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Maxima Manual

38.2 Definitions for Program Flow

Variable: BACKTRACE

default: [] (when DEBUGMODE:ALL has been done) has as value a list of all functions currently entered.

special operator: DO

- The DO statement is used for performing iteration. Due to its great generality the DO statement will be described in two parts. First the usual form will be given which is analogous to that used in several other programming languages (FORTRAN, ALGOL, PL/I, etc.); then the other features will be mentioned. 1. There are three variants of this form that differ only in their terminating conditions. They are:

• (a) FOR variable : initial-value STEP increment THRU limit DO body
• (b) FOR variable : initial-value STEP increment WHILE condition DO body
• (c) FOR variable : initial-value STEP increment UNLESS condition DO body

(Alternatively, the STEP may be given after the termination condition or limit.) The initial-value, increment, limit, and body can be any expressions. If the increment is 1 then "STEP 1" may be omitted. The execution of the DO statement proceeds by first assigning the initial-value to the variable (henceforth called the control-variable). Then: (1) If the control-variable has exceeded the limit of a THRU specification, or if the condition of the UNLESS is TRUE, or if the condition of the WHILE is FALSE then the DO terminates. (2) The body is evaluated. (3) The increment is added to the control-variable. The process from (1) to (3) is performed repeatedly until the termination condition is satisfied. One may also give several termination conditions in which case the DO terminates when any of them is satisfied. In general the THRU test is satisfied when the control-variable is greater than the limit if the increment was non-negative, or when the control-variable is less than the limit if the increment was negative. The increment and limit may be non-numeric expressions as long as this inequality can be determined. However, unless the increment is syntactically negative (e.g. is a negative number) at the time the DO statement is input, MACSYMA assumes it will be positive when the DO is executed. If it is not positive, then the DO may not terminate properly. Note that the limit, increment, and termination condition are evaluated each time through the loop. Thus if any of these involve much computation, and yield a result that does not change during all the executions of the body, then it is more efficient to set a variable to their value prior to the DO and use this variable in the DO form. The value normally returned by a DO statement is the atom DONE, as every statement in MACSYMA returns a value. However, the function RETURN may be used inside the body to exit the DO prematurely and give it any desired value. Note however that a RETURN within a DO that occurs in a BLOCK will exit only the DO and not the BLOCK. Note also that the GO function may not be used to exit from a DO into a surrounding BLOCK. The control-variable is always local to the DO and thus any variable may be used without affecting the value of a variable with the same name outside of the DO. The control-variable is unbound after the DO terminates.

(C1) FOR A:-3 THRU 26 STEP 7 DO LDISPLAY(A)$ (E1) A = -3 (E2) A = 4 (E3) A = 11 (E4) A = 18 (E5) A = 25

The function LDISPLAY generates intermediate labels; DISPLAY does not.

(C6) S:0$ (C7) FOR I:1 WHILE I<=10 DO S:S+I; (D7) DONE (C8) S; (D8) 55

Note that the condition in C7 is equivalent to UNLESS I > 10 and also THRU 10

(C9) SERIES:1$ (C10) TERM:EXP(SIN(X))$ (C11) FOR P:1 UNLESS P>7 DO (TERM:DIFF(TERM,X)/P, SERIES:SERIES+SUBST(X=0,TERM)*X^P)$ (C12) SERIES; 7 6 5 4 2 (D12) X X X X X -- - --- - -- - -- + -- + X + 1 96 240 15 8 2 which gives 8 terms of the Taylor series for e^sin(x). (C13) POLY:0$ (C14) FOR I:1 THRU 5 DO FOR J:I STEP -1 THRU 1 DO POLY:POLY+I*X^J$ (C15) POLY; 5 4 3 2 (D15) 5 X + 9 X + 12 X + 14 X + 15 X (C16) GUESS:-3.0$ (C17) FOR I:1 THRU 10 DO (GUESS:SUBST(GUESS,X,.5*(X+10/X)), IF ABS(GUESS^2-10)<.00005 THEN RETURN(GUESS)); (D17) - 3.1622807

This example computes the negative square root of 10 using the Newton- Raphson iteration a maximum of 10 times. Had the convergence criterion not been met the value returned would have been "DONE". Additional Forms of the DO Statement Instead of always adding a quantity to the control-variable one may sometimes wish to change it in some other way for each iteration. In this case one may use "NEXT expression" instead of "STEP increment". This will cause the control-variable to be set to the result of evaluating expression each time through the loop.

(C1) FOR COUNT:2 NEXT 3*COUNT THRU 20 DO DISPLAY(COUNT)$ COUNT = 2 COUNT = 6 COUNT = 18

As an alternative to FOR variable:value ...DO... the syntax FOR variable FROM value ...DO... may be used. This permits the "FROM value" to be placed after the step or next value or after the termination condition. If "FROM value" is omitted then 1 is used as the initial value. Sometimes one may be interested in performing an iteration where the control-variable is never actually used. It is thus permissible to give only the termination conditions omitting the initialization and updating information as in the following example to compute the square-root of 5 using a poor initial guess.

(C1) X:1000; (C2) THRU 10 WHILE X#0.0 DO X:.5*(X+5.0/X)$ (C3) X; (D3) 2.236068

If it is desired one may even omit the termination conditions entirely and just give "DO body" which will continue to evaluate the body indefinitely. In this case the function RETURN should be used to terminate execution of the DO.

(C1) NEWTON(F,GUESS):= BLOCK([NUMER,Y], LOCAL(DF), NUMER:TRUE, DEFINE(DF(X),DIFF(F(X),X)), DO (Y:DF(GUESS), IF Y=0.0 THEN ERROR("Derivative at:",GUESS," is zero."), GUESS:GUESS-F(GUESS)/Y, IF ABS(F(GUESS))<5.0E-6 THEN RETURN(GUESS)))$ (C2) SQR(X):=X^2-5.0$ (C3) NEWTON(SQR,1000); (D3) 2.236068

(Note that RETURN, when executed, causes the current value of GUESS to be returned as the value of the DO. The BLOCK is exited and this value of the DO is returned as the value of the BLOCK because the DO is the last statement in the block.) One other form of the DO is available in MACSYMA. The syntax is:

FOR variable IN list [end-tests] DO body

The members of the list are any expressions which will successively be assigned to the variable on each iteration of the body. The optional end-tests can be used to terminate execution of the DO; otherwise it will terminate when the list is exhausted or when a RETURN is executed in the body. (In fact, list may be any non-atomic expression, and successive parts are taken.)

(C1) FOR F IN [LOG, RHO, ATAN] DO LDISP(F(1))$ (E1) 0 (E2) RHO(1) %PI (E3) --- 4 (C4) EV(E3,NUMER); (D4) 0.78539816

Function: ERRCATCH (exp1, exp2, ...)

evaluates its arguments one by one and returns a list of the value of the last one if no error occurs. If an error occurs in the evaluation of any arguments, ERRCATCH "catches" the error and immediately returns [] (the empty list). This function is useful in BATCH files where one suspects an error might occur which would otherwise have terminated the BATCH if the error weren't caught.

Variable: ERREXP

default: [ERREXP] When an error occurs in the course of a computation, MACSYMA prints out an error message and terminates the computation. ERREXP is set to the offending expression and the message "ERREXP contains the offending expression" is printed. The user can then type ERREXP; to see this and hopefully find the problem.

Function: ERROR (arg1, arg2, ...)

will evaluate and print its arguments and then will cause an error return to top level MACSYMA or to the nearest enclosing ERRCATCH. This is useful for breaking out of nested functions if an error condition is detected, or wherever one can't type control-^. The variable ERROR is set to a list describing the error, the first of it being a string of text, and the rest the objects in question. ERRORMSG(); is the preferred way to see the last error message. ERRORFUN default: [FALSE] - if set to the name of a function of no arguments will cause that function to be executed whenever an error occurs. This is useful in BATCH files where the user may want his MACSYMA killed or his terminal logged out if an error occurs. In these cases ERRORFUN would be set to QUIT or LOGOUT.

Variable: ERRORFUN

default: [FALSE] - if set to the name of a function of no arguments will cause that function to be executed whenever an error occurs. This is useful in BATCH files where the user may want his MACSYMA killed or his terminal logged out if an error occurs. In these cases ERRORFUN would be set to QUIT or LOGOUT.

Function: ERRORMSG ()

reprints the last error message. This is very helpful if you are using a display console and the message has gone off the screen. The variable ERROR is set to a list describing the error, the first of it being a string of text, and the rest the objects in question. TTYINTFUN:LAMBDA([],ERRORMSG(),PRINT(""))$ will set up the user-interrupt character (^U) to reprint the message.

special operator: FOR

- Used in iterations, do DESCRIBE("DO"); for a description of MACSYMA's iteration facilities.

Function: GO (tag)

is used within a BLOCK to transfer control to the statement of the block which is tagged with the argument to GO. To tag a statement, precede it by an atomic argument as another statement in the BLOCK. For example:

BLOCK([X],X:1,LOOP,X+1,...,GO(LOOP),...)

. The argument to GO must be the name of a tag appearing in the same BLOCK. One cannot use GO to transfer to tag in a BLOCK other than the one containing the GO.

special operator: IF

- The IF statement is used for conditional execution. The syntax is:

IF condition THEN expression1 ELSE expression2.

The result of an IF statement is expression1 if condition is true and expression2 if it is false. expression1 and expression2 are any MACSYMA expressions (including nested IF statements), and condition is an expression which evaluates to TRUE or FALSE and is composed of relational and logical operators which are as follows:

Operator name Symbol Type greater than > relational infix equal to = , EQUAL " " not equal to # " " less than < " " greater than >= or equal to " " less than <= or equal to " " and AND logical infix or OR " " not NOT logical prefix

Function: LISPDEBUGMODE ()

LISPDEBUGMODE(); DEBUGPRINTMODE(); and DEBUG(); make available to the user debugging features used by systems programmers. These tools are powerful, and although some conventions are different from the usual macsyma level it is felt their use is very intuitive. [Some printout may be verbose for slow terminals, there are switches for controlling this.] These commands were designed for the user who must debug translated macsyma code, as such they are a boon. See MACDOC;TRDEBG USAGE for more information. For more help, consult GJC.

Function: MAP (fn, exp1, exp2, ...)

returns an expression whose leading operator is the same as that of the expi but whose subparts are the results of applying fn to the corresponding subparts of the expi. Fn is either the name of a function of n arguments (where n is the number of expi) or is a LAMBDA form of n arguments. MAPERROR[TRUE] - if FALSE will cause all of the mapping functions to (1) stop when they finish going down the shortest expi if not all of the expi are of the same length and (2) apply fn to [exp1, exp2,...] if the expi are not all the same type of object. If MAPERROR is TRUE then an error message will be given in the above two instances. One of the uses of this function is to MAP a function (e.g. PARTFRAC) onto each term of a very large expression where it ordinarily wouldn't be possible to use the function on the entire expression due to an exhaustion of list storage space in the course of the computation.

(C1) MAP(F,X+A*Y+B*Z); (D1) F(B Z) + F(A Y) + F(X) (C2) MAP(LAMBDA([U],PARTFRAC(U,X)),X+1/(X^3+4*X^2+5*X+2)); 1 1 1 (D2) ----- - ----- + -------- + X X + 2 X + 1 2 (X + 1) (C3) MAP(RATSIMP, X/(X^2+X)+(Y^2+Y)/Y); 1 (D3) Y + ----- + 1 X + 1 (C4) MAP("=",[A,B],[-0.5,3]); (D4) [A = - 0.5, B = 3]

Function: MAPATOM (expr)

is TRUE if and only if expr is treated by the MAPping routines as an "atom", a unit. "Mapatoms" are atoms, numbers (including rational numbers), and subscripted variables.

Variable: MAPERROR

default: [TRUE] - if FALSE will cause all of the mapping functions, for example

MAP(fn,exp1,exp2,...))

to (1) stop when they finish going down the shortest expi if not all of the expi are of the same length and (2) apply fn to [exp1, exp2,...] if the expi are not all the same type of object. If MAPERROR is TRUE then an error message will be given in the above two instances.

Function: MAPLIST (fn, exp1, exp2, ...)

yields a list of the applications of fn to the parts of the expi. This differs from MAP(fn,exp1,exp2,...) which returns an expression with the same main operator as expi has (except for simplifications and the case where MAP does an APPLY). Fn is of the same form as in MAP.

Variable: PREDERROR

default: [TRUE] - If TRUE, an error message is signalled whenever the predicate of an IF statement or an IS function fails to evaluate to either TRUE or FALSE. If FALSE, UNKNOWN is returned instead in this case. The PREDERROR:FALSE mode is not supported in translated code.

Function: RETURN (value)

may be used to exit explicitly from a BLOCK, bringing its argument. Do DESCRIBE(BLOCK); for more information.

Function: SCANMAP (function,exp)

recursively applies function to exp, in a "top down" manner. This is most useful when "complete" factorization is desired, for example:

(C1) EXP:(A^2+2*A+1)*Y + X^2$ (C2) SCANMAP(FACTOR,EXP); 2 2 (D2) (A + 1) Y + X

Note the way in which SCANMAP applies the given function FACTOR to the constituent subexpressions of exp; if another form of exp is presented to SCANMAP then the result may be different. Thus, D2 is not recovered when SCANMAP is applied to the expanded form of exp:

(C3) SCANMAP(FACTOR,EXPAND(EXP)); 2 2 (D3) A Y + 2 A Y + Y + X

Here is another example of the way in which SCANMAP recursively applies a given function to all subexpressions, including exponents:

(C4) EXPR : U*V^(A*X+B) + C$ (C5) SCANMAP('F, EXPR); F(F(F(A) F(X)) + F(B)) (D5) F(F(F(U) F(F(V) )) + F(C))

SCANMAP(function,expression,BOTTOMUP) applies function to exp in a "bottom-up" manner. E.g., for undefined F,

SCANMAP(F,A*X+B) -> F(A*X+B) -> F(F(A*X)+F(B)) -> F(F(F(A)*F(X))+F(B)) SCANMAP(F,A*X+B,BOTTOMUP) -> F(A)*F(X)+F(B) -> F(F(A)*F(X))+F(B) -> F(F(F(A)*F(X))+F(B))

In this case, you get the same answer both ways.

Function: THROW (exp)

evaluates exp and throws the value back to the most recent CATCH. THROW is used with CATCH as a structured nonlocal exit mechanism.