80 romberg

80.1 Functions and Variables for romberg

Function: romberg
    romberg (expr, x, a, b)
    romberg (F, a, b)

Computes a numerical integration by Romberg’s method.

romberg(expr, x, a, b) returns an estimate of the integral integrate(expr, x, a, b). expr must be an expression which evaluates to a floating point value when x is bound to a floating point value.

romberg(F, a, b) returns an estimate of the integral integrate(F(x), x, a, b) where x represents the unnamed, sole argument of F; the actual argument is not named x. F must be a Maxima or Lisp function which returns a floating point value when the argument is a floating point value. F may name a translated or compiled Maxima function.

The accuracy of romberg is governed by the global variables rombergabs and rombergtol. romberg terminates successfully when the absolute difference between successive approximations is less than rombergabs, or the relative difference in successive approximations is less than rombergtol. Thus when rombergabs is 0.0 (the default) only the relative error test has any effect on romberg.

romberg halves the stepsize at most rombergit times before it gives up; the maximum number of function evaluations is therefore 2^rombergit. If the error criterion established by rombergabs and rombergtol is not satisfied, romberg prints an error message. romberg always makes at least rombergmin iterations; this is a heuristic intended to prevent spurious termination when the integrand is oscillatory.

romberg repeatedly evaluates the integrand after binding the variable of integration to a specific value (and not before). This evaluation policy makes it possible to nest calls to romberg, to compute multidimensional integrals. However, the error calculations do not take the errors of nested integrations into account, so errors may be underestimated. Also, methods devised especially for multidimensional problems may yield the same accuracy with fewer function evaluations.

See also Introduction to QUADPACK, a collection of numerical integration functions.

Examples:

A 1-dimensional integration.

(%i1) f(x) := 1/((x - 1)^2 + 1/100) + 1/((x - 2)^2 + 1/1000)
              + 1/((x - 3)^2 + 1/200);
                    1                 1                1
(%o1) f(x) := -------------- + --------------- + --------------
                     2    1           2    1            2    1
              (x - 1)  + ---   (x - 2)  + ----   (x - 3)  + ---
                         100              1000              200

(%i2) rombergtol : 1e-6;
(%o2)                 9.999999999999999e-7

(%i3) rombergit : 15;
(%o3)                          15

(%i4) estimate : romberg (f(x), x, -5, 5);
(%o4)                   173.6730736617464

(%i5) exact : integrate (f(x), x, -5, 5);
        3/2          3/2      3/2          3/2
(%o5) 10    atan(7 10   ) + 10    atan(3 10   )
      3/2         9/2       3/2         5/2
 + 5 2    atan(5 2   ) + 5 2    atan(5 2   ) + 10 atan(60)
 + 10 atan(40)

(%i6) abs (estimate - exact) / exact, numer;
(%o6)                 7.552722451569877e-11

A 2-dimensional integration, implemented by nested calls to romberg.

(%i1) g(x, y) := x*y / (x + y);
                                    x y
(%o1)                   g(x, y) := -----
                                   x + y

(%i2) rombergtol : 1e-6;
(%o2)                 9.999999999999999e-7

(%i3) estimate : romberg (romberg (g(x, y), y, 0, x/2), x, 1, 3);
(%o3)                  0.8193023962835647

(%i4) assume (x > 0);
(%o4)                        [x > 0]

(%i5) integrate (integrate (g(x, y), y, 0, x/2), x, 1, 3);
                                           3
                                     2 log(-) - 1
                    9                      2        9
(%o5)      (- 9 log(-)) + 9 log(3) + ------------ + -
                    2                     6         2

(%i6) exact : radcan (%);
                    26 log(3) - 26 log(2) - 13
(%o6)             - --------------------------
                                3

(%i7) abs (estimate - exact) / exact, numer;
(%o7)                 1.371197987185102e-10

Categories:  Package romberg Numerical methods

Option variable: rombergabs

Default value: 0.0

The accuracy of romberg is governed by the global variables rombergabs and rombergtol. romberg terminates successfully when the absolute difference between successive approximations is less than rombergabs, or the relative difference in successive approximations is less than rombergtol. Thus when rombergabs is 0.0 (the default) only the relative error test has any effect on romberg.

See also rombergit and rombergmin.

Categories:  Package romberg

Option variable: rombergit

Default value: 11

romberg halves the stepsize at most rombergit times before it gives up; the maximum number of function evaluations is therefore 2^rombergit. romberg always makes at least rombergmin iterations; this is a heuristic intended to prevent spurious termination when the integrand is oscillatory.

See also rombergabs and rombergtol.

Categories:  Package romberg

Option variable: rombergmin

Default value: 0

romberg always makes at least rombergmin iterations; this is a heuristic intended to prevent spurious termination when the integrand is oscillatory.

See also rombergit, rombergabs, and rombergtol.

Categories:  Package romberg

Option variable: rombergtol

Default value: 1e-4

The accuracy of romberg is governed by the global variables rombergabs and rombergtol. romberg terminates successfully when the absolute difference between successive approximations is less than rombergabs, or the relative difference in successive approximations is less than rombergtol. Thus when rombergabs is 0.0 (the default) only the relative error test has any effect on romberg.

See also rombergit and rombergmin.

Categories:  Package romberg