Calculator variables are really just Lisp variables that contain numbers
or formulas in a form that Calc can understand. The commands in this
section allow you to manipulate variables conveniently. Commands related
to variables use the `s` prefix key.

The `s s` (`calc-store`

) command stores the value at the top of
the stack into a specified variable. It prompts you to enter the
name of the variable. If you press a single digit, the value is stored
immediately in one of the "quick" variables `var-q0`

through
`var-q9`

. Or you can enter any variable name. The prefix ``var-'`
is supplied for you; when a name appears in a formula (as in ``a+q2'`)
the prefix ``var-'` is also supplied there, so normally you can simply
forget about ``var-'` everywhere. Its only purpose is to enable you to
use Calc variables without fear of accidentally clobbering some variable in
another Emacs package. If you really want to store in an arbitrary Lisp
variable, just backspace over the ``var-'`.

The `s s` command leaves the stored value on the stack. There is
also an `s t` (`calc-store-into`

) command, which removes a
value from the stack and stores it in a variable.

If the top of stack value is an equation ``a = 7'` or assignment
``a := 7'` with a variable on the lefthand side, then Calc will
assign that variable with that value by default, i.e., if you type
`s s RET` or

`b`

.)
In fact, the top of stack value can be a vector of equations or assignments with different variables on their lefthand sides; the default will be to store all the variables with their corresponding righthand sides simultaneously.

It is also possible to type an equation or assignment directly at
the prompt for the `s s` or `s t` command: `s s foo = 7`.
In this case the expression to the right of the `=` or `:=`
symbol is evaluated as if by the `=` command, and that value is
stored in the variable. No value is taken from the stack; `s s`
and `s t` are equivalent when used in this way.

The prefix keys `s` and `t` may be followed immediately by a
digit; `s 9` is equivalent to `s s 9`, and `t 9` is
equivalent to `s t 9`. (The `t` prefix is otherwise used
for trail and time/date commands.)

There are also several "arithmetic store" commands. For example,
`s +` removes a value from the stack and adds it to the specified
variable. The other arithmetic stores are `s -`, `s *`, `s /`,
`s ^`, and `s |` (vector concatenation), plus `s n` and
`s &` which negate or invert the value in a variable, and `s [`
and `s ]` which decrease or increase a variable by one.

All the arithmetic stores accept the Inverse prefix to reverse the
order of the operands. If v represents the contents of the
variable, and a is the value drawn from the stack, then regular
`s -` assigns @c{$v \coloneq v - a$}
v := v - a, but `I s -` assigns
v := a - v. While `I s *` might seem pointless, it is
useful if matrix multiplication is involved. Actually, all the
arithmetic stores use formulas designed to behave usefully both
forwards and backwards:

s + v := v + a v := a + v s - v := v - a v := a - v s * v := v * a v := a * v s / v := v / a v := a / v s ^ v := v ^ a v := a ^ v s | v := v | a v := a | v s n v := v / (-1) v := (-1) / v s & v := v ^ (-1) v := (-1) ^ v s [ v := v - 1 v := 1 - v s ] v := v - (-1) v := (-1) - v

In the last four cases, a numeric prefix argument will be used in
place of the number one. (For example, `M-2 s ]` increases
a variable by 2, and `M-2 I s ]` replaces a variable by
minus-two minus the variable.

The first six arithmetic stores can also be typed `s t +`, `s t -`,
etc. The commands `s s +`, `s s -`, and so on are analogous
arithmetic stores that don't remove the value a from the stack.

All arithmetic stores report the new value of the variable in the
Trail for your information. They signal an error if the variable
previously had no stored value. If default simplifications have been
turned off, the arithmetic stores temporarily turn them on for numeric
arguments only (i.e., they temporarily do an `m N` command).
See section Simplification Modes. Large vectors put in the trail by
these commands always use abbreviated (`t .`) mode.

The `s m` command is a general way to adjust a variable's value
using any Calc function. It is a "mapping" command analogous to
`V M`, `V R`, etc. See section Reducing and Mapping Vectors, to see
how to specify a function for a mapping command. Basically,
all you do is type the Calc command key that would invoke that
function normally. For example, `s m n` applies the `n`
key to negate the contents of the variable, so `s m n` is
equivalent to `s n`. Also, `s m Q` takes the square root
of the value stored in a variable, `s m v v` uses `v v` to
reverse the vector stored in the variable, and `s m H I S`
takes the hyperbolic arcsine of the variable contents.

If the mapping function takes two or more arguments, the additional
arguments are taken from the stack; the old value of the variable
is provided as the first argument. Thus `s m -` with a
on the stack computes v - a, just like `s -`. With the
Inverse prefix, the variable's original value becomes the *last*
argument instead of the first. Thus `I s m -` is also
equivalent to `I s -`.

The `s x` (`calc-store-exchange`

) command exchanges the value
of a variable with the value on the top of the stack. Naturally, the
variable must already have a stored value for this to work.

You can type an equation or assignment at the `s x` prompt. The
command `s x a=6` takes no values from the stack; instead, it
pushes the old value of ``a'` on the stack and stores ``a = 6'`.

Until you store something in them, variables are "void," that is, they
contain no value at all. If they appear in an algebraic formula they
will be left alone even if you press `=` (`calc-evaluate`

).
The `s u` (`calc-unstore`

) command returns a variable to the
void state.

The only variables with predefined values are the "special constants"
`pi`

, `e`

, `i`

, `phi`

, and `gamma`

. You are free
to unstore these variables or to store new values into them if you like,
although some of the algebraic-manipulation functions may assume these
variables represent their standard values. Calc displays a warning if
you change the value of one of these variables, or of one of the other
special variables `inf`

, `uinf`

, and `nan`

(which are
normally void).

Note that `var-pi`

doesn't actually have 3.14159265359 stored
in it, but rather a special magic value that evaluates to @c{$\pi$}
pi
at the current precision. Likewise `var-e`

, `var-i`

, and
`var-phi`

evaluate according to the current precision or polar mode.
If you recall a value from `pi`

and store it back, this magic
property will be lost.

The `s c` (`calc-copy-variable`

) command copies the stored
value of one variable to another. It differs from a simple `s r`
followed by an `s t` in two important ways. First, the value never
goes on the stack and thus is never rounded, evaluated, or simplified
in any way; it is not even rounded down to the current precision.
Second, the "magic" contents of a variable like `var-e`

can
be copied into another variable with this command, perhaps because
you need to unstore `var-e`

right now but you wish to put it
back when you're done. The `s c` command is the only way to
manipulate these magic values intact.

The most straightforward way to extract the stored value from a variable
is to use the `s r` (`calc-recall`

) command. This command prompts
for a variable name (similarly to `calc-store`

), looks up the value
of the specified variable, and pushes that value onto the stack. It is
an error to try to recall a void variable.

It is also possible to recall the value from a variable by evaluating a
formula containing that variable. For example, `' a RET =` is
the same as

The `r` prefix may be followed by a digit, so that `r 9` is
equivalent to `s r 9`. (The `r` prefix is otherwise unused
in the current version of Calc.)

The `s e` (`calc-edit-variable`

) command edits the stored
value of a variable without ever putting that value on the stack
or simplifying or evaluating the value. It prompts for the name of
the variable to edit. If the variable has no stored value, the
editing buffer will start out empty. If the editing buffer is
empty when you press `M-# M-#` to finish, the variable will
be made void. See section Editing Stack Entries, for a general
description of editing.

The `s e` command is especially useful for creating and editing
rewrite rules which are stored in variables. Sometimes these rules
contain formulas which must not be evaluated until the rules are
actually used. (For example, they may refer to ``deriv(x,y)'`,
where `x`

will someday become some expression involving `y`

;
if you let Calc evaluate the rule while you are defining it, Calc will
replace ``deriv(x,y)'` with 0 because the formula `x`

does
not itself refer to `y`

.) By contrast, recalling the variable,
editing with ```, and storing will evaluate the variable's value
as a side effect of putting the value on the stack.

There are several special-purpose variable-editing commands that
use the `s` prefix followed by a shifted letter:

`s A`-
Edit
`AlgSimpRules`

. See section Algebraic Simplifications. `s D`-
Edit
`Decls`

. See section Declarations. `s E`-
Edit
`EvalRules`

. See section Default Simplifications. `s F`-
Edit
`FitRules`

. See section Curve Fitting. `s G`-
Edit
`GenCount`

. See section Solving Equations. `s H`-
Edit
`Holidays`

. See section Business Days. `s I`-
Edit
`IntegLimit`

. See section Calculus. `s L`-
Edit
`LineStyles`

. See section Graphics. `s P`-
Edit
`PointStyles`

. See section Graphics. `s R`-
Edit
`PlotRejects`

. See section Graphics. `s T`-
Edit
`TimeZone`

. See section Time Zones. `s U`-
Edit
`Units`

. See section User-Defined Units. `s X`-
Edit
`ExtSimpRules`

. See section "Unsafe" Simplifications.

These commands are just versions of `s e` that use fixed variable
names rather than prompting for the variable name.

The `s p` (`calc-permanent-variable`

) command saves a
variable's value permanently in your ``.emacs'` file, so that its
value will still be available in future Emacs sessions. You can
re-execute `s p` later on to update the saved value, but the
only way to remove a saved variable is to edit your ``.emacs'` file
by hand. (See section General Mode Commands, for a way to tell Calc to
use a different file instead of ``.emacs'`.)

If you do not specify the name of a variable to save (i.e.,
`s p RET`), all

`pi`

, `e`

,
`i`

, `phi`

, and `gamma`

; the variables `TimeZone`

and `PlotRejects`

;
`FitRules`

, `DistribRules`

, and other built-in rewrite
rules; and `PlotData``n`

variables generated
by the graphics commands. (You can still save these variables by
explicitly naming them in an
The `s i` (`calc-insert-variables`

) command writes
the values of all ``var-'` variables into a specified buffer.
The variables are written in the form of Lisp `setq`

commands
which store the values in string form. You can place these commands
in your ``.emacs'` buffer if you wish, though in this case it
would be easier to use `s p RET`. (Note that

If you have an expression like ``a+b^2'` on the stack and you wish to
compute its value where b=3, you can simply store 3 in b and
then press `=` to reevaluate the formula. This has the side-effect
of leaving the stored value of 3 in b for future operations.

The `s l` (`calc-let`

) command evaluates a formula under a
*temporary* assignment of a variable. It stores the value on the
top of the stack into the specified variable, then evaluates the
second-to-top stack entry, then restores the original value (or lack of one)
in the variable. Thus after `' a+b^2 RET 3 s l b RET`,
the stack will contain the formula

The value on the top of the stack may be an equation or assignment, or
a vector of equations or assignments, in which case the default will be
analogous to the case of `s t RET`. See section Storing Variables.

Also, you can answer the variable-name prompt with an equation or
assignment: `s l b=3 RET` is the same as storing 3 on the stack
and typing `s l b RET`.

The `a b` (`calc-substitute`

) command is another way to substitute
a variable with a value in a formula. It does an actual substitution
rather than temporarily assigning the variable and evaluating. For
example, letting n=2 in ``f(n pi)'` with `a b` will
produce ``f(2 pi)'`, whereas `s l` would give ``f(6.28)'`
since the evaluation step will also evaluate `pi`

.

The special algebraic symbol ``=>'` is known as the **evaluates-to
operator**. (It will show up as an `evalto`

function call in
other language modes like Pascal and TeX.) This is a binary
operator, that is, it has a lefthand and a righthand argument,
although it can be entered with the righthand argument omitted.

A formula like `` a => b'` is evaluated by Calc as
follows: First,

For example, suppose you enter the algebraic formula ``2 + 3 => 17'`.
The number 17 is ignored, and the lefthand argument is left in its
unevaluated form; the result is the formula ``2 + 3 => 5'`.

You can enter an ``=>'` formula either directly using algebraic
entry (in which case the righthand side may be omitted since it is
going to be replaced right away anyhow), or by using the `s =`
(`calc-evalto`

) command, which takes `a` from the stack
and replaces it with `` a => b'`.

Calc keeps track of all ``=>'` operators on the stack, and
recomputes them whenever anything changes that might affect their
values, i.e., a mode setting or variable value. This occurs only
if the ``=>'` operator is at the top level of the formula, or
if it is part of a top-level vector. In other words, pushing
``2 + (a => 17)'` will change the 17 to the actual value of
``a'` when you enter the formula, but the result will not be
dynamically updated when ``a'` is changed later because the
``=>'` operator is buried inside a sum. However, a vector
of ``=>'` operators will be recomputed, since it is convenient
to push a vector like ``[a =>, b =>, c =>]'` on the stack to
make a concise display of all the variables in your problem.
(Another way to do this would be to use ``[a, b, c] =>'`,
which provides a slightly different format of display. You
can use whichever you find easiest to read.)

The `m C` (`calc-auto-recompute`

) command allows you to
turn this automatic recomputation on or off. If you turn
recomputation off, you must explicitly recompute an ``=>'`
operator on the stack in one of the usual ways, such as by
pressing `=`. Turning recomputation off temporarily can save
a lot of time if you will be changing several modes or variables
before you look at the ``=>'` entries again.

Most commands are not especially useful with ``=>'` operators
as arguments. For example, given ``x + 2 => 17'`, it won't
work to type `1 +` to get ``x + 3 => 18'`. If you want
to operate on the lefthand side of the ``=>'` operator on
the top of the stack, type `j 1` (that's the digit "one")
to select the lefthand side, execute your commands, then type
`j u` to unselect.

All current modes apply when an ``=>'` operator is computed,
including the current simplification mode. Recall that the
formula ``x + y + x'` is not handled by Calc's default
simplifications, but the `a s` command will reduce it to
the simpler form ``y + 2 x'`. You can also type `m A`
to enable an algebraic-simplification mode in which the
equivalent of `a s` is used on all of Calc's results.
If you enter ``x + y + x =>'` normally, the result will
be ``x + y + x => x + y + x'`. If you change to
algebraic-simplification mode, the result will be
``x + y + x => y + 2 x'`. However, just pressing `a s`
once will have no effect on ``x + y + x => x + y + x'`,
because the righthand side depends only on the lefthand side
and the current mode settings, and the lefthand side is not
affected by commands like `a s`.

The "let" command (`s l`) has an interesting interaction
with the ``=>'` operator. The `s l` command evaluates the
second-to-top stack entry with the top stack entry supplying
a temporary value for a given variable. As you might expect,
if that stack entry is an ``=>'` operator its righthand
side will temporarily show this value for the variable. In
fact, all ``=>'`s on the stack will be updated if they refer
to that variable. But this change is temporary in the sense
that the next command that causes Calc to look at those stack
entries will make them revert to the old variable value.

2: a => a 2: a => 17 2: a => a 1: a + 1 => a + 1 1: a + 1 => 18 1: a + 1 => a + 1 . . . 17 s l a RET p 8 RET

Here the `p 8` command changes the current precision,
thus causing the ``=>'` forms to be recomputed after the
influence of the "let" is gone. The `d SPC` command
(`calc-refresh`

) is a handy way to force the ``=>'`
operators on the stack to be recomputed without any other
side effects.

Embedded Mode also uses ``=>'` operators. In embedded mode,
the lefthand side of an ``=>'` operator can refer to variables
assigned elsewhere in the file by ``:='` operators. The
assignment operator ``a := 17'` does not actually do anything
by itself. But Embedded Mode recognizes it and marks it as a sort
of file-local definition of the variable. You can enter ``:='`
operators in algebraic mode, or by using the `s :`
(`calc-assign`

) [`assign`

] command which takes a variable
and value from the stack and replaces them with an assignment.

See section TeX Language Mode, for the way ``=>'` appears in
TeX language output. The **eqn** mode gives similar
treatment to ``=>'`.

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