uclibc-ng/libm/double

/* 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
*
*
*/