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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
{result interval (for a component of the result of evaluating a complex function)} 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. {maximum relative error (for a component of the result of evaluating a complex function)} 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. {maximum box error (for a component of the result of evaluating a complex 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. {Overflow_Check [partial]} {check, language-defined (Overflow_Check)} 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
5
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 below. 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 OperatorNature of
Result
Nature of
Bound
Error Bound
Modulusrealmax. rel. error3.0
Argumentrealmax. rel. error4.0
Compose_From_Polarcomplexmax. rel. error3.0
"*" (both operands complex)complexmax. box error5.0
"/" (right operand complex)complexmax. box error13.0
Sqrtcomplexmax. rel. error6.0
Logcomplexmax. box error13.0
Exp (complex parameter)complexmax. rel. error7.0
Exp (imaginary parameter)complexmax. rel. error2.0
Sin, Cos, Sinh, and Coshcomplexmax. rel. error11.0
Tan, Cot, Tanh, and Cothcomplexmax. rel. error35.0
inverse trigonometriccomplexmax. rel. error14.0
inverse hyperboliccomplexmax. rel. error14.0
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
11
12
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 to 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. 

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