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RootOfMTP

The tool "RootOfMTP" isolates all the real roots of a mixed trigonometric-polynomial (MTP) based on Maple. Algorithm details can be viewed in our paper.

Code

All code is in "RootOfMTP.mpl" and "mainIBR.mpl".

  • The procedure "RootOfMTP" is used for ``isolating" all the real roots of an MTP.

Dependency

Maple 2021

Using

We illustrate how to use RootOfMTP by a simple example. Suppose we want to isolate all the real roots of the following MTP by RootOfMTP: $$x\sin{x} + \cos{x} - 1.$$ We only need to run the following commands in Maple2021:

read ".../RootOfMTP.mpl";
RootOfMTP(x*sin(x)+cos(x)-1,x,1);

Herein, the first command is to read the file, and the inputs of the second command are an MTP, the variable of the MTP and a rational number $\epsilon>0$ which specifies the maximal length of isolating intervals. The output is

For every k <= -1 (k in Z), 2kPi+(0) is a real root with multiplicity 1.
For every k >= 1 (k in Z), 2kPi+(0) is a real root with multiplicity 1.
There is 1 real root with multiplicity 2 at 0.
For every k >= 2 (k in Z), there is 1 real root with multiplicity 1 
in (2kPi+(2*arctan(63/16)), 2kPi+(Pi)).
For every k <= -2 (k in Z), there is 1 real root with multiplicity 1 
in (2kPi+(-Pi), 2kPi+(-2*arctan(63/16))).
There is 1 real root with multiplicity 1 in each open interval of the list
[[-5/2*Pi-2*arctan(29666650363354128505/36893488147419103232),
-5/2*Pi-2*arctan(7242537696610193/9007199254740992)],
[-1/2*Pi-2*arctan(7030038563941/17592186044416),
-1/2*Pi-2*arctan(14741934773129570377/36893488147419103232)],
[1/2*Pi+2*arctan(14741934773129570377/36893488147419103232),
1/2*Pi+2*arctan(7030038563941/17592186044416)],
[5/2*Pi+2*arctan(7242537696610193/9007199254740992),
5/2*Pi+2*arctan(29666650363354128505/36893488147419103232)]].

Examples

  • Testing examples in "bc_chenMTP.mpl" are from 1 and 2.
  • Testing examples in "bc_haokun1.mpl" and "bc_haokun2.mpl" are generated randomly.

Footnotes

  1. =https://sysmath.com/jweb_xtkxysx/CN/10.12341/jssms12871 ↩

  2. =https://arxiv.org/abs/2204.01481 ↩