# Andre-Louis Cholesky

### Quick Info

Born
15 October 1875
Montguyon, Charentes Maritime, France
Died
31 August 1918
North France

Summary
André-Louis Cholesky was a French military officer and mathematician who devised a clever method of dealing with geodesic measurements.

### Biography

Andre-Louis Cholesky was born in Montguyon in the Charentes Maritime region of France north of Bordeaux. He attended the Lycée in Bordeaux and was awarded the first part of his baccalaureat on 14 November 1892 and second part on 24 July 1893. Cholesky entered l'École Polytechnique on 15 October 1895 being placed 88th out of 223 entrant students in that year. His professors at l'École Polytechnique included Camille Jordan, and Henri Becquerel the famous physicist who discovered radioactivity. After two successful years he took his final examinations from l'École Polytechnique in 1897. He had improved his position to be placed 38th out of 222 students in these examinations. He then joined the army becoming a second lieutenant, and went to study at the school d'Application de l'Artillerie et du Génie starting in October 1897. He completed his course in 1899 and he maintained his steady improvement for now he was placed 5th out of 86 students who qualified in that year.

Tunisia was at that time still officially part of the Ottoman Empire but it had been a French protectorate following the Convention of Al-Marsa which was signed in 1883 by Tunisia and the French government. The French stabilized the economy and established modern communications. From January to June 1902, Cholesky carried out a mission in Tunisia, then he was sent on a second mission from November 1902 to May 1903. On 31 December 1903 Cholesky began a period of service in Algeria. That country had received much poorer treatment from France than had Tunisia. The French established domination over the country and its native inhabitants, but developed new communications, hospitals and medical services. He left Algeria on 6 June 1904.

The most significant step in Cholesky's career was whan he moved to the Geodesic Section of the Army Geographic Service in June 1905 where he was reported to have:-
... a sharp intelligence and a great facility for mathematical work, having an inquiring spirit and original ideas.
Delambre had completed his baseline measurements in the spring of 1798 as part of his contribution to defining the metre. In 1882 French experts returned to Delambre's baseline but did not re-measure it, preferring to check Delambre's calculations indirectly by triangulation. In the first decade of the 20th century, following the revision of the Paris Meridian, a new triangulation of France was planned to take place. The problem of adjusting the grid greatly worried officers in the Geographical Service, who were anxious to find a method which was simple, fast and precise. To solve the condition equations in the method of least squares, Cholesky invented a very ingenious computational procedure which immediately proved extremely useful: it is now known as the Method of Cholesky and we describe it below. On 26 September 1905 he became First Lieutenant and he married two years later on 10 May 1907. Cholesky and his wife had three children; a son and two daughters.

Crete had been a part of the Ottoman Empire for 200 years but as the Empire became weaker the major countries in Europe aimed to become involved in Crete's future. After a revolution in Crete against the Turks in 1896, following which Greece tried to take control, the Great Powers imposed a settlement. The French took responsibility for the Sitia sector, the Italians for the Ierapetra sector, the British for the Iraklion sector, and the Russians for the Rethymnon sector, while the four jointly looked after the Chania sector. Colonel Lubanski, the commanding office of the French troops in Crete, was a former geodesist and wanted mapping work to be carried out on the island. Lieutenant Colonel Bourgeois, Head of the Geodesic Section, made a preliminary reconnaissance in March and April of 1906. Cholesky was sent to Crete on 7 November 1907 as one of the three officers assigned to carry out the task. He measured a baseline in the Kavousi plain and used astronomical measurements to determine the precise position of the southern end of his baseline. After his fellow officers were dispatched to other duties, Cholesky carried out a triangulation as well as making a topographical survey of the French and British sectors. He succeeded in making the triangulation under extremely difficult conditions for winter in Crete saw considerable falls of snow on the high mountains. Political events, however, prevented land surveys being carried out and Cholesky left Crete on 25 June 1908.

He was named Captain with the 2nd regiment of Infantry on 25 March 1909 and from September 1909 to September 1911 he carried out a tour of duty with his unit. Later in 1911 he returned to the Geodesic Section of the Army Geographic Service and was assigned work in Algeria and Tunisia where he had already gained experience seven years earlier. From 27 October 1911 to 24 April 1912, then again from 23 October 1912 to 17 April 1913, he undertook precision levelling work in Algeria and Tunisia. This work was intended to prepare for the building of a railway line. The construction of a grid in Tunisia was completed during the winter of 1913-1914 after which there was a period of checking and readjustment. Similar work had to be carried out in Morocco and Cholesky devised methods to allow the work to be undertaken more quickly yet still retaining the necessary accuracy. After testing his procedures they were used in Morocco beginning in July 1912 with the work being completed in January 1913.

On 25 May 1913, Cholesky was assigned to the Ministry of Foreign Affairs and he was named head of Topographical Service of the Regency of Tunis. He began work there but after World War I broke out in the autumn of 1914 he was transferred to other duties. In January 1915 be began to organise artillery firing and in February he was sent to work with a unit in the Vosges. In September 1916 he was sent on a military mission to Romania where he took on the role of director of the geographical service of the Romanian army. He was promoted in July 1917 and continued to serve in Romania until February 1918.

Cholesky died from wounds received on the battle field on 31 August 1918 at 5 o'clock in the morning in the North of France. After his death one of his fellow officers, Commandant Benoit, published Cholesky's method of computing solutions to the normal equations for some least squares data fitting problems in Note sur une méthode de resolution des équations normales provenant de l'application de la méthode des moindres carrés à un système d'équations lineaires en nombre inferieure à celui des inconnues. Application de la méthode à la resolution d'un système defini d'équations lineaires (Procede du Commandant Cholesky) , published in the Bulletin géodesique in 1924.

The Cholesky Factorization (or Cholesky Decomposition) takes a symmetric positive definite matrix $A$ and writes it as $A = LL'$ where $L$ is a lower triangular matrix with positive diagonal entries (sometimes called the Cholesky triangle), and $L'$ is the transpose of $L$. To $solve Ax = b$ one now needs to solve $LL'x = b$ so put $y = L'x$ which gives $Ly = b$ which is solved for $y$, then $y = L'x$ is solved for $x$ to obtain the solution. The beauty of the method is that it is trivial to solve equations of the type $Mx = b$ when $M$ is a triangular matrix.

The method received little attention after its publication in 1924 but Jack Todd included it in his analysis courses in King's College, London, during World War II. In 1948 the method was analysed in a paper by Fox, Huskey and Wilkinson while in the same year Turing published a paper on the stability of the method.

### References (show)

1. C Brezinski, D Tournès, André-Louis Cholesky (1875-1918), Mathematician, Topographer and Army Officer. (Birkhäuser, Basel) 2014.
2. J H Maindonald, Statistical Computation (Wiley, 1984).
3. C Brezinski, André Louis Cholesky (1875-1918) (French), Numerical analysis, Louvain-la-Neuve, 1995, Bull. Belg. Math. Soc. Simon Stevin (1996), suppl., 45-50.
4. Commandant Benoit, Note sur une méthode de résolution des équations normales provenant de l'application de la méthode des moindres carrés à un système d'équations linéaires en nombre inférieur à celui des inconnues (Procédé du Commandant Cholesky), Bulletin géodésique 2 (1924), 67-77.
5. L Fox, H D Huskey and J H Wilkinson, Notes on the solutions of algebraic linear simultaneous équations, Quart. J. Mech. Appl. Math. 1 (1948), 149-173.
6. A M Turing, Rounding-off errors in matrix processes, Quart. J. Mech. Appl. Math. 1 (1948), 287-308.

### Cross-references (show)

Written by J J O'Connor and E F Robertson
Last Update August 2005