Jim Wilkinson's mother was Kathleen Charlotte Hardy and his father was James William Wilkinson who was in the dairy business. The family were not well off but were close and supported each other. There were five children in the family and Jim was the third child, having one brother and three sisters. Jim showed a great ability to carry out complicated arithmetical calculations as a child, and whats more he enjoyed doing so. He won a Foundation Scholarship to Sir Joseph Williamson's Mathematical School, Rochester, at the age of eleven. This was fortunate, for his parent's dairy business collapsed in the 1930s and the family were left extremely poorly off. At the age of sixteen he won a scholarship to Trinity College, Cambridge and matriculated there shortly after his seventeenth birthday. At Cambridge he was taught by Hardy, Littlewood and Besicovitch. His final examination results were sent to him by Besicovitch who wrote:-
Easily at the top of the First Class. Heartiest Congratulations.
At this stage Wilkinson seemed set to become a classical analyst but the Second World War changed the direction of his research. In 1940, rather than going into the infantry, Wilkinson began war work on the thermodynamics of explosions, ballistics, supersonic flow, and fragmentation of shells. At first he tried to solve analytically the problems he was presented with, which after all was the way he had learnt at university to solve problems, but he soon realised that he would have to use approximate numerical methods if he was to obtain useful results. He began to put his greatest efforts into the numerical solution of hyperbolic partial differential equations, using finite difference methods and the method of characteristics. He carried out the calculations on a mechanical calculating machine which he operated by turning a handle.
In 1945 Wilkinson married Heather Nora Ware, the daughter of a buyer for a drapery warehouse; they had one daughter and one son.
In  Wilkinson makes the interesting comment that if the war had only lasted for three years he would almost certainly have returned to Cambridge and resumed his research on classical analysis. However, he spent six years involved with numerical computing and by this time he was completely fascinated by the new area of research. As a child he had been fascinated by calculating, but his course at Cambridge had taken his interests towards analytic techniques. His war work had taken him back to the love of his childhood, namely calculating, and now he wanted to push forward with the development of computational mathematics. He became Turing's assistant at the National Physical Laboratory in London in May 1946.
At the National Physical Laboratory he worked on the ACE (Automatic Computing Engine) computer project and, although no computer existed at that time, he wrote subroutines to do floating-point arithmetic. His role at this time was described as follows:-
Turing provided the blueprint, but could not get on with others. Wilkie [the name by which Wilkinson was known at that time] was able to cooperate as well as anyone with Turing, and had the tact and wisdom necessary to get things done. He grasped all the technical details very quickly, shared the lectures on ACE with Turing and wrote the definitive description of the design.
In 1948 Turing left the ACE project and Wilkinson :-
... took a leading role in the development of a modified machine, known as Pilot ACE; this proved highly successful from its inception in May 1950.
Pilot ACE was intended to demonstrate the potential of the technology. In  Wilkinson describes the public demonstration of Pilot ACE in 1950, at which the machine, incredibly, ran much better and for much longer than it ever had before. After the successful demonstration, Wilkinson realised for the first time that he would now be expected to do useful work on the Pilot ACE computer, which, until that moment, he had only though of as a means of convincing others that the ACE project had potential and should be continued.
He continued work becoming more involved in writing many high quality papers on numerical analysis, particularly numerical linear algebra. Having written subroutines to do floating-point arithmetic before a computer had been built to run them on, he was now in the fortunate position of being able to progress rapidly, gaining experience with floating-point computing. In numerical linear algebra he developed backward error analysis methods. He worked on numerical methods for solving systems of linear equations and eigenvalue problems. He wrote over 100 papers and was best known for his books Rounding Errors in Algebraic Processes (1963) and the impressive The Algebraic Eigenvalue Problem (1965).
As well as the large numbers of papers on his theoretical work on numerical analysis, Wilkinson developed computer software, working on the production of libraries of numerical routines. The NAG (Numerical Algorithms Group) began work in 1970 and much of the linear algebra routines were due to Wilkinson.
He received many awards for his outstanding work. He was elected a Fellow of the Royal Society in 1969. He received the A M Turing award from the Association of Computing Machinery and the J von Neumann award from the Society for Industrial and Applied Mathematics both in 1970. He was awarded honorary doctorates by Brunel (1971), Heriot-Watt (1973), Waterloo (1978), and Essex (1979).
George Forsythe wrote in 1960:-
In my opinion Wilkinson is single-handedly responsible for the creation of almost all of the current body of scientific knowledge about the computer solution of the problems of linear algebra.
He is described in  as having extrovert characteristics but being a reserved and private man:-
Discussion with him was not always easy because the competitive instinct led him to introduce topics about which he had read and remembered a great deal.
He was described by a colleague as:-
... always optimistic and jovial, very fair and impartial in his evaluations, liked virtually everybody, had a nearly overpowering enthusiasm for many things outside mathematics. ... he was always competitive.
His main interest outside mathematics was music, on which he was extremely knowledgeable, and he was also an expert on wines. He died at home from a heart attack.
Article by: J J O'Connor and E F Robertson