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
.