The ACM Special Interest Group on Programming Languages, SIGPLAN, expects to hold the fourth in a series of conferences on the History of Programming Languages in 2020, see HOPL-IV. The first drafts of papers are to be submitted by August, 2018. That long lead time gives me the opportunity to write a detailed history of MATLAB. I plan to write the paper in sections, which I'll post in this blog as they are available.
This is the fourth such installment. MATLAB takes the first steps from a primitive matrix calculator to a full-fledged technical computing environment.
I spent the 1979-80 academic year at Stanford, as a Visiting Professor of Computer Science. In the fall I taught CS237a, the graduate course in Numerical Analysis. I introduced the class to the matrix calculator that I called Historic MATLAB in my previous post about MATLAB history.
There were maybe fifteen to twenty students in the class. About half of them were Math and CS students. As I remember, they were not impressed with MATLAB. It was not a particularly sophisticated programming language. It was not numerical analysis research.
But the other half of the students were from engineering at Stanford. They loved MATLAB. They were studying subjects like control theory and signal processing that, back then, I knew nothing about. Matrices were a central part of the mathematics in these subjects. The students had been doing small matrix problems by hand and larger problems by writing Fortran programs. MATLAB was immediately useful.
Jack Little studied electrical engineering at MIT in the late 1970's and then came west to Stanford for grad school. He didn't take my CS237 course, but a friend of his did. The friend showed him MATLAB and Little immediately adopted it for his own work in control systems and signal processing.
In 1983 Little suggested the creation of a commercial product based on MATLAB. I said I thought that was a good idea, but I didn't join him initially. The IBM PC had been introduced only two years earlier and was barely powerful enough to run something like MATLAB, but Little anticipated its evolution. He left his job, bought a Compaq PC clone at Sears, moved into the hills behind Stanford, and, with my encouragement, spent a year and a half creating a new and extended version of MATLAB written in C. A friend, Steve Bangert, joined the project and worked on the new MATLAB in his spare time.
The three of us -- Little, Bangert, and myself -- formed MathWorks in California in 1984. PC-MATLAB made its debut in December 1984 at the IEEE Conference on Decision and Control in Las Vegas.
Little and Bangert made many important modifications and improvements to Historic MATLAB when they created the new and extended version. The most significant were functions, toolboxes and graphics
The MATLAB function naming mechanism uses the underlying computer file system. A script or function is a body of MATLAB code stored in a file with the extension .m. If the file begins with the keyword function, then it is a function, otherwise it is a script. The name of the file, minus the .m, is the name of the script or function. For example, a file named hello.m containing the single line
is a MATLAB script. Typing the file name without the .m extension at the command line produces the traditional greeting.
Functions usually have input and output arguments. Here is a simple example that uses a vector outer product to generate the multiplication table you memorized in elementary school
function T = mults(n) j = 1:n; T = j'*j; end
This statement produces a 10-by-10 multiplication table.
T = mults(10)
T = 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100
Input arguments to MATLAB functions are "passed by value" or, more precisely, passed by "copy on write". It is not necessary to make copy of an input argument if the function does not change it. For example, the function
function C = commuator(A,B) C = A*B - B*A; end
does not change A or B, so it does not copy either.
Toolboxes are collections of functions written in MATLAB that can be read, examined, and possibly modified, by users. The matlab toolbox, which is an essential part of every installation, has several dozen functions that provide capabilities beyond the built-in functions in the core.
Other toolboxes are available from MathWorks and many other sources. The first two specialized toolboxes were control and signal.
Technical computer graphics have been an essential part of MATLAB since the first MathWorks version. Let's look at two examples from that first version. Notice that neither one has anything to do with numerical linear algebra, but they both rely on vector notation and array operations. One demo was called simply wow.
% Lets define a peculiar vector, and then plot % a growing cosine wave versus a growing sine: t = 0:(.99*pi/2):500; x = t.*cos(t); y = t.*sin(t); plot(x,y,'k') axis square
The second example investigates the Gibbs phenomena by plotting the successive partial sums of the Fourier series for a square wave.
% For a finale, we can go from the fundamental to the 19th harmonic % and create vectors of successively more harmonics, saving all % intermediate steps as rows in a matrix. t = 0:pi/128:pi; y = zeros(10,max(size(t))); x = sin(t); y(1,:) = x; i = 1; for k = 3:2:19 i = i + 1; x = x + sin(k*t)/k; y(i,:) = x; end % We can plot these successively on an overplot, showing the transition % to a square wave. Note that Gibb's effect says that you will never % really get there. % Or we can plot this as a 3-d mesh surface. mesh(y) colormap([0 0 0]) % Black and white. axis tight axis off view([-27 39])
I cannot find a disc for the very first version of PC-MATLAB. The oldest I can find is version 1.3. Here are all the functions, key words, and operators in that version.
Information and assistance:
help - help facility (help intro, help index) demo - run demonstrations browse - browse utility who - lists current variables in memory what - lists .M files on disk dir - directory list of files on disk casesen- turn off case sensitivity format - set output format pack - memory garbage collection and compaction edit - invoke editor
Attributes and Array Manipulation:
max - maximum value in vector min - minimum value in vector sum - sum of elements in vector prod - product of elements in vector cumsum - cumulative sum of elements in vector cumprod - cumulative product of elements in vector mean - mean value in a vector median - median value in a vector std - standard deviation in a vector size - row and column dimensions of an array length - length of a vector : - array subscripting, selection, and removing sort - sorting find - find array indices of logical values hist - histograms
Array Building Functions:
diag - creates diagonal arrays, removes diagonal vectors eye - creates identity matrices ones - generates an array of all ones zeros - generates an array of all zeros rand - random numbers and arrays magic - generates a magic square tril - lower triangular part triu - upper triangular part hilb - generates Hilbert matrix invhilb - generates inverse Hilbert matrix
cond - condition number in 2-norm det - determinant norm - 1-norm, 2-norm, F-norm, infinity norm rank - rank rcond - condition estimate
inv - inverse expm - matrix exponential logm - matrix logarithm sqrtm - matrix square root funm - arbitrary matrix functions pinv - pseudoinverse poly - characteristic polynomial kron - Kronecker tensor product
Matrix Decompositions and Factorizations:
chol - Cholesky factorization eig - eigenvalues and eigenvectors hess - Hessenberg form lu - factors from Gaussian elimination orth - orthogonalization qr - orthogonal-triangular decomposition schur - Schur decomposition svd - singular value decomposition
poly - characteristic polynomial roots - find polynomial roots polyval - evaluate polynomial conv - multiplication
filter - digital filter fft - fast Fourier transform ifft - inverse fast Fourier transform conv - convolution length - length of a sequence
ans - answer when expression is not assigned eps - machine epsilon, floating point relative accuracy pi - 3.14159265358979323846264338327950288419716939937511... inf - infinity nan - Not-a-Number nargin - number of input arguments in a function nargout - number of output arguments in a function
if s conditionally execute statements elseif s else end for i=v,.. end repeat statements a specific number of times while s,..,end do while break break out of a for or while loop return return from function
Graphics and Plotting:
shg - show graphics screen cla - clear alpha screen clg - clear graphics screen plot - linear X-Y plot loglog - loglog X-Y plot semilogx - semi-log X-Y plot semilogy - semi-log X-Y plot polar - polar plot mesh - 3-dimensional mesh surface title - plot title xlabel - x-axis label ylabel - y-axis label grid - draw grid lines axis - manual axis scaling print - hardcopy disp - compact matrix display
User-defined Functions, Procedures, and Programming:
type - list a Function or file what - lists .M files on disk input - get a number from the user pause - pause until <cr> entered return - return from a user defined Function eval - interpret text in a variable setstr - set flag indicating matrix is a string echo - enables command echoing nargin - number of input arguments in a function nargout - number of output arguments in a function exist - checks if a variable or function exists
diary - diary of the session in a disk file load - loads variables from a file save - saves variables on a file type - list a file dir - directory list of files on disk delete - delete files chdir - change working directory
Elementary Math Functions:
abs - absolute value or complex magnitude conj - complex conjugate fix - round towards zero imag - imaginary part real - real part round - round to nearest integer sign - signum function sqrt - square root rem - remainder sin, cos, tan asin, acos, atan atan2 - four quadrant arctangent exp - exponential base e log - natural logarithm log10 - log base 10 bessel - Bessel functions rat - rational number approximation
= - assignment statement [ - used to form vectors and matrices ] - see [ ( - arithmetic expression precedence ) - see ( . - Decimal point .. - continue a statement to the next line , - separates matrix subscripts and function arguments ; - Ends rows. Suppresses the display of a variable % - comment statement : - subscripting, vector generation, data selection ' - transpose, string delimiter
Element-by-element arithmetic operators:
+ - addition - - subtraction .* - multiplication ./ - division .\ - left division .^ - raise to a power .' - transpose
* - multiplication / - division \ - left division ^ - raise to a power ' - matrix conjugate transpose
< - less than <= - less than or equals > - greater than >= - greater than or equals == - equals ~= - not equals
+ - OR .* - AND ~ - NOT (complement)
Get the MATLAB code
Published with MATLAB® R2018a
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Dear Mr. Moler,
I first encounter MATLAB in September 1993 (then it needed a mathematical co-processor – if I remember right…), at my graduate studies in Electrical Engineering (Electronics), on the third year. Before that, I heard something about it from a bit older students, how great it is, and they use it in DSP – among other things, etc… When I tried it, I was literally blown away! If someone ever tried to do something pretty simple (by hand) – as to find an inverse matrix, he or she have saw that it is a pretty tedious work, and there is always (!) a high possibility to make an small or big error in calculation. If someone done it for a square matrix, let’s say up to order 7-8, You still have a chance to do it correctly. BUT, if You try some “incredible” orders, like 20,30 or more, there is practically no way You can compute it effectively and correctly for a frequent projects/use. There come’s in MATLAB. What I was doing for hours, MATLAB solved in couple of milliseconds! It was unbelievable when I saw it. And, all that in a user friendly environment, with logical commands… Incredible. Not to mention solving (even more in later editions of MATLAB…) non-linear differential equation, or partial differential equations, d.e. unboundary solutions, etc… You need to spend hours, sometimes days – if You cannot figure out the solution – for some/whatever reason (or You make a not so obvious mistake – at least speaking in my name…), and MATLAB does it in a matter of few seconds! Please, for those who don’t know: then, at the time, 25 or more years ago, there wasn’t internet, You “dig” Your answers and knowledge from many books in library, looking for an appropriate solution – or method to solve a problem, keep trying and looking for solution, and for that – You need time, lots of time (often the whole night till the morning hours…), and the result is always questionable – is it correctly defined and calculated…etc…
I strongly believe, that MATLAB has accelerated the piece of science in the world (!) by a great, great amount, not to mention that it was a pivot – a example project, for other similar programs that came latter, but those other programs never (!) were so efficient, fast and user friendly as MATLAB is/was… And, I mentioned a tiny, tiny fraction of fractions what MATLAB is capable of doing…
For me personally and professionally, it is, and it will always be: MATLAB is the “eight” wonder of the world!
A, great and respectful – Thank You, in name of humanity and science.
Showing my age, I remember my first glimpse of PC-Matlab when I believe Cleve was showing it to Paul Swarztruber during a “lab” period at a Super Computing Summer Course at NCAR. I believe it was 1984. Back at what was then AFGL (now Air Force Research Lab), I remember procuring probably PC-Matlab 1 and then various version ever since. Having used early versions of IDL at the University of Colorado in the 1979-1981 time frame, I was quick to pick up on its interactive data analysis abilities and its vector abilities and especially easy graphics.
My first Matlab encounter was with version 4.1 as a graduate student.Fell in love with it immediately as it did not require any effort to learn and more importantly, apply to solving control design problems. It has been 21 years and still Matlab is my go-to software!
I used “classic” MATLAB on the mainframe computer at UofI during my graduate studies. Because of that, after I became an assistant professor, I used the small kitty of money available to me, and I purchased PC-MATLAB in early 1986 from John Little in California. Back in those days, technical assistance meant a phone call. John answered the phone. Being a FORTRAN guy, I had your 1977 Computer Methods book, so I translated fzero from FORTRAN to MATLAB code. I sent it to John via a floppy disk. It was added to MATLAB shortly thereafter. For a while, I was given credit in the M-file for the submission, but then after so many tweaks and changes my name appropriately disappeared (after all it is your work). Then I was asked to write the book to accompany the Student Edition of MATLAB that Prentice Hall published for a number of years. That work led to the creation of the book Mastering MATLAB that went on to do five editions. The second edition, Mastering MATLAB 5, was purchased in bulk by The Mathworks. At one time I also used SIMULAB on an SE30 Macintosh. That name did not stick around for long (trademark infringement?). MATLAB has been very good to me. You started a very good thing. It is difficult to imagine where science and technology would be today without the productivity that so many people benefit from because of MATLAB. I even had a license plate “MATLAB” for about ten years. Cheers!
Hi Duane — It’s great to hear from you. It’s been a while. Your books were certainly important in establishing the MATLAB book collection. Thanks.
I also started with the classic. I did probably the first network installation (Novel 3.11) as I convinced all faculty to use it in Scientific Programming and Numerical Analysis courses, then by all engineering and math courses offered, most of which I taught as well. Its biggest ingenuity is that a variable by definition is a double precision, complex vector and this gives you the ability to implement worry free vectorization among other tricks. Duane’s fzero story brought many memories as I could count at least 100 versions of an .m that Prof. E. Ozismir and I exchanged. Cleve thank you for inventing the engineering tool of the century(ies) that turned to be the top universal computational and simulation platform.
Thanks, George. — Cleve