Force balance at three joints of a light truss gives the linear system for the member…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Numerical and computational mathematics

Error, Chance and the Machine, Ch Linear systems, least squares and the singular value decomposition

Problem

Force balance at three joints of a light truss gives the linear system Ax=b\mathsf{A}\mathbf{x} = \mathbf{b} for the member tensions x1,x2,x3x_1, x_2, x_3 (in kN): A=(021111231),b=(549).\mathsf{A} = \begin{pmatrix}0 & 2 & 1\\ 1 & 1 & 1\\ 2 & 3 & 1\end{pmatrix}, \qquad \mathbf{b} = \begin{pmatrix}5\\ 4\\ 9\end{pmatrix}. ((a)) Factorise A\mathsf{A} as PA=LU\mathsf{P}\mathsf{A} = \mathsf{L}\mathsf{U} by hand, choosing at each step the pivot of largest magnitude in the current column (partial pivoting), and state the permutation P\mathsf{P} applied. ((b)) Use the factors to solve for x1,x2,x3x_1, x_2, x_3. ((c)) State the number of floating-point operations the factorisation costs for a general n×nn \times n matrix, and the extra cost of solving for one more right-hand side once L\mathsf{L}, U\mathsf{U} are known.

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