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numapprox

 chebmult
 multiply two Chebyshev series

 Calling Sequence chebmult(p, q)

Parameters

 p, q - two expressions assumed to be Chebyshev series

Description

 • Given polynomials p and q expressed in a Chebyshev basis, form the product $pq$ expressed in a Chebyshev basis.
 • All Chebyshev basis polynomials $T\left(k,x\right)$ which appear must have the same second argument x (which can be any expression).
 • The input polynomials must be in expanded form (i.e. a sum of products). Normally, each term in the sum contains one and only one $T\left(k,x\right)$ factor except that if there are terms in the sum containing no $T\left(k,x\right)$ factor then each such term t is interpreted to represent $tT\left(0,x\right)$ provided that t and x have no variables in common.
 • If no $T\left(k,x\right)$ factor appears in p or in q then the ordinary product $pq$ is returned.
 • The command with(numapprox,chebmult) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{numapprox}\right):$
 > $\mathrm{Digits}≔3:$
 > $a≔\mathrm{chebyshev}\left(\mathrm{sin}\left(x\right),x\right)$
 ${a}{≔}{0.880}{}{T}{}\left({1}{,}{x}\right){-}{0.0391}{}{T}{}\left({3}{,}{x}\right){+}{0.000500}{}{T}{}\left({5}{,}{x}\right)$ (1)
 > $b≔\mathrm{chebyshev}\left({ⅇ}^{x},x\right)$
 ${b}{≔}{1.26}{}{T}{}\left({0}{,}{x}\right){+}{1.13}{}{T}{}\left({1}{,}{x}\right){+}{0.271}{}{T}{}\left({2}{,}{x}\right){+}{0.0443}{}{T}{}\left({3}{,}{x}\right){+}{0.00547}{}{T}{}\left({4}{,}{x}\right){+}{0.000543}{}{T}{}\left({5}{,}{x}\right)$ (2)
 > $\mathrm{chebmult}\left(a,b\right)$
 ${0.496}{}{T}{}\left({0}{,}{x}\right){+}{1.22}{}{T}{}\left({1}{,}{x}\right){+}{0.494}{}{T}{}\left({2}{,}{x}\right){+}{0.0718}{}{T}{}\left({3}{,}{x}\right){-}{0.00212}{}{T}{}\left({4}{,}{x}\right){-}{0.00227}{}{T}{}\left({5}{,}{x}\right){-}{0.000344}{}{T}{}\left({6}{,}{x}\right){-}{0.0000390}{}{T}{}\left({7}{,}{x}\right){+}{5.}{}{{10}}^{{-7}}{}{T}{}\left({8}{,}{x}\right){+}{0.00000137}{}{T}{}\left({9}{,}{x}\right){+}{1.36}{}{{10}}^{{-7}}{}{T}{}\left({10}{,}{x}\right)$ (3)
 > $c≔\mathrm{c0}T\left(0,x\right)+\mathrm{c1}T\left(1,x\right)$
 ${c}{≔}{\mathrm{c0}}{}{T}{}\left({0}{,}{x}\right){+}{\mathrm{c1}}{}{T}{}\left({1}{,}{x}\right)$ (4)
 > $d≔\mathrm{d0}T\left(0,x\right)+\mathrm{d1}T\left(1,x\right)$
 ${d}{≔}{\mathrm{d0}}{}{T}{}\left({0}{,}{x}\right){+}{\mathrm{d1}}{}{T}{}\left({1}{,}{x}\right)$ (5)
 > $\mathrm{chebmult}\left(c,d\right)$
 $\left({\mathrm{c0}}{}{\mathrm{d0}}{+}\frac{{1}}{{2}}{}{\mathrm{d1}}{}{\mathrm{c1}}\right){}{T}{}\left({0}{,}{x}\right){+}\left({\mathrm{c0}}{}{\mathrm{d1}}{+}{\mathrm{d0}}{}{\mathrm{c1}}\right){}{T}{}\left({1}{,}{x}\right){+}\frac{{1}}{{2}}{}{\mathrm{d1}}{}{\mathrm{c1}}{}{T}{}\left({2}{,}{x}\right)$ (6)
 > $\mathrm{chebmult}\left(T\left(j,x\right),T\left(k,x\right)\right)$
 $\frac{{1}}{{2}}{}{T}{}\left(\left|{-}{k}{+}{j}\right|{,}{x}\right){+}\frac{{1}}{{2}}{}{T}{}\left({k}{+}{j}{,}{x}\right)$ (7)
 > $\mathrm{assume}\left(0
 > $\mathrm{chebmult}\left(\mathrm{c0}+\mathrm{cj}T\left(j,x\right),T\left(k,x\right)\right)$
 $\frac{{1}}{{2}}{}{\mathrm{cj}}{}{T}{}\left({\mathrm{k~}}{-}{\mathrm{j~}}{,}{x}\right){+}{\mathrm{c0}}{}{T}{}\left({\mathrm{k~}}{,}{x}\right){+}\frac{{1}}{{2}}{}{\mathrm{cj}}{}{T}{}\left({\mathrm{j~}}{+}{\mathrm{k~}}{,}{x}\right)$ (8)
 > $\mathrm{assume}\left(5
 > $e≔a+\mathrm{ck}T\left(k,x\right)$
 ${e}{≔}{0.880}{}{T}{}\left({1}{,}{x}\right){-}{0.0391}{}{T}{}\left({3}{,}{x}\right){+}{0.000500}{}{T}{}\left({5}{,}{x}\right){+}{\mathrm{ck}}{}{T}{}\left({\mathrm{k~}}{,}{x}\right)$ (9)
 > $\mathrm{chebmult}\left(e,T\left(j,x\right)\right)$
 ${0.500}{}{\mathrm{ck}}{}{T}{}\left({\mathrm{k~}}{-}{\mathrm{j~}}{,}{x}\right){+}{0.000250}{}{T}{}\left({\mathrm{j~}}{-}{5}{,}{x}\right){-}{0.0196}{}{T}{}\left({\mathrm{j~}}{-}{3}{,}{x}\right){+}{0.440}{}{T}{}\left({\mathrm{j~}}{-}{1}{,}{x}\right){+}{0.440}{}{T}{}\left({1}{+}{\mathrm{j~}}{,}{x}\right){-}{0.0196}{}{T}{}\left({3}{+}{\mathrm{j~}}{,}{x}\right){+}{0.000250}{}{T}{}\left({5}{+}{\mathrm{j~}}{,}{x}\right){+}{0.500}{}{\mathrm{ck}}{}{T}{}\left({\mathrm{k~}}{+}{\mathrm{j~}}{,}{x}\right)$ (10)