Volume 61 Issue 2 (2011) - Годишник на ТУ - София - Технически ...

Roughly this modification consists of the fact that one of the values of the points x iwhich runs k -th iteration is replaced with a value of x in which the error k 1 ( x)haslocal minimum, but only if this value is not "very close" to some node of the splines.In the examples from this section it is used multi-point Remez algorithm, i.e. on eachstep not just one all values x i are changed. This approach is preferred because multipointRamez algorithm is faster convergent a the single-point algorithm. In the examplesystem (6) has always nonzero main determinant and the condition for outputfrom the algorithm is always satisfied. The latter delivers convergence of the algorithmto an element of best uniform approximation. If for other similar examplessome of the above conditions is not fullfilled then the modified Remez algorithm proposedby Nurnberg in [5] has to be used.As a result of applying the algorithm of Remez the coefficients of generalized polynomialof best uniform approximation are found*p ( x) a i ( x), (5)iZwhere for the basis function i (x)(3) and (4) are fulfilled. Because of linearity of the0scheme of Chaikin the data from zero-level f i from which Chaikin algorithm mustbe started to obtain the element of best uniform approximation are the coefficients a i0of (5), i.e. fi ai| i Z. These coefficients must be found and memorized and thefunction f (x)is "memorized".Example 1. To "memorize" function f ( x) sin( x)in the interval 0, . Basic2functions have a limited support therefore the functions involved in approximation off (x) in 0, are only 4. So approximation polynomial corresponding to (5), in this2case looks likep( x) a00 ( x) a10 ( x 1) a20 ( x 2) a30 ( x 3)The algorithm of Remez is applied. In short it is the following. Starting pointsx i | i 0,1,2,3,4 are chosen and the system is solvedip( xi) ( 1)h f ( xi) (6)The values of the coefficients a i | i 0,1,2,3,4 and the value of h are obtained. Thesecoefficients are substituted in p (x)and the points x i where the error ( x) f ( x) p(x)has minimum are found. These are the starting points x i from which starts second iterationof the algorithm. Iterations stop when differenceh imaxx[0, / 2]| |becomes sufficiently small. Numerical experiments42