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(a) Consider the following system of linear equations 2x +6y - z= 18 x+ y+5z =-1.5 7x +2y + z=11 Write the equations in diagonally dominant form. Hence use 3 iterations of the Gauss-Seidel method to find approximate solutions for x, y, z. Use initial estimates of x, = y, = z, =0 and work to 4 decimal places of accuracy throughout.

(a) Consider the following system of linear equations 2x +6y - z= 18 x+ y+5z =-1.5 7x +2y + z=11 Write the equations in diagonally dominant form. Hence use 3 iterations of the Gauss-Seidel method to find approximate solutions for x, y, z. Use initial estimates of x, = y, = z, =0 and work to 4 decimal places of accuracy throughout.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 16EQ
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Q4. (a) Consider the following system of linear equations 2x +6y - z= 18 X+ y+5z =-1.5 7x+2y + z=11 Write the equations in diagonally dominant form. Hence use 3 iterations of the Gauss-Seidel method to find approximate solutions for x, y, z. Use initial estimates of x, = Yo = z, = 0 and work to 4 decimal places of accuracy throughout. (b) The heat conduction equation which describes the temperature in °C, T(x,t), of a particular metal rod at position x and time t is ат 0.15 at The rod is of length 2m and is held at 0°C until timet =0. Two heat sources are introduced at time t = 0, one of 5°C at the end x = 0 and one of 70°C at the end x = 2m. (1) Approximate the above heat conduction equation with finite differences, using a uniform time step of k and uniform spacing h in the x-direction. Show that with a spacing of h=1m and time steps of k = 2s, the difference scheme is T.j = 0.3 (T + T,)+ 0.47, (ii) x = 1m when t= 4s. Use this difference scheme to estimate the temperature at
Transcribed Image Text:Q4. (a) Consider the following system of linear equations 2x +6y - z= 18 X+ y+5z =-1.5 7x+2y + z=11 Write the equations in diagonally dominant form. Hence use 3 iterations of the Gauss-Seidel method to find approximate solutions for x, y, z. Use initial estimates of x, = Yo = z, = 0 and work to 4 decimal places of accuracy throughout. (b) The heat conduction equation which describes the temperature in °C, T(x,t), of a particular metal rod at position x and time t is ат 0.15 at The rod is of length 2m and is held at 0°C until timet =0. Two heat sources are introduced at time t = 0, one of 5°C at the end x = 0 and one of 70°C at the end x = 2m. (1) Approximate the above heat conduction equation with finite differences, using a uniform time step of k and uniform spacing h in the x-direction. Show that with a spacing of h=1m and time steps of k = 2s, the difference scheme is T.j = 0.3 (T + T,)+ 0.47, (ii) x = 1m when t= 4s. Use this difference scheme to estimate the temperature at
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