src/parsec/disk-image/parsec/parsec-benchmark/pkgs/libs/gsl/src/doc/specfunc-ellint.texi - public/gem5-resources - Git at Google

 @cindex elliptic integrals

 The functions described in this section are declared in the header
 file @file{gsl_sf_ellint.h}.  Further information about the elliptic
 integrals can be found in Abramowitz & Stegun, Chapter 17.

 @menu
 * Definition of Legendre Forms::
 * Definition of Carlson Forms::
 * Legendre Form of Complete Elliptic Integrals::
 * Legendre Form of Incomplete Elliptic Integrals::
 * Carlson Forms::
 @end menu

 @node Definition of Legendre Forms
 @subsection Definition of Legendre Forms
 @cindex Legendre forms of elliptic integrals
 The Legendre forms of elliptic integrals @math{F(\phi,k)},
 @math{E(\phi,k)} and @math{\Pi(\phi,k,n)} are defined by,
 @tex
 \beforedisplay
 $$
 \eqalign{
 F(\phi,k)   &= \int_0^\phi dt {1 \over \sqrt{(1 - k^2 \sin^2(t))}}\cr
 E(\phi,k)   &= \int_0^\phi dt   \sqrt{(1 - k^2 \sin^2(t))}\cr
 \Pi(\phi,k,n) &= \int_0^\phi dt {1 \over (1 + n \sin^2(t)) \sqrt{1 - k^2 \sin^2(t)}}
 }
 $$
 \afterdisplay
 @end tex
 @ifinfo

 @example
   F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t)))

   E(\phi,k) = \int_0^\phi dt   \sqrt((1 - k^2 \sin^2(t)))

 Pi(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t)))
 @end example

 @end ifinfo
 @noindent
 The complete Legendre forms are denoted by @math{K(k) = F(\pi/2, k)} and
 @math{E(k) = E(\pi/2, k)}.

 The notation used here is based on Carlson, @cite{Numerische
 Mathematik} 33 (1979) 1 and differs slightly from that used by
 Abramowitz & Stegun, where the functions are given in terms of the
 parameter @math{m = k^2} and @math{n} is replaced by @math{-n}.

 @node Definition of Carlson Forms
 @subsection Definition of Carlson Forms
 @cindex Carlson forms of Elliptic integrals
 The Carlson symmetric forms of elliptical integrals @math{RC(x,y)},
 @math{RD(x,y,z)}, @math{RF(x,y,z)} and @math{RJ(x,y,z,p)} are defined
 by,
 @tex
 \beforedisplay
 $$
 \eqalign{
 RC(x,y)   &= 1/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1}\cr
 RD(x,y,z) &= 3/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-3/2}\cr
 RF(x,y,z) &= 1/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-1/2}\cr
 RJ(x,y,z,p) &= 3/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-1/2} (t+p)^{-1}
 }
 $$
 \afterdisplay
 @end tex
 @ifinfo

 @example
     RC(x,y) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1)

   RD(x,y,z) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2)

   RF(x,y,z) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2)

 RJ(x,y,z,p) = 3/2 \int_0^\infty dt
                  (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1)
 @end example
 @end ifinfo

 @node Legendre Form of Complete Elliptic Integrals
 @subsection Legendre Form of Complete Elliptic Integrals

 @deftypefun double gsl_sf_ellint_Kcomp (double @var{k}, gsl_mode_t @var{mode})
 @deftypefunx int gsl_sf_ellint_Kcomp_e (double @var{k}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
 These routines compute the complete elliptic integral @math{K(k)} to
 the accuracy specified by the mode variable @var{mode}.
 Note that Abramowitz & Stegun define this function in terms of the
 parameter @math{m = k^2}.
 @comment Exceptional Return Values:  GSL_EDOM
 @end deftypefun

 @deftypefun double gsl_sf_ellint_Ecomp (double @var{k}, gsl_mode_t @var{mode})
 @deftypefunx int gsl_sf_ellint_Ecomp_e (double @var{k}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
 These routines compute the complete elliptic integral @math{E(k)} to the
 accuracy specified by the mode variable @var{mode}.
 Note that Abramowitz & Stegun define this function in terms of the
 parameter @math{m = k^2}.
 @comment Exceptional Return Values:  GSL_EDOM
 @end deftypefun

 @deftypefun double gsl_sf_ellint_Pcomp (double @var{k}, double @var{n}, gsl_mode_t @var{mode})
 @deftypefunx int gsl_sf_ellint_Pcomp_e (double @var{k}, double @var{n},  gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
 These routines compute the complete elliptic integral @math{\Pi(k,n)} to the
 accuracy specified by the mode variable @var{mode}.
 Note that Abramowitz & Stegun define this function in terms of the
 parameters @math{m = k^2} and @math{\sin^2(\alpha) = k^2}, with the
 change of sign @math{n \to -n}.
 @comment Exceptional Return Values:  GSL_EDOM
 @end deftypefun

 @node Legendre Form of Incomplete Elliptic Integrals
 @subsection Legendre Form of Incomplete Elliptic Integrals

 @deftypefun double gsl_sf_ellint_F (double @var{phi}, double @var{k}, gsl_mode_t @var{mode})
 @deftypefunx int gsl_sf_ellint_F_e (double @var{phi}, double @var{k}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
 These routines compute the incomplete elliptic integral @math{F(\phi,k)}
 to the accuracy specified by the mode variable @var{mode}.
 Note that Abramowitz & Stegun define this function in terms of the
 parameter @math{m = k^2}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun

 @deftypefun double gsl_sf_ellint_E (double @var{phi}, double @var{k}, gsl_mode_t @var{mode})
 @deftypefunx int gsl_sf_ellint_E_e (double @var{phi}, double @var{k}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
 These routines compute the incomplete elliptic integral @math{E(\phi,k)}
 to the accuracy specified by the mode variable @var{mode}.
 Note that Abramowitz & Stegun define this function in terms of the
 parameter @math{m = k^2}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun

 @deftypefun double gsl_sf_ellint_P (double @var{phi}, double @var{k}, double @var{n}, gsl_mode_t @var{mode})
 @deftypefunx int gsl_sf_ellint_P_e (double @var{phi}, double @var{k}, double @var{n}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
 These routines compute the incomplete elliptic integral @math{\Pi(\phi,k,n)}
 to the accuracy specified by the mode variable @var{mode}.
 Note that Abramowitz & Stegun define this function in terms of the
 parameters @math{m = k^2} and @math{\sin^2(\alpha) = k^2}, with the
 change of sign @math{n \to -n}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun

 @deftypefun double gsl_sf_ellint_D (double @var{phi}, double @var{k}, double @var{n}, gsl_mode_t @var{mode})
 @deftypefunx int gsl_sf_ellint_D_e (double @var{phi}, double @var{k}, double @var{n}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
 These functions compute the incomplete elliptic integral
 @math{D(\phi,k)} which is defined through the Carlson form @math{RD(x,y,z)}
 by the following relation,
 @tex
 \beforedisplay
 $$
 D(\phi,k,n) = {1 \over 3} (\sin \phi)^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).
 $$
 \afterdisplay
 @end tex
 @ifinfo

 @example
 D(\phi,k,n) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).
 @end example
 @end ifinfo
 The argument @var{n} is not used and will be removed in a future release.

 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun


 @node Carlson Forms
 @subsection Carlson Forms

 @deftypefun double gsl_sf_ellint_RC (double @var{x}, double @var{y}, gsl_mode_t @var{mode})
 @deftypefunx int gsl_sf_ellint_RC_e (double @var{x}, double @var{y}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
 These routines compute the incomplete elliptic integral @math{RC(x,y)}
 to the accuracy specified by the mode variable @var{mode}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun

 @deftypefun double gsl_sf_ellint_RD (double @var{x}, double @var{y}, double @var{z}, gsl_mode_t @var{mode})
 @deftypefunx int gsl_sf_ellint_RD_e (double @var{x}, double @var{y}, double @var{z}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
 These routines compute the incomplete elliptic integral @math{RD(x,y,z)}
 to the accuracy specified by the mode variable @var{mode}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun

 @deftypefun double gsl_sf_ellint_RF (double @var{x}, double @var{y}, double @var{z}, gsl_mode_t @var{mode})
 @deftypefunx int gsl_sf_ellint_RF_e (double @var{x}, double @var{y}, double @var{z}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
 These routines compute the incomplete elliptic integral @math{RF(x,y,z)}
 to the accuracy specified by the mode variable @var{mode}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun

 @deftypefun double gsl_sf_ellint_RJ (double @var{x}, double @var{y}, double @var{z}, double @var{p}, gsl_mode_t @var{mode})
 @deftypefunx int gsl_sf_ellint_RJ_e (double @var{x}, double @var{y}, double @var{z}, double @var{p}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
 These routines compute the incomplete elliptic integral @math{RJ(x,y,z,p)}
 to the accuracy specified by the mode variable @var{mode}.
 @comment Exceptional Return Values: GSL_EDOM
 @end deftypefun
	@cindex elliptic integrals

	The functions described in this section are declared in the header
	file @file{gsl_sf_ellint.h}. Further information about the elliptic
	integrals can be found in Abramowitz & Stegun, Chapter 17.

	@menu
	* Definition of Legendre Forms::
	* Definition of Carlson Forms::
	* Legendre Form of Complete Elliptic Integrals::
	* Legendre Form of Incomplete Elliptic Integrals::
	* Carlson Forms::
	@end menu

	@node Definition of Legendre Forms
	@subsection Definition of Legendre Forms
	@cindex Legendre forms of elliptic integrals
	The Legendre forms of elliptic integrals @math{F(\phi,k)},
	@math{E(\phi,k)} and @math{\Pi(\phi,k,n)} are defined by,
	@tex
	\beforedisplay
	$$
	\eqalign{
	F(\phi,k) &= \int_0^\phi dt {1 \over \sqrt{(1 - k^2 \sin^2(t))}}\cr
	E(\phi,k) &= \int_0^\phi dt \sqrt{(1 - k^2 \sin^2(t))}\cr
	\Pi(\phi,k,n) &= \int_0^\phi dt {1 \over (1 + n \sin^2(t)) \sqrt{1 - k^2 \sin^2(t)}}
	}
	$$
	\afterdisplay
	@end tex
	@ifinfo

	@example
	F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t)))

	E(\phi,k) = \int_0^\phi dt \sqrt((1 - k^2 \sin^2(t)))

	Pi(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t)))
	@end example

	@end ifinfo
	@noindent
	The complete Legendre forms are denoted by @math{K(k) = F(\pi/2, k)} and
	@math{E(k) = E(\pi/2, k)}.

	The notation used here is based on Carlson, @cite{Numerische
	Mathematik} 33 (1979) 1 and differs slightly from that used by
	Abramowitz & Stegun, where the functions are given in terms of the
	parameter @math{m = k^2} and @math{n} is replaced by @math{-n}.

	@node Definition of Carlson Forms
	@subsection Definition of Carlson Forms
	@cindex Carlson forms of Elliptic integrals
	The Carlson symmetric forms of elliptical integrals @math{RC(x,y)},
	@math{RD(x,y,z)}, @math{RF(x,y,z)} and @math{RJ(x,y,z,p)} are defined
	by,
	@tex
	\beforedisplay
	$$
	\eqalign{
	RC(x,y) &= 1/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1}\cr
	RD(x,y,z) &= 3/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-3/2}\cr
	RF(x,y,z) &= 1/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-1/2}\cr
	RJ(x,y,z,p) &= 3/2 \int_0^\infty dt (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-1/2} (t+p)^{-1}
	}
	$$
	\afterdisplay
	@end tex
	@ifinfo

	@example
	RC(x,y) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1)

	RD(x,y,z) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2)

	RF(x,y,z) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2)

	RJ(x,y,z,p) = 3/2 \int_0^\infty dt
	(t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1)
	@end example
	@end ifinfo

	@node Legendre Form of Complete Elliptic Integrals
	@subsection Legendre Form of Complete Elliptic Integrals

	@deftypefun double gsl_sf_ellint_Kcomp (double @var{k}, gsl_mode_t @var{mode})
	@deftypefunx int gsl_sf_ellint_Kcomp_e (double @var{k}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
	These routines compute the complete elliptic integral @math{K(k)} to
	the accuracy specified by the mode variable @var{mode}.
	Note that Abramowitz & Stegun define this function in terms of the
	parameter @math{m = k^2}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun double gsl_sf_ellint_Ecomp (double @var{k}, gsl_mode_t @var{mode})
	@deftypefunx int gsl_sf_ellint_Ecomp_e (double @var{k}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
	These routines compute the complete elliptic integral @math{E(k)} to the
	accuracy specified by the mode variable @var{mode}.
	Note that Abramowitz & Stegun define this function in terms of the
	parameter @math{m = k^2}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun double gsl_sf_ellint_Pcomp (double @var{k}, double @var{n}, gsl_mode_t @var{mode})
	@deftypefunx int gsl_sf_ellint_Pcomp_e (double @var{k}, double @var{n}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
	These routines compute the complete elliptic integral @math{\Pi(k,n)} to the
	accuracy specified by the mode variable @var{mode}.
	Note that Abramowitz & Stegun define this function in terms of the
	parameters @math{m = k^2} and @math{\sin^2(\alpha) = k^2}, with the
	change of sign @math{n \to -n}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@node Legendre Form of Incomplete Elliptic Integrals
	@subsection Legendre Form of Incomplete Elliptic Integrals

	@deftypefun double gsl_sf_ellint_F (double @var{phi}, double @var{k}, gsl_mode_t @var{mode})
	@deftypefunx int gsl_sf_ellint_F_e (double @var{phi}, double @var{k}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
	These routines compute the incomplete elliptic integral @math{F(\phi,k)}
	to the accuracy specified by the mode variable @var{mode}.
	Note that Abramowitz & Stegun define this function in terms of the
	parameter @math{m = k^2}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun double gsl_sf_ellint_E (double @var{phi}, double @var{k}, gsl_mode_t @var{mode})
	@deftypefunx int gsl_sf_ellint_E_e (double @var{phi}, double @var{k}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
	These routines compute the incomplete elliptic integral @math{E(\phi,k)}
	to the accuracy specified by the mode variable @var{mode}.
	Note that Abramowitz & Stegun define this function in terms of the
	parameter @math{m = k^2}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun double gsl_sf_ellint_P (double @var{phi}, double @var{k}, double @var{n}, gsl_mode_t @var{mode})
	@deftypefunx int gsl_sf_ellint_P_e (double @var{phi}, double @var{k}, double @var{n}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
	These routines compute the incomplete elliptic integral @math{\Pi(\phi,k,n)}
	to the accuracy specified by the mode variable @var{mode}.
	Note that Abramowitz & Stegun define this function in terms of the
	parameters @math{m = k^2} and @math{\sin^2(\alpha) = k^2}, with the
	change of sign @math{n \to -n}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun double gsl_sf_ellint_D (double @var{phi}, double @var{k}, double @var{n}, gsl_mode_t @var{mode})
	@deftypefunx int gsl_sf_ellint_D_e (double @var{phi}, double @var{k}, double @var{n}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
	These functions compute the incomplete elliptic integral
	@math{D(\phi,k)} which is defined through the Carlson form @math{RD(x,y,z)}
	by the following relation,
	@tex
	\beforedisplay
	$$
	D(\phi,k,n) = {1 \over 3} (\sin \phi)^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).
	$$
	\afterdisplay
	@end tex
	@ifinfo

	@example
	D(\phi,k,n) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).
	@end example
	@end ifinfo
	The argument @var{n} is not used and will be removed in a future release.

	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun


	@node Carlson Forms
	@subsection Carlson Forms

	@deftypefun double gsl_sf_ellint_RC (double @var{x}, double @var{y}, gsl_mode_t @var{mode})
	@deftypefunx int gsl_sf_ellint_RC_e (double @var{x}, double @var{y}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
	These routines compute the incomplete elliptic integral @math{RC(x,y)}
	to the accuracy specified by the mode variable @var{mode}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun double gsl_sf_ellint_RD (double @var{x}, double @var{y}, double @var{z}, gsl_mode_t @var{mode})
	@deftypefunx int gsl_sf_ellint_RD_e (double @var{x}, double @var{y}, double @var{z}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
	These routines compute the incomplete elliptic integral @math{RD(x,y,z)}
	to the accuracy specified by the mode variable @var{mode}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun double gsl_sf_ellint_RF (double @var{x}, double @var{y}, double @var{z}, gsl_mode_t @var{mode})
	@deftypefunx int gsl_sf_ellint_RF_e (double @var{x}, double @var{y}, double @var{z}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
	These routines compute the incomplete elliptic integral @math{RF(x,y,z)}
	to the accuracy specified by the mode variable @var{mode}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun

	@deftypefun double gsl_sf_ellint_RJ (double @var{x}, double @var{y}, double @var{z}, double @var{p}, gsl_mode_t @var{mode})
	@deftypefunx int gsl_sf_ellint_RJ_e (double @var{x}, double @var{y}, double @var{z}, double @var{p}, gsl_mode_t @var{mode}, gsl_sf_result * @var{result})
	These routines compute the incomplete elliptic integral @math{RJ(x,y,z,p)}
	to the accuracy specified by the mode variable @var{mode}.
	@comment Exceptional Return Values: GSL_EDOM
	@end deftypefun