Systems of Linear Equations

P. M. Dew and K. R. James
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P. M. Dew: University of Leeds, Department of Computer Studies
K. R. James: University of Leeds, Department of Computer Studies

Chapter 7 in Introduction to Numerical Computation in Pascal, 1983, pp 140-188 from Springer

Abstract: Abstract One of the commonest problems of numerical computation is to solve a system of simultaneous linear equations (7.1) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe % qaaiaadggadaWgaaWcbaGaaGymaiaaigdaaeqaaOGaamiEamaaBaaa % leaacaaIXaaabeaakiabgUcaRiaadggadaWgaaWcbaGaaGymaiaaik % daaeqaaOGaamiEamaaBaaaleaacaaIYaaabeaakiabgUcaRiaac6ca % caGGUaGaaiOlaiabgUcaRiaadggadaWgaaWcbaGaaGymaiaad6gaae % qaaOGaamiEamaaBaaaleaacaWGUbaabeaakiabg2da9iaadkgadaWg % aaWcbaGaaGymaaqabaaakeaacaWGHbWaaSbaaSqaaiaaikdacaaIXa % aabeaakiaadIhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGHbWa % aSbaaSqaaiaaikdacaaIYaaabeaakiaadIhadaWgaaWcbaGaaGOmaa % qabaGccqGHRaWkcaGGUaGaaiOlaiaac6cacqGHRaWkcaWGHbWaaSba % aSqaaiaaikdacaWGUbaabeaakiaadIhadaWgaaWcbaGaamOBaaqaba % GccqGH9aqpcaWGIbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaaeiiaiaa % bccacaqGGaGaaiOlaiaabccacaqGGaGaaeiiaiaabccacaGGUaGaae % iiaiaabccacaqGGaGaaeiiaiaabccacaGGUaGaaeiiaiaabccacaqG % GaGaaeiiaiaabccacaGGUaGaaeiiaiaabccacaqGGaGaaeiiaiaabc % cacaqGGaGaaiOlaiaabccacaqGGaGaaeiiaiaabccacaGGUaGaaeii % aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaGGUa % aabaGaaeiiaiaabccacaqGGaGaaiOlaiaabccacaqGGaGaaeiiaiaa % bccacaGGUaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaGGUaGaae % iiaiaabccacaqGGaGaaeiiaiaabccacaGGUaGaaeiiaiaabccacaqG % GaGaaeiiaiaabccacaqGGaGaaiOlaiaabccacaqGGaGaaeiiaiaabc % cacaGGUaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii % aiaabccacaGGUaaabaGaamyyamaaBaaaleaacaWGUbGaaGymaaqaba % GccaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamyyamaaBaaa % leaacaWGUbGaaGOmaaqabaGccaWG4bWaaSbaaSqaaiaaikdaaeqaaO % Gaey4kaSIaaiOlaiaac6cacaGGUaGaey4kaSIaamyyamaaBaaaleaa % caWGUbGaamOBaaqabaGccaWG4bWaaSbaaSqaaiaad6gaaeqaaOGaey % ypa0JaamOyamaaBaaaleaacaWGUbaabeaaaaGccaGL9baaaaa!AEF1! $$ \left. \matrix {a_{11}}{x_1} + {a_{12}}{x_2} + ... + {a_{1n}}{x_n} = {b_1} \hfill \cr {a_{21}}{x_1} + {a_{22}}{x_2} + ... + {a_{2n}}{x_n} = {b_2} \hfill \cr {\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}. \hfill \cr {\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}.{\text{ }}. \hfill \cr {a_{n1}}{x_1} + {a_{n2}}{x_2} + ... + {a_{nn}}{x_n} = {b_n} \hfill \cr \endmatrix \right\} $$ The coefficients aij for 1≤ i,j ≤ n and the right hand sides bi for 1≤ I ≤ n are given; the problem is to find numerical values for the unknowns x1,,xn which satisfy the n equations.

Keywords: Coefficient Matrix; Conjugate Gradient Method; Gaussian Elimination; Residual Vector; Triangular System (search for similar items in EconPapers)
Date: 1983
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DOI: 10.1007/978-1-4757-3940-4_7

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