Relaxation Method

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Numerical Computting by Shahid Qureshi Syed Abbas Shamim Rizvi MCS STUDENT

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Object:- Relaxation Method for Solving System Linear Equations Developed By R.V. Southwell.

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Richard Vynne Southwell (born: Norwich 1888 , died 1970) was an English mathematician who specialized in applied mechanics. He studied in Cambridge, and was a Trinity College Fellow. During WWI, he was in the Royal Naval Air Service. After the war, he worked at the Royal Aircraft Establishment in Farnborough, then at the National Physical Laboratory (1920-1925). In 1925, he returned to Cambridge University as a mathematics lecturer. In 1929, he became a professor of engineering at Oxford University. In 1942, he became rector of Imperial College. He retired in 1948. Well As a scientist, Southwell developed Relaxation methods

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Method to Solve System of Linear Equation. How to Solve System of Linear Equation by Relaxation Method. Example :- 123

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First of all, Transform or Re-arrange there Equation in terms of largest Co-efficients of Respectively i.e. After this, Divide 1st,2nd and 3rd Equation by the Co-efficients of Respectively, and make constant term positive.

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,and we get following Equation, are called Residual. Here

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For getting magnitude of Residual, putting Zero in all Equations.

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So, we get (Note that is R 1st the Highest Residual.)

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Now we construct a Table and Multiplying by 1000.

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And finally We add all the values of in column and Dividing by 1000, we get Solution. (i.e. root of Equation).

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Insert all the values of in any Equation. Hence satisfy the 1st system of linear equation. We take first equation. Checking of Results