Manuel Novoa III 417a99ec21 Fix syntax error in comment. 23 rokov pred
..
Makefile 384a7491ad Oops... forgot to remove rint.c from the makefile. 23 rokov pred
README.txt 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
acosh.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
airy.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
arcdot.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
asin.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
asinh.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
atan.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
atanh.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
bdtr.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
bernum.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
beta.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
btdtr.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
cbrt.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
chbevl.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
chdtr.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
cheby.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
clog.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
cmplx.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
coil.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
const.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
cosh.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
cpmul.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
dawsn.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
dcalc.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
dcalc.h 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
dtestvec.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
ei.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
eigens.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
ellie.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
ellik.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
ellpe.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
ellpj.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
ellpk.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
eltst.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
euclid.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
exp.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
exp10.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
exp2.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
expn.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
fabs.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
fac.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
fdtr.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
fftr.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
floor.c 4ab8b2dd6e Remove Erik's broken implementation of rint(). Replace it by one "less broken". 23 rokov pred
fltest.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
fltest2.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
fltest3.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
fresnl.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
gamma.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
gdtr.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
gels.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
hyp2f1.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
hyperg.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
i0.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
i1.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
igam.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
igami.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
incbet.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
incbi.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
isnan.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
iv.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
j0.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
j1.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
jn.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
jv.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
k0.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
k1.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
kn.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
kolmogorov.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
levnsn.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
log.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
log10.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
log2.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
lrand.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
lsqrt.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
ltstd.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
minv.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
mod2pi.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
monot.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
mtherr.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
mtransp.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
mtst.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
nbdtr.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
ndtr.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
ndtri.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
noncephes.c ca3c705bbb Added file for non-Cephes double routines; currently only fmod and modf. 23 rokov pred
paranoia.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
pdtr.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
planck.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
polevl.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
polmisc.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
polrt.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
polylog.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
polyn.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
polyr.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
pow.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
powi.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
psi.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
revers.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
rgamma.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
round.c 417a99ec21 Fix syntax error in comment. 23 rokov pred
setprec.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
shichi.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
sici.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
simpsn.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
simq.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
sin.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
sincos.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
sindg.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
sinh.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
spence.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
sqrt.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
stdtr.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
struve.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
tan.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
tandg.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
tanh.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
time-it.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
unity.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
yn.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
zeta.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred
zetac.c 1077fa4d77 uClibc now has a math library. muahahahaha! 23 rokov pred

README.txt

/* acosh.c
*
* Inverse hyperbolic cosine
*
*
*
* SYNOPSIS:
*
* double x, y, acosh();
*
* y = acosh( x );
*
*
*
* DESCRIPTION:
*
* Returns inverse hyperbolic cosine of argument.
*
* If 1 <= x < 1.5, a rational approximation
*
* sqrt(z) * P(z)/Q(z)
*
* where z = x-1, is used. Otherwise,
*
* acosh(x) = log( x + sqrt( (x-1)(x+1) ).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 1,3 30000 4.2e-17 1.1e-17
* IEEE 1,3 30000 4.6e-16 8.7e-17
*
*
* ERROR MESSAGES:
*
* message condition value returned
* acosh domain |x| < 1 NAN
*
*/

/* airy.c
*
* Airy function
*
*
*
* SYNOPSIS:
*
* double x, ai, aip, bi, bip;
* int airy();
*
* airy( x, _&ai, _&aip, _&bi, _&bip );
*
*
*
* DESCRIPTION:
*
* Solution of the differential equation
*
* y"(x) = xy.
*
* The function returns the two independent solutions Ai, Bi
* and their first derivatives Ai'(x), Bi'(x).
*
* Evaluation is by power series summation for small x,
* by rational minimax approximations for large x.
*
*
*
* ACCURACY:
* Error criterion is absolute when function <= 1, relative
* when function > 1, except * denotes relative error criterion.
* For large negative x, the absolute error increases as x^1.5.
* For large positive x, the relative error increases as x^1.5.
*
* Arithmetic domain function # trials peak rms
* IEEE -10, 0 Ai 10000 1.6e-15 2.7e-16
* IEEE 0, 10 Ai 10000 2.3e-14* 1.8e-15*
* IEEE -10, 0 Ai' 10000 4.6e-15 7.6e-16
* IEEE 0, 10 Ai' 10000 1.8e-14* 1.5e-15*
* IEEE -10, 10 Bi 30000 4.2e-15 5.3e-16
* IEEE -10, 10 Bi' 30000 4.9e-15 7.3e-16
* DEC -10, 0 Ai 5000 1.7e-16 2.8e-17
* DEC 0, 10 Ai 5000 2.1e-15* 1.7e-16*
* DEC -10, 0 Ai' 5000 4.7e-16 7.8e-17
* DEC 0, 10 Ai' 12000 1.8e-15* 1.5e-16*
* DEC -10, 10 Bi 10000 5.5e-16 6.8e-17
* DEC -10, 10 Bi' 7000 5.3e-16 8.7e-17
*
*/

/* asin.c
*
* Inverse circular sine
*
*
*
* SYNOPSIS:
*
* double x, y, asin();
*
* y = asin( x );
*
*
*
* DESCRIPTION:
*
* Returns radian angle between -pi/2 and +pi/2 whose sine is x.
*
* A rational function of the form x + x**3 P(x**2)/Q(x**2)
* is used for |x| in the interval [0, 0.5]. If |x| > 0.5 it is
* transformed by the identity
*
* asin(x) = pi/2 - 2 asin( sqrt( (1-x)/2 ) ).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -1, 1 40000 2.6e-17 7.1e-18
* IEEE -1, 1 10^6 1.9e-16 5.4e-17
*
*
* ERROR MESSAGES:
*
* message condition value returned
* asin domain |x| > 1 NAN
*
*/
/* acos()
*
* Inverse circular cosine
*
*
*
* SYNOPSIS:
*
* double x, y, acos();
*
* y = acos( x );
*
*
*
* DESCRIPTION:
*
* Returns radian angle between 0 and pi whose cosine
* is x.
*
* Analytically, acos(x) = pi/2 - asin(x). However if |x| is
* near 1, there is cancellation error in subtracting asin(x)
* from pi/2. Hence if x < -0.5,
*
* acos(x) = pi - 2.0 * asin( sqrt((1+x)/2) );
*
* or if x > +0.5,
*
* acos(x) = 2.0 * asin( sqrt((1-x)/2) ).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -1, 1 50000 3.3e-17 8.2e-18
* IEEE -1, 1 10^6 2.2e-16 6.5e-17
*
*
* ERROR MESSAGES:
*
* message condition value returned
* asin domain |x| > 1 NAN
*/

/* asinh.c
*
* Inverse hyperbolic sine
*
*
*
* SYNOPSIS:
*
* double x, y, asinh();
*
* y = asinh( x );
*
*
*
* DESCRIPTION:
*
* Returns inverse hyperbolic sine of argument.
*
* If |x| < 0.5, the function is approximated by a rational
* form x + x**3 P(x)/Q(x). Otherwise,
*
* asinh(x) = log( x + sqrt(1 + x*x) ).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -3,3 75000 4.6e-17 1.1e-17
* IEEE -1,1 30000 3.7e-16 7.8e-17
* IEEE 1,3 30000 2.5e-16 6.7e-17
*
*/

/* atan.c
*
* Inverse circular tangent
* (arctangent)
*
*
*
* SYNOPSIS:
*
* double x, y, atan();
*
* y = atan( x );
*
*
*
* DESCRIPTION:
*
* Returns radian angle between -pi/2 and +pi/2 whose tangent
* is x.
*
* Range reduction is from three intervals into the interval
* from zero to 0.66. The approximant uses a rational
* function of degree 4/5 of the form x + x**3 P(x)/Q(x).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10, 10 50000 2.4e-17 8.3e-18
* IEEE -10, 10 10^6 1.8e-16 5.0e-17
*
*/
/* atan2()
*
* Quadrant correct inverse circular tangent
*
*
*
* SYNOPSIS:
*
* double x, y, z, atan2();
*
* z = atan2( y, x );
*
*
*
* DESCRIPTION:
*
* Returns radian angle whose tangent is y/x.
* Define compile time symbol ANSIC = 1 for ANSI standard,
* range -PI < z <= +PI, args (y,x); else ANSIC = 0 for range
* 0 to 2PI, args (x,y).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -10, 10 10^6 2.5e-16 6.9e-17
* See atan.c.
*
*/

/* atanh.c
*
* Inverse hyperbolic tangent
*
*
*
* SYNOPSIS:
*
* double x, y, atanh();
*
* y = atanh( x );
*
*
*
* DESCRIPTION:
*
* Returns inverse hyperbolic tangent of argument in the range
* MINLOG to MAXLOG.
*
* If |x| < 0.5, the rational form x + x**3 P(x)/Q(x) is
* employed. Otherwise,
* atanh(x) = 0.5 * log( (1+x)/(1-x) ).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -1,1 50000 2.4e-17 6.4e-18
* IEEE -1,1 30000 1.9e-16 5.2e-17
*
*/

/* bdtr.c
*
* Binomial distribution
*
*
*
* SYNOPSIS:
*
* int k, n;
* double p, y, bdtr();
*
* y = bdtr( k, n, p );
*
* DESCRIPTION:
*
* Returns the sum of the terms 0 through k of the Binomial
* probability density:
*
* k
* -- ( n ) j n-j
* > ( ) p (1-p)
* -- ( j )
* j=0
*
* The terms are not summed directly; instead the incomplete
* beta integral is employed, according to the formula
*
* y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ).
*
* The arguments must be positive, with p ranging from 0 to 1.
*
* ACCURACY:
*
* Tested at random points (a,b,p), with p between 0 and 1.
*
* a,b Relative error:
* arithmetic domain # trials peak rms
* For p between 0.001 and 1:
* IEEE 0,100 100000 4.3e-15 2.6e-16
* See also incbet.c.
*
* ERROR MESSAGES:
*
* message condition value returned
* bdtr domain k < 0 0.0
* n < k
* x < 0, x > 1
*/
/* bdtrc()
*
* Complemented binomial distribution
*
*
*
* SYNOPSIS:
*
* int k, n;
* double p, y, bdtrc();
*
* y = bdtrc( k, n, p );
*
* DESCRIPTION:
*
* Returns the sum of the terms k+1 through n of the Binomial
* probability density:
*
* n
* -- ( n ) j n-j
* > ( ) p (1-p)
* -- ( j )
* j=k+1
*
* The terms are not summed directly; instead the incomplete
* beta integral is employed, according to the formula
*
* y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ).
*
* The arguments must be positive, with p ranging from 0 to 1.
*
* ACCURACY:
*
* Tested at random points (a,b,p).
*
* a,b Relative error:
* arithmetic domain # trials peak rms
* For p between 0.001 and 1:
* IEEE 0,100 100000 6.7e-15 8.2e-16
* For p between 0 and .001:
* IEEE 0,100 100000 1.5e-13 2.7e-15
*
* ERROR MESSAGES:
*
* message condition value returned
* bdtrc domain x<0, x>1, n */
/* bdtri()
*
* Inverse binomial distribution
*
*
*
* SYNOPSIS:
*
* int k, n;
* double p, y, bdtri();
*
* p = bdtr( k, n, y );
*
* DESCRIPTION:
*
* Finds the event probability p such that the sum of the
* terms 0 through k of the Binomial probability density
* is equal to the given cumulative probability y.
*
* This is accomplished using the inverse beta integral
* function and the relation
*
* 1 - p = incbi( n-k, k+1, y ).
*
* ACCURACY:
*
* Tested at random points (a,b,p).
*
* a,b Relative error:
* arithmetic domain # trials peak rms
* For p between 0.001 and 1:
* IEEE 0,100 100000 2.3e-14 6.4e-16
* IEEE 0,10000 100000 6.6e-12 1.2e-13
* For p between 10^-6 and 0.001:
* IEEE 0,100 100000 2.0e-12 1.3e-14
* IEEE 0,10000 100000 1.5e-12 3.2e-14
* See also incbi.c.
*
* ERROR MESSAGES:
*
* message condition value returned
* bdtri domain k < 0, n <= k 0.0
* x < 0, x > 1
*/

/* beta.c
*
* Beta function
*
*
*
* SYNOPSIS:
*
* double a, b, y, beta();
*
* y = beta( a, b );
*
*
*
* DESCRIPTION:
*
* - -
* | (a) | (b)
* beta( a, b ) = -----------.
* -
* | (a+b)
*
* For large arguments the logarithm of the function is
* evaluated using lgam(), then exponentiated.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0,30 1700 7.7e-15 1.5e-15
* IEEE 0,30 30000 8.1e-14 1.1e-14
*
* ERROR MESSAGES:
*
* message condition value returned
* beta overflow log(beta) > MAXLOG 0.0
* a or b <0 integer 0.0
*
*/

/* btdtr.c
*
* Beta distribution
*
*
*
* SYNOPSIS:
*
* double a, b, x, y, btdtr();
*
* y = btdtr( a, b, x );
*
*
*
* DESCRIPTION:
*
* Returns the area from zero to x under the beta density
* function:
*
*
* x
* - -
* | (a+b) | | a-1 b-1
* P(x) = ---------- | t (1-t) dt
* - - | |
* | (a) | (b) -
* 0
*
*
* This function is identical to the incomplete beta
* integral function incbet(a, b, x).
*
* The complemented function is
*
* 1 - P(1-x) = incbet( b, a, x );
*
*
* ACCURACY:
*
* See incbet.c.
*
*/

/* cbrt.c
*
* Cube root
*
*
*
* SYNOPSIS:
*
* double x, y, cbrt();
*
* y = cbrt( x );
*
*
*
* DESCRIPTION:
*
* Returns the cube root of the argument, which may be negative.
*
* Range reduction involves determining the power of 2 of
* the argument. A polynomial of degree 2 applied to the
* mantissa, and multiplication by the cube root of 1, 2, or 4
* approximates the root to within about 0.1%. Then Newton's
* iteration is used three times to converge to an accurate
* result.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,10 200000 1.8e-17 6.2e-18
* IEEE 0,1e308 30000 1.5e-16 5.0e-17
*
*/

/* chbevl.c
*
* Evaluate Chebyshev series
*
*
*
* SYNOPSIS:
*
* int N;
* double x, y, coef[N], chebevl();
*
* y = chbevl( x, coef, N );
*
*
*
* DESCRIPTION:
*
* Evaluates the series
*
* N-1
* - '
* y = > coef[i] T (x/2)
* - i
* i=0
*
* of Chebyshev polynomials Ti at argument x/2.
*
* Coefficients are stored in reverse order, i.e. the zero
* order term is last in the array. Note N is the number of
* coefficients, not the order.
*
* If coefficients are for the interval a to b, x must
* have been transformed to x -> 2(2x - b - a)/(b-a) before
* entering the routine. This maps x from (a, b) to (-1, 1),
* over which the Chebyshev polynomials are defined.
*
* If the coefficients are for the inverted interval, in
* which (a, b) is mapped to (1/b, 1/a), the transformation
* required is x -> 2(2ab/x - b - a)/(b-a). If b is infinity,
* this becomes x -> 4a/x - 1.
*
*
*
* SPEED:
*
* Taking advantage of the recurrence properties of the
* Chebyshev polynomials, the routine requires one more
* addition per loop than evaluating a nested polynomial of
* the same degree.
*
*/

/* chdtr.c
*
* Chi-square distribution
*
*
*
* SYNOPSIS:
*
* double df, x, y, chdtr();
*
* y = chdtr( df, x );
*
*
*
* DESCRIPTION:
*
* Returns the area under the left hand tail (from 0 to x)
* of the Chi square probability density function with
* v degrees of freedom.
*
*
* inf.
* -
* 1 | | v/2-1 -t/2
* P( x | v ) = ----------- | t e dt
* v/2 - | |
* 2 | (v/2) -
* x
*
* where x is the Chi-square variable.
*
* The incomplete gamma integral is used, according to the
* formula
*
* y = chdtr( v, x ) = igam( v/2.0, x/2.0 ).
*
*
* The arguments must both be positive.
*
*
*
* ACCURACY:
*
* See igam().
*
* ERROR MESSAGES:
*
* message condition value returned
* chdtr domain x < 0 or v < 1 0.0
*/
/* chdtrc()
*
* Complemented Chi-square distribution
*
*
*
* SYNOPSIS:
*
* double v, x, y, chdtrc();
*
* y = chdtrc( v, x );
*
*
*
* DESCRIPTION:
*
* Returns the area under the right hand tail (from x to
* infinity) of the Chi square probability density function
* with v degrees of freedom:
*
*
* inf.
* -
* 1 | | v/2-1 -t/2
* P( x | v ) = ----------- | t e dt
* v/2 - | |
* 2 | (v/2) -
* x
*
* where x is the Chi-square variable.
*
* The incomplete gamma integral is used, according to the
* formula
*
* y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ).
*
*
* The arguments must both be positive.
*
*
*
* ACCURACY:
*
* See igamc().
*
* ERROR MESSAGES:
*
* message condition value returned
* chdtrc domain x < 0 or v < 1 0.0
*/
/* chdtri()
*
* Inverse of complemented Chi-square distribution
*
*
*
* SYNOPSIS:
*
* double df, x, y, chdtri();
*
* x = chdtri( df, y );
*
*
*
*
* DESCRIPTION:
*
* Finds the Chi-square argument x such that the integral
* from x to infinity of the Chi-square density is equal
* to the given cumulative probability y.
*
* This is accomplished using the inverse gamma integral
* function and the relation
*
* x/2 = igami( df/2, y );
*
*
*
*
* ACCURACY:
*
* See igami.c.
*
* ERROR MESSAGES:
*
* message condition value returned
* chdtri domain y < 0 or y > 1 0.0
* v < 1
*
*/

/* clog.c
*
* Complex natural logarithm
*
*
*
* SYNOPSIS:
*
* void clog();
* cmplx z, w;
*
* clog( &z, &w );
*
*
*
* DESCRIPTION:
*
* Returns complex logarithm to the base e (2.718...) of
* the complex argument x.
*
* If z = x + iy, r = sqrt( x**2 + y**2 ),
* then
* w = log(r) + i arctan(y/x).
*
* The arctangent ranges from -PI to +PI.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 7000 8.5e-17 1.9e-17
* IEEE -10,+10 30000 5.0e-15 1.1e-16
*
* Larger relative error can be observed for z near 1 +i0.
* In IEEE arithmetic the peak absolute error is 5.2e-16, rms
* absolute error 1.0e-16.
*/

/* cexp()
*
* Complex exponential function
*
*
*
* SYNOPSIS:
*
* void cexp();
* cmplx z, w;
*
* cexp( &z, &w );
*
*
*
* DESCRIPTION:
*
* Returns the exponential of the complex argument z
* into the complex result w.
*
* If
* z = x + iy,
* r = exp(x),
*
* then
*
* w = r cos y + i r sin y.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 8700 3.7e-17 1.1e-17
* IEEE -10,+10 30000 3.0e-16 8.7e-17
*
*/
/* csin()
*
* Complex circular sine
*
*
*
* SYNOPSIS:
*
* void csin();
* cmplx z, w;
*
* csin( &z, &w );
*
*
*
* DESCRIPTION:
*
* If
* z = x + iy,
*
* then
*
* w = sin x cosh y + i cos x sinh y.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 8400 5.3e-17 1.3e-17
* IEEE -10,+10 30000 3.8e-16 1.0e-16
* Also tested by csin(casin(z)) = z.
*
*/
/* ccos()
*
* Complex circular cosine
*
*
*
* SYNOPSIS:
*
* void ccos();
* cmplx z, w;
*
* ccos( &z, &w );
*
*
*
* DESCRIPTION:
*
* If
* z = x + iy,
*
* then
*
* w = cos x cosh y - i sin x sinh y.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 8400 4.5e-17 1.3e-17
* IEEE -10,+10 30000 3.8e-16 1.0e-16
*/
/* ctan()
*
* Complex circular tangent
*
*
*
* SYNOPSIS:
*
* void ctan();
* cmplx z, w;
*
* ctan( &z, &w );
*
*
*
* DESCRIPTION:
*
* If
* z = x + iy,
*
* then
*
* sin 2x + i sinh 2y
* w = --------------------.
* cos 2x + cosh 2y
*
* On the real axis the denominator is zero at odd multiples
* of PI/2. The denominator is evaluated by its Taylor
* series near these points.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 5200 7.1e-17 1.6e-17
* IEEE -10,+10 30000 7.2e-16 1.2e-16
* Also tested by ctan * ccot = 1 and catan(ctan(z)) = z.
*/
/* ccot()
*
* Complex circular cotangent
*
*
*
* SYNOPSIS:
*
* void ccot();
* cmplx z, w;
*
* ccot( &z, &w );
*
*
*
* DESCRIPTION:
*
* If
* z = x + iy,
*
* then
*
* sin 2x - i sinh 2y
* w = --------------------.
* cosh 2y - cos 2x
*
* On the real axis, the denominator has zeros at even
* multiples of PI/2. Near these points it is evaluated
* by a Taylor series.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 3000 6.5e-17 1.6e-17
* IEEE -10,+10 30000 9.2e-16 1.2e-16
* Also tested by ctan * ccot = 1 + i0.
*/
/* casin()
*
* Complex circular arc sine
*
*
*
* SYNOPSIS:
*
* void casin();
* cmplx z, w;
*
* casin( &z, &w );
*
*
*
* DESCRIPTION:
*
* Inverse complex sine:
*
* 2
* w = -i clog( iz + csqrt( 1 - z ) ).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 10100 2.1e-15 3.4e-16
* IEEE -10,+10 30000 2.2e-14 2.7e-15
* Larger relative error can be observed for z near zero.
* Also tested by csin(casin(z)) = z.
*/

/* cacos()
*
* Complex circular arc cosine
*
*
*
* SYNOPSIS:
*
* void cacos();
* cmplx z, w;
*
* cacos( &z, &w );
*
*
*
* DESCRIPTION:
*
*
* w = arccos z = PI/2 - arcsin z.
*
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 5200 1.6e-15 2.8e-16
* IEEE -10,+10 30000 1.8e-14 2.2e-15
*/
/* catan()
*
* Complex circular arc tangent
*
*
*
* SYNOPSIS:
*
* void catan();
* cmplx z, w;
*
* catan( &z, &w );
*
*
*
* DESCRIPTION:
*
* If
* z = x + iy,
*
* then
* 1 ( 2x )
* Re w = - arctan(-----------) + k PI
* 2 ( 2 2)
* (1 - x - y )
*
* ( 2 2)
* 1 (x + (y+1) )
* Im w = - log(------------)
* 4 ( 2 2)
* (x + (y-1) )
*
* Where k is an arbitrary integer.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 5900 1.3e-16 7.8e-18
* IEEE -10,+10 30000 2.3e-15 8.5e-17
* The check catan( ctan(z) ) = z, with |x| and |y| < PI/2,
* had peak relative error 1.5e-16, rms relative error
* 2.9e-17. See also clog().
*/

/* cmplx.c
*
* Complex number arithmetic
*
*
*
* SYNOPSIS:
*
* typedef struct {
* double r; real part
* double i; imaginary part
* }cmplx;
*
* cmplx *a, *b, *c;
*
* cadd( a, b, c ); c = b + a
* csub( a, b, c ); c = b - a
* cmul( a, b, c ); c = b * a
* cdiv( a, b, c ); c = b / a
* cneg( c ); c = -c
* cmov( b, c ); c = b
*
*
*
* DESCRIPTION:
*
* Addition:
* c.r = b.r + a.r
* c.i = b.i + a.i
*
* Subtraction:
* c.r = b.r - a.r
* c.i = b.i - a.i
*
* Multiplication:
* c.r = b.r * a.r - b.i * a.i
* c.i = b.r * a.i + b.i * a.r
*
* Division:
* d = a.r * a.r + a.i * a.i
* c.r = (b.r * a.r + b.i * a.i)/d
* c.i = (b.i * a.r - b.r * a.i)/d
* ACCURACY:
*
* In DEC arithmetic, the test (1/z) * z = 1 had peak relative
* error 3.1e-17, rms 1.2e-17. The test (y/z) * (z/y) = 1 had
* peak relative error 8.3e-17, rms 2.1e-17.
*
* Tests in the rectangle {-10,+10}:
* Relative error:
* arithmetic function # trials peak rms
* DEC cadd 10000 1.4e-17 3.4e-18
* IEEE cadd 100000 1.1e-16 2.7e-17
* DEC csub 10000 1.4e-17 4.5e-18
* IEEE csub 100000 1.1e-16 3.4e-17
* DEC cmul 3000 2.3e-17 8.7e-18
* IEEE cmul 100000 2.1e-16 6.9e-17
* DEC cdiv 18000 4.9e-17 1.3e-17
* IEEE cdiv 100000 3.7e-16 1.1e-16
*/

/* cabs()
*
* Complex absolute value
*
*
*
* SYNOPSIS:
*
* double cabs();
* cmplx z;
* double a;
*
* a = cabs( &z );
*
*
*
* DESCRIPTION:
*
*
* If z = x + iy
*
* then
*
* a = sqrt( x**2 + y**2 ).
*
* Overflow and underflow are avoided by testing the magnitudes
* of x and y before squaring. If either is outside half of
* the floating point full scale range, both are rescaled.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -30,+30 30000 3.2e-17 9.2e-18
* IEEE -10,+10 100000 2.7e-16 6.9e-17
*/
/* csqrt()
*
* Complex square root
*
*
*
* SYNOPSIS:
*
* void csqrt();
* cmplx z, w;
*
* csqrt( &z, &w );
*
*
*
* DESCRIPTION:
*
*
* If z = x + iy, r = |z|, then
*
* 1/2
* Im w = [ (r - x)/2 ] ,
*
* Re w = y / 2 Im w.
*
*
* Note that -w is also a square root of z. The root chosen
* is always in the upper half plane.
*
* Because of the potential for cancellation error in r - x,
* the result is sharpened by doing a Heron iteration
* (see sqrt.c) in complex arithmetic.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -10,+10 25000 3.2e-17 9.6e-18
* IEEE -10,+10 100000 3.2e-16 7.7e-17
*
* 2
* Also tested by csqrt( z ) = z, and tested by arguments
* close to the real axis.
*/

/* const.c
*
* Globally declared constants
*
*
*
* SYNOPSIS:
*
* extern double nameofconstant;
*
*
*
*
* DESCRIPTION:
*
* This file contains a number of mathematical constants and
* also some needed size parameters of the computer arithmetic.
* The values are supplied as arrays of hexadecimal integers
* for IEEE arithmetic; arrays of octal constants for DEC
* arithmetic; and in a normal decimal scientific notation for
* other machines. The particular notation used is determined
* by a symbol (DEC, IBMPC, or UNK) defined in the include file
* math.h.
*
* The default size parameters are as follows.
*
* For DEC and UNK modes:
* MACHEP = 1.38777878078144567553E-17 2**-56
* MAXLOG = 8.8029691931113054295988E1 log(2**127)
* MINLOG = -8.872283911167299960540E1 log(2**-128)
* MAXNUM = 1.701411834604692317316873e38 2**127
*
* For IEEE arithmetic (IBMPC):
* MACHEP = 1.11022302462515654042E-16 2**-53
* MAXLOG = 7.09782712893383996843E2 log(2**1024)
* MINLOG = -7.08396418532264106224E2 log(2**-1022)
* MAXNUM = 1.7976931348623158E308 2**1024
*
* The global symbols for mathematical constants are
* PI = 3.14159265358979323846 pi
* PIO2 = 1.57079632679489661923 pi/2
* PIO4 = 7.85398163397448309616E-1 pi/4
* SQRT2 = 1.41421356237309504880 sqrt(2)
* SQRTH = 7.07106781186547524401E-1 sqrt(2)/2
* LOG2E = 1.4426950408889634073599 1/log(2)
* SQ2OPI = 7.9788456080286535587989E-1 sqrt( 2/pi )
* LOGE2 = 6.93147180559945309417E-1 log(2)
* LOGSQ2 = 3.46573590279972654709E-1 log(2)/2
* THPIO4 = 2.35619449019234492885 3*pi/4
* TWOOPI = 6.36619772367581343075535E-1 2/pi
*
* These lists are subject to change.
*/

/* cosh.c
*
* Hyperbolic cosine
*
*
*
* SYNOPSIS:
*
* double x, y, cosh();
*
* y = cosh( x );
*
*
*
* DESCRIPTION:
*
* Returns hyperbolic cosine of argument in the range MINLOG to
* MAXLOG.
*
* cosh(x) = ( exp(x) + exp(-x) )/2.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC +- 88 50000 4.0e-17 7.7e-18
* IEEE +-MAXLOG 30000 2.6e-16 5.7e-17
*
*
* ERROR MESSAGES:
*
* message condition value returned
* cosh overflow |x| > MAXLOG MAXNUM
*
*
*/

/* cpmul.c
*
* Multiply two polynomials with complex coefficients
*
*
*
* SYNOPSIS:
*
* typedef struct
* {
* double r;
* double i;
* }cmplx;
*
* cmplx a[], b[], c[];
* int da, db, dc;
*
* cpmul( a, da, b, db, c, &dc );
*
*
*
* DESCRIPTION:
*
* The two argument polynomials are multiplied together, and
* their product is placed in c.
*
* Each polynomial is represented by its coefficients stored
* as an array of complex number structures (see the typedef).
* The degree of a is da, which must be passed to the routine
* as an argument; similarly the degree db of b is an argument.
* Array a has da + 1 elements and array b has db + 1 elements.
* Array c must have storage allocated for at least da + db + 1
* elements. The value da + db is returned in dc; this is
* the degree of the product polynomial.
*
* Polynomial coefficients are stored in ascending order; i.e.,
* a(x) = a[0]*x**0 + a[1]*x**1 + ... + a[da]*x**da.
*
*
* If desired, c may be the same as either a or b, in which
* case the input argument array is replaced by the product
* array (but only up to terms of degree da + db).
*
*/

/* dawsn.c
*
* Dawson's Integral
*
*
*
* SYNOPSIS:
*
* double x, y, dawsn();
*
* y = dawsn( x );
*
*
*
* DESCRIPTION:
*
* Approximates the integral
*
* x
* -
* 2 | | 2
* dawsn(x) = exp( -x ) | exp( t ) dt
* | |
* -
* 0
*
* Three different rational approximations are employed, for
* the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,10 10000 6.9e-16 1.0e-16
* DEC 0,10 6000 7.4e-17 1.4e-17
*
*
*/

/* drand.c
*
* Pseudorandom number generator
*
*
*
* SYNOPSIS:
*
* double y, drand();
*
* drand( &y );
*
*
*
* DESCRIPTION:
*
* Yields a random number 1.0 <= y < 2.0.
*
* The three-generator congruential algorithm by Brian
* Wichmann and David Hill (BYTE magazine, March, 1987,
* pp 127-8) is used. The period, given by them, is
* 6953607871644.
*
* Versions invoked by the different arithmetic compile
* time options DEC, IBMPC, and MIEEE, produce
* approximately the same sequences, differing only in the
* least significant bits of the numbers. The UNK option
* implements the algorithm as recommended in the BYTE
* article. It may be used on all computers. However,
* the low order bits of a double precision number may
* not be adequately random, and may vary due to arithmetic
* implementation details on different computers.
*
* The other compile options generate an additional random
* integer that overwrites the low order bits of the double
* precision number. This reduces the period by a factor of
* two but tends to overcome the problems mentioned.
*
*/

/* eigens.c
*
* Eigenvalues and eigenvectors of a real symmetric matrix
*
*
*
* SYNOPSIS:
*
* int n;
* double A[n*(n+1)/2], EV[n*n], E[n];
* void eigens( A, EV, E, n );
*
*
*
* DESCRIPTION:
*
* The algorithm is due to J. vonNeumann.
*
* A[] is a symmetric matrix stored in lower triangular form.
* That is, A[ row, column ] = A[ (row*row+row)/2 + column ]
* or equivalently with row and column interchanged. The
* indices row and column run from 0 through n-1.
*
* EV[] is the output matrix of eigenvectors stored columnwise.
* That is, the elements of each eigenvector appear in sequential
* memory order. The jth element of the ith eigenvector is
* EV[ n*i+j ] = EV[i][j].
*
* E[] is the output matrix of eigenvalues. The ith element
* of E corresponds to the ith eigenvector (the ith row of EV).
*
* On output, the matrix A will have been diagonalized and its
* orginal contents are destroyed.
*
* ACCURACY:
*
* The error is controlled by an internal parameter called RANGE
* which is set to 1e-10. After diagonalization, the
* off-diagonal elements of A will have been reduced by
* this factor.
*
* ERROR MESSAGES:
*
* None.
*
*/

/* ellie.c
*
* Incomplete elliptic integral of the second kind
*
*
*
* SYNOPSIS:
*
* double phi, m, y, ellie();
*
* y = ellie( phi, m );
*
*
*
* DESCRIPTION:
*
* Approximates the integral
*
*
* phi
* -
* | |
* | 2
* E(phi_\m) = | sqrt( 1 - m sin t ) dt
* |
* | |
* -
* 0
*
* of amplitude phi and modulus m, using the arithmetic -
* geometric mean algorithm.
*
*
*
* ACCURACY:
*
* Tested at random arguments with phi in [-10, 10] and m in
* [0, 1].
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0,2 2000 1.9e-16 3.4e-17
* IEEE -10,10 150000 3.3e-15 1.4e-16
*
*
*/

/* ellik.c
*
* Incomplete elliptic integral of the first kind
*
*
*
* SYNOPSIS:
*
* double phi, m, y, ellik();
*
* y = ellik( phi, m );
*
*
*
* DESCRIPTION:
*
* Approximates the integral
*
*
*
* phi
* -
* | |
* | dt
* F(phi_\m) = | ------------------
* | 2
* | | sqrt( 1 - m sin t )
* -
* 0
*
* of amplitude phi and modulus m, using the arithmetic -
* geometric mean algorithm.
*
*
*
*
* ACCURACY:
*
* Tested at random points with m in [0, 1] and phi as indicated.
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -10,10 200000 7.4e-16 1.0e-16
*
*
*/

/* ellpe.c
*
* Complete elliptic integral of the second kind
*
*
*
* SYNOPSIS:
*
* double m1, y, ellpe();
*
* y = ellpe( m1 );
*
*
*
* DESCRIPTION:
*
* Approximates the integral
*
*
* pi/2
* -
* | | 2
* E(m) = | sqrt( 1 - m sin t ) dt
* | |
* -
* 0
*
* Where m = 1 - m1, using the approximation
*
* P(x) - x log x Q(x).
*
* Though there are no singularities, the argument m1 is used
* rather than m for compatibility with ellpk().
*
* E(1) = 1; E(0) = pi/2.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0, 1 13000 3.1e-17 9.4e-18
* IEEE 0, 1 10000 2.1e-16 7.3e-17
*
*
* ERROR MESSAGES:
*
* message condition value returned
* ellpe domain x<0, x>1 0.0
*
*/

/* ellpj.c
*
* Jacobian Elliptic Functions
*
*
*
* SYNOPSIS:
*
* double u, m, sn, cn, dn, phi;
* int ellpj();
*
* ellpj( u, m, _&sn, _&cn, _&dn, _&phi );
*
*
*
* DESCRIPTION:
*
*
* Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m),
* and dn(u|m) of parameter m between 0 and 1, and real
* argument u.
*
* These functions are periodic, with quarter-period on the
* real axis equal to the complete elliptic integral
* ellpk(1.0-m).
*
* Relation to incomplete elliptic integral:
* If u = ellik(phi,m), then sn(u|m) = sin(phi),
* and cn(u|m) = cos(phi). Phi is called the amplitude of u.
*
* Computation is by means of the arithmetic-geometric mean
* algorithm, except when m is within 1e-9 of 0 or 1. In the
* latter case with m close to 1, the approximation applies
* only for phi < pi/2.
*
* ACCURACY:
*
* Tested at random points with u between 0 and 10, m between
* 0 and 1.
*
* Absolute error (* = relative error):
* arithmetic function # trials peak rms
* DEC sn 1800 4.5e-16 8.7e-17
* IEEE phi 10000 9.2e-16* 1.4e-16*
* IEEE sn 50000 4.1e-15 4.6e-16
* IEEE cn 40000 3.6e-15 4.4e-16
* IEEE dn 10000 1.3e-12 1.8e-14
*
* Peak error observed in consistency check using addition
* theorem for sn(u+v) was 4e-16 (absolute). Also tested by
* the above relation to the incomplete elliptic integral.
* Accuracy deteriorates when u is large.
*
*/

/* ellpk.c
*
* Complete elliptic integral of the first kind
*
*
*
* SYNOPSIS:
*
* double m1, y, ellpk();
*
* y = ellpk( m1 );
*
*
*
* DESCRIPTION:
*
* Approximates the integral
*
*
*
* pi/2
* -
* | |
* | dt
* K(m) = | ------------------
* | 2
* | | sqrt( 1 - m sin t )
* -
* 0
*
* where m = 1 - m1, using the approximation
*
* P(x) - log x Q(x).
*
* The argument m1 is used rather than m so that the logarithmic
* singularity at m = 1 will be shifted to the origin; this
* preserves maximum accuracy.
*
* K(0) = pi/2.
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0,1 16000 3.5e-17 1.1e-17
* IEEE 0,1 30000 2.5e-16 6.8e-17
*
* ERROR MESSAGES:
*
* message condition value returned
* ellpk domain x<0, x>1 0.0
*
*/

/* euclid.c
*
* Rational arithmetic routines
*
*
*
* SYNOPSIS:
*
*
* typedef struct
* {
* double n; numerator
* double d; denominator
* }fract;
*
* radd( a, b, c ) c = b + a
* rsub( a, b, c ) c = b - a
* rmul( a, b, c ) c = b * a
* rdiv( a, b, c ) c = b / a
* euclid( &n, &d ) Reduce n/d to lowest terms,
* return greatest common divisor.
*
* Arguments of the routines are pointers to the structures.
* The double precision numbers are assumed, without checking,
* to be integer valued. Overflow conditions are reported.
*/

/* exp.c
*
* Exponential function
*
*
*
* SYNOPSIS:
*
* double x, y, exp();
*
* y = exp( x );
*
*
*
* DESCRIPTION:
*
* Returns e (2.71828...) raised to the x power.
*
* Range reduction is accomplished by separating the argument
* into an integer k and fraction f such that
*
* x k f
* e = 2 e.
*
* A Pade' form 1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
* of degree 2/3 is used to approximate exp(f) in the basic
* interval [-0.5, 0.5].
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC +- 88 50000 2.8e-17 7.0e-18
* IEEE +- 708 40000 2.0e-16 5.6e-17
*
*
* Error amplification in the exponential function can be
* a serious matter. The error propagation involves
* exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ),
* which shows that a 1 lsb error in representing X produces
* a relative error of X times 1 lsb in the function.
* While the routine gives an accurate result for arguments
* that are exactly represented by a double precision
* computer number, the result contains amplified roundoff
* error for large arguments not exactly represented.
*
*
* ERROR MESSAGES:
*
* message condition value returned
* exp underflow x < MINLOG 0.0
* exp overflow x > MAXLOG INFINITY
*
*/

/* exp10.c
*
* Base 10 exponential function
* (Common antilogarithm)
*
*
*
* SYNOPSIS:
*
* double x, y, exp10();
*
* y = exp10( x );
*
*
*
* DESCRIPTION:
*
* Returns 10 raised to the x power.
*
* Range reduction is accomplished by expressing the argument
* as 10**x = 2**n 10**f, with |f| < 0.5 log10(2).
* The Pade' form
*
* 1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
*
* is used to approximate 10**f.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -307,+307 30000 2.2e-16 5.5e-17
* Test result from an earlier version (2.1):
* DEC -38,+38 70000 3.1e-17 7.0e-18
*
* ERROR MESSAGES:
*
* message condition value returned
* exp10 underflow x < -MAXL10 0.0
* exp10 overflow x > MAXL10 MAXNUM
*
* DEC arithmetic: MAXL10 = 38.230809449325611792.
* IEEE arithmetic: MAXL10 = 308.2547155599167.
*
*/

/* exp2.c
*
* Base 2 exponential function
*
*
*
* SYNOPSIS:
*
* double x, y, exp2();
*
* y = exp2( x );
*
*
*
* DESCRIPTION:
*
* Returns 2 raised to the x power.
*
* Range reduction is accomplished by separating the argument
* into an integer k and fraction f such that
* x k f
* 2 = 2 2.
*
* A Pade' form
*
* 1 + 2x P(x**2) / (Q(x**2) - x P(x**2) )
*
* approximates 2**x in the basic range [-0.5, 0.5].
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -1022,+1024 30000 1.8e-16 5.4e-17
*
*
* See exp.c for comments on error amplification.
*
*
* ERROR MESSAGES:
*
* message condition value returned
* exp underflow x < -MAXL2 0.0
* exp overflow x > MAXL2 MAXNUM
*
* For DEC arithmetic, MAXL2 = 127.
* For IEEE arithmetic, MAXL2 = 1024.
*/

/* expn.c
*
* Exponential integral En
*
*
*
* SYNOPSIS:
*
* int n;
* double x, y, expn();
*
* y = expn( n, x );
*
*
*
* DESCRIPTION:
*
* Evaluates the exponential integral
*
* inf.
* -
* | | -xt
* | e
* E (x) = | ---- dt.
* n | n
* | | t
* -
* 1
*
*
* Both n and x must be nonnegative.
*
* The routine employs either a power series, a continued
* fraction, or an asymptotic formula depending on the
* relative values of n and x.
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0, 30 5000 2.0e-16 4.6e-17
* IEEE 0, 30 10000 1.7e-15 3.6e-16
*
*/

/* fabs.c
*
* Absolute value
*
*
*
* SYNOPSIS:
*
* double x, y;
*
* y = fabs( x );
*
*
*
* DESCRIPTION:
*
* Returns the absolute value of the argument.
*
*/

/* fac.c
*
* Factorial function
*
*
*
* SYNOPSIS:
*
* double y, fac();
* int i;
*
* y = fac( i );
*
*
*
* DESCRIPTION:
*
* Returns factorial of i = 1 * 2 * 3 * ... * i.
* fac(0) = 1.0.
*
* Due to machine arithmetic bounds the largest value of
* i accepted is 33 in DEC arithmetic or 170 in IEEE
* arithmetic. Greater values, or negative ones,
* produce an error message and return MAXNUM.
*
*
*
* ACCURACY:
*
* For i < 34 the values are simply tabulated, and have
* full machine accuracy. If i > 55, fac(i) = gamma(i+1);
* see gamma.c.
*
* Relative error:
* arithmetic domain peak
* IEEE 0, 170 1.4e-15
* DEC 0, 33 1.4e-17
*
*/

/* fdtr.c
*
* F distribution
*
*
*
* SYNOPSIS:
*
* int df1, df2;
* double x, y, fdtr();
*
* y = fdtr( df1, df2, x );
*
* DESCRIPTION:
*
* Returns the area from zero to x under the F density
* function (also known as Snedcor's density or the
* variance ratio density). This is the density
* of x = (u1/df1)/(u2/df2), where u1 and u2 are random
* variables having Chi square distributions with df1
* and df2 degrees of freedom, respectively.
*
* The incomplete beta integral is used, according to the
* formula
*
* P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ).
*
*
* The arguments a and b are greater than zero, and x is
* nonnegative.
*
* ACCURACY:
*
* Tested at random points (a,b,x).
*
* x a,b Relative error:
* arithmetic domain domain # trials peak rms
* IEEE 0,1 0,100 100000 9.8e-15 1.7e-15
* IEEE 1,5 0,100 100000 6.5e-15 3.5e-16
* IEEE 0,1 1,10000 100000 2.2e-11 3.3e-12
* IEEE 1,5 1,10000 100000 1.1e-11 1.7e-13
* See also incbet.c.
*
*
* ERROR MESSAGES:
*
* message condition value returned
* fdtr domain a<0, b<0, x<0 0.0
*
*/
/* fdtrc()
*
* Complemented F distribution
*
*
*
* SYNOPSIS:
*
* int df1, df2;
* double x, y, fdtrc();
*
* y = fdtrc( df1, df2, x );
*
* DESCRIPTION:
*
* Returns the area from x to infinity under the F density
* function (also known as Snedcor's density or the
* variance ratio density).
*
*
* inf.
* -
* 1 | | a-1 b-1
* 1-P(x) = ------ | t (1-t) dt
* B(a,b) | |
* -
* x
*
*
* The incomplete beta integral is used, according to the
* formula
*
* P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ).
*
*
* ACCURACY:
*
* Tested at random points (a,b,x) in the indicated intervals.
* x a,b Relative error:
* arithmetic domain domain # trials peak rms
* IEEE 0,1 1,100 100000 3.7e-14 5.9e-16
* IEEE 1,5 1,100 100000 8.0e-15 1.6e-15
* IEEE 0,1 1,10000 100000 1.8e-11 3.5e-13
* IEEE 1,5 1,10000 100000 2.0e-11 3.0e-12
* See also incbet.c.
*
* ERROR MESSAGES:
*
* message condition value returned
* fdtrc domain a<0, b<0, x<0 0.0
*
*/
/* fdtri()
*
* Inverse of complemented F distribution
*
*
*
* SYNOPSIS:
*
* int df1, df2;
* double x, p, fdtri();
*
* x = fdtri( df1, df2, p );
*
* DESCRIPTION:
*
* Finds the F density argument x such that the integral
* from x to infinity of the F density is equal to the
* given probability p.
*
* This is accomplished using the inverse beta integral
* function and the relations
*
* z = incbi( df2/2, df1/2, p )
* x = df2 (1-z) / (df1 z).
*
* Note: the following relations hold for the inverse of
* the uncomplemented F distribution:
*
* z = incbi( df1/2, df2/2, p )
* x = df2 z / (df1 (1-z)).
*
* ACCURACY:
*
* Tested at random points (a,b,p).
*
* a,b Relative error:
* arithmetic domain # trials peak rms
* For p between .001 and 1:
* IEEE 1,100 100000 8.3e-15 4.7e-16
* IEEE 1,10000 100000 2.1e-11 1.4e-13
* For p between 10^-6 and 10^-3:
* IEEE 1,100 50000 1.3e-12 8.4e-15
* IEEE 1,10000 50000 3.0e-12 4.8e-14
* See also fdtrc.c.
*
* ERROR MESSAGES:
*
* message condition value returned
* fdtri domain p <= 0 or p > 1 0.0
* v < 1
*
*/

/* fftr.c
*
* FFT of Real Valued Sequence
*
*
*
* SYNOPSIS:
*
* double x[], sine[];
* int m;
*
* fftr( x, m, sine );
*
*
*
* DESCRIPTION:
*
* Computes the (complex valued) discrete Fourier transform of
* the real valued sequence x[]. The input sequence x[] contains
* n = 2**m samples. The program fills array sine[k] with
* n/4 + 1 values of sin( 2 PI k / n ).
*
* Data format for complex valued output is real part followed
* by imaginary part. The output is developed in the input
* array x[].
*
* The algorithm takes advantage of the fact that the FFT of an
* n point real sequence can be obtained from an n/2 point
* complex FFT.
*
* A radix 2 FFT algorithm is used.
*
* Execution time on an LSI-11/23 with floating point chip
* is 1.0 sec for n = 256.
*
*
*
* REFERENCE:
*
* E. Oran Brigham, The Fast Fourier Transform;
* Prentice-Hall, Inc., 1974
*
*/

/* ceil()
* floor()
* frexp()
* ldexp()
* signbit()
* isnan()
* isfinite()
*
* Floating point numeric utilities
*
*
*
* SYNOPSIS:
*
* double ceil(), floor(), frexp(), ldexp();
* int signbit(), isnan(), isfinite();
* double x, y;
* int expnt, n;
*
* y = floor(x);
* y = ceil(x);
* y = frexp( x, &expnt );
* y = ldexp( x, n );
* n = signbit(x);
* n = isnan(x);
* n = isfinite(x);
*
*
*
* DESCRIPTION:
*
* All four routines return a double precision floating point
* result.
*
* floor() returns the largest integer less than or equal to x.
* It truncates toward minus infinity.
*
* ceil() returns the smallest integer greater than or equal
* to x. It truncates toward plus infinity.
*
* frexp() extracts the exponent from x. It returns an integer
* power of two to expnt and the significand between 0.5 and 1
* to y. Thus x = y * 2**expn.
*
* ldexp() multiplies x by 2**n.
*
* signbit(x) returns 1 if the sign bit of x is 1, else 0.
*
* These functions are part of the standard C run time library
* for many but not all C compilers. The ones supplied are
* written in C for either DEC or IEEE arithmetic. They should
* be used only if your compiler library does not already have
* them.
*
* The IEEE versions assume that denormal numbers are implemented
* in the arithmetic. Some modifications will be required if
* the arithmetic has abrupt rather than gradual underflow.
*/

/* fresnl.c
*
* Fresnel integral
*
*
*
* SYNOPSIS:
*
* double x, S, C;
* void fresnl();
*
* fresnl( x, _&S, _&C );
*
*
* DESCRIPTION:
*
* Evaluates the Fresnel integrals
*
* x
* -
* | |
* C(x) = | cos(pi/2 t**2) dt,
* | |
* -
* 0
*
* x
* -
* | |
* S(x) = | sin(pi/2 t**2) dt.
* | |
* -
* 0
*
*
* The integrals are evaluated by a power series for x < 1.
* For x >= 1 auxiliary functions f(x) and g(x) are employed
* such that
*
* C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 )
* S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 )
*
*
*
* ACCURACY:
*
* Relative error.
*
* Arithmetic function domain # trials peak rms
* IEEE S(x) 0, 10 10000 2.0e-15 3.2e-16
* IEEE C(x) 0, 10 10000 1.8e-15 3.3e-16
* DEC S(x) 0, 10 6000 2.2e-16 3.9e-17
* DEC C(x) 0, 10 5000 2.3e-16 3.9e-17
*/

/* gamma.c
*
* Gamma function
*
*
*
* SYNOPSIS:
*
* double x, y, gamma();
* extern int sgngam;
*
* y = gamma( x );
*
*
*
* DESCRIPTION:
*
* Returns gamma function of the argument. The result is
* correctly signed, and the sign (+1 or -1) is also
* returned in a global (extern) variable named sgngam.
* This variable is also filled in by the logarithmic gamma
* function lgam().
*
* Arguments |x| <= 34 are reduced by recurrence and the function
* approximated by a rational function of degree 6/7 in the
* interval (2,3). Large arguments are handled by Stirling's
* formula. Large negative arguments are made positive using
* a reflection formula.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -34, 34 10000 1.3e-16 2.5e-17
* IEEE -170,-33 20000 2.3e-15 3.3e-16
* IEEE -33, 33 20000 9.4e-16 2.2e-16
* IEEE 33, 171.6 20000 2.3e-15 3.2e-16
*
* Error for arguments outside the test range will be larger
* owing to error amplification by the exponential function.
*
*/
/* lgam()
*
* Natural logarithm of gamma function
*
*
*
* SYNOPSIS:
*
* double x, y, lgam();
* extern int sgngam;
*
* y = lgam( x );
*
*
*
* DESCRIPTION:
*
* Returns the base e (2.718...) logarithm of the absolute
* value of the gamma function of the argument.
* The sign (+1 or -1) of the gamma function is returned in a
* global (extern) variable named sgngam.
*
* For arguments greater than 13, the logarithm of the gamma
* function is approximated by the logarithmic version of
* Stirling's formula using a polynomial approximation of
* degree 4. Arguments between -33 and +33 are reduced by
* recurrence to the interval [2,3] of a rational approximation.
* The cosecant reflection formula is employed for arguments
* less than -33.
*
* Arguments greater than MAXLGM return MAXNUM and an error
* message. MAXLGM = 2.035093e36 for DEC
* arithmetic or 2.556348e305 for IEEE arithmetic.
*
*
*
* ACCURACY:
*
*
* arithmetic domain # trials peak rms
* DEC 0, 3 7000 5.2e-17 1.3e-17
* DEC 2.718, 2.035e36 5000 3.9e-17 9.9e-18
* IEEE 0, 3 28000 5.4e-16 1.1e-16
* IEEE 2.718, 2.556e305 40000 3.5e-16 8.3e-17
* The error criterion was relative when the function magnitude
* was greater than one but absolute when it was less than one.
*
* The following test used the relative error criterion, though
* at certain points the relative error could be much higher than
* indicated.
* IEEE -200, -4 10000 4.8e-16 1.3e-16
*
*/

/* gdtr.c
*
* Gamma distribution function
*
*
*
* SYNOPSIS:
*
* double a, b, x, y, gdtr();
*
* y = gdtr( a, b, x );
*
*
*
* DESCRIPTION:
*
* Returns the integral from zero to x of the gamma probability
* density function:
*
*
* x
* b -
* a | | b-1 -at
* y = ----- | t e dt
* - | |
* | (b) -
* 0
*
* The incomplete gamma integral is used, according to the
* relation
*
* y = igam( b, ax ).
*
*
* ACCURACY:
*
* See igam().
*
* ERROR MESSAGES:
*
* message condition value returned
* gdtr domain x < 0 0.0
*
*/
/* gdtrc.c
*
* Complemented gamma distribution function
*
*
*
* SYNOPSIS:
*
* double a, b, x, y, gdtrc();
*
* y = gdtrc( a, b, x );
*
*
*
* DESCRIPTION:
*
* Returns the integral from x to infinity of the gamma
* probability density function:
*
*
* inf.
* b -
* a | | b-1 -at
* y = ----- | t e dt
* - | |
* | (b) -
* x
*
* The incomplete gamma integral is used, according to the
* relation
*
* y = igamc( b, ax ).
*
*
* ACCURACY:
*
* See igamc().
*
* ERROR MESSAGES:
*
* message condition value returned
* gdtrc domain x < 0 0.0
*
*/

/*
C
C ..................................................................
C
C SUBROUTINE GELS
C
C PURPOSE
C TO SOLVE A SYSTEM OF SIMULTANEOUS LINEAR EQUATIONS WITH
C SYMMETRIC COEFFICIENT MATRIX UPPER TRIANGULAR PART OF WHICH
C IS ASSUMED TO BE STORED COLUMNWISE.
C
C USAGE
C CALL GELS(R,A,M,N,EPS,IER,AUX)
C
C DESCRIPTION OF PARAMETERS
C R - M BY N RIGHT HAND SIDE MATRIX. (DESTROYED)
C ON RETURN R CONTAINS THE SOLUTION OF THE EQUATIONS.
C A - UPPER TRIANGULAR PART OF THE SYMMETRIC
C M BY M COEFFICIENT MATRIX. (DESTROYED)
C M - THE NUMBER OF EQUATIONS IN THE SYSTEM.
C N - THE NUMBER OF RIGHT HAND SIDE VECTORS.
C EPS - AN INPUT CONSTANT WHICH IS USED AS RELATIVE
C TOLERANCE FOR TEST ON LOSS OF SIGNIFICANCE.
C IER - RESULTING ERROR PARAMETER CODED AS FOLLOWS
C IER=0 - NO ERROR,
C IER=-1 - NO RESULT BECAUSE OF M LESS THAN 1 OR
C PIVOT ELEMENT AT ANY ELIMINATION STEP
C EQUAL TO 0,
C IER=K - WARNING DUE TO POSSIBLE LOSS OF SIGNIFI-
C CANCE INDICATED AT ELIMINATION STEP K+1,
C WHERE PIVOT ELEMENT WAS LESS THAN OR
C EQUAL TO THE INTERNAL TOLERANCE EPS TIMES
C ABSOLUTELY GREATEST MAIN DIAGONAL
C ELEMENT OF MATRIX A.
C AUX - AN AUXILIARY STORAGE ARRAY WITH DIMENSION M-1.
C
C REMARKS
C UPPER TRIANGULAR PART OF MATRIX A IS ASSUMED TO BE STORED
C COLUMNWISE IN M*(M+1)/2 SUCCESSIVE STORAGE LOCATIONS, RIGHT
C HAND SIDE MATRIX R COLUMNWISE IN N*M SUCCESSIVE STORAGE
C LOCATIONS. ON RETURN SOLUTION MATRIX R IS STORED COLUMNWISE
C TOO.
C THE PROCEDURE GIVES RESULTS IF THE NUMBER OF EQUATIONS M IS
C GREATER THAN 0 AND PIVOT ELEMENTS AT ALL ELIMINATION STEPS
C ARE DIFFERENT FROM 0. HOWEVER WARNING IER=K - IF GIVEN -
C INDICATES POSSIBLE LOSS OF SIGNIFICANCE. IN CASE OF A WELL
C SCALED MATRIX A AND APPROPRIATE TOLERANCE EPS, IER=K MAY BE
C INTERPRETED THAT MATRIX A HAS THE RANK K. NO WARNING IS
C GIVEN IN CASE M=1.
C ERROR PARAMETER IER=-1 DOES NOT NECESSARILY MEAN THAT
C MATRIX A IS SINGULAR, AS ONLY MAIN DIAGONAL ELEMENTS
C ARE USED AS PIVOT ELEMENTS. POSSIBLY SUBROUTINE GELG (WHICH
C WORKS WITH TOTAL PIVOTING) WOULD BE ABLE TO FIND A SOLUTION.
C
C SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
C NONE
C
C METHOD
C SOLUTION IS DONE BY MEANS OF GAUSS-ELIMINATION WITH
C PIVOTING IN MAIN DIAGONAL, IN ORDER TO PRESERVE
C SYMMETRY IN REMAINING COEFFICIENT MATRICES.
C
C ..................................................................
C
*/

/* hyp2f1.c
*
* Gauss hypergeometric function F
* 2 1
*
*
* SYNOPSIS:
*
* double a, b, c, x, y, hyp2f1();
*
* y = hyp2f1( a, b, c, x );
*
*
* DESCRIPTION:
*
*
* hyp2f1( a, b, c, x ) = F ( a, b; c; x )
* 2 1
*
* inf.
* - a(a+1)...(a+k) b(b+1)...(b+k) k+1
* = 1 + > ----------------------------- x .
* - c(c+1)...(c+k) (k+1)!
* k = 0
*
* Cases addressed are
* Tests and escapes for negative integer a, b, or c
* Linear transformation if c - a or c - b negative integer
* Special case c = a or c = b
* Linear transformation for x near +1
* Transformation for x < -0.5
* Psi function expansion if x > 0.5 and c - a - b integer
* Conditionally, a recurrence on c to make c-a-b > 0
*
* |x| > 1 is rejected.
*
* The parameters a, b, c are considered to be integer
* valued if they are within 1.0e-14 of the nearest integer
* (1.0e-13 for IEEE arithmetic).
*
* ACCURACY:
*
*
* Relative error (-1 < x < 1):
* arithmetic domain # trials peak rms
* IEEE -1,7 230000 1.2e-11 5.2e-14
*
* Several special cases also tested with a, b, c in
* the range -7 to 7.
*
* ERROR MESSAGES:
*
* A "partial loss of precision" message is printed if
* the internally estimated relative error exceeds 1^-12.
* A "singularity" message is printed on overflow or
* in cases not addressed (such as x < -1).
*/

/* hyperg.c
*
* Confluent hypergeometric function
*
*
*
* SYNOPSIS:
*
* double a, b, x, y, hyperg();
*
* y = hyperg( a, b, x );
*
*
*
* DESCRIPTION:
*
* Computes the confluent hypergeometric function
*
* 1 2
* a x a(a+1) x
* F ( a,b;x ) = 1 + ---- + --------- + ...
* 1 1 b 1! b(b+1) 2!
*
* Many higher transcendental functions are special cases of
* this power series.
*
* As is evident from the formula, b must not be a negative
* integer or zero unless a is an integer with 0 >= a > b.
*
* The routine attempts both a direct summation of the series
* and an asymptotic expansion. In each case error due to
* roundoff, cancellation, and nonconvergence is estimated.
* The result with smaller estimated error is returned.
*
*
*
* ACCURACY:
*
* Tested at random points (a, b, x), all three variables
* ranging from 0 to 30.
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0,30 2000 1.2e-15 1.3e-16
* IEEE 0,30 30000 1.8e-14 1.1e-15
*
* Larger errors can be observed when b is near a negative
* integer or zero. Certain combinations of arguments yield
* serious cancellation error in the power series summation
* and also are not in the region of near convergence of the
* asymptotic series. An error message is printed if the
* self-estimated relative error is greater than 1.0e-12.
*
*/

/* i0.c
*
* Modified Bessel function of order zero
*
*
*
* SYNOPSIS:
*
* double x, y, i0();
*
* y = i0( x );
*
*
*
* DESCRIPTION:
*
* Returns modified Bessel function of order zero of the
* argument.
*
* The function is defined as i0(x) = j0( ix ).
*
* The range is partitioned into the two intervals [0,8] and
* (8, infinity). Chebyshev polynomial expansions are employed
* in each interval.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0,30 6000 8.2e-17 1.9e-17
* IEEE 0,30 30000 5.8e-16 1.4e-16
*
*/
/* i0e.c
*
* Modified Bessel function of order zero,
* exponentially scaled
*
*
*
* SYNOPSIS:
*
* double x, y, i0e();
*
* y = i0e( x );
*
*
*
* DESCRIPTION:
*
* Returns exponentially scaled modified Bessel function
* of order zero of the argument.
*
* The function is defined as i0e(x) = exp(-|x|) j0( ix ).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,30 30000 5.4e-16 1.2e-16
* See i0().
*
*/

/* i1.c
*
* Modified Bessel function of order one
*
*
*
* SYNOPSIS:
*
* double x, y, i1();
*
* y = i1( x );
*
*
*
* DESCRIPTION:
*
* Returns modified Bessel function of order one of the
* argument.
*
* The function is defined as i1(x) = -i j1( ix ).
*
* The range is partitioned into the two intervals [0,8] and
* (8, infinity). Chebyshev polynomial expansions are employed
* in each interval.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0, 30 3400 1.2e-16 2.3e-17
* IEEE 0, 30 30000 1.9e-15 2.1e-16
*
*
*/
/* i1e.c
*
* Modified Bessel function of order one,
* exponentially scaled
*
*
*
* SYNOPSIS:
*
* double x, y, i1e();
*
* y = i1e( x );
*
*
*
* DESCRIPTION:
*
* Returns exponentially scaled modified Bessel function
* of order one of the argument.
*
* The function is defined as i1(x) = -i exp(-|x|) j1( ix ).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0, 30 30000 2.0e-15 2.0e-16
* See i1().
*
*/

/* igam.c
*
* Incomplete gamma integral
*
*
*
* SYNOPSIS:
*
* double a, x, y, igam();
*
* y = igam( a, x );
*
* DESCRIPTION:
*
* The function is defined by
*
* x
* -
* 1 | | -t a-1
* igam(a,x) = ----- | e t dt.
* - | |
* | (a) -
* 0
*
*
* In this implementation both arguments must be positive.
* The integral is evaluated by either a power series or
* continued fraction expansion, depending on the relative
* values of a and x.
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,30 200000 3.6e-14 2.9e-15
* IEEE 0,100 300000 9.9e-14 1.5e-14
*/
/* igamc()
*
* Complemented incomplete gamma integral
*
*
*
* SYNOPSIS:
*
* double a, x, y, igamc();
*
* y = igamc( a, x );
*
* DESCRIPTION:
*
* The function is defined by
*
*
* igamc(a,x) = 1 - igam(a,x)
*
* inf.
* -
* 1 | | -t a-1
* = ----- | e t dt.
* - | |
* | (a) -
* x
*
*
* In this implementation both arguments must be positive.
* The integral is evaluated by either a power series or
* continued fraction expansion, depending on the relative
* values of a and x.
*
* ACCURACY:
*
* Tested at random a, x.
* a x Relative error:
* arithmetic domain domain # trials peak rms
* IEEE 0.5,100 0,100 200000 1.9e-14 1.7e-15
* IEEE 0.01,0.5 0,100 200000 1.4e-13 1.6e-15
*/

/* igami()
*
* Inverse of complemented imcomplete gamma integral
*
*
*
* SYNOPSIS:
*
* double a, x, p, igami();
*
* x = igami( a, p );
*
* DESCRIPTION:
*
* Given p, the function finds x such that
*
* igamc( a, x ) = p.
*
* Starting with the approximate value
*
* 3
* x = a t
*
* where
*
* t = 1 - d - ndtri(p) sqrt(d)
*
* and
*
* d = 1/9a,
*
* the routine performs up to 10 Newton iterations to find the
* root of igamc(a,x) - p = 0.
*
* ACCURACY:
*
* Tested at random a, p in the intervals indicated.
*
* a p Relative error:
* arithmetic domain domain # trials peak rms
* IEEE 0.5,100 0,0.5 100000 1.0e-14 1.7e-15
* IEEE 0.01,0.5 0,0.5 100000 9.0e-14 3.4e-15
* IEEE 0.5,10000 0,0.5 20000 2.3e-13 3.8e-14
*/

/* incbet.c
*
* Incomplete beta integral
*
*
* SYNOPSIS:
*
* double a, b, x, y, incbet();
*
* y = incbet( a, b, x );
*
*
* DESCRIPTION:
*
* Returns incomplete beta integral of the arguments, evaluated
* from zero to x. The function is defined as
*
* x
* - -
* | (a+b) | | a-1 b-1
* ----------- | t (1-t) dt.
* - - | |
* | (a) | (b) -
* 0
*
* The domain of definition is 0 <= x <= 1. In this
* implementation a and b are restricted to positive values.
* The integral from x to 1 may be obtained by the symmetry
* relation
*
* 1 - incbet( a, b, x ) = incbet( b, a, 1-x ).
*
* The integral is evaluated by a continued fraction expansion
* or, when b*x is small, by a power series.
*
* ACCURACY:
*
* Tested at uniformly distributed random points (a,b,x) with a and b
* in "domain" and x between 0 and 1.
* Relative error
* arithmetic domain # trials peak rms
* IEEE 0,5 10000 6.9e-15 4.5e-16
* IEEE 0,85 250000 2.2e-13 1.7e-14
* IEEE 0,1000 30000 5.3e-12 6.3e-13
* IEEE 0,10000 250000 9.3e-11 7.1e-12
* IEEE 0,100000 10000 8.7e-10 4.8e-11
* Outputs smaller than the IEEE gradual underflow threshold
* were excluded from these statistics.
*
* ERROR MESSAGES:
* message condition value returned
* incbet domain x<0, x>1 0.0
* incbet underflow 0.0
*/

/* incbi()
*
* Inverse of imcomplete beta integral
*
*
*
* SYNOPSIS:
*
* double a, b, x, y, incbi();
*
* x = incbi( a, b, y );
*
*
*
* DESCRIPTION:
*
* Given y, the function finds x such that
*
* incbet( a, b, x ) = y .
*
* The routine performs interval halving or Newton iterations to find the
* root of incbet(a,b,x) - y = 0.
*
*
* ACCURACY:
*
* Relative error:
* x a,b
* arithmetic domain domain # trials peak rms
* IEEE 0,1 .5,10000 50000 5.8e-12 1.3e-13
* IEEE 0,1 .25,100 100000 1.8e-13 3.9e-15
* IEEE 0,1 0,5 50000 1.1e-12 5.5e-15
* VAX 0,1 .5,100 25000 3.5e-14 1.1e-15
* With a and b constrained to half-integer or integer values:
* IEEE 0,1 .5,10000 50000 5.8e-12 1.1e-13
* IEEE 0,1 .5,100 100000 1.7e-14 7.9e-16
* With a = .5, b constrained to half-integer or integer values:
* IEEE 0,1 .5,10000 10000 8.3e-11 1.0e-11
*/

/* iv.c
*
* Modified Bessel function of noninteger order
*
*
*
* SYNOPSIS:
*
* double v, x, y, iv();
*
* y = iv( v, x );
*
*
*
* DESCRIPTION:
*
* Returns modified Bessel function of order v of the
* argument. If x is negative, v must be integer valued.
*
* The function is defined as Iv(x) = Jv( ix ). It is
* here computed in terms of the confluent hypergeometric
* function, according to the formula
*
* v -x
* Iv(x) = (x/2) e hyperg( v+0.5, 2v+1, 2x ) / gamma(v+1)
*
* If v is a negative integer, then v is replaced by -v.
*
*
* ACCURACY:
*
* Tested at random points (v, x), with v between 0 and
* 30, x between 0 and 28.
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0,30 2000 3.1e-15 5.4e-16
* IEEE 0,30 10000 1.7e-14 2.7e-15
*
* Accuracy is diminished if v is near a negative integer.
*
* See also hyperg.c.
*
*/

/* j0.c
*
* Bessel function of order zero
*
*
*
* SYNOPSIS:
*
* double x, y, j0();
*
* y = j0( x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of order zero of the argument.
*
* The domain is divided into the intervals [0, 5] and
* (5, infinity). In the first interval the following rational
* approximation is used:
*
*
* 2 2
* (w - r ) (w - r ) P (w) / Q (w)
* 1 2 3 8
*
* 2
* where w = x and the two r's are zeros of the function.
*
* In the second interval, the Hankel asymptotic expansion
* is employed with two rational functions of degree 6/6
* and 7/7.
*
*
*
* ACCURACY:
*
* Absolute error:
* arithmetic domain # trials peak rms
* DEC 0, 30 10000 4.4e-17 6.3e-18
* IEEE 0, 30 60000 4.2e-16 1.1e-16
*
*/
/* y0.c
*
* Bessel function of the second kind, order zero
*
*
*
* SYNOPSIS:
*
* double x, y, y0();
*
* y = y0( x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of the second kind, of order
* zero, of the argument.
*
* The domain is divided into the intervals [0, 5] and
* (5, infinity). In the first interval a rational approximation
* R(x) is employed to compute
* y0(x) = R(x) + 2 * log(x) * j0(x) / PI.
* Thus a call to j0() is required.
*
* In the second interval, the Hankel asymptotic expansion
* is employed with two rational functions of degree 6/6
* and 7/7.
*
*
*
* ACCURACY:
*
* Absolute error, when y0(x) < 1; else relative error:
*
* arithmetic domain # trials peak rms
* DEC 0, 30 9400 7.0e-17 7.9e-18
* IEEE 0, 30 30000 1.3e-15 1.6e-16
*
*/

/* j1.c
*
* Bessel function of order one
*
*
*
* SYNOPSIS:
*
* double x, y, j1();
*
* y = j1( x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of order one of the argument.
*
* The domain is divided into the intervals [0, 8] and
* (8, infinity). In the first interval a 24 term Chebyshev
* expansion is used. In the second, the asymptotic
* trigonometric representation is employed using two
* rational functions of degree 5/5.
*
*
*
* ACCURACY:
*
* Absolute error:
* arithmetic domain # trials peak rms
* DEC 0, 30 10000 4.0e-17 1.1e-17
* IEEE 0, 30 30000 2.6e-16 1.1e-16
*
*
*/
/* y1.c
*
* Bessel function of second kind of order one
*
*
*
* SYNOPSIS:
*
* double x, y, y1();
*
* y = y1( x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of the second kind of order one
* of the argument.
*
* The domain is divided into the intervals [0, 8] and
* (8, infinity). In the first interval a 25 term Chebyshev
* expansion is used, and a call to j1() is required.
* In the second, the asymptotic trigonometric representation
* is employed using two rational functions of degree 5/5.
*
*
*
* ACCURACY:
*
* Absolute error:
* arithmetic domain # trials peak rms
* DEC 0, 30 10000 8.6e-17 1.3e-17
* IEEE 0, 30 30000 1.0e-15 1.3e-16
*
* (error criterion relative when |y1| > 1).
*
*/

/* jn.c
*
* Bessel function of integer order
*
*
*
* SYNOPSIS:
*
* int n;
* double x, y, jn();
*
* y = jn( n, x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of order n, where n is a
* (possibly negative) integer.
*
* The ratio of jn(x) to j0(x) is computed by backward
* recurrence. First the ratio jn/jn-1 is found by a
* continued fraction expansion. Then the recurrence
* relating successive orders is applied until j0 or j1 is
* reached.
*
* If n = 0 or 1 the routine for j0 or j1 is called
* directly.
*
*
*
* ACCURACY:
*
* Absolute error:
* arithmetic range # trials peak rms
* DEC 0, 30 5500 6.9e-17 9.3e-18
* IEEE 0, 30 5000 4.4e-16 7.9e-17
*
*
* Not suitable for large n or x. Use jv() instead.
*
*/

/* jv.c
*
* Bessel function of noninteger order
*
*
*
* SYNOPSIS:
*
* double v, x, y, jv();
*
* y = jv( v, x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of order v of the argument,
* where v is real. Negative x is allowed if v is an integer.
*
* Several expansions are included: the ascending power
* series, the Hankel expansion, and two transitional
* expansions for large v. If v is not too large, it
* is reduced by recurrence to a region of best accuracy.
* The transitional expansions give 12D accuracy for v > 500.
*
*
*
* ACCURACY:
* Results for integer v are indicated by *, where x and v
* both vary from -125 to +125. Otherwise,
* x ranges from 0 to 125, v ranges as indicated by "domain."
* Error criterion is absolute, except relative when |jv()| > 1.
*
* arithmetic v domain x domain # trials peak rms
* IEEE 0,125 0,125 100000 4.6e-15 2.2e-16
* IEEE -125,0 0,125 40000 5.4e-11 3.7e-13
* IEEE 0,500 0,500 20000 4.4e-15 4.0e-16
* Integer v:
* IEEE -125,125 -125,125 50000 3.5e-15* 1.9e-16*
*
*/

/* k0.c
*
* Modified Bessel function, third kind, order zero
*
*
*
* SYNOPSIS:
*
* double x, y, k0();
*
* y = k0( x );
*
*
*
* DESCRIPTION:
*
* Returns modified Bessel function of the third kind
* of order zero of the argument.
*
* The range is partitioned into the two intervals [0,8] and
* (8, infinity). Chebyshev polynomial expansions are employed
* in each interval.
*
*
*
* ACCURACY:
*
* Tested at 2000 random points between 0 and 8. Peak absolute
* error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15.
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0, 30 3100 1.3e-16 2.1e-17
* IEEE 0, 30 30000 1.2e-15 1.6e-16
*
* ERROR MESSAGES:
*
* message condition value returned
* K0 domain x <= 0 MAXNUM
*
*/
/* k0e()
*
* Modified Bessel function, third kind, order zero,
* exponentially scaled
*
*
*
* SYNOPSIS:
*
* double x, y, k0e();
*
* y = k0e( x );
*
*
*
* DESCRIPTION:
*
* Returns exponentially scaled modified Bessel function
* of the third kind of order zero of the argument.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0, 30 30000 1.4e-15 1.4e-16
* See k0().
*
*/

/* k1.c
*
* Modified Bessel function, third kind, order one
*
*
*
* SYNOPSIS:
*
* double x, y, k1();
*
* y = k1( x );
*
*
*
* DESCRIPTION:
*
* Computes the modified Bessel function of the third kind
* of order one of the argument.
*
* The range is partitioned into the two intervals [0,2] and
* (2, infinity). Chebyshev polynomial expansions are employed
* in each interval.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0, 30 3300 8.9e-17 2.2e-17
* IEEE 0, 30 30000 1.2e-15 1.6e-16
*
* ERROR MESSAGES:
*
* message condition value returned
* k1 domain x <= 0 MAXNUM
*
*/
/* k1e.c
*
* Modified Bessel function, third kind, order one,
* exponentially scaled
*
*
*
* SYNOPSIS:
*
* double x, y, k1e();
*
* y = k1e( x );
*
*
*
* DESCRIPTION:
*
* Returns exponentially scaled modified Bessel function
* of the third kind of order one of the argument:
*
* k1e(x) = exp(x) * k1(x).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0, 30 30000 7.8e-16 1.2e-16
* See k1().
*
*/

/* kn.c
*
* Modified Bessel function, third kind, integer order
*
*
*
* SYNOPSIS:
*
* double x, y, kn();
* int n;
*
* y = kn( n, x );
*
*
*
* DESCRIPTION:
*
* Returns modified Bessel function of the third kind
* of order n of the argument.
*
* The range is partitioned into the two intervals [0,9.55] and
* (9.55, infinity). An ascending power series is used in the
* low range, and an asymptotic expansion in the high range.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0,30 3000 1.3e-9 5.8e-11
* IEEE 0,30 90000 1.8e-8 3.0e-10
*
* Error is high only near the crossover point x = 9.55
* between the two expansions used.
*/


/* Re Kolmogorov statistics, here is Birnbaum and Tingey's formula for the
distribution of D+, the maximum of all positive deviations between a
theoretical distribution function P(x) and an empirical one Sn(x)
from n samples.

+
D = sup [ P(x) - Sn(x) ]
n -inf < x < inf


[n(1-e)]
+ - v-1 n-v
Pr{D > e} = > C e (e + v/n) (1 - e - v/n)
n - n v
v=0
[n(1-e)] is the largest integer not exceeding n(1-e).
nCv is the number of combinations of n things taken v at a time.

Exact Smirnov statistic, for one-sided test:
double
smirnov (n, e)
int n;
double e;

Kolmogorov's limiting distribution of two-sided test, returns
probability that sqrt(n) * max deviation > y,
or that max deviation > y/sqrt(n).
The approximation is useful for the tail of the distribution
when n is large.
double
kolmogorov (y)
double y;


Functional inverse of Smirnov distribution
finds e such that smirnov(n,e) = p.
double
smirnovi (n, p)
int n;
double p;

Functional inverse of Kolmogorov statistic for two-sided test.
Finds y such that kolmogorov(y) = p.
If e = smirnovi (n,p), then kolmogi(2 * p) / sqrt(n) should
be close to e.
double
kolmogi (p)
double p;
*/

/* Levnsn.c */
/* Levinson-Durbin LPC
*
* | R0 R1 R2 ... RN-1 | | A1 | | -R1 |
* | R1 R0 R1 ... RN-2 | | A2 | | -R2 |
* | R2 R1 R0 ... RN-3 | | A3 | = | -R3 |
* | ... | | ...| | ... |
* | RN-1 RN-2... R0 | | AN | | -RN |
*
* Ref: John Makhoul, "Linear Prediction, A Tutorial Review"
* Proc. IEEE Vol. 63, PP 561-580 April, 1975.
*
* R is the input autocorrelation function. R0 is the zero lag
* term. A is the output array of predictor coefficients. Note
* that a filter impulse response has a coefficient of 1.0 preceding
* A1. E is an array of mean square error for each prediction order
* 1 to N. REFL is an output array of the reflection coefficients.
*/

/* log.c
*
* Natural logarithm
*
*
*
* SYNOPSIS:
*
* double x, y, log();
*
* y = log( x );
*
*
*
* DESCRIPTION:
*
* Returns the base e (2.718...) logarithm of x.
*
* The argument is separated into its exponent and fractional
* parts. If the exponent is between -1 and +1, the logarithm
* of the fraction is approximated by
*
* log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
*
* Otherwise, setting z = 2(x-1)/x+1),
*
* log(x) = z + z**3 P(z)/Q(z).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0.5, 2.0 150000 1.44e-16 5.06e-17
* IEEE +-MAXNUM 30000 1.20e-16 4.78e-17
* DEC 0, 10 170000 1.8e-17 6.3e-18
*
* In the tests over the interval [+-MAXNUM], the logarithms
* of the random arguments were uniformly distributed over
* [0, MAXLOG].
*
* ERROR MESSAGES:
*
* log singularity: x = 0; returns -INFINITY
* log domain: x < 0; returns NAN
*/

/* log10.c
*
* Common logarithm
*
*
*
* SYNOPSIS:
*
* double x, y, log10();
*
* y = log10( x );
*
*
*
* DESCRIPTION:
*
* Returns logarithm to the base 10 of x.
*
* The argument is separated into its exponent and fractional
* parts. The logarithm of the fraction is approximated by
*
* log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0.5, 2.0 30000 1.5e-16 5.0e-17
* IEEE 0, MAXNUM 30000 1.4e-16 4.8e-17
* DEC 1, MAXNUM 50000 2.5e-17 6.0e-18
*
* In the tests over the interval [1, MAXNUM], the logarithms
* of the random arguments were uniformly distributed over
* [0, MAXLOG].
*
* ERROR MESSAGES:
*
* log10 singularity: x = 0; returns -INFINITY
* log10 domain: x < 0; returns NAN
*/

/* log2.c
*
* Base 2 logarithm
*
*
*
* SYNOPSIS:
*
* double x, y, log2();
*
* y = log2( x );
*
*
*
* DESCRIPTION:
*
* Returns the base 2 logarithm of x.
*
* The argument is separated into its exponent and fractional
* parts. If the exponent is between -1 and +1, the base e
* logarithm of the fraction is approximated by
*
* log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
*
* Otherwise, setting z = 2(x-1)/x+1),
*
* log(x) = z + z**3 P(z)/Q(z).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0.5, 2.0 30000 2.0e-16 5.5e-17
* IEEE exp(+-700) 40000 1.3e-16 4.6e-17
*
* In the tests over the interval [exp(+-700)], the logarithms
* of the random arguments were uniformly distributed.
*
* ERROR MESSAGES:
*
* log2 singularity: x = 0; returns -INFINITY
* log2 domain: x < 0; returns NAN
*/

/* lrand.c
*
* Pseudorandom number generator
*
*
*
* SYNOPSIS:
*
* long y, drand();
*
* drand( &y );
*
*
*
* DESCRIPTION:
*
* Yields a long integer random number.
*
* The three-generator congruential algorithm by Brian
* Wichmann and David Hill (BYTE magazine, March, 1987,
* pp 127-8) is used. The period, given by them, is
* 6953607871644.
*
*
*/

/* lsqrt.c
*
* Integer square root
*
*
*
* SYNOPSIS:
*
* long x, y;
* long lsqrt();
*
* y = lsqrt( x );
*
*
*
* DESCRIPTION:
*
* Returns a long integer square root of the long integer
* argument. The computation is by binary long division.
*
* The largest possible result is lsqrt(2,147,483,647)
* = 46341.
*
* If x < 0, the square root of |x| is returned, and an
* error message is printed.
*
*
* ACCURACY:
*
* An extra, roundoff, bit is computed; hence the result
* is the nearest integer to the actual square root.
* NOTE: only DEC arithmetic is currently supported.
*
*/

/* minv.c
*
* Matrix inversion
*
*
*
* SYNOPSIS:
*
* int n, errcod;
* double A[n*n], X[n*n];
* double B[n];
* int IPS[n];
* int minv();
*
* errcod = minv( A, X, n, B, IPS );
*
*
*
* DESCRIPTION:
*
* Finds the inverse of the n by n matrix A. The result goes
* to X. B and IPS are scratch pad arrays of length n.
* The contents of matrix A are destroyed.
*
* The routine returns nonzero on error; error messages are printed
* by subroutine simq().
*
*/

/* mmmpy.c
*
* Matrix multiply
*
*
*
* SYNOPSIS:
*
* int r, c;
* double A[r*c], B[c*r], Y[r*r];
*
* mmmpy( r, c, A, B, Y );
*
*
*
* DESCRIPTION:
*
* Y = A B
* c-1
* --
* Y[i][j] = > A[i][k] B[k][j]
* --
* k=0
*
* Multiplies an r (rows) by c (columns) matrix A on the left
* by a c (rows) by r (columns) matrix B on the right
* to produce an r by r matrix Y.
*
*
*/

/* mtherr.c
*
* Library common error handling routine
*
*
*
* SYNOPSIS:
*
* char *fctnam;
* int code;
* int mtherr();
*
* mtherr( fctnam, code );
*
*
*
* DESCRIPTION:
*
* This routine may be called to report one of the following
* error conditions (in the include file math.h).
*
* Mnemonic Value Significance
*
* DOMAIN 1 argument domain error
* SING 2 function singularity
* OVERFLOW 3 overflow range error
* UNDERFLOW 4 underflow range error
* TLOSS 5 total loss of precision
* PLOSS 6 partial loss of precision
* EDOM 33 Unix domain error code
* ERANGE 34 Unix range error code
*
* The default version of the file prints the function name,
* passed to it by the pointer fctnam, followed by the
* error condition. The display is directed to the standard
* output device. The routine then returns to the calling
* program. Users may wish to modify the program to abort by
* calling exit() under severe error conditions such as domain
* errors.
*
* Since all error conditions pass control to this function,
* the display may be easily changed, eliminated, or directed
* to an error logging device.
*
* SEE ALSO:
*
* math.h
*
*/

/* mtransp.c
*
* Matrix transpose
*
*
*
* SYNOPSIS:
*
* int n;
* double A[n*n], T[n*n];
*
* mtransp( n, A, T );
*
*
*
* DESCRIPTION:
*
*
* T[r][c] = A[c][r]
*
*
* Transposes the n by n square matrix A and puts the result in T.
* The output, T, may occupy the same storage as A.
*
*
*
*/

/* mvmpy.c
*
* Matrix times vector
*
*
*
* SYNOPSIS:
*
* int r, c;
* double A[r*c], V[c], Y[r];
*
* mvmpy( r, c, A, V, Y );
*
*
*
* DESCRIPTION:
*
* c-1
* --
* Y[j] = > A[j][k] V[k] , j = 1, ..., r
* --
* k=0
*
* Multiplies the r (rows) by c (columns) matrix A on the left
* by column vector V of dimension c on the right
* to produce a (column) vector Y output of dimension r.
*
*
*
*
*/

/* nbdtr.c
*
* Negative binomial distribution
*
*
*
* SYNOPSIS:
*
* int k, n;
* double p, y, nbdtr();
*
* y = nbdtr( k, n, p );
*
* DESCRIPTION:
*
* Returns the sum of the terms 0 through k of the negative
* binomial distribution:
*
* k
* -- ( n+j-1 ) n j
* > ( ) p (1-p)
* -- ( j )
* j=0
*
* In a sequence of Bernoulli trials, this is the probability
* that k or fewer failures precede the nth success.
*
* The terms are not computed individually; instead the incomplete
* beta integral is employed, according to the formula
*
* y = nbdtr( k, n, p ) = incbet( n, k+1, p ).
*
* The arguments must be positive, with p ranging from 0 to 1.
*
* ACCURACY:
*
* Tested at random points (a,b,p), with p between 0 and 1.
*
* a,b Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,100 100000 1.7e-13 8.8e-15
* See also incbet.c.
*
*/
/* nbdtrc.c
*
* Complemented negative binomial distribution
*
*
*
* SYNOPSIS:
*
* int k, n;
* double p, y, nbdtrc();
*
* y = nbdtrc( k, n, p );
*
* DESCRIPTION:
*
* Returns the sum of the terms k+1 to infinity of the negative
* binomial distribution:
*
* inf
* -- ( n+j-1 ) n j
* > ( ) p (1-p)
* -- ( j )
* j=k+1
*
* The terms are not computed individually; instead the incomplete
* beta integral is employed, according to the formula
*
* y = nbdtrc( k, n, p ) = incbet( k+1, n, 1-p ).
*
* The arguments must be positive, with p ranging from 0 to 1.
*
* ACCURACY:
*
* Tested at random points (a,b,p), with p between 0 and 1.
*
* a,b Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,100 100000 1.7e-13 8.8e-15
* See also incbet.c.
*/

/* nbdtrc
*
* Complemented negative binomial distribution
*
*
*
* SYNOPSIS:
*
* int k, n;
* double p, y, nbdtrc();
*
* y = nbdtrc( k, n, p );
*
* DESCRIPTION:
*
* Returns the sum of the terms k+1 to infinity of the negative
* binomial distribution:
*
* inf
* -- ( n+j-1 ) n j
* > ( ) p (1-p)
* -- ( j )
* j=k+1
*
* The terms are not computed individually; instead the incomplete
* beta integral is employed, according to the formula
*
* y = nbdtrc( k, n, p ) = incbet( k+1, n, 1-p ).
*
* The arguments must be positive, with p ranging from 0 to 1.
*
* ACCURACY:
*
* See incbet.c.
*/
/* nbdtri
*
* Functional inverse of negative binomial distribution
*
*
*
* SYNOPSIS:
*
* int k, n;
* double p, y, nbdtri();
*
* p = nbdtri( k, n, y );
*
* DESCRIPTION:
*
* Finds the argument p such that nbdtr(k,n,p) is equal to y.
*
* ACCURACY:
*
* Tested at random points (a,b,y), with y between 0 and 1.
*
* a,b Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,100 100000 1.5e-14 8.5e-16
* See also incbi.c.
*/

/* ndtr.c
*
* Normal distribution function
*
*
*
* SYNOPSIS:
*
* double x, y, ndtr();
*
* y = ndtr( x );
*
*
*
* DESCRIPTION:
*
* Returns the area under the Gaussian probability density
* function, integrated from minus infinity to x:
*
* x
* -
* 1 | | 2
* ndtr(x) = --------- | exp( - t /2 ) dt
* sqrt(2pi) | |
* -
* -inf.
*
* = ( 1 + erf(z) ) / 2
* = erfc(z) / 2
*
* where z = x/sqrt(2). Computation is via the functions
* erf and erfc.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -13,0 8000 2.1e-15 4.8e-16
* IEEE -13,0 30000 3.4e-14 6.7e-15
*
*
* ERROR MESSAGES:
*
* message condition value returned
* erfc underflow x > 37.519379347 0.0
*
*/
/* erf.c
*
* Error function
*
*
*
* SYNOPSIS:
*
* double x, y, erf();
*
* y = erf( x );
*
*
*
* DESCRIPTION:
*
* The integral is
*
* x
* -
* 2 | | 2
* erf(x) = -------- | exp( - t ) dt.
* sqrt(pi) | |
* -
* 0
*
* The magnitude of x is limited to 9.231948545 for DEC
* arithmetic; 1 or -1 is returned outside this range.
*
* For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise
* erf(x) = 1 - erfc(x).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0,1 14000 4.7e-17 1.5e-17
* IEEE 0,1 30000 3.7e-16 1.0e-16
*
*/
/* erfc.c
*
* Complementary error function
*
*
*
* SYNOPSIS:
*
* double x, y, erfc();
*
* y = erfc( x );
*
*
*
* DESCRIPTION:
*
*
* 1 - erf(x) =
*
* inf.
* -
* 2 | | 2
* erfc(x) = -------- | exp( - t ) dt
* sqrt(pi) | |
* -
* x
*
*
* For small x, erfc(x) = 1 - erf(x); otherwise rational
* approximations are computed.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0, 9.2319 12000 5.1e-16 1.2e-16
* IEEE 0,26.6417 30000 5.7e-14 1.5e-14
*
*
* ERROR MESSAGES:
*
* message condition value returned
* erfc underflow x > 9.231948545 (DEC) 0.0
*
*
*/

/* ndtri.c
*
* Inverse of Normal distribution function
*
*
*
* SYNOPSIS:
*
* double x, y, ndtri();
*
* x = ndtri( y );
*
*
*
* DESCRIPTION:
*
* Returns the argument, x, for which the area under the
* Gaussian probability density function (integrated from
* minus infinity to x) is equal to y.
*
*
* For small arguments 0 < y < exp(-2), the program computes
* z = sqrt( -2.0 * log(y) ); then the approximation is
* x = z - log(z)/z - (1/z) P(1/z) / Q(1/z).
* There are two rational functions P/Q, one for 0 < y < exp(-32)
* and the other for y up to exp(-2). For larger arguments,
* w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0.125, 1 5500 9.5e-17 2.1e-17
* DEC 6e-39, 0.135 3500 5.7e-17 1.3e-17
* IEEE 0.125, 1 20000 7.2e-16 1.3e-16
* IEEE 3e-308, 0.135 50000 4.6e-16 9.8e-17
*
*
* ERROR MESSAGES:
*
* message condition value returned
* ndtri domain x <= 0 -MAXNUM
* ndtri domain x >= 1 MAXNUM
*
*/

/* pdtr.c
*
* Poisson distribution
*
*
*
* SYNOPSIS:
*
* int k;
* double m, y, pdtr();
*
* y = pdtr( k, m );
*
*
*
* DESCRIPTION:
*
* Returns the sum of the first k terms of the Poisson
* distribution:
*
* k j
* -- -m m
* > e --
* -- j!
* j=0
*
* The terms are not summed directly; instead the incomplete
* gamma integral is employed, according to the relation
*
* y = pdtr( k, m ) = igamc( k+1, m ).
*
* The arguments must both be positive.
*
*
*
* ACCURACY:
*
* See igamc().
*
*/
/* pdtrc()
*
* Complemented poisson distribution
*
*
*
* SYNOPSIS:
*
* int k;
* double m, y, pdtrc();
*
* y = pdtrc( k, m );
*
*
*
* DESCRIPTION:
*
* Returns the sum of the terms k+1 to infinity of the Poisson
* distribution:
*
* inf. j
* -- -m m
* > e --
* -- j!
* j=k+1
*
* The terms are not summed directly; instead the incomplete
* gamma integral is employed, according to the formula
*
* y = pdtrc( k, m ) = igam( k+1, m ).
*
* The arguments must both be positive.
*
*
*
* ACCURACY:
*
* See igam.c.
*
*/
/* pdtri()
*
* Inverse Poisson distribution
*
*
*
* SYNOPSIS:
*
* int k;
* double m, y, pdtr();
*
* m = pdtri( k, y );
*
*
*
*
* DESCRIPTION:
*
* Finds the Poisson variable x such that the integral
* from 0 to x of the Poisson density is equal to the
* given probability y.
*
* This is accomplished using the inverse gamma integral
* function and the relation
*
* m = igami( k+1, y ).
*
*
*
*
* ACCURACY:
*
* See igami.c.
*
* ERROR MESSAGES:
*
* message condition value returned
* pdtri domain y < 0 or y >= 1 0.0
* k < 0
*
*/

/* polevl.c
* p1evl.c
*
* Evaluate polynomial
*
*
*
* SYNOPSIS:
*
* int N;
* double x, y, coef[N+1], polevl[];
*
* y = polevl( x, coef, N );
*
*
*
* DESCRIPTION:
*
* Evaluates polynomial of degree N:
*
* 2 N
* y = C + C x + C x +...+ C x
* 0 1 2 N
*
* Coefficients are stored in reverse order:
*
* coef[0] = C , ..., coef[N] = C .
* N 0
*
* The function p1evl() assumes that coef[N] = 1.0 and is
* omitted from the array. Its calling arguments are
* otherwise the same as polevl().
*
*
* SPEED:
*
* In the interest of speed, there are no checks for out
* of bounds arithmetic. This routine is used by most of
* the functions in the library. Depending on available
* equipment features, the user may wish to rewrite the
* program in microcode or assembly language.
*
*/

/* polmisc.c
* Square root, sine, cosine, and arctangent of polynomial.
* See polyn.c for data structures and discussion.
*/

/* polrt.c
*
* Find roots of a polynomial
*
*
*
* SYNOPSIS:
*
* typedef struct
* {
* double r;
* double i;
* }cmplx;
*
* double xcof[], cof[];
* int m;
* cmplx root[];
*
* polrt( xcof, cof, m, root )
*
*
*
* DESCRIPTION:
*
* Iterative determination of the roots of a polynomial of
* degree m whose coefficient vector is xcof[]. The
* coefficients are arranged in ascending order; i.e., the
* coefficient of x**m is xcof[m].
*
* The array cof[] is working storage the same size as xcof[].
* root[] is the output array containing the complex roots.
*
*
* ACCURACY:
*
* Termination depends on evaluation of the polynomial at
* the trial values of the roots. The values of multiple roots
* or of roots that are nearly equal may have poor relative
* accuracy after the first root in the neighborhood has been
* found.
*
*/

/* polyn.c
* polyr.c
* Arithmetic operations on polynomials
*
* In the following descriptions a, b, c are polynomials of degree
* na, nb, nc respectively. The degree of a polynomial cannot
* exceed a run-time value MAXPOL. An operation that attempts
* to use or generate a polynomial of higher degree may produce a
* result that suffers truncation at degree MAXPOL. The value of
* MAXPOL is set by calling the function
*
* polini( maxpol );
*
* where maxpol is the desired maximum degree. This must be
* done prior to calling any of the other functions in this module.
* Memory for internal temporary polynomial storage is allocated
* by polini().
*
* Each polynomial is represented by an array containing its
* coefficients, together with a separately declared integer equal
* to the degree of the polynomial. The coefficients appear in
* ascending order; that is,
*
* 2 na
* a(x) = a[0] + a[1] * x + a[2] * x + ... + a[na] * x .
*
*
*
* sum = poleva( a, na, x ); Evaluate polynomial a(t) at t = x.
* polprt( a, na, D ); Print the coefficients of a to D digits.
* polclr( a, na ); Set a identically equal to zero, up to a[na].
* polmov( a, na, b ); Set b = a.
* poladd( a, na, b, nb, c ); c = b + a, nc = max(na,nb)
* polsub( a, na, b, nb, c ); c = b - a, nc = max(na,nb)
* polmul( a, na, b, nb, c ); c = b * a, nc = na+nb
*
*
* Division:
*
* i = poldiv( a, na, b, nb, c ); c = b / a, nc = MAXPOL
*
* returns i = the degree of the first nonzero coefficient of a.
* The computed quotient c must be divided by x^i. An error message
* is printed if a is identically zero.
*
*
* Change of variables:
* If a and b are polynomials, and t = a(x), then
* c(t) = b(a(x))
* is a polynomial found by substituting a(x) for t. The
* subroutine call for this is
*
* polsbt( a, na, b, nb, c );
*
*
* Notes:
* poldiv() is an integer routine; poleva() is double.
* Any of the arguments a, b, c may refer to the same array.
*
*/

/* pow.c
*
* Power function
*
*
*
* SYNOPSIS:
*
* double x, y, z, pow();
*
* z = pow( x, y );
*
*
*
* DESCRIPTION:
*
* Computes x raised to the yth power. Analytically,
*
* x**y = exp( y log(x) ).
*
* Following Cody and Waite, this program uses a lookup table
* of 2**-i/16 and pseudo extended precision arithmetic to
* obtain an extra three bits of accuracy in both the logarithm
* and the exponential.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -26,26 30000 4.2e-16 7.7e-17
* DEC -26,26 60000 4.8e-17 9.1e-18
* 1/26 < x < 26, with log(x) uniformly distributed.
* -26 < y < 26, y uniformly distributed.
* IEEE 0,8700 30000 1.5e-14 2.1e-15
* 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
*
*
* ERROR MESSAGES:
*
* message condition value returned
* pow overflow x**y > MAXNUM INFINITY
* pow underflow x**y < 1/MAXNUM 0.0
* pow domain x<0 and y noninteger 0.0
*
*/

/* powi.c
*
* Real raised to integer power
*
*
*
* SYNOPSIS:
*
* double x, y, powi();
* int n;
*
* y = powi( x, n );
*
*
*
* DESCRIPTION:
*
* Returns argument x raised to the nth power.
* The routine efficiently decomposes n as a sum of powers of
* two. The desired power is a product of two-to-the-kth
* powers of x. Thus to compute the 32767 power of x requires
* 28 multiplications instead of 32767 multiplications.
*
*
*
* ACCURACY:
*
*
* Relative error:
* arithmetic x domain n domain # trials peak rms
* DEC .04,26 -26,26 100000 2.7e-16 4.3e-17
* IEEE .04,26 -26,26 50000 2.0e-15 3.8e-16
* IEEE 1,2 -1022,1023 50000 8.6e-14 1.6e-14
*
* Returns MAXNUM on overflow, zero on underflow.
*
*/

/* psi.c
*
* Psi (digamma) function
*
*
* SYNOPSIS:
*
* double x, y, psi();
*
* y = psi( x );
*
*
* DESCRIPTION:
*
* d -
* psi(x) = -- ln | (x)
* dx
*
* is the logarithmic derivative of the gamma function.
* For integer x,
* n-1
* -
* psi(n) = -EUL + > 1/k.
* -
* k=1
*
* This formula is used for 0 < n <= 10. If x is negative, it
* is transformed to a positive argument by the reflection
* formula psi(1-x) = psi(x) + pi cot(pi x).
* For general positive x, the argument is made greater than 10
* using the recurrence psi(x+1) = psi(x) + 1/x.
* Then the following asymptotic expansion is applied:
*
* inf. B
* - 2k
* psi(x) = log(x) - 1/2x - > -------
* - 2k
* k=1 2k x
*
* where the B2k are Bernoulli numbers.
*
* ACCURACY:
* Relative error (except absolute when |psi| < 1):
* arithmetic domain # trials peak rms
* DEC 0,30 2500 1.7e-16 2.0e-17
* IEEE 0,30 30000 1.3e-15 1.4e-16
* IEEE -30,0 40000 1.5e-15 2.2e-16
*
* ERROR MESSAGES:
* message condition value returned
* psi singularity x integer <=0 MAXNUM
*/

/* revers.c
*
* Reversion of power series
*
*
*
* SYNOPSIS:
*
* extern int MAXPOL;
* int n;
* double x[n+1], y[n+1];
*
* polini(n);
* revers( y, x, n );
*
* Note, polini() initializes the polynomial arithmetic subroutines;
* see polyn.c.
*
*
* DESCRIPTION:
*
* If
*
* inf
* - i
* y(x) = > a x
* - i
* i=1
*
* then
*
* inf
* - j
* x(y) = > A y ,
* - j
* j=1
*
* where
* 1
* A = ---
* 1 a
* 1
*
* etc. The coefficients of x(y) are found by expanding
*
* inf inf
* - - i
* x(y) = > A > a x
* - j - i
* j=1 i=1
*
* and setting each coefficient of x , higher than the first,
* to zero.
*
*
*
* RESTRICTIONS:
*
* y[0] must be zero, and y[1] must be nonzero.
*
*/

/* rgamma.c
*
* Reciprocal gamma function
*
*
*
* SYNOPSIS:
*
* double x, y, rgamma();
*
* y = rgamma( x );
*
*
*
* DESCRIPTION:
*
* Returns one divided by the gamma function of the argument.
*
* The function is approximated by a Chebyshev expansion in
* the interval [0,1]. Range reduction is by recurrence
* for arguments between -34.034 and +34.84425627277176174.
* 1/MAXNUM is returned for positive arguments outside this
* range. For arguments less than -34.034 the cosecant
* reflection formula is applied; lograrithms are employed
* to avoid unnecessary overflow.
*
* The reciprocal gamma function has no singularities,
* but overflow and underflow may occur for large arguments.
* These conditions return either MAXNUM or 1/MAXNUM with
* appropriate sign.
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -30,+30 4000 1.2e-16 1.8e-17
* IEEE -30,+30 30000 1.1e-15 2.0e-16
* For arguments less than -34.034 the peak error is on the
* order of 5e-15 (DEC), excepting overflow or underflow.
*/

/* round.c
*
* Round double to nearest or even integer valued double
*
*
*
* SYNOPSIS:
*
* double x, y, round();
*
* y = round(x);
*
*
*
* DESCRIPTION:
*
* Returns the nearest integer to x as a double precision
* floating point result. If x ends in 0.5 exactly, the
* nearest even integer is chosen.
*
*
*
* ACCURACY:
*
* If x is greater than 1/(2*MACHEP), its closest machine
* representation is already an integer, so rounding does
* not change it.
*/

/* shichi.c
*
* Hyperbolic sine and cosine integrals
*
*
*
* SYNOPSIS:
*
* double x, Chi, Shi, shichi();
*
* shichi( x, &Chi, &Shi );
*
*
* DESCRIPTION:
*
* Approximates the integrals
*
* x
* -
* | | cosh t - 1
* Chi(x) = eul + ln x + | ----------- dt,
* | | t
* -
* 0
*
* x
* -
* | | sinh t
* Shi(x) = | ------ dt
* | | t
* -
* 0
*
* where eul = 0.57721566490153286061 is Euler's constant.
* The integrals are evaluated by power series for x < 8
* and by Chebyshev expansions for x between 8 and 88.
* For large x, both functions approach exp(x)/2x.
* Arguments greater than 88 in magnitude return MAXNUM.
*
*
* ACCURACY:
*
* Test interval 0 to 88.
* Relative error:
* arithmetic function # trials peak rms
* DEC Shi 3000 9.1e-17
* IEEE Shi 30000 6.9e-16 1.6e-16
* Absolute error, except relative when |Chi| > 1:
* DEC Chi 2500 9.3e-17
* IEEE Chi 30000 8.4e-16 1.4e-16
*/

/* sici.c
*
* Sine and cosine integrals
*
*
*
* SYNOPSIS:
*
* double x, Ci, Si, sici();
*
* sici( x, &Si, &Ci );
*
*
* DESCRIPTION:
*
* Evaluates the integrals
*
* x
* -
* | cos t - 1
* Ci(x) = eul + ln x + | --------- dt,
* | t
* -
* 0
* x
* -
* | sin t
* Si(x) = | ----- dt
* | t
* -
* 0
*
* where eul = 0.57721566490153286061 is Euler's constant.
* The integrals are approximated by rational functions.
* For x > 8 auxiliary functions f(x) and g(x) are employed
* such that
*
* Ci(x) = f(x) sin(x) - g(x) cos(x)
* Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x)
*
*
* ACCURACY:
* Test interval = [0,50].
* Absolute error, except relative when > 1:
* arithmetic function # trials peak rms
* IEEE Si 30000 4.4e-16 7.3e-17
* IEEE Ci 30000 6.9e-16 5.1e-17
* DEC Si 5000 4.4e-17 9.0e-18
* DEC Ci 5300 7.9e-17 5.2e-18
*/

/* simpsn.c */
* Numerical integration of function tabulated
* at equally spaced arguments
*/

/* simq.c
*
* Solution of simultaneous linear equations AX = B
* by Gaussian elimination with partial pivoting
*
*
*
* SYNOPSIS:
*
* double A[n*n], B[n], X[n];
* int n, flag;
* int IPS[];
* int simq();
*
* ercode = simq( A, B, X, n, flag, IPS );
*
*
*
* DESCRIPTION:
*
* B, X, IPS are vectors of length n.
* A is an n x n matrix (i.e., a vector of length n*n),
* stored row-wise: that is, A(i,j) = A[ij],
* where ij = i*n + j, which is the transpose of the normal
* column-wise storage.
*
* The contents of matrix A are destroyed.
*
* Set flag=0 to solve.
* Set flag=-1 to do a new back substitution for different B vector
* using the same A matrix previously reduced when flag=0.
*
* The routine returns nonzero on error; messages are printed.
*
*
* ACCURACY:
*
* Depends on the conditioning (range of eigenvalues) of matrix A.
*
*
* REFERENCE:
*
* Computer Solution of Linear Algebraic Systems,
* by George E. Forsythe and Cleve B. Moler; Prentice-Hall, 1967.
*
*/

/* sin.c
*
* Circular sine
*
*
*
* SYNOPSIS:
*
* double x, y, sin();
*
* y = sin( x );
*
*
*
* DESCRIPTION:
*
* Range reduction is into intervals of pi/4. The reduction
* error is nearly eliminated by contriving an extended precision
* modular arithmetic.
*
* Two polynomial approximating functions are employed.
* Between 0 and pi/4 the sine is approximated by
* x + x**3 P(x**2).
* Between pi/4 and pi/2 the cosine is represented as
* 1 - x**2 Q(x**2).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0, 10 150000 3.0e-17 7.8e-18
* IEEE -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17
*
* ERROR MESSAGES:
*
* message condition value returned
* sin total loss x > 1.073741824e9 0.0
*
* Partial loss of accuracy begins to occur at x = 2**30
* = 1.074e9. The loss is not gradual, but jumps suddenly to
* about 1 part in 10e7. Results may be meaningless for
* x > 2**49 = 5.6e14. The routine as implemented flags a
* TLOSS error for x > 2**30 and returns 0.0.
*/
/* cos.c
*
* Circular cosine
*
*
*
* SYNOPSIS:
*
* double x, y, cos();
*
* y = cos( x );
*
*
*
* DESCRIPTION:
*
* Range reduction is into intervals of pi/4. The reduction
* error is nearly eliminated by contriving an extended precision
* modular arithmetic.
*
* Two polynomial approximating functions are employed.
* Between 0 and pi/4 the cosine is approximated by
* 1 - x**2 Q(x**2).
* Between pi/4 and pi/2 the sine is represented as
* x + x**3 P(x**2).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17
* DEC 0,+1.07e9 17000 3.0e-17 7.2e-18
*/

/* sincos.c
*
* Circular sine and cosine of argument in degrees
* Table lookup and interpolation algorithm
*
*
*
* SYNOPSIS:
*
* double x, sine, cosine, flg, sincos();
*
* sincos( x, &sine, &cosine, flg );
*
*
*
* DESCRIPTION:
*
* Returns both the sine and the cosine of the argument x.
* Several different compile time options and minimax
* approximations are supplied to permit tailoring the
* tradeoff between computation speed and accuracy.
*
* Since range reduction is time consuming, the reduction
* of x modulo 360 degrees is also made optional.
*
* sin(i) is internally tabulated for 0 <= i <= 90 degrees.
* Approximation polynomials, ranging from linear interpolation
* to cubics in (x-i)**2, compute the sine and cosine
* of the residual x-i which is between -0.5 and +0.5 degree.
* In the case of the high accuracy options, the residual
* and the tabulated values are combined using the trigonometry
* formulas for sin(A+B) and cos(A+B).
*
* Compile time options are supplied for 5, 11, or 17 decimal
* relative accuracy (ACC5, ACC11, ACC17 respectively).
* A subroutine flag argument "flg" chooses betwen this
* accuracy and table lookup only (peak absolute error
* = 0.0087).
*
* If the argument flg = 1, then the tabulated value is
* returned for the nearest whole number of degrees. The
* approximation polynomials are not computed. At
* x = 0.5 deg, the absolute error is then sin(0.5) = 0.0087.
*
* An intermediate speed and precision can be obtained using
* the compile time option LINTERP and flg = 1. This yields
* a linear interpolation using a slope estimated from the sine
* or cosine at the nearest integer argument. The peak absolute
* error with this option is 3.8e-5. Relative error at small
* angles is about 1e-5.
*
* If flg = 0, then the approximation polynomials are computed
* and applied.
*
*
*
* SPEED:
*
* Relative speed comparisons follow for 6MHz IBM AT clone
* and Microsoft C version 4.0. These figures include
* software overhead of do loop and function calls.
* Since system hardware and software vary widely, the
* numbers should be taken as representative only.
*
* flg=0 flg=0 flg=1 flg=1
* ACC11 ACC5 LINTERP Lookup only
* In-line 8087 (/FPi)
* sin(), cos() 1.0 1.0 1.0 1.0
*
* In-line 8087 (/FPi)
* sincos() 1.1 1.4 1.9 3.0
*
* Software (/FPa)
* sin(), cos() 0.19 0.19 0.19 0.19
*
* Software (/FPa)
* sincos() 0.39 0.50 0.73 1.7
*
*
*
* ACCURACY:
*
* The accurate approximations are designed with a relative error
* criterion. The absolute error is greatest at x = 0.5 degree.
* It decreases from a local maximum at i+0.5 degrees to full
* machine precision at each integer i degrees. With the
* ACC5 option, the relative error of 6.3e-6 is equivalent to
* an absolute angular error of 0.01 arc second in the argument
* at x = i+0.5 degrees. For small angles < 0.5 deg, the ACC5
* accuracy is 6.3e-6 (.00063%) of reading; i.e., the absolute
* error decreases in proportion to the argument. This is true
* for both the sine and cosine approximations, since the latter
* is for the function 1 - cos(x).
*
* If absolute error is of most concern, use the compile time
* option ABSERR to obtain an absolute error of 2.7e-8 for ACC5
* precision. This is about half the absolute error of the
* relative precision option. In this case the relative error
* for small angles will increase to 9.5e-6 -- a reasonable
* tradeoff.
*/

/* sindg.c
*
* Circular sine of angle in degrees
*
*
*
* SYNOPSIS:
*
* double x, y, sindg();
*
* y = sindg( x );
*
*
*
* DESCRIPTION:
*
* Range reduction is into intervals of 45 degrees.
*
* Two polynomial approximating functions are employed.
* Between 0 and pi/4 the sine is approximated by
* x + x**3 P(x**2).
* Between pi/4 and pi/2 the cosine is represented as
* 1 - x**2 P(x**2).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC +-1000 3100 3.3e-17 9.0e-18
* IEEE +-1000 30000 2.3e-16 5.6e-17
*
* ERROR MESSAGES:
*
* message condition value returned
* sindg total loss x > 8.0e14 (DEC) 0.0
* x > 1.0e14 (IEEE)
*
*/
/* cosdg.c
*
* Circular cosine of angle in degrees
*
*
*
* SYNOPSIS:
*
* double x, y, cosdg();
*
* y = cosdg( x );
*
*
*
* DESCRIPTION:
*
* Range reduction is into intervals of 45 degrees.
*
* Two polynomial approximating functions are employed.
* Between 0 and pi/4 the cosine is approximated by
* 1 - x**2 P(x**2).
* Between pi/4 and pi/2 the sine is represented as
* x + x**3 P(x**2).
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC +-1000 3400 3.5e-17 9.1e-18
* IEEE +-1000 30000 2.1e-16 5.7e-17
* See also sin().
*
*/

/* sinh.c
*
* Hyperbolic sine
*
*
*
* SYNOPSIS:
*
* double x, y, sinh();
*
* y = sinh( x );
*
*
*
* DESCRIPTION:
*
* Returns hyperbolic sine of argument in the range MINLOG to
* MAXLOG.
*
* The range is partitioned into two segments. If |x| <= 1, a
* rational function of the form x + x**3 P(x)/Q(x) is employed.
* Otherwise the calculation is sinh(x) = ( exp(x) - exp(-x) )/2.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC +- 88 50000 4.0e-17 7.7e-18
* IEEE +-MAXLOG 30000 2.6e-16 5.7e-17
*
*/

/* spence.c
*
* Dilogarithm
*
*
*
* SYNOPSIS:
*
* double x, y, spence();
*
* y = spence( x );
*
*
*
* DESCRIPTION:
*
* Computes the integral
*
* x
* -
* | | log t
* spence(x) = - | ----- dt
* | | t - 1
* -
* 1
*
* for x >= 0. A rational approximation gives the integral in
* the interval (0.5, 1.5). Transformation formulas for 1/x
* and 1-x are employed outside the basic expansion range.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,4 30000 3.9e-15 5.4e-16
* DEC 0,4 3000 2.5e-16 4.5e-17
*
*
*/

/* sqrt.c
*
* Square root
*
*
*
* SYNOPSIS:
*
* double x, y, sqrt();
*
* y = sqrt( x );
*
*
*
* DESCRIPTION:
*
* Returns the square root of x.
*
* Range reduction involves isolating the power of two of the
* argument and using a polynomial approximation to obtain
* a rough value for the square root. Then Heron's iteration
* is used three times to converge to an accurate value.
*
*
*
* ACCURACY:
*
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0, 10 60000 2.1e-17 7.9e-18
* IEEE 0,1.7e308 30000 1.7e-16 6.3e-17
*
*
* ERROR MESSAGES:
*
* message condition value returned
* sqrt domain x < 0 0.0
*
*/

/* stdtr.c
*
* Student's t distribution
*
*
*
* SYNOPSIS:
*
* double t, stdtr();
* short k;
*
* y = stdtr( k, t );
*
*
* DESCRIPTION:
*
* Computes the integral from minus infinity to t of the Student
* t distribution with integer k > 0 degrees of freedom:
*
* t
* -
* | |
* - | 2 -(k+1)/2
* | ( (k+1)/2 ) | ( x )
* ---------------------- | ( 1 + --- ) dx
* - | ( k )
* sqrt( k pi ) | ( k/2 ) |
* | |
* -
* -inf.
*
* Relation to incomplete beta integral:
*
* 1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z )
* where
* z = k/(k + t**2).
*
* For t < -2, this is the method of computation. For higher t,
* a direct method is derived from integration by parts.
* Since the function is symmetric about t=0, the area under the
* right tail of the density is found by calling the function
* with -t instead of t.
*
* ACCURACY:
*
* Tested at random 1 <= k <= 25. The "domain" refers to t.
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -100,-2 50000 5.9e-15 1.4e-15
* IEEE -2,100 500000 2.7e-15 4.9e-17
*/

/* stdtri.c
*
* Functional inverse of Student's t distribution
*
*
*
* SYNOPSIS:
*
* double p, t, stdtri();
* int k;
*
* t = stdtri( k, p );
*
*
* DESCRIPTION:
*
* Given probability p, finds the argument t such that stdtr(k,t)
* is equal to p.
*
* ACCURACY:
*
* Tested at random 1 <= k <= 100. The "domain" refers to p:
* Relative error:
* arithmetic domain # trials peak rms
* IEEE .001,.999 25000 5.7e-15 8.0e-16
* IEEE 10^-6,.001 25000 2.0e-12 2.9e-14
*/

/* struve.c
*
* Struve function
*
*
*
* SYNOPSIS:
*
* double v, x, y, struve();
*
* y = struve( v, x );
*
*
*
* DESCRIPTION:
*
* Computes the Struve function Hv(x) of order v, argument x.
* Negative x is rejected unless v is an integer.
*
* This module also contains the hypergeometric functions 1F2
* and 3F0 and a routine for the Bessel function Yv(x) with
* noninteger v.
*
*
*
* ACCURACY:
*
* Not accurately characterized, but spot checked against tables.
*
*/

/* tan.c
*
* Circular tangent
*
*
*
* SYNOPSIS:
*
* double x, y, tan();
*
* y = tan( x );
*
*
*
* DESCRIPTION:
*
* Returns the circular tangent of the radian argument x.
*
* Range reduction is modulo pi/4. A rational function
* x + x**3 P(x**2)/Q(x**2)
* is employed in the basic interval [0, pi/4].
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC +-1.07e9 44000 4.1e-17 1.0e-17
* IEEE +-1.07e9 30000 2.9e-16 8.1e-17
*
* ERROR MESSAGES:
*
* message condition value returned
* tan total loss x > 1.073741824e9 0.0
*
*/
/* cot.c
*
* Circular cotangent
*
*
*
* SYNOPSIS:
*
* double x, y, cot();
*
* y = cot( x );
*
*
*
* DESCRIPTION:
*
* Returns the circular cotangent of the radian argument x.
*
* Range reduction is modulo pi/4. A rational function
* x + x**3 P(x**2)/Q(x**2)
* is employed in the basic interval [0, pi/4].
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE +-1.07e9 30000 2.9e-16 8.2e-17
*
*
* ERROR MESSAGES:
*
* message condition value returned
* cot total loss x > 1.073741824e9 0.0
* cot singularity x = 0 INFINITY
*
*/

/* tandg.c
*
* Circular tangent of argument in degrees
*
*
*
* SYNOPSIS:
*
* double x, y, tandg();
*
* y = tandg( x );
*
*
*
* DESCRIPTION:
*
* Returns the circular tangent of the argument x in degrees.
*
* Range reduction is modulo pi/4. A rational function
* x + x**3 P(x**2)/Q(x**2)
* is employed in the basic interval [0, pi/4].
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0,10 8000 3.4e-17 1.2e-17
* IEEE 0,10 30000 3.2e-16 8.4e-17
*
* ERROR MESSAGES:
*
* message condition value returned
* tandg total loss x > 8.0e14 (DEC) 0.0
* x > 1.0e14 (IEEE)
* tandg singularity x = 180 k + 90 MAXNUM
*/
/* cotdg.c
*
* Circular cotangent of argument in degrees
*
*
*
* SYNOPSIS:
*
* double x, y, cotdg();
*
* y = cotdg( x );
*
*
*
* DESCRIPTION:
*
* Returns the circular cotangent of the argument x in degrees.
*
* Range reduction is modulo pi/4. A rational function
* x + x**3 P(x**2)/Q(x**2)
* is employed in the basic interval [0, pi/4].
*
*
* ERROR MESSAGES:
*
* message condition value returned
* cotdg total loss x > 8.0e14 (DEC) 0.0
* x > 1.0e14 (IEEE)
* cotdg singularity x = 180 k MAXNUM
*/

/* tanh.c
*
* Hyperbolic tangent
*
*
*
* SYNOPSIS:
*
* double x, y, tanh();
*
* y = tanh( x );
*
*
*
* DESCRIPTION:
*
* Returns hyperbolic tangent of argument in the range MINLOG to
* MAXLOG.
*
* A rational function is used for |x| < 0.625. The form
* x + x**3 P(x)/Q(x) of Cody _& Waite is employed.
* Otherwise,
* tanh(x) = sinh(x)/cosh(x) = 1 - 2/(exp(2x) + 1).
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC -2,2 50000 3.3e-17 6.4e-18
* IEEE -2,2 30000 2.5e-16 5.8e-17
*
*/

/* unity.c
*
* Relative error approximations for function arguments near
* unity.
*
* log1p(x) = log(1+x)
* expm1(x) = exp(x) - 1
* cosm1(x) = cos(x) - 1
*
*/

/* yn.c
*
* Bessel function of second kind of integer order
*
*
*
* SYNOPSIS:
*
* double x, y, yn();
* int n;
*
* y = yn( n, x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of order n, where n is a
* (possibly negative) integer.
*
* The function is evaluated by forward recurrence on
* n, starting with values computed by the routines
* y0() and y1().
*
* If n = 0 or 1 the routine for y0 or y1 is called
* directly.
*
*
*
* ACCURACY:
*
*
* Absolute error, except relative
* when y > 1:
* arithmetic domain # trials peak rms
* DEC 0, 30 2200 2.9e-16 5.3e-17
* IEEE 0, 30 30000 3.4e-15 4.3e-16
*
*
* ERROR MESSAGES:
*
* message condition value returned
* yn singularity x = 0 MAXNUM
* yn overflow MAXNUM
*
* Spot checked against tables for x, n between 0 and 100.
*
*/

/* zeta.c
*
* Riemann zeta function of two arguments
*
*
*
* SYNOPSIS:
*
* double x, q, y, zeta();
*
* y = zeta( x, q );
*
*
*
* DESCRIPTION:
*
*
*
* inf.
* - -x
* zeta(x,q) = > (k+q)
* -
* k=0
*
* where x > 1 and q is not a negative integer or zero.
* The Euler-Maclaurin summation formula is used to obtain
* the expansion
*
* n
* - -x
* zeta(x,q) = > (k+q)
* -
* k=1
*
* 1-x inf. B x(x+1)...(x+2j)
* (n+q) 1 - 2j
* + --------- - ------- + > --------------------
* x-1 x - x+2j+1
* 2(n+q) j=1 (2j)! (n+q)
*
* where the B2j are Bernoulli numbers. Note that (see zetac.c)
* zeta(x,1) = zetac(x) + 1.
*
*
*
* ACCURACY:
*
*
*
* REFERENCE:
*
* Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals,
* Series, and Products, p. 1073; Academic Press, 1980.
*
*/

/* zetac.c
*
* Riemann zeta function
*
*
*
* SYNOPSIS:
*
* double x, y, zetac();
*
* y = zetac( x );
*
*
*
* DESCRIPTION:
*
*
*
* inf.
* - -x
* zetac(x) = > k , x > 1,
* -
* k=2
*
* is related to the Riemann zeta function by
*
* Riemann zeta(x) = zetac(x) + 1.
*
* Extension of the function definition for x < 1 is implemented.
* Zero is returned for x > log2(MAXNUM).
*
* An overflow error may occur for large negative x, due to the
* gamma function in the reflection formula.
*
* ACCURACY:
*
* Tabulated values have full machine accuracy.
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 1,50 10000 9.8e-16 1.3e-16
* DEC 1,50 2000 1.1e-16 1.9e-17
*
*
*/