**Disclaimer:** This calculator is not very efficient
and indeed rather slow.
It has in fact been designed intentionally to be
used in an undergraduate computer laboratory where a
large number of people are working, and where
speed is less important than politeness.
But in any event, this calculator is intended to
demonstrate basic principles of mathematical algorithms,
not to make complicated practical calculations.
Have patience with it. Even a calculator
as simple as this one can make
manipulations with vectors much more pleasant and rapid than
can an average hand-held calculator.

Warning!
*If you edit the calculator and then
exit the page, reload it, or resize it,
your text will be replaced by the original.*

Please!
I have tried very hard to make this calculator foolproof
and bug-free, and to give an intuitive interface,
but of course I cannot guarantee anything.
*If you encounter bizarre behaviour of any kind, please
report it to me,
explaining in as much detail as you can what the circumstances were.*

Clicking in the program window will always reset
the calculator, too - the calculator assumes that a click
in this window means
you are about to change the program.
In a separate window
labelled **Stack** the full stack is displayed
(upside down). You can toggle the display of
this window on and off by pressing the **Stack** button.
Similarly, you can toggle the display of a graphics window with
the **Graphics** button.

There is one extra quirk to be patient with, and that is that the cursor doesn't seem to appear regularly in the source window, even though it is still apparently functioning. Dunno what to do about this problem.

**Normally,
results are not displayed when calculated**.
If you want to display them, you can
type **=**, which will display the item at the top of
the stack without removing it.
If you use **!** instead you will both
display and remove it. (So there are two ways to
output the item at the top of the stack:
**!** which is destructive and **=**
which is non-destructive.)
Thus you would type **6 7 + =** to calculate
6+7=13 and display the result, leaving it on the stack.

The `backwards' behaviour of the calculator may seem peculiar at first, but it is extremely efficient in a chain of complicated calculations, and you should get comfortable with it in time.

**Input:**

6 7 8 * + !

**Output:**

62

**Remark:**
What is going on here inside the stack?
First we enter 6, 7, and 8.
At this point the stack has three items on it.
Then we replace 7 and 8 by 7*8=56, leaving 6 and 56 on the stack.
Finally, we replace 6 and 56 by 6+56 = 62
and display the result destructively.
At the end, the stack is empty.
Here is a sequence of pictures of the stack as the calculation proceeds:

6 6 7 6 7 8 6 56 62

**Task: Calculate the sum of vectors [1 2] and [1 -3].**

**Input:**

[ 1 2 ][ 1 -3 ] + =

**Output:**

[ 2 -1 ]

**Remark:** Here the result is left on the stack.

**Task: Define variables x = 6, y = 7, set z = x + y,
and display z.**

**Input:**

6 @x def 7 @y def x y + @z def z !

**Output:**

13

**Task:
Calculate and display
the sum of the first 10 squares 1 + 4 + 9 + ...**

**Input:**

0 @n def 0 @s def 10 { 1 n + @n def n dup * s + @s def } repeat s !

**Output:**

385

**Task: Construct a procedure called
average which has just one argument, a vector, and returns
the average of its coordinates.
**

**Input:**

{ @v def v dim @n def 0 @s def 0 @i def n { v(i) s + @s def i 1 + @i def } repeat s n / } @average def

**Remark:** This is not as efficient as it might be.
Cleverer stack manipulations could do better.
Note that **v(i)** is
the **i**-th coordinate of **v** if **v** is
a vector.

**Task: Calculate numerically the integral
of y = exp(-x^2) from
0 to 1 by applying the trapezoidal rule with 10 intervals.
**

**Input:**

# Define the function to be integrated # Here f(x) = exp(-x*x) { dup * -1 * exp } @f def # Do the sums for the trapezoidal rule # Each term = (f(x) + f(x+h))*0.5*h 10 @N def 0 @x def 1 N / @h def 0 @s def N { x f x h + @x def x f + 0.5 * h * s + @s def } repeat # display the result s !

**Output:**

0.7462

**Remark:**
This is more complicated than other examples.
First we define the variable f to be the **procedure**
or function which takes the variable
x off the stack and then places exp(-x*x) on the stack.
Just to be sure you get the point, I'll
repeat it: **you can define variables to be equal
to procedures as well as ordinary constants**.
And almost always functions defined in the calculator
will do something like this one - remove
some items on the stack as its arguments, and place
something on the stack as its return value.
Incidentally, the command **dup** used here just
makes an extra copy of what is on top of the stack.
Also, this function is not as efficient as it might be.
With a little care you can get away with
only one function evaluation in each loop.

**Task: Construct a function which takes
a single argument which is a vector, and returns its length.
**

Left as an exercise.

**Task: Construct a function which takes
two arguments which are vectors and returns the angle between them.
**

I'll leave this as an exercise, too. It will use ***** to calculate
the dot product of two vectors, the **length** function from the
previous exercise, and the function **acos** (inverse cosine).
You'll have to recall a formula from linear algebra relating
the dot product to angles.

**Task: Evaluate the polynomial
z^{2} + z + 1 for 10 values of the complex variable z
distributed uniformly on the circle |z| = 1, and display the values
you get.
**

{ local @z def z z * z + 1 + endlocal } @poly def 0 @T def 2 pi * 10 / @dT def 10 { < T cos T sin > poly ! T dT + @T def } repeat

**Remark:**
The nested pair **local**, **endlocal**
arranges things so that all variables in between are local,
which means in this case that
all assignments to *z* within the procedure **poly** do not
affect the values of any other variable *z* used outside the procedure.
Complex numbers are expressed as a pair of real numbers betwen
angle brackets like **< 1 2 >** for *1 + 2i*.
The variable **i** is defined by default to
be the square root of *-1*, so you can
also enter *1 + 2i* as **1 2 i * +**.

An implementation of Newton's method

A printable version of this file

A dictionary of calculator commands

A more efficient version can be run on any computer with
a Java interpreter installed. If you install your own copy,
and you have installed Java on
your computer (which you can do without cost
through Sun Microsystems' home page)
then you have another option.
If **java** is in your execution
path and the directory above `rpn`
is in your Java class path, you can run the calculator
through standard input in any UNIX terminal or MSDOS window by
typing `java rpn.vc.vc`.
Typing `java rpn.vc.vc x`
will run the calculator with the file **x** as input.
You can also run the file ca.html under the
program **appletviewer**.

The calculator applet and this page were constructed by Bill Casselman.