7 18 you can replace the supremum of a polynomial

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Unformatted text preview: true , the true value of c, in the vector c_true. Use this to give the two relative errors ctrue − c 2 ˆ ctrue − cls 2 ˆ , . ctrue 2 ctrue 2 5.14 Spectrum analysis with quantized measurements. A sample is made up of n compounds, in quantities qi ≥ 0, for i = 1, . . . , n. Each compound has a (nonnegative) spectrum, which we represent as a vector s(i) ∈ Rm , for i = 1, . . . , n. (Precisely what s(i) means won’t matter to us.) The spectrum + of the sample is given by s = n qi s(i) . We can write this more compactly as s = Sq , where i=1 S ∈ Rm×n is a matrix whose columns are s(1) , . . . , s(n) . Measurement of the spectrum of the sample gives us an interval for each spectrum value, i.e., l, u ∈ Rm for which + li ≤ si ≤ ui , i = 1, . . . , m. (We don’t directly get s.) This occurs, for example, if our measurements are quantized. Given l and u (and S ), we cannot in general deduce q exactly. Instead, we ask you to do the following. For each compound i, find the range of...
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