/* (kill (all), load (romberg), float_approx_equal_tolerance : rombergtol, 0); */
(load (romberg), float_approx_equal_tolerance : rombergtol, 0);
0;

(I : [2*x - -(log((4 + %e)/(2*%pi)))*(((4 + %e)/(2*%pi))^x), x, -1, 0],
 apply (romberg, I));
''(first (apply (quad_qags, I)));

(I : [2*x - cos((%e + %pi)*x), x, 0, 1],
 apply (romberg, I));
''(apply (integrate, I),
 ev (%%, numer, %enumer));

(expr : x^2 - 5,
 romberg (expr, x, 0, 10));
''(integrate (x^2 - 5, x, 0, 10),
 ev (%%, numer));

/* verify that romberg expression is returned for non-numeric expression or bounds
 */
(kill (a, b), romberg (x^a - 5, x, 0, 10));
romberg (x^a - 5, x, 0.0, 10.0);

romberg (x^3 - 5, x, a, b);
romberg (x^3 - 5, x, a, b);

/* verify that romberg nested inside another function call is OK */
(quad_qags (romberg (sin(a)*cos(x) - a*x, x, 0, 10), a, 1, 3),
 first (%%));
''(integrate (integrate (sin(a)*cos(x) - a*x, x, 0, 10), a, 1, 3),
 ev (%%, numer));

romberg (romberg (a^2 - x, a, 0, x) - 7, x, 0, 100);
7999300.0;

/* verify that symbolic function name is OK */
(foo (a) := 3^a - 5,
 bar : foo,
 romberg (bar, 0, 10));
''(integrate (bar (a), a, 0, 10),
 ev (%%, numer));

/* misc examples */

(expr : t - (297 * exp ((1000000 * t) / 33) - 330) / 10000000,
 romberg (expr, t, 1e-9, 0.003));
''(float ((-9801*%e^(1000/11)+9801*%e^(1/33000)+9197999933999/200000) /10000000000000));

(expr : 6096 * tan((2 * atan(c/(2 * fl))) / r) / (tan((1/60) * (%pi/180))),
 ev (romberg (expr-6096, fl, 1, 10), c=7.176, r=3264));
''(first (ev (quad_qags (expr-6096, fl, 1, 10), c=7.176, r=3264)));

/* more fun with evaluation */

(g (a) := romberg (f (x, a), x, 0, 200),
 f (x, a) := x^a - 5,
 0);
0;

g (0.5);
''(integrate (f (x, 1/2), x, 0, 200), ev (%%, numer));

expr : g (z + z);
romberg (x^(2 * z) - 5, x, 0.0, 200.0);

''(at (expr, z=0.25));
''(integrate (f (x, 1/2), x, 0, 200), ev (%%, numer));

(quad_qags (g (z), z, 1, 3),
 first (%%));
''(assume (z > 0),
 integrate (f (x, z), x, 0, 200),
 quad_qags (%%, z, 1, 3),
 first (%%));

/* verify that symbolic constants (in integrand or limits of integration) evaluate to numbers */

(f(x) := cos (sin (%e) + %e^%pi + %pi^%e) * sin(x) - x/2, 0);
0;

[romberg (cos (sin (%e) + %e^%pi + %pi^%e) * sin(x) - x/2, x, 0.1, %pi),
 romberg (f(x), x, 0.1, %pi),
 romberg (f, 0.1, %pi)];
''(integrate (f (x), x, a, b),
 ev (%%, a=0.1, b=%pi, numer, %enumer),
 [%%, %%, %%]);

[romberg (f, 1/(%pi*%e), 2*%pi*sin(%e)),
 romberg (f, log(%pi), %e^%pi),
 romberg (f, sin(%e), %pi^%e),
 romberg (f, exp(1/5), exp(cos(%e + %pi))),
 romberg (f, cos(exp(2))/10, 10*cos(exp(2)))];
''([integrate (f (x), x, 1/(%pi*%e), 2*%pi*sin(%e)),
 integrate (f (x), x, log(%pi), %e^%pi),
 integrate (f (x), x, sin(%e), %pi^%e),
 integrate (f (x), x, exp(1/5), exp(cos(%e + %pi))),
 integrate (f (x), x, cos(exp(2))/10, 10*cos(exp(2)))],
 ev (%%, numer, %enumer));

/* adapted from the mailing list 2007/06/10
 * charfun2 copied from the interpol share package
 */

/* KNOWN FAILURE: romberg computes a result but relative error is greater than rombergtol */

(charfun2 (z, l1, l2) := charfun (l1 <= z and z < l2),
 e1 : (-.329*x^3+.494*x^2 +.559*x+.117),
 e2 : (.215*x^3-1.94*x^2 +4.85*x-2.77),
 e3 : (.0933*x^3-1.02*x^2+2.56*x-.866),
 e4 : (.0195*x^3-.581*x^2+1.67*x-.275),
 e5 : (.00117*x^3-.498*x^2 +1.55*x -.213),
 expr : e1 *charfun2(x,minf,1.0)
    +e2 *charfun2(x,2.5,inf)
    +e3 *charfun2(x,2.0,2.5)
    +e4 *charfun2(x,1.5,2.0)
    +e5 *charfun2(x,1.0,1.5),
 romberg (expr, x, 0, 4));
''(ev (integrate (e1, x,0,1)
 + integrate (e2, x,5/2,4)
 + integrate (e3, x,2,5/2)
 + integrate (e4, x,3/2,2)
 + integrate (e5, x,1,3/2), keepfloat));

(reset(float_approx_equal_tolerance),0);
0$
