A.N.Krylov |

Alexei Nikolaevich Krylov was born on August 3, 1863 in a village of the Simbirsk Region in Russia. The name A.N. Krylov is probably one of the most widely known and recognized in Russia among outstanding scientists in the 20th century. Although his research interests were incredibly broad (at least for a scientist of the 20th century), his professional activity could apparently be best characterized by saying that he was a very good Maritime Engineer. This should necessarily mean that he was a very good Applied Mathematician too.

Note that in the word "Krylov" the second syllable is stressed and that "y" should be pronounced in the same way as in "Chebyshev". Another possible spelling of "Krylov" is "Kriloff".

At the age of 15, Krylov entered a Naval College in 1878 and finished it with distinction in 1884. After spending several years at the Main Hydrographic Administration and at a shipbuilding plant, in 1888 he continued his study in the Naval Academy in Saint-Petersburg. He was a talented and promising student and after ahead-of-schedule graduation from the Academy in 1890 he could stay there as Mathematics and Ship-theory lecturer.

The fame came to him in the 1890s, when his pioneering "Theory of oscillating motions of the ship", significantly extending R.E. Froude's rolling theory, became internationally known. This was the first comprehensive theoretical study in the field. In 1898 Krylov received a Gold Medal from the Royal Institution of Naval Architects (it was the first time the prize was awarded to a foreigner).

After 1900 Krylov actively collaborated with S.O. Makarov, admiral and
maritime scientist, working on the ship floodability problem. The results
of this work soon became classic, they are widely used nowadays over the
world. Many years later, Krylov wrote about one of the early ideas of Makarov
to fight the heel of a sinking ship by flooding its undamaged compartments:
"This appeared to be such a great nonsense [to the naval officials] that
it took 35 years ...to convince [them] that the ideas of the 22-years-old
Makarov are of great practical value".

Krylov knew what he was writing about: his own ideas were often equally
"welcome". To promote his innovations, he often had to fight against stagnation
and rigid views of the top officials. Once, giving a speech at an important
meeting before a large audience, Krylov addressed naval officers asking
them for their support in his "fight against the rut in the shipbuilding".
Being a naval officer at that time, he got an official reprimand for this
speech.

Another incident also quite remarkably illustrates Krylov's personality:
on a sitting of a high-rank technical committee Krylov once took with him
several technicians directly from the ships so that they could support
his opinion in the debates.

Krylov wrote about 300 papers and books. They span a wide range of topics,
including shipbuilding, magnetism, artillery, mathematics, astronomy, and
geodesy. His "floodability tables" have been used worldwide.

In 1904 he built the first machine in Russia for integrating ODEs.

In 1931 he published a paper on what is now called the "Krylov subspace".
The title of this paper is not always translated correctly; this is the
case, for instance, in B. Parlett's "The Symmetric Eigenvalue Problem".
The title of the paper should be:
*"On the numerical solution of the equation by which, in technical
matters, frequencies of small oscillations of material systems are determined"*.

The paper deals with eigenvalue problems, namely, with computation
of the characteristic polynomial coefficients of a given matrix. In his
book, Parlett calls this an "unfortunate goal". However, they
were not so unfortunate in that time.
Krylov, very much restricted by the computational
possibilities in 1931,
was interested in problems of size approx. 6 only.
Ill-conditioning does not play such a big role for these
small systems.

"It is clear", wrote Krylov, "that, if for *k*=2 and *k*=3
it is easy to compose this [secular] equation, then for *k*=4 the
laying-out becomes cumbersome, and for values *k* more than 5 this
is completely un-realizable in a direct way. Therefore one should use methods
where the full development of the determinant is avoided."

He continues, "The aim of the paper... is to present simple methods
of composition of the secular equation in the developed form, after which,
its solution, i.e. numerical computation of its roots, does not present
any difficulty".

The paper is really fascinating, it is written in a bright and appealing
manner and contains a lot of historical details:

"Before we describe these methods, it is good to return a bit back
and consider how the first creators of these methods,
**Lagrange** and
**Laplace**, and then such a great astronomer as **Leverrier** and
such a great mathematician as **Jacobi** proceeded..." Speaking about
Jacobi, Krylov mentions his 1845 paper "Ueber ein leichtes Verfahren, die
in der Theorie der Secularstrorungen vorkommenden Gleichungen numerisch
aufzulosen" published in the "Crelle" journal: "...The sum of the squares
of the off-diagonal elements decreases after each transformation [of the
Jacobi method] and can be made smaller than any a-priori prescribed value..."
This work of Jacobi has received more attention recently in connection
with the Jacobi-Davidson method of
G. Sleijpen
and
H. van der Vorst.

Krylov is concerned with efficient computations and, as a real computational
scientist, he counts the work as number of "separate numerical
multiplications"--something
not very typical for a 1931 mathematical paper. Krylov begins with a careful
comparison of the existing methods that includes the worst-case-scenario
estimate of the computational work in the Jacobi method. After that, he
presents his own method which is superior to the known methods of that
time. In his method, the initial matrix determinant with eigenvalues *lambda*
along the main diagonal is reduced to an equivalent system with the *lambda*'s
present in the first column of the matrix only.

For further discussion of this paper we refer to the translated book

D.K.Faddeev, V.N.Faddeeva, *"Computational methods of linear algebra"*,
Freeman publ., 1963.

Krylov published the first Russian translation of Isaac Newton, "Philosophić Naturalis Principia Mathematica" (1915 & later editions).

Alexei Nikolaevich Krylov died in Saint-Petersburg (by that time Leningrad) on October 26, 1945, shortly after the end of the World War II, so he fortunately could witness what Russians call "the Victory". He is buried on the Belkovo cemetery, not far from the physiologist Ivan P. Pavlov and the chemist Dmitri I. Mendeleev.

...In one of his autobiographical papers, Krylov describes his activity as "shipbuilding, i.e. application of Mathematics to various Maritime problems..."

Michele Benzi added the following information:

"There are at least two other Russian mathematicians named Krylov who made important contributions to Numerical Analysis.

One was Nikolai Mitrofanovich Krylov (1879-1955), who was among the first to study the convergence of Rayleigh/Ritz approximations to differential equations (under the general heading of "direct methods in the calculus of variations"). He also studied approximation of functions and initiated the modern theory of nonlinear ODE's. He is definitely the most famous Krylov (at least among mathematicians), and it would be easy to think that he is the Krylov of Krylov sequences and subspaces.

Another one was a V. I. Krylov who coauthored a famous 1962 monograph with L. Kantorovich, titled "Approximate methods of higher analysis", which was for many years the most influential book on numerical analysis in the Soviet Union."

References

1. A.N. Krylov, *On the numerical solution of the equation by which
in technical questions frequencies of small oscillations of material systems
are determined*, Izvestija AN SSSR (News of Academy of Sciences of the
USSR), Otdel. mat. i estest. nauk, 1931, VII, Nr.4, 491-539 (in Russian).

2. A.I. Balkashin, *Outstanding scientist and ship-builder Alexei
Nikolaevich Krylov*,
http://www.navy.ru/history/b-krylov.htm
(in Russian).

3. Great Soviet Encyclopedia, 1973 (a Macmillan English Translation
of the third edition).

Mike Botchev

Last modified: Mon Apr 8 15:23:46 MET DST 2002

Acknowledgements: Prof. T.A. Tibilov, Garry J. Tee, Michele Benzi |