STEP 32

NUMERICAL INTEGRATION 3

Gauss integration formulae

The numerical integration procedures, discussed so far (i.e., the trapezoidal rule and Simpson's Rule) involve equally spaced argument values. However, for a fixed number of points, accuracy may be increased, if we do not insist on equidistant points. This is the background of an alternative integration process, due to Gauss. Thus, assuming that we can compute a specified number of values of the integrand (at arbitrary points), we shall construct a formula by selecting arguments (or abscissae) within the range of integration, in order to arrive at a most accurate integration rule.

1. Gauss two-point integration formula

   Consider any two-point formula of the form

   ,

   where the weights w_l, w₂ and the abscissae x_l, x₂ are to be determined such that the formula integrates exactly 1, x, x², and x³ (and hence all cubic functions). Then the following four conditions are imposed on the four unknowns:

   1. f (x) = 1 integrates exactly, ,
      
   2. f (x) = x integrates exactly, ,
      
   3. f (x) = x² integrates exactly, ,
      
   4. f (x) = x³ integrates exactly, /

   It is easily verified that

   

   satisfies the four equations 1 - 4. Thus, we arrive at the readily programmable Gauss two-point integration formula (PSEUDO-CODES):

   .

   The following change of variable makes this formula applicable to any integration interval.

   ,

   so that the evaluated integral becomes, say,

   ,

   If we write

   

   then:

   ,

   since .

   Note that the Gauss two-point formula is exact for cubic polynomials, and hence may be compared in accuracy with Simson's Rule. (In fact, the error for the Gauss formula is about 2/3 that for Simpson's Rule.). Since the Gauss formula requires one less evaluation of function values, it may be preferred, provided function evaluations at irrational abscissae values are possible.

2. Other Gauss formulae

   The Gauss two-point integration formula discussed is but one of a large family of such formulae. For example, the Gauss three-point formula

   ,

   exact for fifth degree polynomials, involves an error smaller than

   .

   These two formulae are the first in the series of so called Gauss-Legendre formulae, because of their association with Legendre polynomials.

   There are yet further formulae, associated with other orthogonal polynomials (Laguerre, Hermite, Chebyshev, etc.) of the general form

   ,

   where W(x) is referred to as the weight function, (x₁, x₂, . . . , x₂} is a set of points in the integration range and the weights w_i are known constants.

3. Application of Gauss quadrature

   The sets {x_i} and (w_i} are tabulated in reference books, so that application of Gauss quadrature is easy.

   As an demonstration of the Gauss-Legendre two-point and four-point formulae we will evaluate the integral:

   .

   The two-point formula

   ,

   after the change of variable

   ,

   yields:

   .

   Letting , we find g(-0.577 350 27) = 0.325 886 and g(0.577 350 27) = 0.945 409, whence:

   .

   The four-point formula

   .

   yields:

   ,

   a result correct to 7D. This accuracy is impressive enough; Simpson's Rule with 64 points yields 0.999 999 83!

Checkpoint

1. What is one disadvantage of integration formulae which employ equally spaced values of the argument?
   
2. What is the general form of the Gauss integration formula?
   
3. How accurate are the Gauss-Legendre two-point and three-point formulae?

   EXERCISE

   Apply the Gauss two-point and four-point formulae to evaluate the integral

   .

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