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| 67.1 Functions and Variables for romberg |
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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.
load(romberg) loads this function.
See also Einführung in QUADPACK, a collection of numerical integration functions.
Examples:
A 1-dimensional integration.
(%i1) load (romberg);
(%o1) /usr/share/maxima/5.11.0/share/numeric/romberg.lisp
(%i2) f(x) := 1/((x - 1)^2 + 1/100) + 1/((x - 2)^2 + 1/1000)
+ 1/((x - 3)^2 + 1/200);
1 1 1
(%o2) f(x) := -------------- + --------------- + --------------
2 1 2 1 2 1
(x - 1) + --- (x - 2) + ---- (x - 3) + ---
100 1000 200
(%i3) rombergtol : 1e-6;
(%o3) 9.9999999999999995E-7
(%i4) rombergit : 15;
(%o4) 15
(%i5) estimate : romberg (f(x), x, -5, 5);
(%o5) 173.6730736617464
(%i6) exact : integrate (f(x), x, -5, 5);
(%o6) 10 sqrt(10) atan(70 sqrt(10))
+ 10 sqrt(10) atan(30 sqrt(10)) + 10 sqrt(2) atan(80 sqrt(2))
+ 10 sqrt(2) atan(20 sqrt(2)) + 10 atan(60) + 10 atan(40)
(%i7) abs (estimate - exact) / exact, numer;
(%o7) 7.5527060865060088E-11
A 2-dimensional integration, implemented by nested calls to romberg.
(%i1) load (romberg);
(%o1) /usr/share/maxima/5.11.0/share/numeric/romberg.lisp
(%i2) g(x, y) := x*y / (x + y);
x y
(%o2) g(x, y) := -----
x + y
(%i3) rombergtol : 1e-6;
(%o3) 9.9999999999999995E-7
(%i4) estimate : romberg (romberg (g(x, y), y, 0, x/2), x, 1, 3);
(%o4) 0.81930239628356
(%i5) assume (x > 0);
(%o5) [x > 0]
(%i6) integrate (integrate (g(x, y), y, 0, x/2), x, 1, 3);
3
2 log(-) - 1
9 2 9
(%o6) - 9 log(-) + 9 log(3) + ------------ + -
2 6 2
(%i7) exact : radcan (%);
26 log(3) - 26 log(2) - 13
(%o7) - --------------------------
3
(%i8) abs (estimate - exact) / exact, numer;
(%o8) 1.3711979871851024E-10
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.
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.
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.
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.
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