Control flow functions¶

MaxEvalDepth(n)¶

set the maximum evaluation depth

Parameters:	n – new maximum evaluation depth

Use this command to set the maximum evaluation depth to n. The default value is 1000.

The point of having a maximum evaluation depth is to catch any infinite recursion. For example, after the definition f(x) := f(x), evaluating the expression f(x) would call f(x), which would call f(x), etc. The interpreter will halt if the maximum evaluation depth is reached. Also indirect recursion, e.g. the pair of definitions f(x) := g(x) and g(x) := f(x), will be caught.

An example of an infinite recursion, caught because the maximum evaluation depth is reached

In> f(x) := f(x)
Out> True;
In> f(x)

Error on line 1 in file [CommandLine]
Max evaluation stack depth reached.
Please use MaxEvalDepth to increase the stack
size as needed.

However, a long calculation may cause the maximum evaluation depth to be reached without the presence of infinite recursion. The function MaxEvalDepth() is meant for these cases

In> 10 # g(0) <-- 1;
Out> True;
In> 20 # g(n_IsPositiveInteger) <-- \
2 * g(n-1);
Out> True;
In> g(1001);
Error on line 1 in file [CommandLine]
Max evaluation stack depth reached.
Please use MaxEvalDepth to increase the stack
size as needed.
In> MaxEvalDepth(10000);
Out> True;
In> g(1001);
Out> 21430172143725346418968500981200036211228096234
1106721488750077674070210224987224498639675763139171
6255189345835106293650374290571384628087196915514939
7149607869135549648461970842149210124742283755908364
3060929499671638825347975351183310878921541258291423
92955373084335320859663305248773674411336138752;

Hold(expr)¶

keep expression unevaluated

Parameters:	expr – expression to keep unevaluated

The expression expr is returned unevaluated. This is useful to prevent the evaluation of a certain expression in a context in which evaluation normally takes place.

Example:

In> Echo({ Hold(1+1), "=", 1+1 });
1+1 = 2
Out> True;

See also

Eval(), HoldArg(), UnList()

Eval(expr)¶

force evaluation of expression

Parameters:	expr – expression to evaluate

This function explicitly requests an evaluation of the expression expr, and returns the result of this evaluation.

Example:

In> a := x;
Out> x;
In> x := 5;
Out> 5;
In> a;
Out> x;
In> Eval(a);
Out> 5;

The variable a is bound to x, and x is bound to 5. Hence evaluating a will give x. Only when an extra evaluation of a is requested, the value 5 is returned. Note that the behavior would be different if we had exchanged the assignments. If the assignment a := x were given while x had the value 5, the variable a would also get the value 5 because the assignment operator :=() evaluates the right-hand side.

See also

Hold(), HoldArg(), :=()

While(pred) expr¶

loop while a condition is met

Parameters:	pred – predicate deciding whether to keep on looping expr – expression to loop over

Keep on evaluating expr while pred evaluates to True. More precisely, While() evaluates the predicate pred, which should evaluate to either True or False. If the result is True, the expression expr is evaluated and then the predicate pred is evaluated again. If it is still True, the expressions expr and pred are again evaluated and so on until pred evaluates to False. At that point, the loop terminates and While() returns True.

In particular, if pred immediately evaluates to False, the body is never executed. While() is the fundamental looping construct on which all other loop commands are based. It is equivalent to the while command in the programming language C.

Example:

In> x := 0;
Out> 0;
In> While (x! < 10^6) \
[ Echo({x, x!}); x++; ];
0  1
1  1
2  2
3  6
4  24
5  120
6  720
7  5040
8  40320
9  362880
Out> True;

See also

Until(), For()

Until(pred) expr¶

loop until a condition is met

Parameters:	pred – predicate deciding whether to stop expr – expression to loop over

Keep on evaluating expr until pred becomes True. More precisely, Until() first evaluates the expression body. Then the predicate pred is evaluated, which should yield either True or False. In the latter case, the expressions expr and pred are again evaluated and this continues as long as “pred” is False. As soon as pred yields True, the loop terminates and Until() returns True.

The main difference with While() is that Until() always evaluates expr at least once, but While() may not evaluate it at all. Besides, the meaning of the predicate is reversed: While() stops if pred is False while Until() stops if pred is True. The command Until(pred) expr; is equivalent to pred; While(Not pred) body;. In fact, the implementation of Until() is based on the internal command While(). The Until() command can be compared to the do ... while construct in the programming language C.

Example:

In> x := 0;
Out> 0;
In> Until (x! > 10^6) \
[ Echo({x, x!}); x++; ];
0  1
1  1
2  2
3  6
4  24
5  120
6  720
7  5040
8  40320
9  362880
Out> True;

See also

While(), For()

If(pred, then[, else])¶

branch point

Parameters:	pred – predicate to test then – expression to evaluate if `pred` is `True` else – expression to evaluate if `pred` is `False`

This command implements a branch point. The predicate pred is evaluated, which should result in either True or False. In the first case, the expression then is evaluated and returned. If the predicate yields False, the expression else (if present) is evaluated and returned. If there is no else branch, the If() expression returns False.

The sign function is defined to be 1 if its argument is positive and -1 if its argument is negative. A possible implementation is:

In> mysign(x) := If (IsPositiveReal(x), 1, -1);
Out> True;
In> mysign(Pi);
Out> 1;
In> mysign(-2.5);
Out> -1;

Note that this will give incorrect results, if x cannot be numerically approximated:

In> mysign(a);
Out> -1;

Hence a better implementation would be:

In> mysign(_x)_IsNumber(N(x)) <-- If(IsPositiveReal(x), 1, -1);
Out> True;

SystemCall(str)¶

pass a command to the shell

Parameters:	str – the command to call

The command contained in the string str is executed by the underlying operating system. The return value of SystemCall() is True or False according to the exit code of the command.

The SystemCall() function is not allowed in the body of the Secure() command.

In a UNIX environment, the command SystemCall("ls") would print the contents of the current directory:

In> SystemCall("ls")
AUTHORS
COPYING
ChangeLog
... (truncated to save space)
Out> True;

The standard UNIX command test returns success or failure depending on conditions. For example, the following command will check if a directory exists:

In> SystemCall("test -d scripts/")
Out> True;

Check that a file exists:

In> SystemCall("test -f COPYING")
Out> True;
In> SystemCall("test -f nosuchfile.txt")
Out> False;

See also

Secure()

Function() func(args)¶

Function(funcname, {args}) body

declare or define a function

Parameters:	func(args) – function declaration, e.g. `f(x,y)` args – list of atoms, formal arguments to the function body – expression comprising the body of the function

This command can be used to define a new function with named arguments.

The number of arguments of the new function and their names are determined by the list args. If the ellipsis ... follows the last atom in args, a function with a variable number of arguments is declared (using RuleBaseListed()). Note that the ellipsis cannot be the only element of args and must be preceded by an atom.

A function with variable number of arguments can take more arguments than elements in args; in this case, it obtains its last argument as a list containing all extra arguments.

The short form of the Function() call merely declares a RuleBase() for the new function but does not define any function body. This is a convenient shorthand for RuleBase() and RuleBaseListed(), when definitions of the function are to be supplied by rules. If the new function has been already declared with the same number of arguments (with or without variable arguments), Function() returns false and does nothing.

The second, longer form of the Function() call declares a function and also defines a function body. It is equivalent to a single rule such as funcname(_arg1, _arg2) <-- body. The rule will be declared at precedence 1025. Any previous rules associated with funcname (with the same arity) will be discarded. More complicated functions (with more than one body) can be defined by adding more rules.

Example:

This will declare a new function with two or more arguments, but define no rules for it. This is equivalent to RuleBase ("f1", {x, y, ...}):

In> Function() f1(x,y,...);
Out> True;
In> Function() f1(x,y);
Out> False;

This defines a function FirstOf which returns the first element of a list. Equivalent definitions would be FirstOf(_list) <-- list[1] or FirstOf(list) := list[1]:

In> Function("FirstOf", {list})  list[1];
Out> True;
In> FirstOf({a,b,c});
Out> a;

The following function will print all arguments to a string:

In> Function("PrintAll",{x, ...}) If(IsList(x), PrintList(x), ToString()Write(x));
Out> True;
In> PrintAll(1):
Out> " 1";
In> PrintAll(1,2,3);
Out> " 1 2 3";

See also

TemplateFunction(), Rule(), RuleBase(), RuleBaseListed(), :=(), Retract()

Macro() func(args)¶

Macro(funcname, {args}) body

declare or define a macro

Parameters:	func(args) – function declaration, e.g. `f(x,y)` args – list of atoms, formal arguments to the function body – expression comprising the body of the function

This does the same as Function(), but for macros. One can define a macro easily with this function, instead of having to use DefMacroRuleBase().

Example:

The following example defines a looping function

In> Macro("myfor",{init,pred,inc,body}) [@init;While(@pred)[@body;@inc;];True;];
Out> True;
In> a:=10
Out> 10;

Here this new macro myfor is used to loop, using a variable a from the calling environment

In> myfor(i:=1,i<10,i++,Echo(a*i))
10
20
30
40
50
60
70
80
90
Out> True;
In> i
Out> 10;

See also

Function(), DefMacroRuleBase()

For(init, pred, incr) expr¶

C-style for loop

Parameters:	init – expression for performing the initialization pred – predicate deciding whether to continue the loop incr – expression to increment the counter expr – expression to loop over

This commands implements a C style for loop. First of all, the expression init is evaluated. Then the predicate pred is evaluated, which should return True or False. Next, the loop is executed as long as the predicate yields True. One traversal of the loop consists of the subsequent evaluations of expr, incr, and pred. Finally, True is returned.

This command is most often used in a form such as For(i=1, i<=10, i++) expr, which evaluates expr with i subsequently set to 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

The expression For(init, pred, incr) expr is equivalent to init; While(pred) [expr; incr;].

Example:

In> For (i:=1, i<=10, i++) Echo({i, i!});
1
2
6
24
120
720
5040
40320
362880
3628800
Out> True;

See also

While(), Until(), ForEach()

ForEach(var, list) expr¶

loop over all entries in list

Parameters:	var – looping variable list – list of values to assign to `var` expr – expression to evaluate with different values of `var`

The expression expr is evaluated multiple times. The first time, var has the value of the first element of “list”, then it gets the value of the second element and so on. ForEach() returns True.

Example:

In> ForEach(i,{2,3,5,7,11}) Echo({i, i!});
2
6
120
5040
39916800
Out> True;

See also

For()

Apply(fn, arglist)¶

apply a function to arguments

Parameters:	fn – function to apply arglist – list of arguments

This function applies the function “fn” to the arguments in “arglist” and returns the result. The first parameter “fn” can either be a string containing the name of a function or a pure function. Pure functions, modeled after lambda-expressions, have the form “{varlist,body}”, where “varlist” is the list of formal parameters. Upon application, the formal parameters are assigned the values in “arglist” (the second parameter of {Apply}) and the “body” is evaluated.

Another way to define a pure function is with the Lambda construct. Here, in stead of passing in “{varlist,body}”, one can pass in “Lambda(varlist,body)”. Lambda has the advantage that its arguments are not evaluated (using lists can have undesirable effects because lists are evaluated). Lambda can be used everywhere a pure function is expected, in principle, because the function Apply is the only function dealing with pure functions. So all places where a pure function can be passed in will also accept Lambda.

An shorthand for {Apply} is provided by the {@} operator.

Example:

In> Apply("+", {5,9});
Out> 14;
In> Apply({{x,y}, x-y^2}, {Cos(a), Sin(a)});
Out> Cos(a)-Sin(a)^2;
In>  Apply(Lambda({x,y}, x-y^2), {Cos(a), Sin(a)});
Out> Cos(a)-Sin(a)^2
In>  Lambda({x,y}, x-y^2) @ {Cos(a), Sin(a)}
Out> Cos(a)-Sin(a)^2

See also

Map(), MapSingle(), @()

MapArgs(expr, fn)¶

apply a function to all top-level arguments

Parameters:	expr – an expression to work on fn – an operation to perform on each argument

Every top-level argument in expr is substituted by the result of applying fn to this argument. Here fn can be either the name of a function or a pure function (see Apply() for more information on pure functions).

Example:

In> MapArgs(f(x,y,z),"Sin");
Out> f(Sin(x),Sin(y),Sin(z));
In> MapArgs({3,4,5,6}, {{x},x^2});
Out> {9,16,25,36};

See also

MapSingle(), Map(), Apply()

Subst(from, to) expr¶

perform a substitution

Parameters:	from – expression to be substituted to – expression to substitute for “from” expr – expression in which the substitution takes place

This function substitutes every occurrence of from in expr by to. This is a syntactical substitution: only places where from occurs as a subexpression are affected.

Example:

In> Subst(x, Sin(y)) x^2+x+1;
Out> Sin(y)^2+Sin(y)+1;
In> Subst(a+b, x) a+b+c;
Out> x+c;
In> Subst(b+c, x) a+b+c;
Out> a+b+c;

The explanation for the last result is that the expression a+b+c is internally stored as (a+b)+c. Hence a+b is a subexpression, but b+c is not.

See also

WithValue(), /:()

WithValue(var, val, expr)¶

temporary assignment during an evaluation

Parameters:	var – variable to assign to val – value to be assigned to “var” expr – expression to evaluate with “var” equal to “val”

First, the expression “val” is assigned to the variable “var”. Then, the expression “expr” is evaluated and returned. Finally, the assignment is reversed so that the variable “var” has the same value as it had before {WithValue} was evaluated.

The second calling sequence assigns the first element in the list of values to the first element in the list of variables, the second value to the second variable, etc.

Example:

In> WithValue(x, 3, x^2+y^2+1);
Out> y^2+10;
In> WithValue({x,y}, {3,2}, x^2+y^2+1);
Out> 14;

See also

Subst(), /:()

expression /: patterns¶

local simplification rules

Parameters:	expression – an expression patterns – a list of patterns

Sometimes you have an expression, and you want to use specific simplification rules on it that are not done by default. This can be done with the {/:} and the {/::} operators. Suppose we have the expression containing things such as {Ln(a*b)}, and we want to change these into {Ln(a)+Ln(b)}, the easiest way to do this is using the {/:} operator, as follows:

In> Sin(x)*Ln(a*b)
Out> Sin(x)*Ln(a*b);
In> % /: { Ln(_x*_y) <- Ln(x)+Ln(y) }
Out> Sin(x)*(Ln(a)+Ln(b));

A whole list of simplification rules can be built up in the list, and they will be applied to the expression on the left hand side of {/:} .

The forms the patterns can have are one of: ::: pattern <- replacement {pattern,replacement} {pattern,postpredicate,replacement}

Note that for these local rules, {<-} should be used instead of {<–} which would be used in a global rule.

The {/:} operator traverses an expression much as {Subst} does, that is, top down, trying to apply the rules from the beginning of the list of rules to the end of the list of rules. If the rules cannot be applied to an expression, it will try subexpressions of that expression and so on.

It might be necessary sometimes to use the {/::} operator, which repeatedly applies the {/:} operator until the result doesn’t change any more. Caution is required, since rules can contradict each other, which could result in an infinite loop. To detect this situation, just use /: repeatedly on the expression. The repetitive nature should become apparent.

Example:

In> Sin(u)*Ln(a*b) /: {Ln(_x*_y) <- Ln(x)+Ln(y)}
Out> Sin(u)*(Ln(a)+Ln(b));
In> Sin(u)*Ln(a*b) /:: { a <- 2, b <- 3 }
Out> Sin(u)*Ln(6);

See also

Subst()

TraceStack(expression)¶

show calling stack after an error occurs

Parameters:	expression – an expression to evaluate

TraceStack shows the calling stack after an error occurred. It shows the last few items on the stack, not to flood the screen. These are usually the only items of interest on the stack. This is probably by far the most useful debugging function in Yacas. It shows the last few things it did just after an error was generated somewhere.

For each stack frame, it shows if the function evaluated was a built-in function or a user-defined function, and for the user-defined function, the number of the rule it is trying whether it was evaluating the pattern matcher of the rule, or the body code of the rule.

This functionality is not offered by default because it slows down the evaluation code.

Example:

Here is an example of a function calling itself recursively,
causing Yacas to flood its stack:
In> f(x):=f(Sin(x))
Out> True;
In> TraceStack(f(2))
Debug> 982 :  f (Rule # 0 in body)
Debug> 983 :  f (Rule # 0 in body)
Debug> 984 :  f (Rule # 0 in body)
Debug> 985 :  f (Rule # 0 in body)
Debug> 986 :  f (Rule # 0 in body)
Debug> 987 :  f (Rule # 0 in body)
Debug> 988 :  f (Rule # 0 in body)
Debug> 989 :  f (Rule # 0 in body)
Debug> 990 :  f (Rule # 0 in body)
Debug> 991 :  f (Rule # 0 in body)
Debug> 992 :  f (Rule # 0 in body)
Debug> 993 :  f (Rule # 0 in body)
Debug> 994 :  f (Rule # 0 in body)
Debug> 995 :  f (User function)
Debug> 996 :  Sin (Rule # 0 in pattern)
Debug> 997 :  IsList (Internal function)
Error on line 1 in file [CommandLine]
Max evaluation stack depth reached.
Please use MaxEvalDepth to increase the stack
size as needed.

See also

TraceExp(), TraceRule()

TraceExp(expr)¶

evaluate with tracing enabled

Parameters:	expr – expression to trace

The expression “expr” is evaluated with the tracing facility turned on. This means that every subexpression, which is evaluated, is shown before and after evaluation. Before evaluation, it is shown in the form {TrEnter(x)}, where {x} denotes the subexpression being evaluated. After the evaluation the line {TrLeave(x,y)} is printed, where {y} is the result of the evaluation. The indentation shows the nesting level.

Note that this command usually generates huge amounts of output. A more specific form of tracing (eg. {TraceRule}) is probably more useful for all but very simple expressions.

Example:

In> TraceExp(2+3);
TrEnter(2+3);
TrEnter(2);
TrLeave(2, 2);
TrEnter(3);
TrLeave(3, 3);
TrEnter(IsNumber(x));
TrEnter(x);
TrLeave(x, 2);
TrLeave(IsNumber(x),True);
TrEnter(IsNumber(y));
TrEnter(y);
TrLeave(y, 3);
TrLeave(IsNumber(y),True);
TrEnter(True);
TrLeave(True, True);
TrEnter(MathAdd(x,y));
TrEnter(x);
TrLeave(x, 2);
TrEnter(y);
TrLeave(y, 3);
TrLeave(MathAdd(x,y),5);
TrLeave(2+3, 5);
Out> 5;

See also

TraceStack(), TraceRule()

TraceRule(template) expr¶

turn on tracing for a particular function

Parameters:	template – template showing the operator to trace expr – expression to evaluate with tracing on

The tracing facility is turned on for subexpressions of the form “template”, and the expression “expr” is evaluated. The template “template” is an example of the function to trace on. Specifically, all subexpressions with the same top-level operator and arity as “template” are shown. The subexpressions are displayed before (indicated with {TrEnter}) and after ({TrLeave}) evaluation. In between, the arguments are shown before and after evaluation ({TrArg}). Only functions defined in scripts can be traced.

This is useful for tracing a function that is called from within another function. This way you can see how your function behaves in the environment it is used in.

Example:

In> TraceRule(x+y) 2+3*5+4;
TrEnter(2+3*5+4);
TrEnter(2+3*5);
TrArg(2, 2);
TrArg(3*5, 15);
TrLeave(2+3*5, 17);
TrArg(2+3*5, 17);
TrArg(4, 4);
TrLeave(2+3*5+4, 21);
Out> 21;

See also

TraceStack(), TraceExp()

Time(expr)¶

measure the time taken by a function

Parameters:	expr – any expression

The function {Time(expr)} evaluates the expression {expr} and prints the time in seconds needed for the evaluation. The time is printed to the current output stream. The built-in function {GetTime} is used for timing.

The result is the “user time” as reported by the OS, not the real (“wall clock”) time. Therefore, any CPU-intensive processes running alongside Yacas will not significantly affect the result of {Time}.

Example:

In> Time(N(MathLog(1000),40))
0.34 seconds taken
Out> 6.9077552789821370520539743640530926228033;

See also

GetTime()