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# 67. romberg

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## 67.1 Functions and Variables for romberg

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

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
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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