Three mesh currents (in amperes) in a resistor network satisfy with A = pmatrix 1 & 2…

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

Three mesh currents

i_1, i_2, i_3

(in amperes) in a resistor network satisfy

\mathsf{A}\mathbf{i} = \mathbf{v}

with

\mathsf{A} = \begin{pmatrix} 1 & 2 & -1\\ 2 & 4 & 1\\ 3 & -1 & 2\end{pmatrix}\ \Omega.

((a)) Factorise

\mathsf{A}

\mathsf{P}\mathsf{A} = \mathsf{L}\mathsf{U}

by Gaussian elimination with partial pivoting, choosing at each stage the largest available entry (by magnitude) in the current column at or below the diagonal. Give the multipliers

l_{21}

l_{31}

from the first elimination stage (applied after any row exchange) and

l_{32}

from the second stage, and the final pivot

u_{33}

. ((b)) State the number of floating-point operations required by LU factorisation of a general

n\times n

matrix, to leading order, and evaluate it for

n = 3

. Then count the flops exactly for

n = 3

from the sum over the elimination steps in Eq. (2.3), and say why the two numbers differ.

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