Non algebraic equations

eq1:=x^2-x*cos(y)-1: eq2:=y^2-cos(x)-1:for which we want to determine if real roots exist in the range [-5,5], [-5,5]. With a depth level 0 (no bisection is done) the analyzer proposes the following ranges as possibly containing a real root:

Range 1( x y) [-0.86812856880248989722,-0.86812856880248989722] [-1.28306500467802298,-1.28306500467802298] Range 2( x y) [1.219833908062562422,1.219833908062562422] [-1.1592247392497665448,-1.1592247392497665448] Range 3( x y) [1.2198350997243703198,1.2198367316341245381] [-1.159223547587958647,-1.1592195879408853099] Range 4( x y) [-0.86812856880248978619,-0.86812856880248978619] [1.28306500467802298,1.28306500467802298] Range 5( x y) [1.219833908062562422,1.219833908062562422] [1.1592247392497667668,1.1592247392497667668] Range 6( x y) [1.2198352388536839452,1.2198367316341249822] [1.1592195879408850878,1.1592236867169092296]Among this 6 ranges, 4 are reduced to point and are therefore a result of the application of the Newton scheme. The other 4 ranges are very close to one of the solution, but the algorithm has not been able to eliminate them. With a depth level of 1 the algorithm proposes directly the four solutions of this system:

Range 1( x y) [-0.86812741994331155126,-0.86812741994331155126] [-1.2830653469203250339,-1.2830653469203250339] Range 2( x y) [1.2198339926936523359,1.2198339926936523359] [-1.1592245852349318813,-1.1592245852349318813] Range 3( x y) [-0.86812741994331166229,-0.86812741994331166229] [1.2830653469203250339,1.2830653469203250339] Range 4( x y) [1.219833901899016082,1.219833901899016082] [1.159224777165462017,1.159224777165462017]Note that the initial range [-5,5] may be largely expanded without modifying the result and with only a low amount of additional computation time. For example for the range [-1000,1000] for both variables the computation time remains the same.

The previous system may be considered as a special occurrence of the system:

eq1:=x^2-x*cos(y)-x1: eq2:=y^2-cos(x)-1:where the parameter

x1 1Now just by changing the value of

x1 2and modifying the configuration file to include the sentence:

Directory /tmpwe will get the solutions:

Range 1( x y) [-1.2271026453420417202,-1.2271026453420417202] [-1.1562745250557344701,-1.1562745250557344701] Range 2( x y) [1.757647484313144215,1.757647484313144215] [-0.90234927430355182931,-0.90234927430355182931] Range 3( x y) [-1.2271026453420417202,-1.2271026453420417202] [1.1562722309470763182,1.1562722309470763182] Range 4( x y) [1.757647484313144215,1.757647484313144215] [0.90234927430355205136,0.90234927430355205136]