A three-axis positioning stage misses its target by the error vector

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 Rounding, backward error and the sensitivity of a computation

Problem

A three-axis positioning stage misses its target by the error vector d=(0.3, −0.4, 1.2)T mm\mathbf{d}=(0.3,\,-0.4,\,1.2)^{\rm T}\ \mathrm{mm}. A correction step passes the error through the dimensionless coupling matrix A=(2−1013−2−401),\mathsf{A}=\begin{pmatrix}2&-1&0\\ 1&3&-2\\ -4&0&1\end{pmatrix}, so that the error after the step is Ad\mathsf{A}\mathbf{d}. ((a)) Find ∥d∥1\|\mathbf{d}\|_1, ∥d∥2\|\mathbf{d}\|_2 and ∥d∥∞\|\mathbf{d}\|_\infty, and confirm that they are in the order the chapter gives. ((b)) Find the induced norms ∥A∥1\|\mathsf{A}\|_1 and ∥A∥∞\|\mathsf{A}\|_\infty. ((c)) Find Ad\mathsf{A}\mathbf{d}. Then compute ∥Ad∥1\|\mathsf{A}\mathbf{d}\|_1 and ∥Ad∥∞\|\mathsf{A}\mathbf{d}\|_\infty, and check that each is no larger than the bound ∥A∥ ∥d∥\|\mathsf{A}\|\,\|\mathbf{d}\| in the same norm. ((d)) Give a vector x\mathbf{x} for which ∥Ax∥1=∥A∥1∥x∥1\|\mathsf{A}\mathbf{x}\|_1=\|\mathsf{A}\|_1\|\mathbf{x}\|_1, and so show that the 1-norm bound cannot be improved for this matrix.

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