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Mata is a full-blown programming language that compiles what you type into byte-code, optimizes it, and executes it fast. Behind the scenes, many of the new features of Stata 9, such as linear mixed models and multinomial probit, were written in Mata. You can use Mata to implement big systems, or you can use it interactively.To enter Mata, type mata at Stata’s dot prompt. To exit, type end at Mata’s colon prompt:
Mata supports real and complex numbers, binary and text strings (up to 2,147,483,647 characters long), and, for serious programming problems, even pointers!
Mata uses LAPACK routines for its advanced matrix features, such as Cholesky decomposition, LU decomposition, QR decomposition, SV decomposition, eigenvalues and eigenvectors, and solvers and inverters.
Mata supports matrices that are views onto, not copies of, the data. Say you have loaded a dataset of 200,000 observations and 150 variables, and you need a matrix of 80 of those variables in 180,000 of the observations. Rather than requiring 110 megabytes, Mata needs only 640 bytes.
Everybody knows that matrix languages evaluate matrix expressions, such as b=invsym(X'X)*X'y, and Mata is no exception. Because of Mata’s design, however, it is fast enough to work at the element level. Here is Mata’s polynomial solver:
Much of Mata is written in Mata.
numeric rowvector polysolve(numeric vector y, numeric vector x) { numeric rowvector res, c, empty real scalar i, j, n if (cols(y) != cols(x) | rows(y) != rows(x)) _error(3200) if ((n=length(x)) == 0) _error(3200) res = (iscomplex(y) | iscomplex(x) ? 0i : 0) for (j=1; j<=n; j++) { c = (1) for (i=1; i<=n; i++) { if (i != j) { c = polymult(c, (-x[i],1) :/ (x[j]-x[i])) } } res = polyadd(res, y[j] :* c) } while (res[cols(res)]==0) res = res[|1,1 \ 1,cols(res)-1|] return(res) }
Here is how we introduce Mata in the Mata Reference Manual.
[M-1] first — Introduction and first session
Mata is a component of Stata. It is a matrix programming language that can be used interactively or as an extension for do-files and ado-files. ThusMata has something for everybody.
- Mata can be used by users who want to think in matrix terms and perform (not necessarily simple) matrix calculations interactively, and
- Mata can be used by advanced Stata programmers who want to add features to Stata.
Primary features of Mata are that it is fast and that it is C-like.
This introduction is presented under the headingsInvoking Mata
Using Mata
Making mistakes: interpreting error messages
Working with real numbers, complex numbers, and strings
Working with scalars, vectors, and matrices
Working with functions
Distinguishing real and complex values
Working with matrix and scalar functions
Performing element-by-element calculations: colon operators
Writing programs
More functions
Mata environment commands
Exiting Mata
To enter Mata, type mata at Stata’s dot prompt and press enter; to exit Mata, type end at Mata’s colon prompt:
: 2+2 <- type Mata statements at the 4 colon prompt: end <- type end to return to Stata _________________________________ . _ <- you are back to Stata
When you type a statement into Mata, Mata compiles what you typed and, if it compiled without error, executes it:
We typed 2+2, a particular example from the general class of expressions. Mata responded with 4, the evaluation of the expression.
Often what you type are expressions, although you will probably choose more complicated examples. When an expression is not assigned to a variable, the result of the expression is displayed. Assignment is performed by the = operator:
When we type x=2+2, the expression is evaluated and stored in the variable we just named x. The result is not displayed. We can look at the contents of x, however, simply by typing x. From Mata’s perspective, x is not only a variable, it is also an expression, albeit a rather simple one. Just as 2+2 says to load 2, load another 2, and add them, the expression x says to load x and stop there.
As an aside, Mata distinguishes uppercase and lowercase. X is not the same as x:
If you make a mistake, Mata complains, and then you continue on your way. For instance,
2,,3 makes no sense to Mata, so Mata complained. This is an example of what is called a compile-time error; Mata could not make sense out of what we typed.
The other kind of error is called a run-time error. For example, we have no variable called y. Let us ask Mata to show us the contents of y :
In this case, what we typed made perfect sense—show me y—but y has never been defined. This ugly message is called an run-time error message—see [M-2] errors for a complete description—but all that’s important is to understand the difference between
: y <istmt>: 3499 y not found r(3499); : _invalid expressionand<istmt>: 3499 y not foundThe run-time message is prefixed by an identity (<istmt> in this case) and a number (3499 in this case). Mata is telling us, "I was executing your istmt [that’s what everything you type is called] and I got error 3499, the details of which are that I was unable to find y."The compile-time error message is of a simpler form: "invalid expression". When you get such unprefixed error messages, that means Mata could not understand what you typed. When you get the more complicated error message, that means Mata understood what you typed, but there was a problem performing your request.
Another way to tell the difference between compile-time errors and run-time errors is to look at the return code. Compile-time errors have a return code of 3000:
Run-time errors have a return code that might be in the 3000s, but is never 3000 exactly:
: y <istmt>: 3499 y not found r(3499);
As we have seen, Mata works with real numbers:Mata also understands complex numbers; you write the imaginary part by suffixing a lowercase i:
Mata also understands strings, which you write enclosed in double quotes:
Just like Stata, Mata understands simple and compound double quotes:
You can add complex and reals
: 2 + "alpha" <istmt>: 3250 type mismatch ;r(3250);
In addition to understanding scalars—be they real, complex, or string — Mata understands vectors and matrices of real, complex, and string elements:x now contains the row vector (1,2). We can add vectors:
: x=(1,2) : x 1 2 +---------+ 1 | 1 2 | +---------+The "," is the column-join operator; things like (1,2) are expressions, just as (1+2) is an expression:
: x + (3,4) 1 2 +---------+ 1 | 4 6 | +---------+In the above, we could have dispensed with the parentheses and typed "y=3,4" followed by "z=x,y", just as we could using the + operator, although most people find vectors more readable when enclosed in parentheses. "\" is the row-join operator:
: y = (3,4) : z = (x,y) : z 1 2 3 4 +-----------------+ 1 | 1 2 3 4 | +-----------------+
: a = (1\2) : a 1 +-----+ 1 | 1 | 2 | 2 | +-----+ : b = (3\4) : c = (a\b) : c 1 +-----+ 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | +-----+Using the column-join and row-join operators, we can enter matrices:
The use of these operators is not limited to scalars. Remember, x is the row vector (1,2), y is the row vector (3,4), a is the column vector (1\2), and b is the column vector (3\4). Therefore,
: A = (1,2 \ 3,4) : A 1 2 +---------+ 1 | 1 2 | 2 | 3 4 | +---------+
: x\y 1 2 +---------+ 1 | 1 2 | 2 | 3 4 | +---------+ : a,b +---------+ | 1 3 | | 2 4 | +---------+But if we try something nonsensical, we get an error:
We create complex vectors and matrices just as we create real ones, the only difference being that their elements are complex:
: Z = (1+1i, 2+3i \ 3-2i , -1-1i) : Z 1 2 +---------------------+ 1 | 1 + 1i 2 + 3i | 2 | 3 - 2i -1 - 1i | +---------------------+Similarly, we can create string vectors and matrices, which are vectors and matrices with string elements:
In the case of strings, the individual elements can be up to 2,147,437,647 characters long.
: S = ("1st element", "2nd element" \ "another row", "last element") : S 1 2 +-------------------------------+ 1 | 1st element 2nd element | 2 | another row last element | +-------------------------------+
Mata’s expressions also include functions:
When we ask for the square root of -4, Mata replies "." Further, note that . can be stored just like any other number:
"." means missing, that there is no answer to our calculation. Taking the square root of a negative number is not an error; it merely produces missing. To Mata, missing is a number like any other number, and the rules for all the operators have been generalized to understand missing. For instance, the addition rule is generalized such that missing plus anything is missing:
By the way, used with one argument, C() also allows complex, and then it does nothing:
It is virtually impossible to tell the difference between a real value and a complex value with zero imaginary part:
Yet, as we have seen, the difference is important: sqrt(areal) is missing, sqrt(acomplex) is -2i. One solution is the eltype() function:
eltype() can also be used with strings
but this is mostly useful in programming contexts.
Some functions are matrix functions: they require a matrix and return a matrix. Mata’s invsym(X) is an example of such a function. It returns the matrix that is the inverse of symmetric, real matrix X:
: X = (76, 53, 48 \ 53, 88, 46 \ 48, 46, 63) : Xi = invsym(X) : Xi [symmetric] 1 2 3 +----------------------------------------------+ 1 | .0298458083 | 2 | -.0098470272 .0216268926 | 3 | -.0155497706 -.0082885675 .0337724301 | +----------------------------------------------+ : Xi*X 1 2 3 +----------------------------------------------+ 1 | 1 -8.67362e-17 -8.50015e-17 | 2 | -1.38778e-16 1 -1.02349e-16 | 3 | 0 1.11022e-16 1 | +----------------------------------------------+
The last matrix, Xi*X, differs just a little from the identity matrix due to unavoidable computational roundoff error.
Other functions are, mathematically speaking, scalar functions. sqrt() is an example in that it makes no sense to speak of sqrt(X). (That is, it makes no sense to speak of sqrt(X) unless we were speaking of the Cholesky square-root decomposition. Mata has such a matrix function; see help [M-5] cholesky().)
When a function is, mathematically speaking, a scalar function, the corresponding Mata function will usually allow vector and matrix arguments, and in that case, the Mata function makes the calculation on each element individually:
: M = (1,2 \ 3,4 \ 5,6) : M 1 2 +---------+ 1 | 1 2 | 2 | 3 4 | 3 | 5 6 | +---------+ : S = sqrt(M) : S 1 2 +-----------------------------+ 1 | 1 1.414213562 | 2 | 1.732050808 2 | 3 | 2.236067977 2.449489743 | +-----------------------------+ : S[1,2]*S[1,2] 2 : S[2,1]*S[2,1] 3
When a function returns a result calculated in this way, it is said to return an element-by-element result.
Mata’s operators, such as + (addition) and * (multiplication), work as you would expect. In particular, * performs matrix multiplication:The first element of the result was calculated as 1*5+2*7=19.
: A = (1, 2 \ 3, 4) : B = (5, 6 \ 7, 8) : A*B 1 2 +-----------+ 1 | 19 22 | 2 | 43 50 | +-----------+Sometimes, you really want the element-by-element result. When you do, place a colon in front of the operator: Mata’s :* operator performs element-by-element multiplication:
See [M-2] op_colon for more information.
: A:*B 1 2 +-----------+ 1 | 5 12 | 2 | 21 32 | +-----------+
Mata is a complete programming language; it will allow you to create your own functions:: function add(a,b) return(a+b)That single statement creates a new function, although perhaps you would prefer if we typed it as
: function add(a, b) > { > return(a+b) > }
because that makes it obvious that a program can contain many lines. In either case, once defined, we can use the function:
: add(1,2) 3 : add(1+2i,4-1i) 5+1i : add( (1,2), (3,4) ) 1 2 +---------+ 1 | 4 6 | +---------+ : add(x,y) 1 2 +---------+ 1 | 4 6 | +---------+ : add(A,A) 1 2 +---------+ 1 | 2 4 | 2 | 6 8 | +---------+ : Z1 = (1+1i, 1+1i \ 2, 2i) : Z2 = (1+2i, -3+3i \ 6i, -2+2i) : add(Z1, Z2) 1 2 +---------------------+ 1 | 2 + 3i -2 + 4i | 2 | 2 + 6i -2 + 4i | +---------------------+ : add("Alpha","Beta") AlphaBeta : S1 = ("one", "two" \ "three", "four") : S2 = ("abc", "def" \ "ghi", "jkl") : add(S1, S2) 1 2 +-----------------------+ 1 | oneabc twodef | 2 | threeghi fourjkl | +-----------------------+
Of course, our little function add() does not do anything that the + operator does not already do, but we could write a program that did do something different. The following program will allow us to make n x n identity matrices:
The function J() in the program line res = J(n, n, 0) is a Mata built-in function that returns an n x n matrix containing 0s (J(r, c, val) returns an r x c matrix, the elements of which are all equal to val); see [M-5] J(). for (i=1; i<=n; i++) says that starting with i=1 and so long as i<=n do what is inside the braces (set res[i,i] equal to 1) and then (we are back to the for part again), increment i. The final line—return(res)—says to return the matrix we have just created. Actually, just as with add(), we do not need id() because Mata has a built-in function I(n) that makes identity matrices, but it is interesting to see how the problem could be programmed.
: real matrix id (real scalar n) > { > real scalar i > real matrix res > > res = J(n, n, 0) > for (i=1; i<=n; i++) { > res[i,i] = 1 > } > return(res) > } : I3 = id(3) : I3 [symmetric] 1 2 3 +-------------+ 1 | 1 | 2 | 0 1 | 3 | 0 0 1 | +-------------+
Mata has lots of functions already and much of this manual concerns documenting what those functions do; see [M-4] intro. But right now, what is important is that many of the functions are themselves written in Mata!One of those functions is pi();it takes no arguments and returns the value of pi. The code for it reads
real scalar pi() return(3.141592653589793238462643)There is no reason to type the above function because it is already included as part of Mata:: pi() 3.141592654When Mata lists a result, it does not show as many digits, but we could ask to see more:: printf("%17.0g", pi()) 3.14159265358979Other Mata functions include the hyperbolic functions sinh(u), cosh(u), etc. The code for sinh(u), cosh(u), and tanh(u) reads
numeric matrix sinh(numeric matrix u) return((exp(u)-exp(-u)):/2) numeric matrix cosh(numeric matrix u) return((exp(u)+exp(-u)):/2) numeric matrix tanh(numeric matrix u) { numeric matrix eu, emu eu = exp(u) emu = exp(-u) return( (eu-emu):/(eu+emu) ) }See for yourself: at the Stata dot prompt (not the Mata colon prompt), type
When the code for a function was written in Mata (as opposed to having been written in C), viewsource can show you the code; see [M-1] source.
Returning to the functions themselves,
numeric matrix sinh(numeric matrix u) return((exp(u)-exp(-u)):/2) numeric matrix cosh(numeric matrix u) return((exp(u)+exp(-u)):/2) numeric matrix tanh(numeric matrix u) { numeric matrix eu, emu eu = exp(u) emu = exp(-u) return( (eu-emu):/(eu+emu) ) }
this is the first time we have seen the word numeric: it means real or complex. Built-in (previously written) function exp() works like sqrt() in that it allows a real or complex argument and correspondingly returns a real or complex result. Said in Mata jargon, exp() allows a numeric argument and correspondingly returns a numeric result. sinh(), cosh(), and tanh() will also work like sqrt() and exp().
Another characteristic sinh(), cosh() and tanh() share with sqrt() and exp() is element-by-element operation. sinh(), cosh(), and tanh() are element-by-element because exp() is element-by-element and because we were careful to use the :/ (element-by-element) divide operator.
In any case, there is no need to type the above functions because they are already part of Mata. You could learn more about them by seeing their manual entry, [M-5] sin().
At the other extreme, Mata functions can become quite long. Here is Mata’s function to solve AX=B for X when A is lower triangular, placing the result X back into A:
real scalar _solvelower( numeric matrix A, numeric matrix b, |real scalar usertol, numeric scalar userd) { real p; calar tol, rank, a_t, b_t, d_t real scalar n, m, i, im1, complex_case numeric rowvector sum numeric scalar zero, d d = userd if ((n=rows(A))!=cols(A)) _error(3205) if (n != rows(b)) _error(3200) if (isview(b)) _error(3104) m = cols(b) rank = n a_t = iscomplex(A) b_t = iscomplex(b) d_t = d<. ? iscomplex(d) : 0 complex_case = a_t | b_t | d_t if (complex_case) { if (!a_t) A = C(A) if (!b_t) b = C(b) if (d<. & !d_t) d = C(d) zero = 0i } else zero = 0 if (n==0 | m==0) return(0) tol = solve_tol(A, usertol) if (abs(d) >=. ) { if (abs(d=A[1,1])<=tol) { b[1,.] = J(1, m, zero) --rank } else { b[1,.] = b[1,.] :/ d if (missing(d)) rank = . } for (i=2; i<=n; i++) { im1 = i - 1 sum = A[|i,1\i,im1|] * b[|1,1\im1,m|] if (abs(d=A[i,i])<=tol) { b[i,.] = J(1, m, zero) --rank } else { b[i,.] = (b[i,.]-sum) :/ d if (missing(d)) rank = . } } } else { if (abs(d)<=tol) { rank = 0 b = J(rows(b), cols(b), zero) } else { b[1,.] = b[1,.] :/ d for (i=2; i<=n; i++) { im1 = i - 1 sum = A[|i,1\i,im1|] * b[|1,1\im1,m|] b[i,.] = (b[i,.]-sum) :/ d } } } return(rank) }
If the function were not already part of Mata and you wanted to use it, you could type it into a do-file or onto the end of an ado-file (especially good if you just want to use _solvelower() as a subroutine). In those cases, do not forget to enter and exit Mata:
Sharp-eyed readers will notice that we put a colon on the end of the Mata command. That’s a detail and why we did that is explained in [M-3] mata.
In addition to loading functions by putting their code in do- and ado-files, you can also save the compiled versions of functions in .mo files (see [M-3] mata mosave) or into .mlib Mata libraries (see [M-3] mata mlib).
In the case of _solvelower(), it has already been saved into a library, namely Mata’s official library, so you need not do any of this.
When you are using Mata, there is a set of commands that will tell you about and manipulate Mata’s environment.The most useful such command is mata describe:
: mata describe # bytes type name and extent34 transmorphic matrix add() 98 real matrix id() 32 real matrix A[2,2] 32 real matrix B[2,2] 72 real matrix I3[3,3] 48 real matrix M[3,2] 48 real matrix S[3,2] 47 string matrix S1[2,2] 44 string matrix S2[2,2] 72 real matrix X[3,3] 72 real matrix Xi[3,3] 64 complex matrix Z[2,2] 64 complex matrix Z1[2,2] 64 complex matrix Z2[2,2] 16 real colvector a[2] 16 complex scalar acomplex 8 real scalar areal 16 real colvector b[2] 32 real colvector c[4] 8 real scalar findout 16 real rowvector x[2] 16 real rowvector y[2] 32 real rowvector z[4]Another useful command is mata clear, which will clear Mata without disturbing Stata:
There are other useful mata commands; see [M-3] intro. Do not confuse this command mata, which you type at Mata’s colon prompt with Stata’s command mata, which you type at Stata’s dot prompt and which invokes Mata.
: mata clear : mata describe # bytes type name and extent
When you are done using Mata, type end to Mata’s colon prompt:
Exiting Mata does not clear it:
: mata describe # bytes type name and extent34 transmorphic matrix add() 8 real scalar x 16 complex scalar yExiting Stata clears Mata, as does Stata’s clear command; see [R] drop.
Also see Manual: [M-1] first Online: [M-0] intro, [M-1] intro
