Texas Instruments TI-89 User Manual - Page 804

Optionally, you can specify an initial guess for a variable. Each varOrGuess must have the form: variable - or - variable = real or non-real number For example, x is valid and so is x=3+i. If all of the expressions are polynomials and you do NOT specify any initial guesses, cZeros() uses the lexical Gröbner/Buchberger elimination method to attempt to determine all complex zeros. Complex zeros can include both real and non-real zeros, as in the example to the right. Each row of the resulting matrix represents an alternate zero, with the components ordered the same as the varOrGuess list. To extract a row, index the matrix by [row]. Note: The following examples use an underscore so that the variables will be treated as complex. cZeros({u_ùv_ìu_ìv_,v_^2+u_}, {u_,v_}) ¸   1/2 ì 23øi 1/2 + 23øi   1/2 + 23øi 1/2 ì 23øi   0 0 Extract row 2: ans(1)[2] ¸ 1/2 + 23øi 1/2 ì 23øi Simultaneous polynomials can have extra variables that have no values, but represent given numeric values that could be substituted later. cZeros({u_ùv_ìu_ì(c_ùv_), v_^2+u_},{u_,v_}) ¸ ë ( 1ì 4øc_+1)2   4 1ì 4øc_+1 2 1ì 4øc_ì 1)2 ë ( 1ì 4øc_ì 1)   4 2   0 0 You can also include unknown variables that do not appear in the expressions. These zeros show how families of zeros 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. For polynomial systems, computation time or memory exhaustion may depend strongly on the order in which you list unknowns. If your initial choice exhausts memory or your patience, try rearranging the variables in the expressions and/or varOrGuess list. If you do not include any guesses and if any expression is non-polynomial in any variable but all expressions are linear in all unknowns, cZeros() uses Gaussian elimination to attempt to determine all zeros. If a system is neither polynomial in all of its variables nor linear in its unknowns, cZeros() determines at most one zero using an approximate iterative method. To do so, the number of unknowns must equal the number of expressions, and all other variables in the expressions must simplify to numbers. cZeros({u_ùv_ìu_ìv_,v_^2+u_}, {u_,v_,w_}) ¸   1/2 ì 23øi 1/2 + 23øi @1   1/2 + 23øi 1/2 ì 23øi @1   0 0 @1 cZeros({u_+v_ìe^(w_),u_ìv_ìi}, {u_,v_}) ¸ e2w_ +1/2øi e w_ì 2 i  cZeros({e^(z_)ìw_,w_ìz_^2}, {w_,z_}) ¸ [.494... ë.703...] 804 Appendix A: Functions and Instructions