Name: solutions1.1 - Math 307: Problems for section 1.1 1. Use Gaussian elimination 1 2 3 1 2 3 (a) A = 5 6 7 5 6 7 to nd the solution(s) to Ax = b where 1
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Math 307: Problems for section 1.11.Use Gaussian elimination to find the solution(s) toAx=b where(a)A=1234−12−345678−56−78b=1111,(b)A=111111−1−11−1000011b=3101.The process of Gaussian elimination is not unique — there is more than one way to do it.We create the augmented matrixC= [A|b] and perform Gaussian elimination onC(a)12341−12−34156781−56−781(b)1111311−1−111−100000111mapsto→123410408256781−56−781mapsto→1111300−2−2−21−100000111mapsto→1234104082567810120162mapsto→11113000001−100000111mapsto→12341040820−4−8−12−40120162mapsto→111131−10000000000111mapsto→123410408200−8−4−20120162mapsto→111130−2−1−1−30000000111mapsto→123410408200−8−4−2000−8−4mapsto→111130−2−1−1−30011100000The solution to the system will be the same regardless of how the Gaussian elimination is done:(a)x1x2x3x4=0−1/201/2(b)x1x2x3x4=1110+s00−111

2.Use MATLAB/Octave to find the solution(s) toAx=b where(a)A=111111−1−11−100001−1b=1111,(b)A=1032−421650−11−3−11,b=47−5.(a)> A=[1 1 1 1; 1 1 -1 -1 ; 1 -1 0 0; 0 0 1 -1];> b=[1 1 1 1]’;> C=[A b];> rref(C)ans =1 0 0 0 10 1 0 0 00 0 1 0 0.50 0 0 1 -0.5and so the answer isx1x2x3x4=100.5−0.5We could have instead done> A=[1 1 1 1; 1 1 -1 -1 ; 1 -1 0 0; 0 0 1 -1];> b=[1 1 1 1]’;> A\bans =100.5-0.5and found the same answer — be careful though (see next part).(b)> A=[1 0 3 2 -4; 2 1 6 5 0; -1 1 -3 -1 1];> b = [4 7 -5]’;> C=[A b];> rref(C)ans =1 0 3 2 0 40 1 0 1 0 -10 0 0 0 1 0The solution isx1x2x3x4x5=4−1000+s−2−1010+t−301002

If we had instead used> A\bthen MATLAB/Octave would have given only one of the many solutions.3.Compute the time it takes to solveAx=b usingA\bon your computer for five or morerandom problems of increasing size.Make a plot of time vs.size.Repeat using themethodA^(-1)*band plot on the same graph. (If your calculation is taking too long, itcan be interrupted by typing<ctrl> c.)

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Fall

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RICHARDFROESE

solutions1.1 - Math 307: Problems for section 1.1 1. Use...

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