Showing papers on "Function approximation published in 1968"
TL;DR: In this article, the authors compared I?(a:) to P(a;) for x > 2 and showed that I?[a:] appears to be a better fit than Hasting's approximation [4] for x> 2 and E(x) appeared to be superior to P[a;).
Abstract: _ xexpj-A/2) IV + 6a:4 14a:2 28] a; + 2 \\-x + ox — 20a; — 4 -i and the motivation for such a definition may be found in [2] and [3]. It is readily apparent that EOx) and F(x) have the following properties in common: 1. For x > 2, E(x) is real, positive and finite. 2. For x > 2, dE/dx is real, negative and finite. 3. Asa:-» oo,E(x) -» 0. 4. As x —* oo, x exp ix2/2) Eix) —» 1. 5. As x —> oo, dE/dx —* 0. The following table compares I?(a:) to P(a;). P(a;) is a better fit than Hasting's approximation [4] for x > 2 and E(x) appears to be superior to P(a;). The relative error, e^(a;) is the quantity selected as a basis for comparison and
10 citations
01 Jan 1968
TL;DR: Stochastic approximation schemes are applied to the identification of linear distributed parameter systems and the constant parameters multiplying the approximating functions are obtained sequentially by stochastic approximation algorithms that minimize a mean-square error performance criterion.
Abstract: Stochastic approximation schemes are applied to the identification of linear distributed parameter systems The dependent function is assumed piecewise continuous and is approximated by a finite number of functions chosen from a "complete" system of orthonormal functions It is also assumed that noisy measurements of the function are available at random points in space and time Therefore, the identification is performed off-line The constant parameters multiplying the approximating functions are obtained sequentially by stochastic approximation algorithms that minimize a mean-square error performance criterion
6 citations
4 citations
TL;DR: Approximates a funct ion y of x by a half range cosine or sine series of period 2h from values specified at discrete points, not necessarily equally-spaced, in the range (0, h).
Abstract: p r o c e d u r e trigfit (index, n , m, h, e, x, f , rot, a) ; v a l u e index, n, m, h, e; i n t e g e r index, n , m, mr; r e a l h, e; a r r a y x, f, a; c o m m e n t Approximates a funct ion y of x by a half range cosine or sine series of period 2h from values specified at discrete points , not necessarily equally-spaced, in the range (0, h). The i npu t parameters are: i ndex i f index = O, a cosine series is fi t ted, if index = 1, a sine series. No o ther value is permi t ted . n n u m b e r of funct ion-values given. m-o rde r of the highest harlnonic required. h-ha l f -per iod of the fi t ted series. e--used to t e rmina te the process if rounding errors s t a r t to accumulate excessively (see note below). x t h e given values of x are s tored on x[1], x[2], . . , x[n]. f t h e value of y corresponding to x = x[i] is s tored on f[i] ( i = 1 , 2 , . . , n ) . The procedure then cMculates the coefficients a[r] in the approximat ion
3 citations
01 Oct 1968
TL;DR: Constructive existence theorems, function approximations, and computational algorithms developed for optimal control problems have been developed for optimization problems as mentioned in this paper, where the objective is to find the optimal control solution.
Abstract: Constructive existence theorems, function approximations, and computational algorithms developed for optimal control problems