The changing face of Mathcad
Jack Crenshaw - May 27, 2011
Was someone at PTC listening to Jack when they created Mathcad Prime 1.0?
Editor's note: Due to unforeseeable circumstances (including a sick computer), Jack Crenshaw will
finish writing about the Kalman filter in upcoming months. Stay tuned.
Click image to go
to digital edition.
There's some exciting news about a product I've lived with—and sometimes praised, sometimes
cursed—for decades: PTC's premier math analysis tool, Mathcad.
Regular readers know that most of my columns are math-oriented, as is most of my work. For that
reason, I'm a heavy user of Mathcad. For more than 15 years, I've maintained a love-hate
relationship with Mathcad. I've mentioned it many times in my columns, in terms ranging from
wholehearted endorsements to heated, curmudgeonly rants. Although I often curse Mathcad as I use
it, the reality is it remains, by far, the most oft-used tool in my toolbox.
Why do I continue to use Mathcad when it frustrates me so? To quote the old joke, it's the only game
in town. Oh, yes, other math-analysis tools exist—some with more impressive resumes, more
respected pedigrees, and much higher prices. But none of them do quite what I want done. For me,
Mathcad stands alone. The reasons are part of this story.
I've not told just you readers my beefs about Mathcad. I've also communicated them to Mathsoft and
now PTC. I've talked to people from technical support—very excellent people, by the way—to vice
presidents. I've posted my opinions on their web sites and blogs. I was a long-time member of their
"Power Users' Group," and I still post from time to time in the Mathcad Community site. I've given
them so much "feedback," I think they must surely cringe when they hear my cybernetic footsteps.
Do you get the idea that I am not among Mathcad's biggest fans? Then what I have to tell you next
should carry all the more weight.
PTC has recently announced a new version of Mathcad. It's called Mathcad Prime 1.0, and it
represents a bold and dramatic departure from older versions.
No, belay that. Prime 1.0 isn't just yet another version, the 16th of 16. It's a brand new product,
written on a blank sheet of paper and built from scratch to new specifications. You have to admire
the courage of a company willing to invest the time and money in these precarious times.
More importantly for me, the user interface—the part that elicited most of my rants—is radically
new. I don't deceive myself that PTC made the changes just for me, but it's as though they were
reading my mail (hey, maybe they were). As near as I can tell, every one of the idiosyncrasies I used
to fuss about has not just been changed, but changed in just the way I'd hoped for, It's an awesome
turn of events, and I'm very excited about it. Can you tell?
The back story
What is a math aid, anyhow? Well, it's something that makes doing math easier. Until the advent of
digital computers, we all used to do math the same way the Sumerians and Egyptians did it: First
with stylus and clay, then quill and papyrus, later pencil and paper. What alternatives did we have?
When you're doing long math calculations, it's easy to get something wrong. The great astronomer,
Ernest William Brown, developed a theory for the motion of the Moon, in terms of a trigonometric
series with some 1,500 terms. Later, it was discovered that he'd made an error in the 500th term,
give or take, rendering the next 1,000 terms useless.
My own favorite errors were stupid ones; getting a sign wrong, or leaving out a factor of 2. I often
wished I could hire an apprentice, whose only job would be to check my math and catch those
errors. Or, better yet, do the tedious, turn-the-crank manipulations for me. Just say, "start with these
equations, and solve for x." The ideal apprentice would know the rules of algebra, trig, and calculus.
Such an idea became practical with the advent of digital computers. Computers don't just crunch
numbers. They can also deal with concepts. The rules of algebra, operator precedence, and so forth,
are well known and never change. Ditto for the rules of calculus and trig identities. If we can explain
the rules, a computer program can do all the tedious work. And it won't get the sign wrong.
One of the earliest math analysis tools was Macsyma, developed at MIT and refined in various
commercial forms. Versions of Macsyma, highly refined, are still with us today.
You must understand: in those days, the acronym WYSIWYG wasn't even on the radar. We gave
computers data by punching text into IBM cards or paper tape, and they gave stuff back by printing
them on a chain printer (very fast, very loud). In a real sense, a computer program only changed one
text string to another. There was no hope to input formulas as they'd look in a math textbook. If you
wanted to write an equation like the quadratic formula:
(1)
You had to convert it to its equivalent Fortran form:
X1 = (-B + SQRT(B**2 - 4*A*C)/(2*A)
X2 = (-B - SQRT(B**2 - 4*A*C)/(2*A)
Macsyma's forte' was symbolic processing. That is, it could take an expression and transform it into
an equivalent one, using the rules of algebra, trig, and calculus. Macsyma was, in fact, one of the
very first knowledge-based systems. But it could only operate on Fortran inputs, and it could only
give back more of the same. One of its most important applications was in taking the calculus
derivatives of expressions.
I once got to see Macsyma used—or misused. A company building robotics controllers was writing
new embedded software, and they needed to implement the rather complicated vector/matrix math
and Euler angle rotations. The project lead had a most boneheaded idea. Arguing that calling
functions and subroutines would be "too inefficient," he decided to get Macsyma to expand them all
into scalar equations involving a whole lot of trig functions. He chose Macsyma to expand the
equations into Fortran. Pages and pages of it. Then he turned the Fortran over to a bunch of newhire programmers, trained in C. Their job: To turn the Fortran into yet another language: Pascal.
Think that was efficient?
Most computer programs don't yet have the intelligence to say, "You want me to do WHAT?"
Maple and friends
The folks at Waterloo University in Waterloo, Ontario, Canada enjoy a long and honored reputation
in the fields of language translation and symbolic computations. In the 1960s, computer science
departments all over the world were hurting because of all the students running Fortran jobs for
class credit. What the world needed, the educators pleaded, was a Fortran compiler system that
would read the students' programs, compile, link, and run them, all in one batch process. IBM
experts said building such a system was impractical, if not impossible.
Someone forgot to tell the folks at Waterloo, so they just went and built it. They called it Watfor.
A similar situation came up around 1980. The story goes that the Waterloo folks wanted to run
Macsyma, but couldn't afford the large mainframe computer it required. So they wrote a symbolic
engine of their own. They called it Maple. Over the years, Maple has continued to evolve and is
generally considered to be one of the premier symbolic engines of all time. It's commercially
available today, from Maplesoft.
The printed page
While we technical types were learning to computerize the processing of math equations, another
group had been routinely creating more beautiful equations for centuries. These were the book
publishers, who published, among other things, the math textbooks from which we learned our craft.
It was only natural that typesetting systems evolved into computer systems capable of driving
automated typesetting machinery.
Generating print for textbooks is one thing. For a single scientist or small group, generating it for
published papers was quite another. I well remember serious technical papers written on a
typewriter, with all the math equations written by hand. Or, not much better, generated by a typist
skilled in typing subscripts, superscripts, and Greek letters. It was only natural that we average
computer users would yearn to be able to generate book-quality print and equations, from a
computer. In other words, let the text output of a general-purpose computer be the text input to a
typesetting system. One of the early typesetting systems was the Unix program, troff. Another was
Donald Knuth's justly famous TEX, developed ca. 1980. To see the quadratic formula written in TEX,
go to http://en.wikipedia.org/wiki/TeX.
The problem with troff and TEX was that you couldn't actually see their output until it went to the
typesetter or a pixel-capable printer. The input itself was still a mere text string.
WYSIWYG changes everything
All that changed when Apple introduced its ground-breaking MacIntosh computer, with its highresolution pixel graphics and a laser printer to match. Suddenly, an affordable personal computer
could generate output rivaling the book publishers.
Seemingly the next day, Microsoft came out with Word for the Mac, and we learned that we no
longer had to deal with Courier-in, Courier-out text. We could see on the screen the printed page
exactly as it would appear. WYSIWYG was upon us with a vengeance, and we'd never be satisfied
with pure text editing again.
If we could create text and graphics on a personal computer, it was only natural that we'd want to
create math equations as well. Microsoft came through again, with its Microsoft Equation Editor,
based on Design Sciences very excellent MathType. It's Word with MathType that I use for all my
columns you've seen.
Tying things together
As nice as the output of WYSIWYG editors with equation editing was, it was still completely dumb.
Mathtype, like troff and TEX, has no smarts behind it. You have to put the symbols where you want
them, one by one. Symbolic math processing remained a largely text-only operation. Math aids like
MacSyma, Maple, and Mathematica (loosely derived from MacSyma) still used ordinary text strings
for input and output. Mathematica at least made an attempt to pretty-print math equations, by
simply printing their parts on separate lines. But the results were a far cry from book-publishable.
The next step is obvious: Give me a math analysis program that lets me input my equations in a form
that looks just like a book publisher might typeset it, and give me the results the same way.
This is where Mathcad comes in. To my knowledge, it was the very first personal computer program
capable of accepting symbolic inputs in natural mathematical form, performing operations on them
symbolically, and showing the results in the same form. As far as I'm concern, they still remain
alone.
In fairness, Maple too has evolved over the years, and later versions do indeed support mathematical
forms for both inputs and outputs. But, for me at least, the text-oriented nature underpinning Maple
still shows through. The math expressions are basically sleight-of-hand tricks to hide its inherent
nature.
A lot of my work involves deriving the math equations associated with new problems and
documenting the results. For me, it's important that I be able to intermix text with math equations,
as well as the numerical and symbolic computations associated with them. Only Mathcad can do this
seamlessly.
Which is why I use it.
How hard can it be?
While the ability to input equations in WYSIWYG form is Mathcad's strength, you should never get
the idea that it's easy to do. It's not, and that's because the most convenient input device remains
the keyboard. So how do you get the computer to accept text inputs and display the equivalent
WYSIWYG form? Granted, I should be able to type a "Fortranish" character string like:
"x = a+b+2*y"
And get:
x=a+b+2 • y
(2)
But try to get fancier, and things start to go south in a hurry. Type the harmonic series:
"x = 1+1/2+1/3+1/4+…"
And you might be surprised by the result. It'll be:
(3)
Mathcad understands operator precedence but can't possibly know your intentions unless you tell it.
Think about it for awhile, and you'll realize that you can't possibly avoid the need to navigate around
inside an equation, selecting the place where you want to put the next symbol. To do it, you're going
to need to use the navigation keys, plus some special keys for things like superscripts, subscripts,
square root symbols, matrix operations, and the like.
It's this navigation that's the hard part. It gets even harder when you're not just inputting an
expression from scratch, but modifying an existing one. Or, worse yet, editing the output of the
symbolic processor. Over the years, the Mathcad folks have tried several schemes, each with their
pros and cons.
This is the big area of change in Mathcad Prime 1.0. They've completely replaced the user interface,
including the rules for navigating inside expressions. It's the area of the program that elicited by far
my most vociferous comments and under-the-breath curses. And the new interface is exactly what I
had hoped for. It's a dramatic change, and it's the reason I'm so excited about Mathcad 1.0 Prime.
Still incomplete
Now for the bad news: Mathcad Prime 1.0 is still not complete. In developing it, PTC
decided—wisely, in my opinion—not to put things in until they were robust and fully refactored. So
key parts of earlier versions are still missing, notably the symbolic processing.
What's that you say? A symbolic processing program that can't do symbolic processing? Well, don't
forget that symbolics is far from Mathcad's only function. It can also do ordinary numerical
computations, vector/matrix operations, graphing, and much more, just as Mathematica, Maple,
Matlab, and others do. Plus a bewildering array of special functions and numerical methods. It's only
the connection with the symbolic engine that's missing.
To complete the picture, PTC has a plan. First, the purchase price of Prime 1.0 includes a
maintenance contract, providing free upgrades for a year. They expect that, long before the year is
out, the missing features will be included. To cover the missing features in the interim, they're
bundling Prime 1.0 with a new version, Version 15, of the old Mathcad.
Can you tell that I'm excited? I think it's a great plan, and a great product. If you have a need for
math processing, I think you should take a hard look at Mathcad Prime 1.0. If you've been avoiding
Mathcad because of its steep learning curve or my dire warnings, it's time to take another look.
Jack Crenshaw is a systems engineer and the author of Math Toolkit for Real-Time Programming.
He holds a PhD in physics from Auburn University. E-mail him at [email protected].