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# 8. Evaluation

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## 8.1 Functions and Variables for Evaluation

Operator: '

  The single quote operator `'` prevents evaluation.

  Applied to a symbol, the single quote prevents evaluation of the symbol.

  Applied to a function call, the single quote prevents evaluation of the function call, although the arguments of the function are still evaluated (if evaluation is not otherwise prevented). The result is the noun form of the function call.

  Applied to a parenthesized expression, the single quote prevents evaluation of all symbols and function calls in the expression. E.g., `'(f(x))` means do not evaluate the expression `f(x)`. `'f(x)` (with the single quote applied to `f` instead of `f(x)`) means return the noun form of `f` applied to `[x]`.

  The single quote does not prevent simplification.

  When the global flag `noundisp` is `true`, nouns display with a single quote. This switch is always `true` when displaying function definitions.

  See also the quote-quote operator `''` and `nouns`.

  Examples:

  Applied to a symbol, the single quote prevents evaluation of the symbol.

      (%i1) aa: 1024;
      (%o1)                         1024
      (%i2) aa^2;
      (%o2)                        1048576
      (%i3) 'aa^2;
                                       2
      (%o3)                          aa
      (%i4) ''%;
      (%o4)                        1048576

  Applied to a function call, the single quote prevents evaluation of the function call. The result is the noun form of the function call.

      (%i1) x0: 5;
      (%o1)                           5
      (%i2) x1: 7;
      (%o2)                           7
      (%i3) integrate (x^2, x, x0, x1);
                                     218
      (%o3)                          ---
                                      3
      (%i4) 'integrate (x^2, x, x0, x1);
                                   7
                                  /
                                  [   2
      (%o4)                       I  x  dx
                                  ]
                                  /
                                   5
      (%i5) %, nouns;
                                     218
      (%o5)                          ---
                                      3

  Applied to a parenthesized expression, the single quote prevents evaluation of all symbols and function calls in the expression.

      (%i1) aa: 1024;
      (%o1)                         1024
      (%i2) bb: 19;
      (%o2)                          19
      (%i3) sqrt(aa) + bb;
      (%o3)                          51
      (%i4) '(sqrt(aa) + bb);
      (%o4)                     bb + sqrt(aa)
      (%i5) ''%;
      (%o5)                          51

  The single quote does not prevent simplification.

      (%i1) sin (17 * %pi) + cos (17 * %pi);
      (%o1)                          - 1
      (%i2) '(sin (17 * %pi) + cos (17 * %pi));
      (%o2)                          - 1

  Maxima considers floating point operations by its in-built mathematical functions to be a simplification.

      (%i1) sin(1.0);
      (%o1)                          .8414709848078965
      (%i2) '(sin(1.0));
      (%o2)                          .8414709848078965

  ·

  @ref{Category: Evaluation} · @ref{Category: Operators}

Operator: "

  The quote-quote operator `''` (two single quote marks) modifies evaluation in input expressions.

  Applied to a general expression expr, quote-quote causes the value of expr to be substituted for expr in the input expression.

  Applied to the operator of an expression, quote-quote changes the operator from a noun to a verb (if it is not already a verb).

  The quote-quote operator is applied by the input parser; it is not stored as part of a parsed input expression. The quote-quote operator is always applied as soon as it is parsed, and cannot be quoted. Thus quote-quote causes evaluation when evaluation is otherwise suppressed, such as in function definitions, lambda expressions, and expressions quoted by single quote `'`.

  Quote-quote is recognized by `batch` and `load`.

  See also the single-quote operator `'` and `nouns`.

  Examples:

  Applied to a general expression expr, quote-quote causes the value of expr to be substituted for expr in the input expression.

      (%i1) expand ((a + b)^3);
                           3        2      2      3
      (%o1)               b  + 3 a b  + 3 a  b + a
      (%i2) [_, ''_];
                               3    3        2      2      3
      (%o2)     [expand((b + a) ), b  + 3 a b  + 3 a  b + a ]
      (%i3) [%i1, ''%i1];
                               3    3        2      2      3
      (%o3)     [expand((b + a) ), b  + 3 a b  + 3 a  b + a ]
      (%i4) [aa : cc, bb : dd, cc : 17, dd : 29];
      (%o4)                   [cc, dd, 17, 29]
      (%i5) foo_1 (x) := aa - bb * x;
      (%o5)                 foo_1(x) := aa - bb x
      (%i6) foo_1 (10);
      (%o6)                      cc - 10 dd
      (%i7) ''%;
      (%o7)                         - 273
      (%i8) ''(foo_1 (10));
      (%o8)                         - 273
      (%i9) foo_2 (x) := ''aa - ''bb * x;
      (%o9)                 foo_2(x) := cc - dd x
      (%i10) foo_2 (10);
      (%o10)                        - 273
      (%i11) [x0 : x1, x1 : x2, x2 : x3];
      (%o11)                    [x1, x2, x3]
      (%i12) x0;
      (%o12)                         x1
      (%i13) ''x0;
      (%o13)                         x2
      (%i14) '' ''x0;
      (%o14)                         x3

  Applied to the operator of an expression, quote-quote changes the operator from a noun to a verb (if it is not already a verb).

      (%i1) declare (foo, noun);
      (%o1)                         done
      (%i2) foo (x) := x - 1729;
      (%o2)                 ''foo(x) := x - 1729
      (%i3) foo (100);
      (%o3)                       foo(100)
      (%i4) ''foo (100);
      (%o4)                        - 1629

  The quote-quote operator is applied by the input parser; it is not stored as part of a parsed input expression.

      (%i1) [aa : bb, cc : dd, bb : 1234, dd : 5678];
      (%o1)                 [bb, dd, 1234, 5678]
      (%i2) aa + cc;
      (%o2)                        dd + bb
      (%i3) display (_, op (_), args (_));
                                 _ = cc + aa

                               op(cc + aa) = +

                          args(cc + aa) = [cc, aa]

      (%o3)                         done
      (%i4) ''(aa + cc);
      (%o4)                         6912
      (%i5) display (_, op (_), args (_));
                                 _ = dd + bb

                               op(dd + bb) = +

                          args(dd + bb) = [dd, bb]

      (%o5)                         done

  Quote-quote causes evaluation when evaluation is otherwise suppressed, such as in function definitions, lambda expressions, and expressions quoted by single quote `'`.

      (%i1) foo_1a (x) := ''(integrate (log (x), x));
      (%o1)               foo_1a(x) := x log(x) - x
      (%i2) foo_1b (x) := integrate (log (x), x);
      (%o2)           foo_1b(x) := integrate(log(x), x)
      (%i3) dispfun (foo_1a, foo_1b);
      (%t3)               foo_1a(x) := x log(x) - x

      (%t4)           foo_1b(x) := integrate(log(x), x)

      (%o4)                      [%t3, %t4]
      (%i5) integrate (log (x), x);
      (%o5)                     x log(x) - x
      (%i6) foo_2a (x) := ''%;
      (%o6)               foo_2a(x) := x log(x) - x
      (%i7) foo_2b (x) := %;
      (%o7)                    foo_2b(x) := %
      (%i8) dispfun (foo_2a, foo_2b);
      (%t8)               foo_2a(x) := x log(x) - x

      (%t9)                    foo_2b(x) := %

      (%o9)                      [%t7, %t8]
      (%i10) F : lambda ([u], diff (sin (u), u));
      (%o10)             lambda([u], diff(sin(u), u))
      (%i11) G : lambda ([u], ''(diff (sin (u), u)));
      (%o11)                  lambda([u], cos(u))
      (%i12) '(sum (a[k], k, 1, 3) + sum (b[k], k, 1, 3));
      (%o12)         sum(b , k, 1, 3) + sum(a , k, 1, 3)
                          k                  k
      (%i13) '(''(sum (a[k], k, 1, 3)) + ''(sum (b[k], k, 1, 3)));
      (%o13)             b  + a  + b  + a  + b  + a
                          3    3    2    2    1    1

  ·

  @ref{Category: Evaluation} · @ref{Category: Operators}

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.
     - `simp` causes expr to be simplified regardless of the setting of the switch `simp` which inhibits simplification if `false`.
     - `noeval` suppresses 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.
     - `nouns` causes the evaluation of noun forms (typically unevaluated functions such as `'integrate` or `'diff`) in expr.
     - `expand` causes expansion.
     - `expand (m, n)` causes expansion, setting the values of `maxposex` and `maxnegex` to m and n respectively.
     - `detout` causes any matrix inverses computed in expr to have their determinant kept outside of the inverse rather than dividing through each element.
     - `diff` causes all differentiations indicated in expr to be performed.
     - `derivlist (x, y, z, ...)` causes only differentiations with respect to the indicated variables. See also `derivlist`.
     - `risch` causes integrals in expr to be evaluated using the Risch algorithm. See `risch`. The standard integration routine is invoked when using the special symbol `nouns`.
     - `float` causes non-integral rational numbers to be converted to floating point.
     - `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.
     - `pred` causes predicates (expressions which evaluate to `true` or `false`) to be evaluated.
     - `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.
     - `A` where `A` is an atom declared to be an evaluation flag `evflag` causes `A` to be bound to `true` during the evaluation of expr.
     - `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.
     - `F` where `F`, a function name, has been declared to be an evaluation function `evfun` causes `F` to be applied to expr.
     - Any other function names, e.g. `sum`, cause evaluation of occurrences of those names in expr as though they were verbs.
     - 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`.
     - 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`, and `float` 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 (%, numer, expand, diff, x=2, y=1);
                                     2
      (%o2)                cos(w) + w  + 2 w + 2.449599732693821

  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

                                                1
      (%t7)                               y = - -
                                                5

                                               6
      (%t8)                                x = -
                                               5
      (%o8)                            [[%t7, %t8]]
      (%i8) %o6, %o8;
      (%o8)                              - 4 = - 4
      (%i9) x + 1/x > gamma (1/2);
                                         1
      (%o9)                          x + - > sqrt(%pi)
                                         x
      (%i10) %, numer, x=1/2;
      (%o10)                      2.5 > 1.772453850905516
      (%i11) %, pred;
      (%o11)                               true

  ·

  @ref{Category: Evaluation}

Special symbol: eval

  As an argument in a call to `ev (expr)`, `eval` causes an extra evaluation of expr. See `ev`.

  Example:

      (%i1) [a:b,b:c,c:d,d:e];
      (%o1)                            [b, c, d, e]
      (%i2) a;
      (%o2)                                  b
      (%i3) ev(a);
      (%o3)                                  c
      (%i4) ev(a),eval;
      (%o4)                                  e
      (%i5) a,eval,eval;
      (%o5)                                  e

  ·

  @ref{Category: Evaluation flags}

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
         m1pbranch          numer_pbranch   programmode
         radexpand          ratalgdenom     ratfac
         ratmx              ratsimpexpons   simp
         simpproduct        simpsum         sumexpand
         trigexpand

  Examples:

      (%i1) sin (1/2);
                                       1
      (%o1)                        sin(-)
                                       2
      (%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;
                                       2
                                    %pi
      (%o9)                         ----
                                     6
      (%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

  ·

  @ref{Category: Evaluation flags} · @ref{Category: Simplification flags and variables}

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   trigreduce

  Examples:

      (%i1) x^3 - 1;
                                    3
      (%o1)                        x  - 1
      (%i2) x^3 - 1, factor;
                                      2
      (%o2)                 (x - 1) (x  + x + 1)
      (%i3) factor (x^3 - 1);
                                      2
      (%o3)                 (x - 1) (x  + x + 1)
      (%i4) cos(4 * x) / sin(x)^4;
                                  cos(4 x)
      (%o4)                       --------
                                     4
                                  sin (x)
      (%i5) cos(4 * x) / sin(x)^4, trigexpand;
                       4           2       2         4
                    sin (x) - 6 cos (x) sin (x) + cos (x)
      (%o5)         -------------------------------------
                                      4
                                   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))

  ·

  @ref{Category: Evaluation flags}

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.

  ·

  @ref{Category: Evaluation flags}

Special symbol: noeval

  `noeval` suppresses the evaluation phase of `ev`. This is useful in conjunction with other switches and in causing expressions to be resimplified without being reevaluated.

  ·

  @ref{Category: Evaluation flags}

Special symbol: nouns

  `nouns` is an `evflag`. When used as an option to the `ev` command, `nouns` converts all "noun" forms occurring in the expression being `ev`'d to "verbs", i.e., evaluates them. See also `noun`, `nounify`, `verb`, and `verbify`.

  ·

  @ref{Category: Evaluation flags} · @ref{Category: Nouns and verbs}

Special symbol: pred

  As an argument in a call to `ev (expr)`, `pred` causes predicates (expressions which evaluate to `true` or `false`) to be evaluated. See `ev`.

  Example:

      (%i1) 1<2;
      (%o1)                                1 < 2
      (%i2) 1<2,pred;
      (%o2)                                true

  ·

  @ref{Category: Evaluation flags}

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