2012/05/29 DRAFT WG5/N1921

Transcription

1 // DRAFT WG/N Special Mathematical Functions in Fortran ISO/IEC - : x Auxiliary to ISO/IEC : Programming Language Fortran NOTE This paper is intended to suggest some special functions for which standard procedure interfaces might be specified. Whether it is done as part of Clause of -, as -, or as a Technical Report can be decided later. The exact set of procedures can be decided later. Whether the procedures are module procedures or intrinsic procedures can be decided later. If they are module procedures, the module name and whether the module is intrinsic can be decided later. Subclause. describes the same procedures as WG n, plus procedures to compute two additional functions related to the ones described therein that are better behaved. Subclause. proposes additional procedures that are widely used in scientific and engineering calculations. NOTE At the WG meeting in Garching, there was a discussion (see the Tuesday part of the minutes in N) concerning whether WG should produce a standard for interfaces to certain mathematical functions, analogous to those for C (ISO/IEC :) and C++ (ISO/IEC :). The resulting straw vote was: yes - no - undecided.

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3 // DRAFT WG/N Contents Overview Scope Inclusions Exclusions Conformance Notation used in this part of ISO/IEC Applicability of requirements Informative notes Normative references Nonnormative references The module ISO Fortran Special Functions General Mathematical constants Euler s constant γ Summary of the procedures Specifications for the procedures General Proposed additional procedures General Annex A (informative) Processor dependencies

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5 // DRAFT WG/N Information technology Programming languages Fortran Part : Special Mathematical Functions Overview. Scope ISO/IEC is a multipart International Standard; the parts are published separately. This publication, ISO/IEC -, which is the fourth part, describes the standard intrinsic module ISO Fortran Special Functions. The purpose of this part of ISO/IEC is to promote portability, reliability, maintainability, and efficient evaluation of mathematical special functions in Fortran programs, for use on a variety of computing systems. This part is normative, but optional. A processor need not provide support for this part.. Inclusions This part of ISO/IEC specifies the procedures defined by the module ISO Fortran Special Functions, the interface definitions for those procedures, and the mathematical function evaluated by each procedure.. Exclusions This part of ISO/IEC does not specify the methods to evaluate the functions, or the accuracy of the results of the procedures.. Conformance A program conforms to ISO/IEC if it conforms to ISO/IEC - and this part of ISO/IEC. A processor that supports this part of ISO/IEC conforms to it if it executes any standard-conforming program in a manner that fulfills the interpretations herein and in ISO/IEC -, subject to any limitations that the processor may impose upon the range of the arguments of the procedures, and it contains the capability to detect and report the use within a program of argument values outside the ranges specified herein.

6 WG/N DRAFT //. Notation used in this part of ISO/IEC.. Applicability of requirements In this part of ISO/IEC, shall is to be interpreted as a requirement; conversely, shall not is to be interpreted as a prohibition. Except where stated otherwise, such requirements and prohibitions apply to programs rather than processors... Informative notes Informative notes of explanation, rationale, examples, and other material are interspersed with the normative body of this part of ISO/IEC. The informative material is nonnormative; it is identified by being in shaded, framed boxes that have numbered headings beginning with NOTE.. Normative references The following referenced standards are indispensable for the application of this part of ISO/IEC. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced standard (including any amendments) applies. ISO/IEC -:, Information technology Programming languages Fortran Part : Base Language. ISO -:, Quantities and units Part : Mathematical signs and symbols for use in the physical sciences and technology, Clause Special Functions.. Nonnormative references The following referenced materials are useful but not indispensable for the application of this part of ISO/IEC. Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, Charles W. Clark, NIST Handbook of Mathematical Functions, National Institute of Standards and Technology and Cambridge University Press (), ISBN ---- (hardback), ---- (paperback). Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, U. S. National Bureau of Standards (now National Institute of Standards and Technology) Applied Mathematics Series # () LCCCN -. Jerome Spanier and Keith B. Oldham, An Atlas of Functions, Hemisphere Publishing Corporation, New York () ISBN ---.

7 // DRAFT WG/N The module ISO Fortran Special Functions. General The module ISO Fortran Special Functions contains named mathematical constants and the definitions of the interfaces of procedures to evaluate special mathematical functions. The procedures are all generic procedures. For each generic procedure defined here, the processor shall provide specific procedures for all real kinds supported by the processor. It is processor dependent whether the processor provides specific procedures for integer kinds other than default integer. The names of the specific procedures are private identifiers of ISO Fortran Special Functions. The procedures might be separate module procedures (... in ISO/IEC -). If so, the submodule identifiers of the submodules in which the procedures are defined are processor dependent. It is recommended that documentation that accompanies the processor include descriptions of the relationship between the ranges of the values of the arguments of the procedures and the accuracy of the results.. Mathematical constants.. Euler s constant γ Euler s constant γ (sometimes called the Euler-Mascheroni constant) is defined as γ = lim n ( n i= ) i ln n and by other definitions that appear in the references (.,.). The kind of the named constant EULER_GAMMA shall be the kind supported by the processor that provides the most precise representation. The radix of that kind is processor dependent. REAL(kind), PARAMETER :: EULER_GAMMA = & &._kind. Summary of the procedures ASSOC LAGUERRE (N, M, ) ASSOC LEGENDRE (L, M, ) BETA (, Y) CYL BESSEL I (NU, ) CYL BESSEL I (NU,, N) CYL BESSEL J (NU, ) CYL BESSEL J (NU,, N) CYL BESSEL K (NU, ) CYL BESSEL K (NU,, N) CYL NEUMANN (NU, ) CYL NEUMANN (NU,, N) Associated Laguerre polynomials Associated Legendre Polynomials Beta function Regular modified cylindrical Bessel function. Regular modified cylindrical Bessel function. Cylindrical Bessel function. Cylindrical Bessel function. Irregular modified cylindrical Bessel function. Irregular modified cylindrical Bessel function. Cylindrical Neumann function. Cylindrical Neumann function.

8 WG/N DRAFT // ELLINT (K ) ELLINT (K, PHI) ELLINT (K ) ELLINT (K, PHI) ELLINT (K, NU ) ELLINT (K, NU, PHI) EIN () EPINT () HERMITE (N, ) LAGUERRE (N, ) LEGENDRE (N, ) RIEMANN ZETA () SPH BESSEL (N, ) SPH BESSEL (N,, N) SPH LEGENDRE (L, M, THETA) SPH NEUMANN (N, ) SPH NEUMANN (N,, N) Complete elliptic integral of the first kind Incomplete elliptic integral of the first kind Complete elliptic integral of the second kind Incomplete elliptic integral of the second kind Complete elliptic integral of the third kind Complete elliptic integral of the third kind Entire exponential integral Exponential integral Hermite polynomials Laguerre polynomials Legendre polynomials Riemann zeta function Spherical Bessel function of the first kind Spherical Bessel function of the first kind Spherical associated Legendre function Spherical Neumann function Spherical Neumann function. Specifications for the procedures.. General Detailed specifications of the procedures whose interfaces are defined in the module ISO Fortran Special - Functions are provided here in alphabetical order. The types and type parameters of the arguments and function results of these procedures are determined by these specifications. The Argument(s) paragraphs specify requirements on the actual arguments of the procedures. The result characteristics are sometimes specified in terms of the characteristics of dummy arguments. A program shall not invoke one of these procedures under circumstances where a value to be assigned to a subroutine argument or returned as a function result is not representable by objects of the specified type and type parameters. If an IEEE infinity is assigned or returned, the intrinsic module IEEE ARITHMETIC is accessible, and the actual arguments were finite numbers, the flag IEEE OVERFLOW or IEEE DIVIDE BY ZERO shall signal. If an IEEE NaN is assigned or returned, the actual arguments were finite numbers, the intrinsic module IEEE ARITHMETIC is accessible, and the exception IEEE INVALID is supported, the flag IEEE INVALID shall signal. If no IEEE infinity or NaN is assigned or returned, these flags shall have the same status as when the intrinsic procedure was invoked... ASSOC LAGUERRE (N, M, ) Description. Associated Laguerre polynomials. N shall be of type integer and nonnegative. M shall be of type integer and nonnegative. Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the associated La-

9 // DRAFT WG/N guerre polynomial L m n (x) of orders N and M and argument, defined by L m n (x) = n! n i= ( n! m + n i! n i ) ( x) i = ( ) m dm dx m L m+n(x) where L m+n (x) is a Laguerre polynomial (..) Example. ASSOC LAGUERRE (,,. ) has the value. (approximately)... ASSOC LEGENDRE (L, M, ) Description. Associated Legendre polynomials. L shall be of type integer and nonnegative. M shall be of type integer and nonnegative. The absolute value of shall be less than or equal to.. Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the associated Legendre polynomial Pl m (x) of orders L and M and argument, defined by P m l (x) = ( x m/ dm ) dx m P l(x), x where P l (x) is a Legendre polynomial (..). Example. ASSOC LEGENDRE (,,. ) has the value. (approximately)... BETA (, Y) Description. Beta function. Y shall be of type real and nonnegative. shall be of type real with the same kind as. The value of Y shall not be negative. Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the beta function B(x, y) with arguments and Y, defined by B(x, y) = Γ(x)Γ(y) Γ(x + y) = t x ( t) y dt, x >, y > and several other representations that appear in the references (.,.). Example. BETA (.,. ) has the value. (approximately)... CYL BESSEL I (NU, ) or CYL BESSEL I (NU,, N) Description. Regular modified cylindrical Bessel function.

10 WG/N DRAFT // Class. CYL BESSEL I (NU, ) is an elemental function. CYL BESSEL I (NU,, N) is a transformational function. NU N shall be of type real and the same kind as NU. The value of shall not be negative. shall be of type integer and nonnegative. Result Characteristics. Same type and kind as. The result of CYL BESSEL I (NU, ) is scalar. The result of CYL BESSEL I (NU,, N) is a rank-one array with extent N. Result Value. The value of the result of CYL BESSEL I (NU, ) is a processor-dependent approximation to the regular modified cylindrical Bessel function I ν (x) of order NU with argument, defined by ( x ) ν I ν (x) = i ν (x/) k J ν (ix) = k!γ(ν + k + ), x, k= and several other representations that appear in the references (.,.). The value of the i th element of the result of CYL BESSEL I (NU,, N) is the same as the value of the result of CYL BESSEL I (NU+i, ) Example. CYL BESSEL I (.,. ) has the value. (approximately)... CYL BESSEL J (NU, ) or CYL BESSEL J (NU,, N) Description. Cylindrical Bessel function. Class. CYL BESSEL J (NU, ) is an elemental function. CYL BESSEL J (NU,, N) is a transformational function. NU N shall be of type real and the same kind as NU. The value of shall not be negative. shall be of type integer and nonnegative. Result Characteristics. Same type and kind as. The result of CYL BESSEL J (NU, ) is scalar. The result of CYL BESSEL J (NU,, N) is a rank-one array with extent N. Result Value. The value of the result of CYL BESSEL J (NU, ) is a processor-dependent approximation to the cylindrical Bessel function J ν (x) of order NU with argument, defined by ( x ) ν ( ) k (x/) k J ν (x) = k!γ(ν + k + ), x, k=

11 // DRAFT WG/N and several other representations that appear in the references (.,.). The value of the i th element of the result of CYL BESSEL J (NU,, N) is the same as the value of the result of CYL BESSEL J (NU+i, ) NOTE. This is a generalization of the standard intrinsic function BESSEL JN to noninteger order. Example. CYL BESSEL J (.,. ) has the value. (approximately)... CYL BESSEL K (NU, ) or CYL BESSEL K (NU,, N) Description. Irregular modified cylindrical Bessel function. Class. CYL BESSEL K (NU, ) is an elemental function. CYL BESSEL K (NU,, N) is a transformational function. NU N shall be of type real and the same kind as NU. The value of shall not be negative. shall be of type integer and nonnegative. Result Characteristics. Same type and kind as. The result of CYL BESSEL K (NU, ) is scalar. The result of CYL BESSEL K (NU,, N) is a rank-one array with extent N. Result Value. The value of the result of CYL BESSEL K (NU, ) is a processor-dependent approximation to the irregular modified cylindrical Bessel function K ν (x) of order NU with argument, defined by K ν (x) = π iν+ (J ν (ix) + in ν (ix)) = π lim µ ν I µ (x) I µ (x), x sin µx and several other representations that appear in the references (.,.). The value of the i th element of the result of CYL BESSEL K (NU,, N) is the same as the value of the result of CYL BESSEL K (NU+i, ) NOTE. The irregular modified cylindrical Bessel function is also known as the Bassett function. Example. CYL BESSEL K (., HUGE(.) ) has the value. (approximately)... CYL NEUMANN (NU, ) or CYL NEUMANN (NU,, N) Description. Cylindrical Neumann function. Class. CYL NEUMANN (NU, ) is an elemental function. CYL NEUMANN (NU,, N) is a transformational function.

12 WG/N DRAFT // NU N shall be of type real and the same kind as NU. The value of shall not be negative. shall be of type integer and nonnegative. Result Characteristics. Same type and kind as. The result of CYL NEUMANN (NU, ) is scalar. The result of CYL NEUMANN (NU,, N) is a rank-one array with extent N. Result Value. The value of the result of CYL NEUMANN (NU, ) is a processor-dependent approximation to the Neumann function N ν (x) of order NU with argument, defined by J µ (x) cos µx J µ (x) N ν (x) = lim, x µ ν sin µx and several other representations that appear in the references (.,.). The value of the i th element of the result of CYL NEUMANN (NU,, N) is the same as the value of the result of CYL NEUMANN (NU+i, ) NOTE. The Neumann function is also known as the cylindrical Bessel function of the second kind, Y ν (x). NOTE. This is a generalization of the standard intrinsic function BESSEL YN to noninteger order. Example. CYL NEUMANN (.,. ) has the value. (approximately)... ELLINT (K) or ELLINT (K, PHI) Description. Elliptic integral of the first kind. K The absolute value of K shall be less than or equal to.. PHI shall be of type real and the same kind as K. Result Characteristics. Same as K. Result Value. The value of the result of ELLINT (K) is a processor-dependent approximation to the complete elliptic integral of the first kind K(k) with argument K, defined by K(k) = π/ dθ k sin θ = dt t k t, k and several other representations that appear in the references (.,.). The value of the result of ELLINT (K,PHI) is a processor-dependent approximation to the incomplete elliptic integral of the first kind F (k, φ) with arguments K and PHI, defined by F (k, φ) = φ dθ k sin θ, k

13 // DRAFT WG/N and several other representations that appear in the references (.,.). Examples. ELLINT (. ) has the value. (approximately). ELLINT (.,. ) has the value. (approximately)... ELLINT (K) or ELLINT (K, PHI) Description. Elliptic integral of the second kind. K The absolute value of K shall be less than or equal to.. PHI shall be of type real and the same kind as K. Result Characteristics. Same as K. Result Value. The value of the result of ELLINT (K) is a processor-dependent approximation to the complete elliptic integral of the second kind E(k) with argument K, defined by π/ E(k) = k sin k t θ dθ = dt, k t and several other representations that appear in the references (.,.). The value of the result of ELLINT (K, PHI) is a processor-dependent approximation to the incomplete elliptic integral of the second kind E(k, φ) with arguments K and PHI, defined by E(k, φ) = φ k sin θ dθ, k and several other representations that appear in the references (.,.). Examples. ELLINT (. ) has the value. (approximately). ELLINT (.,. ) has the value. (approximately)... ELLINT (NU, K) or ELLINT (NU, K, PHI) Description. Elliptic integral of the third kind. NU K shall be of type real and the same kind as NU. The absolute value of K shall be less than or equal to.. PHI shall be of type real and the same kind as K. Result Characteristics. Same as NU. Result Value. The value of the result of ELLINT (NU, K) is a processor-dependent approximation to the complete elliptic integral of the third kind Π(ν; k) with arguments NU and K, defined by Π(ν; k) = π/ dθ ( + ν sin θ) k sin θ = dt ( + νt ) ( t )( k t ), k and several other representations that appear in the references (.,.).

14 WG/N DRAFT // The value of the result of ELLINT (NU, K, PHI) is a processor-dependent approximation to the incomplete elliptic integral of the third kind Π(ν; k, φ) with arguments NU, K, and PHI, defined by Π(ν; k, φ) = φ dθ ( + ν sin θ) k sin θ, k and several other representations that appear in the references (.,.). Examples. ELLINT (.,. ) has the value. (approximately). ELLINT (.,.,. ) has the value. (approximately)... EIN () Description. Entire exponential integral. Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the entire exponential integral Ein(x) with argument, defined by Ein(x) = x exp( t) dt = γ + ln x + E (x) = γ + ln x R Ei( x), t where γ. is Euler s constant, and several other representations that appear in the references (.,.). Example. EIN (. ) has the value. (approximately)... EPINT () Description. Exponential integral. Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the exponential integral Ei(x) with argument, defined by Ei(x) = x exp(t) dt = t x exp( t) dt. t The integrand is singular at x =, so for x > the integral is interpeted as the Cauchy limit ( x Ei(x) = lim Ei( ɛ) + ɛ + ɛ ) exp(t) dt, x >. t Several other representations appear in the references (.,.).

15 // DRAFT WG/N Example. EPINT (. ) has the value. (approximately)... HERMITE (N, ) Description. Hermite polynomial. N shall be of type integer and nonnegative. Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the Hermite polynomial H n (x) of order N with argument, defined by the Rodrigues formula H n (x) = ( ) n exp(x ) dn dx n exp( x ), and several other representations that appear in the references (.,.). Example. HERMITE (,. ) has the value. (approximately)... LAGUERRE (N, ) Description. Laguerre polynomial. N shall be of type integer and nonnegative. The value of shall not be negative. Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the Laguerre polynomial L n (x) of order N with argument, defined by the Rodrigues formula L n (x) = exp(x) n! d n dx n (xn exp( x)), x, and several other representations that appear in the references (.,.). Example. LAGUERRE (,. ) has the value. (approximately)... LEGENDRE (N, ) Description. Legendre polynomial. N shall be of type integer and nonnegative. Result Characteristics. Same as.

16 WG/N DRAFT // Result Value. The value of the result is a processor-dependent approximation to the Legendre polynomial P n (x) of order N with argument, defined by the Rodrigues formula d n P n (x) = ( x n n! dx n ) n, and several other representations that appear in the references (.,.). Example. LEGENDRE (,. ) has the value. (approximately)... RIEMANN ZETA () Description. Riemann zeta function. Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the Riemann zeta function ζ(x) with argument, defined by k= k x x > ζ(x) = x k= ( )k k x x, x π x sin ( ) πx Γ( x)ζ( x) x < and several other representations that appear in the references (.,.). Example. RIEMANN ZETA (. ) has the value. (approximately)... SPH BESSEL (N, ) or SPH BESSEL (N, N, ) Description. Spherical Bessel function of the first kind. Class. SPH BESSEL (N, ) is an elemental function. SPH BESSEL (N, N, ) is a transformational function. N shall be of type integer and nonnegative. N shall be of type integer and nonnegative. N shall be of type integer and nonnegative. Result Characteristics. Same type and kind as. The result of SPH BESSEL (N, ) is scalar. The result of SPH BESSEL (N, N, ) is a rank-one array with extent MA(N-N+,). Result Value.

17 // DRAFT WG/N The value of the result of SPH BESSEL (N, ) is a processor-dependent approximation to the Spherical Bessel function of the first kind j n (x) of order N with argument, defined by j n (x) = π x J n+/(x), x, where J n+/ is the cylindrical Bessel function. Several other representations appear in the references (.,.). The value of the i th element of the result of SPH BESSEL (N, N, ) is the same as the value of SPH BESSEL (N+i, ) Example. SPH BESSEL (,. ) has the value. (approximately)... SPH LEGENDRE (L, M, THETA) Description. Spherical associated Legendre function. L shall be of type integer. M shall be of type integer. The absolute value of M shall be less than or equal to the value of L. THETA Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the Spherical associated Legendre function Yl m m (θ, ) of order M and L with argument THETA, where Yl (θ, φ) is defined by Y m [ ] / (l + ) l (θ, φ) = ( ) m (l m)! Pl m (cos θ) exp(imφ), m l, π (l + m)! and several other representations that appear in the references (.,.). NOTE. Spherical associated Legendre functions are also known as spherical harmonics. Example. SPH LEGENDRE (,,. ) has the value. (approximately)... SPH NEUMANN (N, ) or SPH NEUMANN (N, N, ) Description. Spherical Neumann function. Class. SPH NEUMANN (N, ) is an elemental function. SPH NEUMANN (N, N, ) is a transformational function. N shall be of type integer and nonnegative. N shall be of type integer and nonnegative.

18 WG/N DRAFT // N shall be of type integer and nonnegative. Result Characteristics. Same type and kind as. Result Value. The result of SPH NEUMANN (N, ) is scalar. The result of SPH NEUMANN (N, N, ) is a rank-one array with extent MA(N- N+,). The value of the result of SPH NEUMANN (N, ) is a processor-dependent approximation to the Spherical Neumann function n n (x) of order N with argument, defined by n n (x) = π x N n+/(x), x, where N n+/ is the Neumann function. Several other representations appear in the references (.,.). The value of the i th element of the result of SPH NEUMANN (N, N, ) is the same as the value of SPH NEUMANN (N+i, ) Example. SPH NEUMANN (,. ) has the value. (approximately).. Proposed additional procedures.. General Procedures to evaluate the functions proposed in subclause. do not appear in ISO -:, ISO/IEC :, or ISO/IEC :. Their use appears sufficiently frequently in practice that their availability would be useful... CHEBYSHEV (N, ) or CHEBYSHEV (N, N, ) Description. Chebyshev polynomial. Class. CHEBYSHEV (N, ) is an elemental function. CHEBYSHEV (N, N, ) is a transformational function. N shall be of type integer and nonnegative. N shall be of type integer and nonnegative. N shall be of type integer and nonnegative. Result Characteristics. Same type and kind as. The result of CHEBYSHEV (N, ) is a scalar. The result of CHEBYSHEV (N, N, ) is a rank-one array with extent max(n-n+,). Result Value.

19 // DRAFT WG/N The value of the result of CHEBYSHEV (N, ) is a processor-dependent approximation to the Chebyshev polynomial T n (x) of order N with argument, defined by T n (x) = cos(n cos x) and other representations that appear in the references (.,.). The i th element of the value of the result of CHEBYSHEV (N, N, ) is the same as the value of CHEBYSHEV (N+i, ). Example. CHEBYSEV (,. ) has the value. (approximately)... CI () Description. Cosine integral. The value of shall not be zero. Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the cosine integral Ci(x) with argument, defined by Ci(x) = x cos t dt = γ + ln x Cin(x), t and other representations that appear in the references (.,.). The integrand is singular at zero, so for x < the integral is interpeted as the Cauchy limit ( ɛ Ci(x) = lim ɛ x ) cos(t) dt + Ci( x) t Example. CI (. ) has the value. (approximately)... CIN () Description. Entire cosine integral. Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the entire cosine integral Cin(x) with argument, defined by Cin(x) = x cos t dt = γ + ln x Ci(x), t and other representations that appear in the references (.,.).

20 WG/N DRAFT // Example. CIN (. ) has the value. (approximately)... DAW () Description. Dawson function. Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the Dawson function daw(x) with argument, defined by daw(x) = x exp(t x ) dt and other representations that appear in the references (.,.). Example. DAW (. ) has the value. (approximately)... ERFCI () Description. Inverse co-error function. Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the value y such that x = erfc(y), that is x = erfc(y) = exp( t ) dt, π y and other representations that appear in the references (.,.). Example. ERFCI (. ) has the value. (approximately)... ERFI () Description. Inverse error function. Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the value y such that x = erf(y), that is x = erf(y) = y exp( t ) dt, π

21 // DRAFT WG/N and other representations that appear in the references (.,.). Example. ERFI (. ) has the value. (approximately)... FRESNEL C () Description. Fresnel cosine integral. Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the Fresnel cosine integral C(x) with argument, defined by C(x) = x cos ( π t) dt, and other representations that appear in the references (.,.). Example. FRESNEL C (. ) has the value. (approximately)... FRESNEL S () Description. Fresnel sine integral. Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the Fresnel sine integral S(x) with argument, defined by S(x) = x sin ( π t) dt, and other representations that appear in the references (.,.). Example. FRESNEL S (. ) has the value. (approximately)... INCOMPLETE GAMMA RATIOS (NU,, P, Q) Description. Incomplete gamma function ratios. Class. Elemental subroutine. NU The value of NU shall be greater than zero. NU is an INTENT(IN) argument. shall be of type real and the same kind as NU. The value of shall be greater than zero. is an INTENT(IN) argument.

22 WG/N DRAFT // P Q shall be of type real and the same kind as NU. P is an INTENT(OUT) argument. shall be of type real and the same kind as NU. Q is an INTENT(OUT) argument. Result Value. The values of the P and Q arguments are processor-dependent approximations to the incomplete gamma ratios P (ν; x) and Q(ν; x) with arguments NU and, defined by γ(ν; x) Γ(ν; x) P (ν; x) = and Q(ν; x) = Γ(ν) Γ(ν), where γ(ν; x) = x t ν exp( t) dt and Γ(ν; x) = x t ν exp( t) dt, x >, ν >, and other representations that appear in the references (.,.). Example. After executing CALL INCOMPLETE GAMMA RATIO (.,., P, Q ), the variables P and Q have the values. and., respectively (approximately). NOTE. P (ν; x) + Q(ν; x) =, but they are not equally well conditioned computationally. In general, when one is small, it should not be computed by subtracting the other from.. When ν x and ν >>, x >> they are both very poorly conditioned... SI () Description. Sine integral. Result Characteristics. Same as. Result Value. The value of the result is a processor-dependent approximation to the sine integral Si(x), defined by Si(x) = x sin t dt, t and other representations that appear in the references (.,.). Example. SI (. ) has the value. (approximately).

23 // DRAFT WG/N Annex A (Informative) Processor dependencies According to this part of ISO/IEC, the following are processor dependent: Whether a processor supports this part of ISO/IEC (.). Whether a processor provides specific procedures for arguments of type integer and kind other than default integer kind (.). Whether the module procedures are separate procedures, and if so, the submodule identifiers of the submodules in which the procedures are defined (.). The kind and radix of the named constant EULER_GAMMA (..). The mathematical approximations used (.,.).