By H. Piaggio
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Extra resources for An Elementary Treatise on Diff. Eqns. and Their Applns.
Before examining the algorithms for filtering which we obtain this way we will prove: THEOREM 1. Proof. e. zfc = Z/l"o"~ hi-kXi where it is understood that hj = 0 unless 0 ^/ ^ n -1. That is 39 40 CHAPTER V F(ra, n] can be written as Hx where H is an m x (m + n — 1) matrix such that Hi,j = hj-\, and x is the (column) vector x = (jco, *i, • • • , jt m + n _ 2 ) T . It is easily verified that pc (H) = m + n — l and by Theorem 1 of Illb we obtain /u, (F(m, n)) ^ m + n — 1. But we have already seen that /I (F(m, n)) = m + n — l, so (j, (F(m, n)) = m + n — l.
For every C~l there exists an algorithm A = A(C~l) computing M(x)y, satisfying n(A) = /tt(M(jt)y) such that its /cth m/d step mk is given by mk = (£/=i «ifc*«)(Z/=i j3/fcy/), where the a,fc's and /S/fc's are in G. (M(x)y) there exists a matrix C~ 1 suchthatA=A(C" 1 ). IVb. Classification of the algorithms. We will use Theorem 1 of the last subsection to exhibit all the algorithms for computing z = x * y using m+n + 1 m/d steps. We first write this system of bilinear forms as M(x)y where M(x) is PRODUCT OF POLYNOMIALS 29 the (m + n +1) x n matrix whose (/, /)th entry is *,•_/ whenever 0 ^ / —/ ^ m, and 0 otherwise.
We partition the matrix of F(24,16) into 8x8 blocks, and denote them by z0, z\, z2,23. Using the F(3, 2) algorithm we have to compute z 0 ~ 22, z\ — z3, z\ + z2, and z^ — z\. This requires 15 + 8-1-15 + 15 = 53 additions. ) Partitioning each of these 8x8 matrices into blocks of 4 x 4 matrices we can use the F(2, 2) algorithm. If we denote the 4 x 4 blocks of each of the 8x8 blocks by XQ,X\, x2 we have to compute x0 — xi and Xi~x2 which uses 7+4 = 11 additions. This last statement is true for Zo — Z2,Zi + zi, and z2 — z\; but the x0 — x\ of z\ — z3 uses only four additions once we have computed the x\—x2 of z0 — z2.
An Elementary Treatise on Diff. Eqns. and Their Applns. by H. Piaggio