Syntax
Int( function , variable , a , b [ , tolerance ] )
Examples
Int( x ^ 2 , x , 1 , 2 ) = 2.333... (all decimals correct)
Description
Gauss integral approximations are precise. The only method that produces better results is symbolic analysis of commercial software,
such as Maple or Mathematica. These use advanced symbolic derivation techniques to represent the integrals and then
evaluate expressions to provide an exact integral value or a very high precision.
The Expression Calculator is no rival to this software. It does integral approximations with Gauss quadrature method combined with
a rather recent (1980-s) technique, developed at the University of Geneva. The value of integrals is an approximation,
even if in most cases all decimals shown are found to be exact.
The Expression Calculator uses the Tegral embedded quadrature formulas method. The main idea is to approximate the functions' integrals
with an order 30 quadrature formula, which is calculated using Legendre's orthogonal polynomials. Two embedded quadrature
formulas are extracted from the order 30, one of order 14 and another of order 6. All formulas are symmetric. The integral
for the elementary interval is calculated from the order 30 quadrature formula, the maximum errors committed during the
calculation using the other two. Thus (and this is very different from classic Gauss integration methods) the program
knows exactly the maximum error it makes and can check the tolerance for the integral's result.
Gauss is calculated for an interval [ a , b ] without using the bounds. If the error committed satisfies the given tolerance,
the integral's value is kept as final. If not, the interval is divided into two equal parts and the Gauss integral is calculated
again. In case the sum of the absolute values of all errors does not match the tolerance, Tegral will look for the interval
that has the worst error and divide it into two parts. The algorithm recurses until a tolerated error value is reached.
The enhancements to Tegral are made through the Epsilon Algorithm (followed by Square Aitken's Delta theory, 1926), which
aims to find a faster parallel series that converges to the integral value. The Epsilon Algorithm is excellent, but huge
matrix determinants are to be calculated. The Expression Calculator uses a discovery of P. Wynn in 1936, which makes all this as simple
as a Pascal triangle. Details and demonstrations for the mathematical theories can be found in
C. Berezinsky, 1977 - "Accélération de la Convergence en Analyse Numérique", Lecture Notes in Mathematics, Nr. 584, Springer-Verlag
.
Notes
The Gauss integrals are never calculated on the interval bounds. For example, the famous Euler's Gamma integral
Int( Log( Log( 1 / x ) ) , x , 0 , 1 ) = -0.577215664901533
yields all decimals correct.
This function tends to infinity in x = 1 and has virtually no convergent values that could be used in
points close to x = 1.
