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A computer program which can carry out mathematical integrations or
prove that the integration cannot be done must be the dream of every A-level
pupil struggling with homework that seems to be impossible. Earlier this
month those dreams came true, in the form of a program called Axiom, which
has been under development for 13 years.

‘You type in the formula,’ says James Davenport, professor of information
technology at the University of Bath, ‘and the program types back the solution
or a proof that it can’t be done.’ He says this type of program allows people
to solve problems that they simply could not attempt before.

An integral is the area on a graph between a curve and one of the graph’s
axes. Knowing this area is important in many disciplines, including physics,
engineering, chemistry and biology. Integrals can be approximated by dividing
the area into blocks with regular shapes and working out the combined area
of all the blocks. The accuracy of the result depends on how finely the
area is divided.

But if the curve can be described as a mathematical formula, integration
can be done algebraically, giving an exact answer. For example, integrating
the curve y=2x with respect to x gives x to the power of +c.

A computer needs a set of instructions, an algorithm, to tell it how
to integrate. The methods used by mathematicians involve a combination of
guesswork and simple rules. ‘This approach is not complete,’ says Davenport,
‘you try 10 rules, and if they don’t work, you look for a bigger textbook
with more rules. It gives you a potentially infinite list of things to try.’

Mathematicians began to look for systematic methods of integrating during
the last century, but no real progress was made until the late 1960s. By
1981 Davenport had worked out a complete, systematic algorithm. If the computer
works through the algorithm and comes up with no solution, there will not
be one. But, according to Malcolm MacCallum, professor of applied mathematics
at Queen Mary and Westfield College in London, Davenport’s algorithm ‘was
so horrible that nobody has implemented it in full’.

Parts of the algorithm have been made into Axiom, developed at IBM’s
research laboratories in Yorktown Heights, New York. Axiom has already been
used to control the camera on the commercial remote sensing satellite Spot.

Control of the satellite is so delicate and complicated that commands
made by its human operators have to be tested before they are carried out.
The effect of a command is turned into a series of mathematical equations,
which are processed by Axiom. It ensures, for example, that pointing the
camera at the Earth does not turn the solar panels away from the Sun, cutting
off the satellite’s power supply.

John Fitch, professor of software engineering at Bath and a colleague
of Davenport’s, said he no longer bothers with traditional integration but
always uses the computer algorithm. He predicts that integration by computer
‘could revolutionise algebra as the calculator and log tables did.’