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2 Methods |
Numerical integration is performed for three main reasons:
The most primitive numerical integration method is suggested by the Riemann integral:
Reasons for Numerical Integration
Methods
The sum on the right is termed a Riemann sum, since it looks like the integral of a step function (see the section on the Riemann integral.) Of course, it may not be possible in general to construe the sum on the right as being either a lower, or upper sum, however, if one can build actual lower and upper sums, then this algorithm can be made to yield an interval of certainty.
The disadvantage of this algorithm is that it has extremely poor convergence properties. In order to get any precision at all, one needs n to be extremely large, even for extremely well behaved functions. For instance, with
We note at this point that the step function in (*) was built by approximating f(x) over each interval by the value of f at x=(k+1)/n, that is, the right-hand side of the interval. Had we chosen the middle of the interval instead, for a large class of functions (at least all twice differentiable functions) the error term of the algorithm would have been at least one order of magnitude better.
Using the function value in the middle of two equidistant points and multiply it by the length of the interval is referred to as the midpoint rule or rectangular rule in numerical integration. In (*) this would mean evaluating the function at f(0.5) and multiply by 1. This is a common method.
Instead of using linear polynomials, one can use polynomials of any degree, these algorithms are called quadrature algorithms.
Interpolation with polynomials on equally-spaced intervals, i.e. where the integration span is divided into equidistant subintervals, correspond to Newton-Cotes formulas. In this group you will find the midpoint rule with polynomial index 0, the trapezoid rule with index 1 and at index 2 a quadratic polynomial interpolating between three points at a time. Using quadratic polynomials leads to Simpson's rule.
If we lose the criteria of equally spaced intervals, we find another group of quadrature formulas, the best of which are known as Gaussian quadrature. (Another such quadrature formula is to interpolate using Chebyshev polynomials.) The error for such quadrature can be shown to be very small, if the function f is very smooth (if it has many derivatives.)
If f does not have many derivatives at all points, or if the derivatives become large, then Gaussian quadrature is often insufficient. In this case, an algorithm similar to the following will perform better:
x:=0
h:=initial_h
accumulator:=0
WHILE x<1.0 DO
IF x+h>1.0 THEN
h=1.0-x
END IF
IF error from quadrature over [x,x+h] for f is too large THEN
Make h smaller
ELSE
accumulator:=accumulator + quadrature of f over [x,x+h]
x:=x+h
IF error from quadrature over [x,x+h] is very small THEN
Make h larger
END IF
END IF
END WHILE
RETURN accumulator
Of course, more sophisticated heuristics are possible.
Let f have a bounded first derivative over [a,b]. The mean value theorem for f, where , gives
This integration method can be combined with interval arithmetic to produce computer proofs and verified calculations.
See also: Runge-Kutta method, Numerical partial differential equations, Finite element method, Monte Carlo simulation.Quadrature
Another very simple choice to approximate the integral on the left hand side of (*) would be to evaluate f(0) and f(1), then to approximate f with a linear polynomial between 0 and 1, and then calculate the exact area under this linear polynomial. This yields
This is called the trapezoid rule of numerical integration. Instead of using a single trapezoid, we can use multiple trapezoids placed at regular intervals, in which case we obtain
The preceding formula is called the iterated trapezoid rule.Adaptive algorithms
// This algorithm calculates the definite integral of a function
// from 0 to 1, adaptively, by choosing smaller steps near
// problematic points.
// Set initial_h to the initial step size.
Some details of the algorithm require careful thought. For many cases, estimating the error from quadrature over an interval for a function f isn't obvious. One popular solution is to use two different rules of quadrature, and use their difference as an estimate of the error from quadrature. The other problem is deciding what "too large" or "very small" signify. A possible criterion for "too large" is that the quadrature error should not be larger than th where t, a real number, is the tolerance we wish to set for global error. Then again, if h is already tiny, it may not be worthwhile to make it even smaller even if the quadrature error is apparently large. This type of error analysis is usually called "a posteriori" since we compute the error after having computed the approximation.Conservative (a priori) error estimation
for some yx in [a,x] depending on x. If we integrate in x from a to b on both sides and the take absolute values, we obtain
We can further approximate the integral on the right-hand side by bringing the absolute value into the integrand, and replacing the term in f' by an upper bound:
(See supremum.) Hence, if we approximate the integral ∫abf(x)dx by the quadrature rule (b-a)f(a) our error is no greater than the right hand side of (**). We can convert this into an error analysis for the Riemann sum (*), giving an upper bound of
for the error term of that particular approximation. (Note that this is precisely the error we calculated for the example .) Using more derivatives, and by tweaking the quadrature, we can do a similar error analysis using a Taylor series (using a partial sum with remainder term) for f. This error analysis gives a strict upper bound on the error, if the derivatives of f are available.