(maxima.info)Functions and Variables for romberg
79.1 Functions and Variables for romberg
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-- 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.
'load(romberg)' loads this function.
See also '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
-- 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'.
-- 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'.
-- 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'.
-- 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'.
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