Texas Instruments TI-89 User Manual - Page 880

If all of the equations are polynomials and if you do NOT specify any initial guesses, solve() uses the lexical Gröbner/Buchberger elimination method to attempt to determine all real solutions. For example, suppose you have a circle of radius r at the origin and another circle of radius r centered where the first circle crosses the positive x-axis. Use solve() to find the intersections. As illustrated by r in the example to the right, simultaneous polynomial equations can have extra variables that have no values, but represent given numeric values that could be substituted later. solve(x^2+y^2=r^2 and (xìr)^2+y^2=r^2,{x,y}) ¸ x= r 2 and y= 3ør 2 or x= r 2 and y= ë 3ør 2 You can also (or instead) include solution variables that do not appear in the equations. For example, you can include z as a solution variable to extend the previous example to two parallel intersecting cylinders of radius r. The cylinder solutions illustrate how families of solutions might contain arbitrary constants of the form @k, where k is an integer suffix from 1 through 255. The suffix resets to 1 when you use ClrHome or ƒ 8:Clear Home. solve(x^2+y^2=r^2 and (xìr)^2+y^2=r^2,{x,y,z}) ¸ x= r 2 and y= 3ør 2 and z=@1 or x= r 2 and y= ë 3ør 2 and z=@1 For polynomial systems, computation time or memory exhaustion may depend strongly on the order in which you list solution variables. If your initial choice exhausts memory or your patience, try rearranging the variables in the equations and/or varOrGuess list. If you do not include any guesses and if any equation is non-polynomial in any variable but all equations are linear in the solution variables, solve() uses Gaussian elimination to attempt to determine all real solutions. solve(x+e^(z)ùy=1 and xìy=sin(z),{x,y}) ¸ x= ezøsin(z)+1 ez +1 and y= ë (sin(z)ì 1) ez +1 If a system is neither polynomial in all of its variables nor linear in its solution variables, solve() determines at most one solution using an approximate iterative method. To do so, the number of solution variables must equal the number of equations, and all other variables in the equations must simplify to numbers. solve(e^(z)ùy=1 and ëy=sin(z),{y,z}) ¸ y=.041... and z=3.183... 880 Appendix A: Functions and Instructions