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-/* acoshl.c
- *
- * Inverse hyperbolic cosine, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, acoshl();
- *
- * y = acoshl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns inverse hyperbolic cosine of argument.
- *
- * If 1 <= x < 1.5, a rational approximation
- *
- * sqrt(2z) * 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
- * IEEE 1,3 30000 2.0e-19 3.9e-20
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * acoshl domain |x| < 1 0.0
- *
- */
-
-/* asinhl.c
- *
- * Inverse hyperbolic sine, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, asinhl();
- *
- * y = asinhl( 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
- * IEEE -3,3 30000 1.7e-19 3.5e-20
- *
- */
-
-/* asinl.c
- *
- * Inverse circular sine, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * double x, y, asinl();
- *
- * y = asinl( 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
- * IEEE -1, 1 30000 2.7e-19 4.8e-20
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * asin domain |x| > 1 0.0
- *
- */
- /* acosl()
- *
- * Inverse circular cosine, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * double x, y, acosl();
- *
- * y = acosl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns radian angle between -pi/2 and +pi/2 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
- * IEEE -1, 1 30000 1.4e-19 3.5e-20
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * asin domain |x| > 1 0.0
- */
-
-/* atanhl.c
- *
- * Inverse hyperbolic tangent, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, atanhl();
- *
- * y = atanhl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns inverse hyperbolic tangent of argument in the range
- * MINLOGL to MAXLOGL.
- *
- * 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
- * IEEE -1,1 30000 1.1e-19 3.3e-20
- *
- */
-
-/* atanl.c
- *
- * Inverse circular tangent, long double precision
- * (arctangent)
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, atanl();
- *
- * y = atanl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns radian angle between -pi/2 and +pi/2 whose tangent
- * is x.
- *
- * Range reduction is from four intervals into the interval
- * from zero to tan( pi/8 ). The approximant uses a rational
- * function of degree 3/4 of the form x + x**3 P(x)/Q(x).
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -10, 10 150000 1.3e-19 3.0e-20
- *
- */
- /* atan2l()
- *
- * Quadrant correct inverse circular tangent,
- * long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, z, atan2l();
- *
- * z = atan2l( 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 60000 1.7e-19 3.2e-20
- * See atan.c.
- *
- */
-
-/* bdtrl.c
- *
- * Binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * long double p, y, bdtrl();
- *
- * y = bdtrl( 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 (k,n,p) with a and b between 0
- * and 10000 and p between 0 and 1.
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,10000 3000 1.6e-14 2.2e-15
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * bdtrl domain k < 0 0.0
- * n < k
- * x < 0, x > 1
- *
- */
- /* bdtrcl()
- *
- * Complemented binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * long double p, y, bdtrcl();
- *
- * y = bdtrcl( 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:
- *
- * See incbet.c.
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * bdtrcl domain x<0, x>1, n<k 0.0
- */
- /* bdtril()
- *
- * Inverse binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * long double p, y, bdtril();
- *
- * p = bdtril( 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:
- *
- * See incbi.c.
- * Tested at random k, n between 1 and 10000. The "domain" refers to p:
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,1 3500 2.0e-15 8.2e-17
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * bdtril domain k < 0, n <= k 0.0
- * x < 0, x > 1
- */
-
-
-/* btdtrl.c
- *
- * Beta distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * long double a, b, x, y, btdtrl();
- *
- * y = btdtrl( 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
- *
- *
- * The mean value of this distribution is a/(a+b). The variance
- * is ab/[(a+b)^2 (a+b+1)].
- *
- * This function is identical to the incomplete beta integral
- * function, incbetl(a, b, x).
- *
- * The complemented function is
- *
- * 1 - P(1-x) = incbetl( b, a, x );
- *
- *
- * ACCURACY:
- *
- * See incbetl.c.
- *
- */
-
-/* cbrtl.c
- *
- * Cube root, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, cbrtl();
- *
- * y = cbrtl( 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
- * IEEE .125,8 80000 7.0e-20 2.2e-20
- * IEEE exp(+-707) 100000 7.0e-20 2.4e-20
- *
- */
-
-/* chdtrl.c
- *
- * Chi-square distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * long double df, x, y, chdtrl();
- *
- * y = chdtrl( 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
- */
- /* chdtrcl()
- *
- * Complemented Chi-square distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * long double v, x, y, chdtrcl();
- *
- * y = chdtrcl( 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
- */
- /* chdtril()
- *
- * Inverse of complemented Chi-square distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * long double df, x, y, chdtril();
- *
- * x = chdtril( 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
- *
- */
-
-/* clogl.c
- *
- * Complex natural logarithm
- *
- *
- *
- * SYNOPSIS:
- *
- * void clogl();
- * cmplxl z, w;
- *
- * clogl( &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.
- */
-
- /* cexpl()
- *
- * Complex exponential function
- *
- *
- *
- * SYNOPSIS:
- *
- * void cexpl();
- * cmplxl z, w;
- *
- * cexpl( &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
- *
- */
- /* csinl()
- *
- * Complex circular sine
- *
- *
- *
- * SYNOPSIS:
- *
- * void csinl();
- * cmplxl z, w;
- *
- * csinl( &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.
- *
- */
- /* ccosl()
- *
- * Complex circular cosine
- *
- *
- *
- * SYNOPSIS:
- *
- * void ccosl();
- * cmplxl z, w;
- *
- * ccosl( &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
- */
- /* ctanl()
- *
- * Complex circular tangent
- *
- *
- *
- * SYNOPSIS:
- *
- * void ctanl();
- * cmplxl z, w;
- *
- * ctanl( &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.
- */
- /* ccotl()
- *
- * Complex circular cotangent
- *
- *
- *
- * SYNOPSIS:
- *
- * void ccotl();
- * cmplxl z, w;
- *
- * ccotl( &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.
- */
-
- /* casinl()
- *
- * Complex circular arc sine
- *
- *
- *
- * SYNOPSIS:
- *
- * void casinl();
- * cmplxl z, w;
- *
- * casinl( &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.
- */
- /* cacosl()
- *
- * Complex circular arc cosine
- *
- *
- *
- * SYNOPSIS:
- *
- * void cacosl();
- * cmplxl z, w;
- *
- * cacosl( &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
- */
-
- /* catanl()
- *
- * Complex circular arc tangent
- *
- *
- *
- * SYNOPSIS:
- *
- * void catanl();
- * cmplxl z, w;
- *
- * catanl( &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().
- */
-
-/* cmplxl.c
- *
- * Complex number arithmetic
- *
- *
- *
- * SYNOPSIS:
- *
- * typedef struct {
- * long double r; real part
- * long double i; imaginary part
- * }cmplxl;
- *
- * cmplxl *a, *b, *c;
- *
- * caddl( a, b, c ); c = b + a
- * csubl( a, b, c ); c = b - a
- * cmull( a, b, c ); c = b * a
- * cdivl( a, b, c ); c = b / a
- * cnegl( c ); c = -c
- * cmovl( 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
- */
-
-/* cabsl()
- *
- * Complex absolute value
- *
- *
- *
- * SYNOPSIS:
- *
- * long double cabsl();
- * cmplxl z;
- * long 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
- */
- /* csqrtl()
- *
- * Complex square root
- *
- *
- *
- * SYNOPSIS:
- *
- * void csqrtl();
- * cmplxl z, w;
- *
- * csqrtl( &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.
- */
-
-/* coshl.c
- *
- * Hyperbolic cosine, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, coshl();
- *
- * y = coshl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns hyperbolic cosine of argument in the range MINLOGL to
- * MAXLOGL.
- *
- * cosh(x) = ( exp(x) + exp(-x) )/2.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE +-10000 30000 1.1e-19 2.8e-20
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * cosh overflow |x| > MAXLOGL MAXNUML
- *
- *
- */
-
-/* elliel.c
- *
- * Incomplete elliptic integral of the second kind
- *
- *
- *
- * SYNOPSIS:
- *
- * long double phi, m, y, elliel();
- *
- * y = elliel( 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
- * IEEE -10,10 50000 2.7e-18 2.3e-19
- *
- *
- */
-
-/* ellikl.c
- *
- * Incomplete elliptic integral of the first kind
- *
- *
- *
- * SYNOPSIS:
- *
- * long double phi, m, y, ellikl();
- *
- * y = ellikl( 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 30000 3.6e-18 4.1e-19
- *
- *
- */
-
-/* ellpel.c
- *
- * Complete elliptic integral of the second kind
- *
- *
- *
- * SYNOPSIS:
- *
- * long double m1, y, ellpel();
- *
- * y = ellpel( 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
- * IEEE 0, 1 10000 1.1e-19 3.5e-20
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * ellpel domain x<0, x>1 0.0
- *
- */
-
-/* ellpjl.c
- *
- * Jacobian Elliptic Functions
- *
- *
- *
- * SYNOPSIS:
- *
- * long double u, m, sn, cn, dn, phi;
- * int ellpjl();
- *
- * ellpjl( 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-12 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
- * IEEE sn 10000 1.7e-18 2.3e-19
- * IEEE cn 20000 1.6e-18 2.2e-19
- * IEEE dn 10000 4.7e-15 2.7e-17
- * IEEE phi 10000 4.0e-19* 6.6e-20*
- *
- * Accuracy deteriorates when u is large.
- *
- */
-
-/* ellpkl.c
- *
- * Complete elliptic integral of the first kind
- *
- *
- *
- * SYNOPSIS:
- *
- * long double m1, y, ellpkl();
- *
- * y = ellpkl( 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
- * IEEE 0,1 10000 1.1e-19 3.3e-20
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * ellpkl domain x<0, x>1 0.0
- *
- */
-
-/* exp10l.c
- *
- * Base 10 exponential function, long double precision
- * (Common antilogarithm)
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, exp10l()
- *
- * y = exp10l( 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 +-4900 30000 1.0e-19 2.7e-20
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * exp10l underflow x < -MAXL10 0.0
- * exp10l overflow x > MAXL10 MAXNUM
- *
- * IEEE arithmetic: MAXL10 = 4932.0754489586679023819
- *
- */
-
-/* exp2l.c
- *
- * Base 2 exponential function, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, exp2l();
- *
- * y = exp2l( 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 +-16300 300000 9.1e-20 2.6e-20
- *
- *
- * See exp.c for comments on error amplification.
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * exp2l underflow x < -16382 0.0
- * exp2l overflow x >= 16384 MAXNUM
- *
- */
-
-/* expl.c
- *
- * Exponential function, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, expl();
- *
- * y = expl( 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 of degree 2/3 is used to approximate exp(f) - 1
- * in the basic range [-0.5 ln 2, 0.5 ln 2].
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE +-10000 50000 1.12e-19 2.81e-20
- *
- *
- * 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 long 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 MAXNUM
- *
- */
-
-/* fabsl.c
- *
- * Absolute value
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y;
- *
- * y = fabsl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the absolute value of the argument.
- *
- */
-
-/* fdtrl.c
- *
- * F distribution, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * int df1, df2;
- * long double x, y, fdtrl();
- *
- * y = fdtrl( 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) = incbetl( df1/2, df2/2, (df1*x/(df2 + df1*x) ).
- *
- *
- * The arguments a and b are greater than zero, and x
- * x is nonnegative.
- *
- * 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 10000 9.3e-18 2.9e-19
- * IEEE 0,1 1,10000 10000 1.9e-14 2.9e-15
- * IEEE 1,5 1,10000 10000 5.8e-15 1.4e-16
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * fdtrl domain a<0, b<0, x<0 0.0
- *
- */
- /* fdtrcl()
- *
- * Complemented F distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int df1, df2;
- * long double x, y, fdtrcl();
- *
- * y = fdtrcl( 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
- *
- * (See fdtr.c.)
- *
- * The incomplete beta integral is used, according to the
- * formula
- *
- * P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ).
- *
- *
- * ACCURACY:
- *
- * See incbet.c.
- * Tested at random points (a,b,x).
- *
- * x a,b Relative error:
- * arithmetic domain domain # trials peak rms
- * IEEE 0,1 0,100 10000 4.2e-18 3.3e-19
- * IEEE 0,1 1,10000 10000 7.2e-15 2.6e-16
- * IEEE 1,5 1,10000 10000 1.7e-14 3.0e-15
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * fdtrcl domain a<0, b<0, x<0 0.0
- *
- */
- /* fdtril()
- *
- * Inverse of complemented F distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int df1, df2;
- * long double x, p, fdtril();
- *
- * x = fdtril( 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:
- *
- * See incbi.c.
- * 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 40000 4.6e-18 2.7e-19
- * IEEE 1,10000 30000 1.7e-14 1.4e-16
- * For p between 10^-6 and .001:
- * IEEE 1,100 20000 1.9e-15 3.9e-17
- * IEEE 1,10000 30000 2.7e-15 4.0e-17
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * fdtril domain p <= 0 or p > 1 0.0
- * v < 1
- */
-
-/* ceill()
- * floorl()
- * frexpl()
- * ldexpl()
- * fabsl()
- *
- * Floating point numeric utilities
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y;
- * long double ceill(), floorl(), frexpl(), ldexpl(), fabsl();
- * int expnt, n;
- *
- * y = floorl(x);
- * y = ceill(x);
- * y = frexpl( x, &expnt );
- * y = ldexpl( x, n );
- * y = fabsl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * All four routines return a long double precision floating point
- * result.
- *
- * floorl() returns the largest integer less than or equal to x.
- * It truncates toward minus infinity.
- *
- * ceill() returns the smallest integer greater than or equal
- * to x. It truncates toward plus infinity.
- *
- * frexpl() 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.
- *
- * ldexpl() multiplies x by 2**n.
- *
- * fabsl() returns the absolute value of its argument.
- *
- * These functions are part of the standard C run time library
- * for some but not all C compilers. The ones supplied are
- * written in C for 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.
- */
-
-/* gammal.c
- *
- * Gamma function
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, gammal();
- * extern int sgngam;
- *
- * y = gammal( 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| <= 13 are reduced by recurrence and the function
- * approximated by a rational function of degree 7/8 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
- * IEEE -40,+40 10000 3.6e-19 7.9e-20
- * IEEE -1755,+1755 10000 4.8e-18 6.5e-19
- *
- * Accuracy for large arguments is dominated by error in powl().
- *
- */
-/* lgaml()
- *
- * Natural logarithm of gamma function
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, lgaml();
- * extern int sgngam;
- *
- * y = lgaml( 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 33, 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 MAXLGML (10^4928) return MAXNUML.
- *
- *
- *
- * ACCURACY:
- *
- *
- * arithmetic domain # trials peak rms
- * IEEE -40, 40 100000 2.2e-19 4.6e-20
- * IEEE 10^-2000,10^+2000 20000 1.6e-19 3.3e-20
- * The error criterion was relative when the function magnitude
- * was greater than one but absolute when it was less than one.
- *
- */
-
-/* gdtrl.c
- *
- * Gamma distribution function
- *
- *
- *
- * SYNOPSIS:
- *
- * long double a, b, x, y, gdtrl();
- *
- * y = gdtrl( 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
- * gdtrl domain x < 0 0.0
- *
- */
- /* gdtrcl.c
- *
- * Complemented gamma distribution function
- *
- *
- *
- * SYNOPSIS:
- *
- * long double a, b, x, y, gdtrcl();
- *
- * y = gdtrcl( 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
- * gdtrcl 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
-*/
-
-/* igamil()
- *
- * Inverse of complemented imcomplete gamma integral
- *
- *
- *
- * SYNOPSIS:
- *
- * long double a, x, y, igamil();
- *
- * x = igamil( a, y );
- *
- *
- *
- * DESCRIPTION:
- *
- * Given y, the function finds x such that
- *
- * igamc( a, x ) = y.
- *
- * Starting with the approximate value
- *
- * 3
- * x = a t
- *
- * where
- *
- * t = 1 - d - ndtri(y) sqrt(d)
- *
- * and
- *
- * d = 1/9a,
- *
- * the routine performs up to 10 Newton iterations to find the
- * root of igamc(a,x) - y = 0.
- *
- *
- * ACCURACY:
- *
- * Tested for a ranging from 0.5 to 30 and x from 0 to 0.5.
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * DEC 0,0.5 3400 8.8e-16 1.3e-16
- * IEEE 0,0.5 10000 1.1e-14 1.0e-15
- *
- */
-
-/* igaml.c
- *
- * Incomplete gamma integral
- *
- *
- *
- * SYNOPSIS:
- *
- * long double a, x, y, igaml();
- *
- * y = igaml( 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
- * DEC 0,30 4000 4.4e-15 6.3e-16
- * IEEE 0,30 10000 3.6e-14 5.1e-15
- *
- */
- /* igamcl()
- *
- * Complemented incomplete gamma integral
- *
- *
- *
- * SYNOPSIS:
- *
- * long double a, x, y, igamcl();
- *
- * y = igamcl( 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:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * DEC 0,30 2000 2.7e-15 4.0e-16
- * IEEE 0,30 60000 1.4e-12 6.3e-15
- *
- */
-
-/* incbetl.c
- *
- * Incomplete beta integral
- *
- *
- * SYNOPSIS:
- *
- * long double a, b, x, y, incbetl();
- *
- * y = incbetl( 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 random points (a,b,x) with x between 0 and 1.
- * arithmetic domain # trials peak rms
- * IEEE 0,5 20000 4.5e-18 2.4e-19
- * IEEE 0,100 100000 3.9e-17 1.0e-17
- * Half-integer a, b:
- * IEEE .5,10000 100000 3.9e-14 4.4e-15
- * Outputs smaller than the IEEE gradual underflow threshold
- * were excluded from these statistics.
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * incbetl domain x<0, x>1 0.0
- */
-
-/* incbil()
- *
- * Inverse of imcomplete beta integral
- *
- *
- *
- * SYNOPSIS:
- *
- * long double a, b, x, y, incbil();
- *
- * x = incbil( a, b, y );
- *
- *
- *
- * DESCRIPTION:
- *
- * Given y, the function finds x such that
- *
- * incbet( a, b, x ) = y.
- *
- * the routine performs up to 10 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 10000 1.1e-14 1.4e-16
- */
-
-/* j0l.c
- *
- * Bessel function of order zero
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, j0l();
- *
- * y = j0l( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of first kind, order zero of the argument.
- *
- * The domain is divided into the intervals [0, 9] and
- * (9, infinity). In the first interval the rational approximation
- * is (x^2 - r^2) (x^2 - s^2) (x^2 - t^2) P7(x^2) / Q8(x^2),
- * where r, s, t are the first three zeros of the function.
- * In the second interval the expansion is in terms of the
- * modulus M0(x) = sqrt(J0(x)^2 + Y0(x)^2) and phase P0(x)
- * = atan(Y0(x)/J0(x)). M0 is approximated by sqrt(1/x)P7(1/x)/Q7(1/x).
- * The approximation to J0 is M0 * cos(x - pi/4 + 1/x P5(1/x^2)/Q6(1/x^2)).
- *
- *
- * ACCURACY:
- *
- * Absolute error:
- * arithmetic domain # trials peak rms
- * IEEE 0, 30 100000 2.8e-19 7.4e-20
- *
- *
- */
- /* y0l.c
- *
- * Bessel function of the second kind, order zero
- *
- *
- *
- * SYNOPSIS:
- *
- * double x, y, y0l();
- *
- * y = y0l( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of the second kind, of order
- * zero, of the argument.
- *
- * The domain is divided into the intervals [0, 5>, [5,9> and
- * [9, infinity). In the first interval a rational approximation
- * R(x) is employed to compute y0(x) = R(x) + 2/pi * log(x) * j0(x).
- *
- * In the second interval, the approximation is
- * (x - p)(x - q)(x - r)(x - s)P7(x)/Q7(x)
- * where p, q, r, s are zeros of y0(x).
- *
- * The third interval uses the same approximations to modulus
- * and phase as j0(x), whence y0(x) = modulus * sin(phase).
- *
- * ACCURACY:
- *
- * Absolute error, when y0(x) < 1; else relative error:
- *
- * arithmetic domain # trials peak rms
- * IEEE 0, 30 100000 3.4e-19 7.6e-20
- *
- */
-
-/* j1l.c
- *
- * Bessel function of order one
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, j1l();
- *
- * y = j1l( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of order one of the argument.
- *
- * The domain is divided into the intervals [0, 9] and
- * (9, infinity). In the first interval the rational approximation
- * is (x^2 - r^2) (x^2 - s^2) (x^2 - t^2) x P8(x^2) / Q8(x^2),
- * where r, s, t are the first three zeros of the function.
- * In the second interval the expansion is in terms of the
- * modulus M1(x) = sqrt(J1(x)^2 + Y1(x)^2) and phase P1(x)
- * = atan(Y1(x)/J1(x)). M1 is approximated by sqrt(1/x)P7(1/x)/Q8(1/x).
- * The approximation to j1 is M1 * cos(x - 3 pi/4 + 1/x P5(1/x^2)/Q6(1/x^2)).
- *
- *
- * ACCURACY:
- *
- * Absolute error:
- * arithmetic domain # trials peak rms
- * IEEE 0, 30 40000 1.8e-19 5.0e-20
- *
- *
- */
- /* y1l.c
- *
- * Bessel function of the second kind, order zero
- *
- *
- *
- * SYNOPSIS:
- *
- * double x, y, y1l();
- *
- * y = y1l( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of the second kind, of order
- * zero, of the argument.
- *
- * The domain is divided into the intervals [0, 4.5>, [4.5,9> and
- * [9, infinity). In the first interval a rational approximation
- * R(x) is employed to compute y0(x) = R(x) + 2/pi * log(x) * j0(x).
- *
- * In the second interval, the approximation is
- * (x - p)(x - q)(x - r)(x - s)P9(x)/Q10(x)
- * where p, q, r, s are zeros of y1(x).
- *
- * The third interval uses the same approximations to modulus
- * and phase as j1(x), whence y1(x) = modulus * sin(phase).
- *
- * ACCURACY:
- *
- * Absolute error, when y0(x) < 1; else relative error:
- *
- * arithmetic domain # trials peak rms
- * IEEE 0, 30 36000 2.7e-19 5.3e-20
- *
- */
-
-/* jnl.c
- *
- * Bessel function of integer order
- *
- *
- *
- * SYNOPSIS:
- *
- * int n;
- * long double x, y, jnl();
- *
- * y = jnl( 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 domain # trials peak rms
- * IEEE -30, 30 5000 3.3e-19 4.7e-20
- *
- *
- * Not suitable for large n or x.
- *
- */
-
-/* ldrand.c
- *
- * Pseudorandom number generator
- *
- *
- *
- * SYNOPSIS:
- *
- * double y;
- * int ldrand();
- *
- * ldrand( &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.
- *
- * Versions invoked by the different arithmetic compile
- * time options IBMPC, and MIEEE, produce the same sequences.
- *
- */
-
-/* log10l.c
- *
- * Common logarithm, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, log10l();
- *
- * y = log10l( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the base 10 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 30000 9.0e-20 2.6e-20
- * IEEE exp(+-10000) 30000 6.0e-20 2.3e-20
- *
- * In the tests over the interval exp(+-10000), the logarithms
- * of the random arguments were uniformly distributed over
- * [-10000, +10000].
- *
- * ERROR MESSAGES:
- *
- * log singularity: x = 0; returns MINLOG
- * log domain: x < 0; returns MINLOG
- */
-
-/* log2l.c
- *
- * Base 2 logarithm, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, log2l();
- *
- * y = log2l( 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 (natural)
- * 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 9.8e-20 2.7e-20
- * IEEE exp(+-10000) 70000 5.4e-20 2.3e-20
- *
- * In the tests over the interval exp(+-10000), the logarithms
- * of the random arguments were uniformly distributed over
- * [-10000, +10000].
- *
- * ERROR MESSAGES:
- *
- * log singularity: x = 0; returns MINLOG
- * log domain: x < 0; returns MINLOG
- */
-
-/* logl.c
- *
- * Natural logarithm, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, logl();
- *
- * y = logl( 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 8.71e-20 2.75e-20
- * IEEE exp(+-10000) 100000 5.39e-20 2.34e-20
- *
- * In the tests over the interval exp(+-10000), the logarithms
- * of the random arguments were uniformly distributed over
- * [-10000, +10000].
- *
- * ERROR MESSAGES:
- *
- * log singularity: x = 0; returns MINLOG
- * log domain: x < 0; returns MINLOG
- */
-
-/* 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 mconf.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:
- *
- * mconf.h
- *
- */
-
-/* nbdtrl.c
- *
- * Negative binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * long double p, y, nbdtrl();
- *
- * y = nbdtrl( 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 (k,n,p) with k and n between 1 and 10,000
- * and p between 0 and 1.
- *
- * arithmetic domain # trials peak rms
- * Absolute error:
- * IEEE 0,10000 10000 9.8e-15 2.1e-16
- *
- */
- /* nbdtrcl.c
- *
- * Complemented negative binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * long double p, y, nbdtrcl();
- *
- * y = nbdtrcl( 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 incbetl.c.
- *
- */
- /* nbdtril
- *
- * Functional inverse of negative binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * long double p, y, nbdtril();
- *
- * p = nbdtril( 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
- * See also incbil.c.
- */
-
-/* ndtril.c
- *
- * Inverse of Normal distribution function
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, ndtril();
- *
- * x = ndtril( 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 log(y) ); then the approximation is
- * x = z - log(z)/z - (1/z) P(1/z) / Q(1/z) .
- * For larger arguments, x/sqrt(2 pi) = w + w^3 R(w^2)/S(w^2)) ,
- * where w = y - 0.5 .
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * Arguments uniformly distributed:
- * IEEE 0, 1 5000 7.8e-19 9.9e-20
- * Arguments exponentially distributed:
- * IEEE exp(-11355),-1 30000 1.7e-19 4.3e-20
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * ndtril domain x <= 0 -MAXNUML
- * ndtril domain x >= 1 MAXNUML
- *
- */
-
-/* ndtril.c
- *
- * Inverse of Normal distribution function
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, ndtril();
- *
- * x = ndtril( 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 log(y) ); then the approximation is
- * x = z - log(z)/z - (1/z) P(1/z) / Q(1/z) .
- * For larger arguments, x/sqrt(2 pi) = w + w^3 R(w^2)/S(w^2)) ,
- * where w = y - 0.5 .
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * Arguments uniformly distributed:
- * IEEE 0, 1 5000 7.8e-19 9.9e-20
- * Arguments exponentially distributed:
- * IEEE exp(-11355),-1 30000 1.7e-19 4.3e-20
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * ndtril domain x <= 0 -MAXNUML
- * ndtril domain x >= 1 MAXNUML
- *
- */
-
-/* pdtrl.c
- *
- * Poisson distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k;
- * long double m, y, pdtrl();
- *
- * y = pdtrl( 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().
- *
- */
- /* pdtrcl()
- *
- * Complemented poisson distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k;
- * long double m, y, pdtrcl();
- *
- * y = pdtrcl( 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.
- *
- */
- /* pdtril()
- *
- * Inverse Poisson distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k;
- * long double m, y, pdtrl();
- *
- * m = pdtril( 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
- *
- */
-
-/* polevll.c
- * p1evll.c
- *
- * Evaluate polynomial
- *
- *
- *
- * SYNOPSIS:
- *
- * int N;
- * long double x, y, coef[N+1], polevl[];
- *
- * y = polevll( 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 p1evll() assumes that coef[N] = 1.0 and is
- * omitted from the array. Its calling arguments are
- * otherwise the same as polevll().
- *
- * This module also contains the following globally declared constants:
- * MAXNUML = 1.189731495357231765021263853E4932L;
- * MACHEPL = 5.42101086242752217003726400434970855712890625E-20L;
- * MAXLOGL = 1.1356523406294143949492E4L;
- * MINLOGL = -1.1355137111933024058873E4L;
- * LOGE2L = 6.9314718055994530941723E-1L;
- * LOG2EL = 1.4426950408889634073599E0L;
- * PIL = 3.1415926535897932384626L;
- * PIO2L = 1.5707963267948966192313L;
- * PIO4L = 7.8539816339744830961566E-1L;
- *
- * 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.
- *
- */
-
-/* powil.c
- *
- * Real raised to integer power, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, powil();
- * int n;
- *
- * y = powil( 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
- * IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18
- * IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18
- * IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17
- *
- * Returns MAXNUM on overflow, zero on underflow.
- *
- */
-
-/* powl.c
- *
- * Power function, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, z, powl();
- *
- * z = powl( 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/32 and pseudo extended precision arithmetic to
- * obtain several extra bits of accuracy in both the logarithm
- * and the exponential.
- *
- *
- *
- * ACCURACY:
- *
- * The relative error of pow(x,y) can be estimated
- * by y dl ln(2), where dl is the absolute error of
- * the internally computed base 2 logarithm. At the ends
- * of the approximation interval the logarithm equal 1/32
- * and its relative error is about 1 lsb = 1.1e-19. Hence
- * the predicted relative error in the result is 2.3e-21 y .
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- *
- * IEEE +-1000 40000 2.8e-18 3.7e-19
- * .001 < x < 1000, with log(x) uniformly distributed.
- * -1000 < y < 1000, y uniformly distributed.
- *
- * IEEE 0,8700 60000 6.5e-18 1.0e-18
- * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * pow overflow x**y > MAXNUM MAXNUM
- * pow underflow x**y < 1/MAXNUM 0.0
- * pow domain x<0 and y noninteger 0.0
- *
- */
-
-/* sinhl.c
- *
- * Hyperbolic sine, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, sinhl();
- *
- * y = sinhl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns hyperbolic sine of argument in the range MINLOGL to
- * MAXLOGL.
- *
- * 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
- * IEEE -2,2 10000 1.5e-19 3.9e-20
- * IEEE +-10000 30000 1.1e-19 2.8e-20
- *
- */
-
-/* sinl.c
- *
- * Circular sine, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, sinl();
- *
- * y = sinl( 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 the Cody
- * and Waite polynomial form
- * x + x**3 P(x**2) .
- * Between pi/4 and pi/2 the cosine is represented as
- * 1 - .5 x**2 + x**4 Q(x**2) .
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE +-5.5e11 200,000 1.2e-19 2.9e-20
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * sin total loss x > 2**39 0.0
- *
- * Loss of precision occurs for x > 2**39 = 5.49755813888e11.
- * The routine as implemented flags a TLOSS error for
- * x > 2**39 and returns 0.0.
- */
- /* cosl.c
- *
- * Circular cosine, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, cosl();
- *
- * y = cosl( 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 - .5 x**2 + x**4 Q(x**2) .
- * Between pi/4 and pi/2 the sine is represented by the Cody
- * and Waite polynomial form
- * x + x**3 P(x**2) .
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE +-5.5e11 50000 1.2e-19 2.9e-20
- */
-
-/* sqrtl.c
- *
- * Square root, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, sqrtl();
- *
- * y = sqrtl( 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.
- *
- * Note, some arithmetic coprocessors such as the 8087 and
- * 68881 produce correctly rounded square roots, which this
- * routine will not.
- *
- * ACCURACY:
- *
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,10 30000 8.1e-20 3.1e-20
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * sqrt domain x < 0 0.0
- *
- */
-
-/* stdtrl.c
- *
- * Student's t distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * long double p, t, stdtrl();
- * int k;
- *
- * p = stdtrl( 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 < -1.6, 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 <= 100. The "domain" refers to t.
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -100,-1.6 10000 5.7e-18 9.8e-19
- * IEEE -1.6,100 10000 3.8e-18 1.0e-19
- */
-
-/* stdtril.c
- *
- * Functional inverse of Student's t distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * long double p, t, stdtril();
- * int k;
- *
- * t = stdtril( k, p );
- *
- *
- * DESCRIPTION:
- *
- * Given probability p, finds the argument t such that stdtrl(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 0,1 3500 4.2e-17 4.1e-18
- */
-
-/* tanhl.c
- *
- * Hyperbolic tangent, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, tanhl();
- *
- * y = tanhl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns hyperbolic tangent of argument in the range MINLOGL to
- * MAXLOGL.
- *
- * 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
- * IEEE -2,2 30000 1.3e-19 2.4e-20
- *
- */
-
-/* tanl.c
- *
- * Circular tangent, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, tanl();
- *
- * y = tanl( 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
- * IEEE +-1.07e9 30000 1.9e-19 4.8e-20
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * tan total loss x > 2^39 0.0
- *
- */
- /* cotl.c
- *
- * Circular cotangent, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, cotl();
- *
- * y = cotl( 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 1.9e-19 5.1e-20
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * cot total loss x > 2^39 0.0
- * cot singularity x = 0 MAXNUM
- *
- */
-
-/* unityl.c
- *
- * Relative error approximations for function arguments near
- * unity.
- *
- * log1p(x) = log(1+x)
- * expm1(x) = exp(x) - 1
- * cos1m(x) = cos(x) - 1
- *
- */
-
-/* ynl.c
- *
- * Bessel function of second kind of integer order
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, ynl();
- * int n;
- *
- * y = ynl( 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
- * y0l() and y1l().
- *
- * If n = 0 or 1 the routine for y0l or y1l is called
- * directly.
- *
- *
- *
- * ACCURACY:
- *
- *
- * Absolute error, except relative error when y > 1.
- * x >= 0, -30 <= n <= +30.
- * arithmetic domain # trials peak rms
- * IEEE -30, 30 10000 1.3e-18 1.8e-19
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * ynl singularity x = 0 MAXNUML
- * ynl overflow MAXNUML
- *
- * Spot checked against tables for x, n between 0 and 100.
- *
- */
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