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 COMMENT
 REDUCE INTERACTIVE LESSON NUMBER 2
 David R. Stoutemyer
 University of Hawaii
 COMMENT This is lesson 2 of 7 REDUCE lessons. Please refrain from
 using variables beginning with the letters F through H during the
 lesson.
 By now you have probably had the experience of generating an
 expression, and then having to repeat the calculation because you
 forgot to assign it to a variable or because you did not expect to
 want to use it later. REDUCE maintains a history of all inputs and
 computation during an interactive session. (Note, this is only for
 interactive sessions.) To use an input expression in a new
 computation, you can say
 INPUT(n)
 where n is the appropriate command number. The evaluated computations
 can be accessed through
 WS(n) or simply WS
 if you wish to refer to the last computation. WS stands for Work Space.
 As with all REDUCE expressions, these can also be used to create new
 expressions:
 (INPUT(n)/WS(n2))**2
 Special characters can be used to make unique REDUCE variable names
 that reduce the chance of accidental interference with any other
 variables. In general, whenever you want to include an otherwise
 forbidden character such as * in a name, merely precede it by an
 exclamation point, which is called the escape character. However,
 pick a character other than "*", which is used for many internal
 REDUCE names. Otherwise, if most of us use "*" the purpose will be
 defeated;
 G+!%H;
 WS;
 PAUSE;
 COMMENT You can also name the expression in the workspace by using
 the command SAVEAS, for example:;
 SAVEAS GPLUSH;
 GPLUSH;
 PAUSE;
 COMMENT You may have noticed that REDUCE imposes its own order on the
 indeterminates and functional forms that appear in results, and that
 this ordering can strongly affect the intelligibility of the results.
 For example:;
 G1:= 2*H*G + E + F1 + F + F**2 + F2 + 5 + LOG(F1) + SIN(F1);
 COMMENT The ORDER declaration permits us to order indeterminates and
 functional forms as we choose. For example, to order F2 before F1,
 and to order F1 before all remaining variables:;
 ORDER F2, F1;
 G1;
 PAUSE;
 COMMENT Now suppose we partially change our mind and decide to
 order LOG(F1) ahead of F1;
 ORDER LOG(F1), F1;
 G1;
 COMMENT Note that any other indeterminates or functional forms under
 the influence of a previous ORDER declaration, such as F2, rank
 before those mentioned in the later declaration. Try to determine
 the default ordering algorithm used in your REDUCE implementation, and
 try to achieve some delicate rearrangements using the ORDER
 declaration.;
 PAUSE;
 COMMENT You may have also noticed that REDUCE factors out any
 number, indeterminate, functional form, or the largest integer power
 thereof which exactly divides every term of a result or every term of
 a parenthesized subexpression of a result. For example:;
 ON EXP, MCD;
 G1:= F**2*(G**2 + 2*G) + F*(G**2+H)/(2*F1);
 COMMENT This process usually leads to more compact expressions and
 reveals important structural information. However, the process can
 yield results which are difficult to interpret if the resulting
 parentheses are nested more than about two levels, and it is often
 desirable to see a fully expanded result to facilitate direct
 comparison of all terms. To suppress this monomial factoring, we can
 turn off an output control switch named ALLFAC;
 OFF ALLFAC;
 G1;
 PAUSE;
 COMMENT The ALLFAC monomialfactorization process is strongly
 dependent upon the ordering. We can achieve a more selective monomial
 factorization by using the FACTOR declaration, which declares a
 variable to have FACTOR status. If any indeterminates or functional
 forms occurring in an expression are in FACTOR status when the
 expression is printed, terms having the same powers of the
 indeterminates or functional forms are collected together, and the
 power is factored out. Terms containing two or more indeterminates or
 functional forms under FACTOR status are not included in this monomial
 factorization process. For example:;
 OFF ALLFAC; FACTOR F; G1;
 FACTOR G; G1; PAUSE;
 COMMENT We can use the REMFAC command to remove items from factor
 status;
 REMFAC F;
 G1;
 COMMENT ALLFAC can still have an effect on the coefficients of the
 monomials that have been factored out under the influence of FACTOR:;
 ON ALLFAC;
 G1;
 PAUSE;
 COMMENT It is often desirable to distribute denominators over all
 factored subexpressions generated under the influence of a FACTOR
 declaration, such as when we wish to view a result as a polynomial or
 as a power series in the factored indeterminates or functional forms,
 with coefficients which are rational functions of any other
 indeterminates or functional forms. (A mnemonic aid is: think RAT
 for RATionalfunction coefficients.) For example:;
 ON RAT;
 G1;
 PAUSE;
 COMMENT RAT has no effect on expressions which have no
 indeterminates or functional forms under the influence of FACTOR.
 The related but different DIV switch permits us to distribute numerical
 and monomial factors of the denominator over every term of the
 numerator, expressing these distributed portions as rationalnumber
 coefficients and negative power factors respectively. (A mnemonic
 aid: DIV DIVides by monomials.) The overall effect can also depend
 strongly on whether the RAT switch is on or off. Series and
 polynomials are often most attractive with RAT and DIV both on;
 ON DIV, RAT;
 G1;
 OFF RAT;
 G1;
 PAUSE;
 REMFAC G;
 G1;
 PAUSE;
 COMMENT With a very complicated result, detailed study of the result
 is often facilitated by having each new term begin on a new line,
 which can be accomplished using the LIST switch:;
 ON LIST;
 G1;
 PAUSE;
 COMMENT In various combinations, ORDER, FACTOR, the computational
 switches EXP, MCD, GCD, and ROUNDED, together with the output control
 switches ALLFAC, RAT, DIV, and LIST provide a variety of output
 alternatives. With experience, it is usually possible to use these
 tools to produce a result in the desired form, or at least in a form
 which is far more acceptable than the one produced by the default
 settings. I encourage you to experiment with various combinations
 while this information is fresh in your mind;
 PAUSE;
 OFF LIST, RAT, DIV, GCD, ROUNDED;
 ON ALLFAC, MCD, EXP;
 COMMENT You may have wondered whether or not an assignment to a
 variable, say F1, automatically updates the value of a bound
 variable, say G1, which was previously assigned an expression
 containing F1. The answer is:
 1. If F1 was a bound variable in the expression when it was set
 to G1, then subsequent changes to the value of F1 have no
 effect on G1 because all traces of F1 in G1 disappeared after
 F1 contributed its value to the formation of G1.
 2. If F1 was an indeterminate in an expression previously
 assigned to G1, then for each subsequent use of G1, F1
 contributes its current value at the time of that use.
 These phenomena are illustrated by the following sequence:;
 PAUSE;
 F2 := F;
 G1 := F1 + F2;
 F2 := G;
 G1;
 F1 := G;
 F1 := H;
 G1;
 F1 := G;
 G1;
 COMMENT Experience indicates that it is well worth studying this
 sequence and experimenting with others until these phenomena are
 thoroughly understood. You might, for example, mimic the above
 example, but with another level of evaluation included by inserting a
 statement analogous to "Q9:=G1" after "F2:=G", and inserting an
 expression analogous to "Q9" at the end, to compare with G1. ;
 PAUSE;
 COMMENT Note also, that if an indeterminate is used directly, or
 indirectly through another expression, in evaluating itself, this will
 lead to an infinite recursion. For example, the following expression
 results in infinite recursion at the first evaluation of H1. On some
 machines (Vax/Unix, IBM) this will cause REDUCE to terminate abnormally.
 H1 := H1 + 1
 You may experiment with this problem, later at your own risk.
 It is often desirable to make an assignment to an indeterminate in a
 previously established expression have a permanent effect, as if the
 assignment were done before forming the expression. This can be done by
 using the substitute function, SUB.
 G1 := F1 + F2;
 H1 := SUB(F1=H, G1);
 F1 := G;
 H1;
 COMMENT Note the use of "=" rather than ":=" in SUB. This function
 is also valuable for achieving the effect of a local assignment
 within a subexpression, without binding the involved indeterminate or
 functional form in the rest of the expression or wherever else it
 occurs. More generally the SUB function can have any number of
 equations of the form "indeterminate or functional form =
 expression", separated by commas, before the expression which is its
 last argument. Try devising a set of examples which reveals whether
 such multiple substitutions are done left to right, right to left, in
 parallel, or unpredictably.
 This is the end of lesson 2. To execute lesson 3, start a fresh
 REDUCE job.
 ;END;
