Function: ev (expr, arg

Function: ev (expr, arg_1, ..., arg_n)

Evaluates the expression expr in the environment specified by the arguments arg_1, ..., arg_n. The arguments are switches (Boolean flags), assignments, equations, and functions. ev returns the result (another expression) of the evaluation.

The evaluation is carried out in steps, as follows.

1. First the environment is set up by scanning the arguments which may be any or all of the following.

o simp causes expr to be simplified regardless of the setting of the switch simp which inhibits simplification if false.

o noeval supresses the evaluation phase of ev (see step (4) below). This is useful in conjunction with the other switches and in causing expr to be resimplified without being reevaluated.

o nouns causes the evaluation of noun forms (typically unevaluated functions such as 'integrate or 'diff) in expr.

o expand causes expansion.

o expand (m, n) causes expansion, setting the values of maxposex and maxnegex to m and n respectively.

o detout causes any matrix inverses computed in expr to have their determinant kept outside of the inverse rather than dividing through each element.

o diff causes all differentiations indicated in expr to be performed.

o derivlist (x, y, z, ...) causes only differentiations with respect to the indicated variables.

o float causes non-integral rational numbers to be converted to floating point.

o numer causes some mathematical functions (including exponentiation) with numerical arguments to be evaluated in floating point. It causes variables in expr which have been given numervals to be replaced by their values. It also sets the float switch on.

o pred causes predicates (expressions which evaluate to true or false) to be evaluated.

o eval causes an extra post-evaluation of expr to occur. (See step (5) below.) eval may occur multiple times. For each instance of eval, the expression is evaluated again.

o A where A is an atom declared to be an evaluation flag (see evflag) causes A to be bound to true during the evaluation of expr.

o V: expression (or alternately V=expression) causes V to be bound to the value of expression during the evaluation of expr. Note that if V is a Maxima option, then expression is used for its value during the evaluation of expr. If more than one argument to ev is of this type then the binding is done in parallel. If V is a non-atomic expression then a substitution rather than a binding is performed.

o F where F, a function name, has been declared to be an evaluation function (see evfun) causes F to be applied to expr.

o Any other function names (e.g., sum) cause evaluation of occurrences of those names in expr as though they were verbs.

o In addition a function occurring in expr (say F(x)) may be defined locally for the purpose of this evaluation of expr by giving F(x) := expression as an argument to ev.

o If an atom not mentioned above or a subscripted variable or subscripted expression was given as an argument, it is evaluated and if the result is an equation or assignment then the indicated binding or substitution is performed. If the result is a list then the members of the list are treated as if they were additional arguments given to ev. This permits a list of equations to be given (e.g. [X=1, Y=A**2]) or a list of names of equations (e.g., [%t1, %t2] where %t1 and %t2 are equations) such as that returned by solve.

The arguments of ev may be given in any order with the exception of substitution equations which are handled in sequence, left to right, and evaluation functions which are composed, e.g., ev (expr, ratsimp, realpart) is handled as realpart (ratsimp (expr)).

The simp, numer, float, and pred switches may also be set locally in a block, or globally in Maxima so that they will remain in effect until being reset.

If expr is a canonical rational expression (CRE), then the expression returned by ev is also a CRE, provided the numer and float switches are not both true.

2. During step (1), a list is made of the non-subscripted variables appearing on the left side of equations in the arguments or in the value of some arguments if the value is an equation. The variables (subscripted variables which do not have associated array functions as well as non-subscripted variables) in the expression expr are replaced by their global values, except for those appearing in this list. Usually, expr is just a label or % (as in %i2 in the example below), so this step simply retrieves the expression named by the label, so that ev may work on it.

3. If any substitutions are indicated by the arguments, they are carried out now.

4. The resulting expression is then re-evaluated (unless one of the arguments was noeval) and simplified according to the arguments. Note that any function calls in expr will be carried out after the variables in it are evaluated and that ev(F(x)) thus may behave like F(ev(x)).

5. For each instance of eval in the arguments, steps (3) and (4) are repeated.

Examples

(%i1) sin(x) + cos(y) + (w+1)^2 + 'diff (sin(w), w);

                                     d                    2

(%o1)              cos(y) + sin(x) + -- (sin(w)) + (w + 1)

dw

(%i2) ev (%, sin, expand, diff, x=2, y=1);

(%o2)           cos(w) + w  + 2 w + cos(1) + 1.909297426825682

An alternate top level syntax has been provided for ev, whereby one may just type in its arguments, without the ev(). That is, one may write simply

expr, arg_1, ..., arg_n

This is not permitted as part of another expression, e.g., in functions, blocks, etc.

Notice the parallel binding process in the following example.

(%i3) programmode: false;

(%o3)                                false

(%i4) x+y, x: a+y, y: 2;

(%o4)                              y + a + 2

(%i5) 2*x - 3*y = 3$

(%i6) -3*x + 2*y = -4$

(%i7) solve ([%o5, %o6]);

Solution

(%t7)                               y = - -

(%t8)                                x = -

(%o8)                            [[%t7, %t8]]

(%i8) %o6, %o8;

(%o8)                              - 4 = - 4

(%i9) x + 1/x > gamma (1/2);

(%o9)                          x + - > sqrt(%pi)

(%i10) %, numer, x=1/2;

(%o10)                      2.5 > 1.772453850905516

(%i11) %, pred;

(%o11)                               true

Property: evflag

When a symbol x has the evflag property, the expressions ev(expr, x) and expr, x (at the interactive prompt) are equivalent to ev(expr, x = true). That is, x is bound to true while expr is evaluated.

The expression declare(x, evflag) gives the evflag property to the variable x.

The flags which have the evflag property by default are the following: algebraic, cauchysum, demoivre, dotscrules, %emode, %enumer, exponentialize, exptisolate, factorflag, float, halfangles, infeval, isolate_wrt_times, keepfloat, letrat, listarith, logabs, logarc, logexpand, lognegint, lognumer, m1pbranch, numer_pbranch, programmode, radexpand, ratalgdenom, ratfac, ratmx, ratsimpexpons, simp, simpsum, sumexpand, and trigexpand.

Examples:

(%i1) sin (1/2);

(%o1)                        sin(-)

(%i2) sin (1/2), float;

(%o2)                   0.479425538604203

(%i3) sin (1/2), float=true;

(%o3)                   0.479425538604203

(%i4) simp : false;

(%o4)                         false

(%i5) 1 + 1;

(%o5)                         1 + 1

(%i6) 1 + 1, simp;

(%o6)                           2

(%i7) simp : true;

(%o7)                         true

(%i8) sum (1/k^2, k, 1, inf);

inf

                            ====

                            \     1

(%o8)                        >    --

                            /      2

                            ====  k

                            k = 1

(%i9) sum (1/k^2, k, 1, inf), simpsum;

%pi

(%o9)                         ----

(%i10) declare (aa, evflag);

(%o10)                        done

(%i11) if aa = true then YES else NO;

(%o11)                         NO

(%i12) if aa = true then YES else NO, aa;

(%o12)                         YES

Property: evfun

When a function F has the evfun property, the expressions ev(expr, F) and expr, F (at the interactive prompt) are equivalent to F(ev(expr)).

If two or more evfun functions F, G, etc., are specified, the functions are applied in the order that they are specified.

The expression declare(F, evfun) gives the evfun property to the function F.

The functions which have the evfun property by default are the following: bfloat, factor, fullratsimp, logcontract, polarform, radcan, ratexpand, ratsimp, rectform, rootscontract, trigexpand, and trigreduce.

Examples:

(%i1) x^3 - 1;

(%o1)                        x  - 1

(%i2) x^3 - 1, factor;

(%o2)                 (x - 1) (x  + x + 1)

(%i3) factor (x^3 - 1);

(%o3)                 (x - 1) (x  + x + 1)

(%i4) cos(4 * x) / sin(x)^4;

                            cos(4 x)

(%o4)                       --------

                            sin (x)

(%i5) cos(4 * x) / sin(x)^4, trigexpand;

                 4           2       2         4

              sin (x) - 6 cos (x) sin (x) + cos (x)

(%o5)         -------------------------------------

                             sin (x)

(%i6) cos(4 * x) / sin(x)^4, trigexpand, ratexpand;

                           2         4

                      6 cos (x)   cos (x)

(%o6)               - --------- + ------- + 1

                          2          4

                       sin (x)    sin (x)

(%i7) ratexpand (trigexpand (cos(4 * x) / sin(x)^4));

                           2         4

                      6 cos (x)   cos (x)

(%o7)               - --------- + ------- + 1

                          2          4

                       sin (x)    sin (x)

(%i8) declare ([F, G], evfun);

(%o8)                         done

(%i9) (aa : bb, bb : cc, cc : dd);

(%o9)                          dd

(%i10) aa;

(%o10)                         bb

(%i11) aa, F;

(%o11)                        F(cc)

(%i12) F (aa);

(%o12)                        F(bb)

(%i13) F (ev (aa));

(%o13)                        F(cc)

(%i14) aa, F, G;

(%o14)                      G(F(cc))

(%i15) G (F (ev (aa)));

(%o15)                      G(F(cc))

Option variable: infeval

Enables "infinite evaluation" mode. ev repeatedly evaluates an expression until it stops changing. To prevent a variable, say X, from being evaluated away in this mode, simply include X='X as an argument to ev. Of course expressions such as ev (X, X=X+1, infeval) will generate an infinite loop.