Direct Methods for Sparse Linear Systems (Fundamentals of by Timothy A. Davis

By Timothy A. Davis

This books offers an honest library of sparse matrix features. notwithstanding, it may be obscure the code every now and then as the writer selected to take advantage of cryptic variable names.

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18) |Xk |ejφk ej2πfk t . y(t) = k=−n CHAPTER 1. REPRESENTATION OF FUNCTION CONTENTS 14 A reminder: The de nitions fk = T1k and ωk = 2πfk may be used to express y(t) in terms of Tk (individual period) or ωk (individual angular frequency) in all forms. Also, when fk = k/T , this fact is commonly recognized wherever fk is used. To convert one form to another, one may use the relationship between the coef cients as summarized below. Relation 1 De n e X0 ≡ 0 when A0 and B0 are missing. For 1 ≤ k ≤ n, Ak ∓ jBk , and f−k = −fk .

6 9 = 3 3 . It can be easily veri ed that y(t + To ) = y(t). , y(t + T ) = y(t). Since we have uniform spacing f = fk+1 − fk = 1/T , we may still plot Ak and Bk versus k with the understanding that k is the index of equispaced fk ; of course, one may plot Ak and Bk versus the values of fk if that is desired. 7. REVIEW OF RESULTS AND TECHNIQUES 13 3. A non-commensurate y(t) is not periodic, although all its components are periodic. For example, the function √ y(t) = sin(2πt) + 5 sin(2 3πt) √ is not periodic because f1 = 1 and f2 = 3 are not commensurate.

We do not know how many cycles x ˜(t) has completed over the interval T . Mathematically, the function x ˜(t) interpolating the two samples is no longer unique if the ˜ frequency f is not speci ed. , the frequency we can resolve for x ˜(t) is f˜ = r˜/T = 1/T . When we deal with discrete samples taken from a composite signal, the so-called aliased frequencies are equivalent in the sense that they contribute the same numerical values at the sample points. 5, the signal y(θ) = cos(θ) + 2 cos(3θ) + 3 cos(5θ) cannot be distinguished from x(θ) = 6 cos(θ) based on the two values sampled at θ1 = 0 and θ2 = π, because y(0) = x(0) = 6 and y(π) = x(π) = −6.

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