J3/05-248
To: J3
From: Dan Nagle
Subject: libm math functions
Date: 2005 August 4
At Delft, it was decided to pursue the libm procedures.
This paper attempts to provide edits to do so.
The specifications are in 05-132r2.
There is one straw vote which should be taken before details
may be finalized, viz, how to handle the signgam external
variable used by the libm C binding to return the sign of gamma
when the log of abs( gamma) is computed. I believe the best
choices are:
1. Have two procedures, say log_gamma_func() and sign_gamma_func(),
and trusting compilers to recognize when the argument
doesn't change between references,
-or-
2. Use a subroutine, say log_gamma_func(), which may be elemental
and allow optional, intent( out) arguments,
-or-
3. Any other ideas,
-or-
4. Undecided.
The design specified by earlier papers used an optional,
intent(out) argument to the log_gamma function, which is not allowed
if log_gamma() is to be a pure function.
As a reminder, the specifications are:
Detailed Specification: Add subsections to Section 13 detailing
the Fortran names for these procedures. (The C names
should not be used due to the common usage, in
Fortran, of names such as j0 etc.)
The functions are (the C names):
Bessel functions (j0, j1, jn, y0, y1, yn)
Error Functions (erf, erfc)
Hypotenuse (hypot)
Gamma and log gamma (tgamma, lgamma)
The detailed mathematical specification of these
procedures is given in the references above.
The intention is to allow the vendor to use
the procedure supplied by libm, so the exact
specification is left to libm, which is most likely
what the applications programmer wants.
Edits would include adding to the list in 13.5.2:
BESSEL_J0
BESSEL_J1
BESSEL_JN
BESSEL_Y0
BESSEL_Y1
BESSEL_YN
COMP_ERROR_FUNC
ERROR_FUNC
GAMMA_FUNC
HYPOT
LOG_GAMMA_FUNC
Edits: (Note that these edits assume branch 2 of the straw vote is taken.)
[Add to the list 13.5.2]
[294:28+] Add
"BESSEL_J0
BESSEL_J1
BESSEL_JN
BESSEL_Y0
BESSEL_Y1
BESSEL_YN
COMP_ERROR_FUNC"
[294:30+] Add
"ERROR_FUNC"
[294:31+] Add
"GAMMA_FUNC
HYPOT"
[294:33+] Add
"LOG_GAMMA_FUNC"
[306:13+] Add
"13.7.15+ BESSEL_J0 (X)
*Description.* Bessel function of the first kind
of order zero.
*Class.* Elemental function.
*Argument.* X shall be of type real. Its value
shall satisfy the inequality X >= 0.
*Result Characteristics.* Same as X.
*Result Value.* The result has a value equal
to a processor-dependent approximation of the Bessel
function of the first kind of the zeroth order of X.
13.7.15+ BESSEL_J1 (X)
*Description.* Bessel function of the first kind
of order one.
*Class.* Elemental function.
*Argument.* X shall be of type real. Its value
shall satisfy the inequality X >= 0.
*Result Characteristics.* Same as X.
*Result Value.* The result has a value equal
to a processor-dependent approximation of the Bessel
function of the first kind of the first order of X.
13.7.15+ BESSEL_JN (N,X)
*Description.* Bessel function of the first kind
of order N.
*Class.* Elemental function.
*Arguments.*
X shall be of type real. Its value
shall satisfy the inequality X >= 0.
N shall be of type integer. Its value
shall satisfy the inequality N >= 0.
It shall be a scalar.
*Result Characteristics.* Same as X.
*Result Value.* The result has a value equal
to a processor-dependent approximation of the Bessel
function of the first kind of the Nth order of X.
13.7.15+ BESSEL_Y0 (X)
*Description.* Bessel function of the second kind
of order zero.
*Class.* Elemental function.
*Argument.* X shall be of type real. Its value
shall satisfy the inequality X > 0.
*Result Characteristics.* Same as X.
*Result Value.* The result has a value equal
to a processor-dependent approximation of the Bessel
function of the second kind of the zeroth order of X.
13.7.15+ BESSEL_Y1 (X)
*Description.* Bessel function of the second kind
of order one.
*Class.* Elemental function.
*Argument.* X shall be of type real. Its value
shall satisfy the inequality X > 0.
*Result Characteristics.* Same as X.
*Result Value.* The result has a value equal
to a processor-dependent approximation of the Bessel
function of the second kind of the first order of X.
13.7.15+ BESSEL_YN (N,X)
*Description.* Bessel function of the second kind
of order N.
*Class.* Elemental function.
*Arguments.*
X shall be of type real. Its value
shall satisfy the inequality X > 0.
N shall be of type integer. Its value
shall satisfy the inequality N >= 0.
It shall be a scalar.
*Result Characteristics.* Same as X.
*Result Value.* The result has a value equal
to a processor-dependent approximation of the Bessel
function of the second kind of the Nth order of X."
[308:20+] Add
"COMP_ERROR_FUNC (X)
*Description.* Complementary error function.
*Class.* Elemental function.
*Argument.* X shall be of type real.
*Result Characteristics.* Same as X.
*Result Value.* The result has a value equal
to a processor-dependent approximation of the
complement (that is, 1.0 - ERROR_FUNC(X)) of the
error function, ERROR_FUNC(X)."
[315:24+] Add
"ERROR_FUNC (X)
*Description.* Error function.
*Class.* Elemental function.
*Argument.* X shall be of type real.
*Result Characteristics.* Same as X.
*Result Value.* The result has a value equal
to a processor-dependent approximation of the
error function,
({2} over {pi} times int {0} {x} exp( -t*t) dt)."
[317:10+] Add
"GAMMA_FUNC (X)
*Description.* Gamma function.
*Class.* Elemental function.
*Argument.* X shall be of type real. Its value
shall satisfy the inequality X >= 0.
*Result Characteristics.* Same as X.
*Result Value.* The result has a value equal
to a processor-dependent approximation
of the gamma function,
(int {0} {inf} exp( -t) t**( x - 1) dt)."
[319:20+] Add
"HYPOT (X,Y)
*Description.* Euclidean distance function
*Class.* Elemental function.
*Argument.*
X shall be of type real.
Y shall be of type real. It shall
have the same kind as X.
*Result Characteristics.* Same as X.
*Result Value.* The result has a value equal
to a processor-dependent approximation of the
Euclidean distance sqrt( x*x + y*y ), taking
precautions against unwarranted overflows."
[329:21+] Add
"LOG_GAMMA_FUNC (X , LOGGAMMA, [, SIGNGAM])
*Description.* log gamma function.
*Class.* Elemental subroutine.
*Argument.*
X shall be of type real. Its value
shall not be a negative integer.
LOGGAMMA shall be of type real and
of the same type kind parameter
as X. It is an INTENT(OUT) argument.
SIGNGAM (optional) shall be of type real and
of the same type kind parameter
as X. It is an INTENT(OUT) argument.
*Result Characteristics.* Same as X.
*Result Value.* The LOGGAMMA has a value equal
to a processor-dependent approximation of the
natural logarithm of the absolute value of the
gamma function,
(int {0} {inf} exp( -t) t**( x - 1) dt).
If present, SIGNGAM is +1.0 if the GAMMA function
is positive, and -1.0 if the GAMMA function
is negative."