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Matrix language

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

 mata (type end to exit)

: sqrt(-4)
  .

: sqrt(-4+0i)
  2i

: end

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:

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)
}
Much of Mata is written in Mata.

Here is how we introduce Mata in the Mata Reference Manual.

help m1 first

Title

[M-1] first — Introduction and first session

Description

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. Thus
  1. Mata can be used by users who want to think in matrix terms and perform (not necessarily simple) matrix calculations interactively, and
  2. Mata can be used by advanced Stata programmers who want to add features to Stata.
Mata has something for everybody.

Primary features of Mata are that it is fast and that it is C-like.

Remarks

This introduction is presented under the headings

Invoking 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

Invoking Mata

To enter Mata, type mata at Stata’s dot prompt and press enter; to exit Mata, type end at Mata’s colon prompt:
. mata                                      <- type mata to enter Mata
  

 mata (type end to exit)

      : 2+2                                 <- type Mata statements at the
        4                                      colon prompt
                                                                              
      : end                                 <- type end to return to Stata
      _________________________________
                                                                           
      . _                                   <- you are back to Stata

Using Mata

When you type a statement into Mata, Mata compiles what you typed and, if it compiled without error, executes it:

: 2+2
  4

: _

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:

: x = 2 + 2

: x
  4

: _
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:

: X = 2 + 3

: x
  4

: X
  5

Making mistakes: interpreting error messages

If you make a mistake, Mata complains, and then you continue on your way. For instance,
: 2,,3
invalid expression
r(3000);

: _

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 :

: y
             <istmt>:   3499  y not found
r(3499);

: _
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
     invalid   expression 
and
     <istmt>:    3499   y not found 
The 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:

: 2,,3
invalid expression
r(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);
Whether the error is compile-time or run-time, once the error message is issued, Mata is ready to continue just as if the error never happened.

Working with real numbers, complex numbers, and strings

As we have seen, Mata works with real numbers:
: 2+3
  5
Mata also understands complex numbers; you write the imaginary part by suffixing a lowercase i:
: 1+2i + 4-1i
  5+1i
For imaginary numbers, you can omit the real part:
: 1+2i - 2i
  1
Whether a number is real or complex, you can use the same computer notation for the imaginary part as you would for the real part:
: 2.5e+3i
  2500i

: 1.25e+2+2.5e+3i          /* i.e., 1.25e+02 + 2.5e+03i */
  125 + 2500i
We purposely wrote the last example in nearly unreadable form just to emphasize that Mata could interpret it.

Mata also understands strings, which you write enclosed in double quotes:

: "Alpha"+"Beta"
  AlphaBeta

Just like Stata, Mata understands simple and compound double quotes:

: `"Alpha"'+`"Beta"'
  AlphaBeta
You can add complex and reals
: 1+2i + 3
  4+2i
but you may not add reals or complex to strings:
: 2 + "alpha"
           <istmt>:  3250  type mismatch
;r(3250);
We got a run-time error. Mata understood 2 + "alpha" alright, it just could not perform our request.

Working with scalars, vectors, and matrices

In addition to understanding scalars—be they real, complex, or string — Mata understands vectors and matrices of real, complex, and string elements:
: x=(1,2)

: x
       1   2
    +---------+
  1 |  1   2  |
    +---------+
x now contains the row vector (1,2). We can add vectors:
: x + (3,4)
       1   2
    +---------+
  1 |  4   6  |
    +---------+
The "," is the column-join operator; things like (1,2) are expressions, just as (1+2) is an expression:
: y = (3,4)

: z = (x,y)

: z

       1   2   3   4
    +-----------------+
  1 |  1   2   3   4  |
    +-----------------+
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:
: 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:

: A = (1,2 \ 3,4)

: A
       1   2
    +---------+
  1 |  1   2  |
  2 |  3   4  |
    +---------+
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,
: 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:

	: a,x
	          <istmt>:   3200 nonconformable matrices

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:

: S = ("1st element", "2nd element" \ "another row", "last element")

: S
                1              2
  +-------------------------------+
1 |   1st element    2nd element  |
2 |   another row   last element  |
  +-------------------------------+
In the case of strings, the individual elements can be up to 2,147,437,647 characters long.

Working with functions

Mata’s expressions also include functions:

: sqrt(4)
  2

: sqrt(-4)
  .

When we ask for the square root of -4, Mata replies "." Further, note that . can be stored just like any other number:

: findout = sqrt(-4)

: findout
  .
"." 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:
	: 2 + .
	  .
Still, it should surprise you that Mata produced missing for the sqrt(-4). We said that Mata understands complex numbers, so should not the answer be 2i? The answer is that is should be if you are working on the complex plane, but otherwise, missing is probably a better answer. Mata attempts to intuit the kind of answer you want by context, and in particular, uses inheritance rules. If you ask for the square root of a real number, you get a real number back. If you ask for the square root of a complex number, you get a complex number back:
: sqrt(-4+0i)
  2i
Here complex means multipart: -4+0i is a complex number, it merely happens to have 0 imaginary part. Thus:
: areal = -4

: acomplex = -4 + 0i

: sqrt(areal)
  .

: sqrt(acomplex)
  2i
If you ever have a real scalar, vector, or matrix, and want to make it complex, use the C() function, which means "convert to complex":
	: sqrt(C(areal))
	  2i
C() is documented in [M-5] C(). C() allows one or two arguments. With one argument, it casts to complex. With two arguments, it makes a complex out of the two real arguments. Thus you could type
: sqrt(-4+2i)
  .485868272 + 2.05817103i
or you could type
: sqrt(C(-4,2))
  .485868272 + 2.05817103i

By the way, used with one argument, C() also allows complex, and then it does nothing:

: sqrt(C(acomplex))
  2i

Distinguishing real and complex values

It is virtually impossible to tell the difference between a real value and a complex value with zero imaginary part:
: areal = -4

: acomplex = -4 + 0i

: areal
  -4

: acomplex
  -4

Yet, as we have seen, the difference is important: sqrt(areal) is missing, sqrt(acomplex) is -2i. One solution is the eltype() function:

: eltype(areal)
  real

: eltype(acomplex)
  complex

eltype() can also be used with strings

: astring = "hello"

: eltype(astring)
  string
but this is mostly useful in programming contexts.

Working with matrix and scalar functions

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.

Performing element-by-element calculations: colon operators

Mata’s operators, such as + (addition) and * (multiplication), work as you would expect. In particular, * performs matrix multiplication:
: A = (1, 2 \ 3, 4)

: B = (5, 6 \ 7, 8)

: A*B
        1    2
    +-----------+
  1 |  19   22  |
  2 |  43   50  |
    +-----------+
The first element of the result was calculated as 1*5+2*7=19.

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:

: A:*B
        1    2
    +-----------+
  1 |   5   12  |
  2 |  21   32  |
    +-----------+
See [M-2] op_colon for more information.

Writing programs

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:

: 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  |
    +-------------+
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.

More functions

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()
	  3.141592654
When Mata lists a result, it does not show as many digits, but we could ask to see more:
	: printf("%17.0g", pi())
	  3.14159265358979
Other 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

. viewsource sinh.mata

. viewsource cosh.mata

. viewsource tanh.mata

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:


 top of file


program  mycommand 
	        ...
	        ado-file code appears here
	        ...
end

mata:
	        _solvelower() code appears here
end


 bottom of file 

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.

Mata environment commands

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 extent

           34   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:

: mata clear

: mata describe

      # bytes   type                       name and extent


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.

Exiting Mata

When you are done using Mata, type end to Mata’s colon prompt:
: end

. _

Exiting Mata does not clear it:

. mata

 mata (type end to exit) 

: x = 2

: y = (3+2i)

: function add(a,b) return(a+b)

: end

. ...

. mata

 mata (type end to exit) 

: mata describe
	
         # bytes   type                       name and extent

              34   transmorphic matrix        add()
               8   real scalar                x
              16   complex scalar             y

: end

Exiting 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

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