Leonid Kantorovich linked pure maths to practical computing

Person: Leonid Kantorovich
Lived: 1912–1986
Field: Mathematical economics
From: Soviet

The story

In the 1930s Leonid Kantorovich, a young professor at Leningrad University, worked in two worlds that seemed to have nothing in common. In pure mathematics he helped build the new field of functional analysis, which studies spaces whose points are functions. In applied mathematics, spurred by the rapid industrialisation of the country, he developed new approximate methods for practical problems, set out in Approximate Methods of Higher Analysis, written with V. I. Krylov in 1936.

His pure research focused on spaces of functions with an ordering, so that some elements can be called larger than others. In 1935 he defined a new type of space that he called a semiordered linear space; such spaces are now known as Kantorovich spaces, or K-spaces, in his honour. Similar ideas were being developed at the same time in the United States, Japan and the Netherlands, and he discussed them with John von Neumann, Garrett Birkhoff and others at the Moscow Topological Congress of 1935. A complete book on the subject, written with colleagues, followed in 1950.

After the Second World War he succeeded in joining his two worlds. In a paper whose title, Functional Analysis and Applied Mathematics, seemed paradoxical at the time, he showed that the ideas of functional analysis could be used to analyse iterative numerical methods. Among the results were the Kantorovich inequalities on how fast the gradient method and Newton's method converge. The Kantorovich theorem, first stated in 1948, gives conditions under which Newton's method, starting from a given point, is guaranteed to find a nearby solution.

The work was awarded a State Prize in 1949 and was later expanded into the book Functional Analysis in Normed Spaces, written with G. P. Akilov and published in 1959. Reviewers welcomed its emphasis on concrete applications to approximation and to differential and integral equations.

Today the Newton-Kantorovich theorem remains a standard result in numerical analysis, and the idea that abstract mathematics can guarantee when a computer method will succeed underlies much of modern scientific computing.

Why it matters

Kantorovich's theorems tell engineers and scientists when computer methods for solving equations will work, a basis of modern numerical computing.

About Leonid Kantorovich

Leningrad mathematician who invented linear programming in 1939 while solving a plywood factory’s puzzle, and the only Soviet economist to win the Nobel Prize, in 1975.

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Leonid Kantorovich

1912–1986 · Mathematical economics

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