1 The functions LINSIMP and DECLARE_LINEAR_OPERATOR provide 2the user with the capability of simplifying expressions that contain 3operators that are linear in one or more arguments. It is possible 4to accomplish something quite similar using the standard MACSYMA 5function DECLARE, but there is no simple way to extend this to operators 6that have more than one argument. The example below illustrates the 7capabilities that are built into MACSYMA: 8 9(C1) DECLARE(F,LINEAR); 10(D1) DONE 11 12(C2) F(2*A); 13(D2) 2 F(A) 14 15(C3) F(2); 16(D3) 2 F(1) 17 18(C4) F(A*B); 19(D4) F(A B) 20 21(C5) DECLARE(B,CONSTANT); 22(D5) DONE 23 24(C6) F(A*B); 25(D6) B F(A) 26 27 28Notice that constant factors are extracted from the argument of F. This 29will often be what is needed, but not always. Operators that have more 30than one argument are treated differently. A complete description is given 31in MACDOC;UPDATE >, in a note describing new features of MACSYMA #261. 32 33 LINSIMP and DECLARE_LINEAR_OPERATOR fill these minor gaps in 34MACSYMA's understanding of linear operators. DECLARE_LINEAR_OPERATOR is 35used to set up the information necessary for appropriate simplifications 36to be carried out, and LINSIMP is used to execute those simplifications. 37The desired simplification rules are not applied automatically, which is 38a disadvantage of this method relative to MACSYMA's built in capability. 39To obtain automatic simplification, one might use DECLIN in conjunction 40with TELLSIMP or TELLSIMPAFTER. 41 42LINSIMP(exp, operator1, operator2, ... ) simplifies exp with respect to 43 the linearity properties of operator1, then operator2, and so on. 44 Any terms belonging to the same sum that can be combined together 45 under the same operator are combined, and any factors that can be 46 extracted from any of these operators are extracted. If any of the 47 operators in the argument list of LINSIMP have not been declared 48 linear using the function DECLARE_LINEAR_OPERATOR, an error will re- 49 sult. 50 51DECLARE_LINEAR_OPERATOR(operator, linear-arguments, separation-predicate) 52 sets up the linearity property of "operator", which must be an atomic 53 symbol. The second argument of DECLARE_LINEAR_OPERATOR, linear-arguments, 54 is a list of one or more elements the entries of which denote the spe- 55 cific arguments of "operator" that participate in the linearity property. 56 "operator" is considered to be a linear function of these specific argu- 57 ments, taken together. For example, let F be linear in its first three 58 arguments. Then: 59 60 F(2*A, 2*B, 2*C, D, E) = 2*F(A, B, C, D, E) 61 62 The third argument of DECLARE_LINEAR_OPERATOR is the separation predicate. 63 It must be a function of one argument which returns TRUE if its argument 64 is not meant to be extracted from "operator", and FALSE if its argument 65 is meant to be extracted. Thus, for example, the built in separation 66 predicate used by MACSYMA for this purpose could be defined as 67 68 NOT_CONSTANTP(EXP):=NOT CONSTANTP(EXP)$ 69 70 With such a definition, the declaration necessary to mimic MACSYMA's 71 built-in capability would be, for the function H of one argument, 72 73 DECLARE_LINEAR_OPERATOR(H, [1], NOT_CONSTANTP)$ 74 75 Subsequent use of LINSIMP would then produce forms similar to those pro- 76 duced by MACSYMA if H had been DECLAREd LINEAR. 77