : Real_Matrix;
Right : Complex_Matrix) return Complex_Matrix;
function "*" (Left : Complex_Matrix;
Right : Real_Matrix) return Complex_Matrix;
120/2 {AI95-00296-01} Each operation provides the standard mathematical
operation for matrix multiplication. The first and second index
ranges of the result are Left'Range(1) and Right'Range(2)
respectively. Constraint_Error is raised if Left'Length(2) is not
equal to Right'Length(1). These operations involve inner products.
121/2 function "*" (Left : Real_Vector;
Right : Complex_Vector) return Complex_Matrix;
function "*" (Left : Complex_Vector;
Right : Real_Vector) return Complex_Matrix;
122/2 {AI95-00296-01} Each operation returns the outer product of a
(column) vector Left by a (row) vector Right using the appropriate
operation "*" in numerics.generic_complex_types for computing the
individual components. The first and second index ranges of the
result are Left'Range and Right'Range respectively.
123/2 function "*" (Left : Real_Vector;
Right : Complex_Matrix) return Complex_Vector;
function "*" (Left : Complex_Vector;
Right : Real_Matrix) return Complex_Vector;
124/2 {AI95-00296-01} Each operation provides the standard mathematical
operation for multiplication of a (row) vector Left by a matrix
Right. The index range of the (row) vector result is
Right'Range(2). Constraint_Error is raised if Left'Length is not
equal to Right'Length(1). These operations involve inner products.
125/2 function "*" (Left : Real_Matrix;
Right : Complex_Vector) return Complex_Vector;
function "*" (Left : Complex_Matrix;
Right : Real_Vector) return Complex_Vector;
126/2 {AI95-00296-01} Each operation provides the standard mathematical
operation for multiplication of a matrix Left by a (column) vector
Right. The index range of the (column) vector result is
Left'Range(1). Constraint_Error is raised if Left'Length(2) is not
equal to Right'Length. These operations involve inner products.
127/2 function "*" (Left : Complex; Right : Complex_Matrix) return Complex_Matrix;
128/2 {AI95-00296-01} This operation returns the result of multiplying
each component of Right by the complex number Left using the
appropriate operation "*" in numerics.generic_complex_types. The
index ranges of the result are those of Right.
129/2 function "*" (Left : Complex_Matrix; Right : Complex) return Complex_Matrix;
function "/" (Left : Complex_Matrix; Right : Complex) return Complex_Matrix;
130/2 {AI95-00296-01} Each operation returns the result of applying the
corresponding operation in numerics.generic_complex_types to each
component of the matrix Left and the complex number Right. The
index ranges of the result are those of Left.
131/2 function "*" (Left : Real'Base;
Right : Complex_Matrix) return Complex_Matrix;
132/2 {AI95-00296-01} This operation returns the result of multiplying
each component of Right by the real number Left using the
appropriate operation "*" in numerics.generic_complex_types. The
index ranges of the result are those of Right.
133/2 function "*" (Left : Complex_Matrix;
Right : Real'Base) return Complex_Matrix;
function "/" (Left : Complex_Matrix;
Right : Real'Base) return Complex_Matrix;
134/2 {AI95-00296-01} Each operation returns the result of applying the
corresponding operation in numerics.generic_complex_types to each
component of the matrix Left and the real number Right. The index
ranges of the result are those of Left.
135/2 function Solve (A : Complex_Matrix; X : Complex_Vector) return Complex_Vector;
136/2 {AI95-00296-01} This function returns a vector Y such that X is
(nearly) equal to A * Y. This is the standard mathematical
operation for solving a single set of linear equations. The index
range of the result is A'Range(2). Constraint_Error is raised if
A'Length(1), A'Length(2), and X'Length are not equal.
Constraint_Error is raised if the matrix A is ill-conditioned.
136.a/2 Discussion: The text says that Y is such that "X is (nearly) equal
to A * Y" rather than "X is equal to A * Y" because rounding
errors may mean that there is no value of Y such that X is exactly
equal to A * Y. On the other hand it does not mean that any old
rough value will do. The algorithm given under
Implementation Advice should be followed.
136.b/2 The requirement to raise Constraint_Error if the matrix is
ill-conditioned is really a reflection of what will happen if the
matrix is ill-conditioned. See Implementation Advice. We do not
make any attempt to define ill-conditioned formally.
136.c/2 These remarks apply to all versions of Solve and Inverse.
137/2 function Solve (A, X : Complex_Matrix) return Complex_Matrix;
138/2 {AI95-00296-01} This function returns a matrix Y such that X is
(nearly) equal to A * Y. This is the standard mathematical
operation for solving several sets of linear equations. The index
ranges of the result are A'Range(2) and X'Range(2).
Constraint_Error is raised if A'Length(1), A'Length(2), and
X'Length(1) are not equal. Constraint_Error is raised if the
matrix A is ill-conditioned.
139/2 function Inverse (A : Complex_Matrix) return Complex_Matrix;
140/2 {AI95-00296-01} This function returns a matrix B such that A * B
is (nearly) equal to the unit matrix. The index ranges of the
result are A'Range(2) and A'Range(1). Constraint_Error is raised
if A'Length(1) is not equal to A'Length(2). Constraint_Error is
raised if the matrix A is ill-conditioned.
141/2 function Determinant (A : Complex_Matrix) return Complex;
142/2 {AI95-00296-01} This function returns the determinant of the
matrix A. Constraint_Error is raised if A'Length(1) is not equal
to A'Length(2).
143/2 function Eigenvalues(A : Complex_Matrix) return Real_Vector;
144/2 {AI95-00296-01} This function returns the eigenvalues of the
Hermitian matrix A as a vector sorted into order with the largest
first. Constraint_Error is raised if A'Length(1) is not equal to
A'Length(2). The index range of the result is A'Range(1).
Argument_Error is raised if the matrix A is not Hermitian.
144.a/2 Discussion: A Hermitian matrix is one whose transpose is equal to
its complex conjugate. The eigenvalues of a Hermitian matrix are
always real. We only support this case because algorithms for
solving the general case are inherently unstable.
145/2 procedure Eigensystem(A : in Complex_Matrix;
Values : out Real_Vector;
Vectors : out Complex_Matrix);
146/3 {AI95-00296-01} {AI05-0047-1} This procedure computes both the
eigenvalues and eigenvectors of the Hermitian matrix A. The out
parameter Values is the same as that obtained by calling the
function Eigenvalues. The out parameter Vectors is a matrix whose
columns are the eigenvectors of the matrix A. The order of the
columns corresponds to the order of the eigenvalues. The
eigenvectors are mutually orthonormal, including when there are
repeated eigenvalues. Constraint_Error is raised if A'Length(1) is
not equal to A'Length(2), or if Values'Range is not equal to
A'Range(1), or if the index ranges of the parameter Vectors are
not equal to those of A. Argument_Error is raised if the matrix A
is not Hermitian. Constraint_Error is also raised in
implementation-defined circumstances if the algorithm used does
not converge quickly enough.
146.a/3 Ramification: {AI05-0047-1} There is no requirement on the
absolute direction of the returned eigenvectors. Thus they might
be multiplied by any complex number whose modulus is 1. It is only
the ratios of the components that matter. This is standard
practice.
147/2 function Unit_Matrix (Order : Positive;
First_1, First_2 : Integer := 1)
return Complex_Matrix;
148/2 {AI95-00296-01} This function returns a square unit matrix with
Order**2 components and lower bounds of First_1 and First_2 (for
the first and second index ranges respectively). All components
are set to (0.0, 0.0) except for the main diagonal, whose
components are set to (1.0, 0.0). Constraint_Error is raised if
First_1 + Order - 1 > Integer'Last or First_2 + Order - 1 >
Integer'Last.
Implementation Requirements
149/2 {AI95-00296-01} Accuracy requirements for the subprograms Solve,
Inverse, Determinant, Eigenvalues and Eigensystem are implementation defined.
149.a/2 Implementation defined: The accuracy requirements for the
subprograms Solve, Inverse, Determinant, Eigenvalues and
Eigensystem for type Complex_Matrix.
150/2 {AI95-00296-01} For operations not involving an inner product, the
accuracy requirements are those of the corresponding operations of the type
Real'Base and Complex in both the strict mode and the relaxed mode (see G.2).
151/2 {AI95-00296-01} For operations involving an inner product, no
requirements are specified in the relaxed mode. In the strict mode the modulus
of the absolute error of the inner product X*Y shall not exceed
g*abs(X)*abs(Y) where g is defined as
152/2 g = X'Length * Real'Machine_Radix**(1 - Real'Model_Mantissa)
for mixed complex and real operands
153/2 g = sqrt(2.0) * X'Length * Real'Machine_Radix**(1 - Real'Model_Mantissa)
for two complex operands
154/2 {AI95-00418-01} For the L2-norm, no accuracy requirements are specified
in the relaxed mode. In the strict mode the relative error on the norm shall
not exceed g / 2.0 + 3.0 * Real'Model_Epsilon where g has the definition
appropriate for two complex operands.
Documentation Requirements
155/2 {AI95-00296-01} Implementations shall document any techniques used to
reduce cancellation errors such as extended precision arithmetic.
155.a/2 Documentation Requirement: Any techniques used to reduce
cancellation errors in Numerics.Generic_Complex_Arrays shall be
documented.
155.b/2 Implementation Note: The above accuracy requirement is met by the
canonical implementation of the inner product by multiplication
and addition using the corresponding operations of type Complex
and performing the cumulative addition using ascending indices.
Note however, that some hardware provides special operations for
the computation of the inner product and although these may be
fast they may not meet the accuracy requirement specified. See
Accuracy and Stability of Numerical Algorithms by N J Higham (ISBN
0-89871-355-2), Sections 3.1 and 3.6.
Implementation Permissions
156/2 {AI95-00296-01} The nongeneric equivalent packages may, but need not, be
actual instantiations of the generic package for the appropriate predefined
type.
157/2 {AI95-00296-01} Although many operations are defined in terms of
operations from numerics.generic_complex_types, they need not be implemented
by calling those operations provided that the effect is the same.
Implementation Advice
158/3 {AI95-00296-01} {AI05-0264-1} Implementations should implement the Solve
and Inverse functions using established techniques. Implementations are
recommended to refine the result by performing an iteration on the residuals;
if this is done, then it should be documented.
158.a/2 Implementation Advice: Solve and Inverse for
Numerics.Generic_Complex_Arrays should be implemented using
established techniques and the result should be refined by an
iteration on the residuals.
159/2 {AI95-00296-01} It is not the intention that any special provision
should be made to determine whether a matrix is ill-conditioned or not. The
naturally occurring overflow (including division by zero) which will result
from executing these functions with an ill-conditioned matrix and thus raise
Constraint_Error is sufficient.
159.a/2 Discussion: There isn't any advice for the implementation to
document with this paragraph.
160/2 {AI95-00296-01} The test that a matrix is Hermitian should use the
equality operator to compare the real components and negation followed by
equality to compare the imaginary components (see G.2.1).
160.a/2 Implementation Advice: The equality and negation operators should
be used to test that a matrix is Hermitian.
160.1/3 An implementation should minimize the circumstances under which the
algorithm used for Eigenvalues and Eigensystem fails to converge.
160.a.1/3 Implementation Advice: An implementation should minimize the
circumstances under which the algorithm used for
Numerics.Generic_Complex_Arrays.Eigenvalues and
Numerics.Generic_Complex_Arrays.Eigensystem fails to converge.
160.b/3 Implementation Note: J. H. Wilkinson is the acknowledged expert in
this area. See for example Wilkinson, J. H., and Reinsch, C. ,
Linear Algebra , vol II of Handbook for Automatic Computation,
Springer-Verlag, or Wilkinson, J. H., The Algebraic Eigenvalue
Problem, Oxford University Press.
161/2 {AI95-00296-01} Implementations should not perform operations on mixed
complex and real operands by first converting the real operand to complex. See
G.1.1.
161.a/2 Implementation Advice: Mixed real and complex operations should
not be performed by converting the real operand to complex.
Extensions to Ada 95
161.b/2 {AI95-00296-01} The package Numerics.Generic_Complex_Arrays and
its nongeneric equivalents are new.
Wording Changes from Ada 2005
161.c/3 {AI05-0047-1} Correction: Corrected various accuracy and
definition issues.
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