[pspp] / doc / matrices.texi

@c PSPP - a program for statistical analysis.
@c Copyright (C) 2017, 2020, 2021 Free Software Foundation, Inc.
@c Permission is granted to copy, distribute and/or modify this document
@c under the terms of the GNU Free Documentation License, Version 1.3
@c or any later version published by the Free Software Foundation;
@c with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
@c A copy of the license is included in the section entitled "GNU
@c Free Documentation License".
@c
@node Matrices
@chapter Matrices

Some @pspp{} procedures work with matrices by producing numeric
matrices that report results of data analysis, or by consuming
matrices as a basis for further analysis.  This chapter documents the
format of data files that store these matrices and commands for
working with them, as well as @pspp{}'s general-purpose facility for
matrix operations.

@node Matrix Files
@section Matrix Files
@vindex Matrix file

A matrix file is an SPSS system file that conforms to the dictionary
and case structure described in this section.  Procedures that read
matrices from files expect them to be in the matrix file format.
Procedures that write matrices also use this format.

Text files that contain matrices can be converted to matrix file
format.  @xref{MATRIX DATA}, for a command to read a text file as a
matrix file.

A matrix file's dictionary must have the following variables in the
specified order:

@enumerate
@item
Zero or more numeric split variables.  These are included by
procedures when @cmd{SPLIT FILE} is active.  @cmd{MATRIX DATA} assigns
split variables format F4.0.

@item
@code{ROWTYPE_}, a string variable with width 8.  This variable
indicates the kind of matrix or vector that a given case represents.
The supported row types are listed below.

@item
Zero or more numeric factor variables.  These are included by
procedures that divide data into cells.  For within-cell data, factor
variables are filled with non-missing values; for pooled data, they
are missing.  @cmd{MATRIX DATA} assigns factor variables format F4.0.

@item
@code{VARNAME_}, a string variable.  Matrix data includes one row per
continuous variable (see below), naming each continuous variable in
order.  This column is blank for vector data.  @cmd{MATRIX DATA} makes
@code{VARNAME_} wide enough for the name of any of the continuous
variables, but at least 8 bytes.

@item
One or more numeric continuous variables.  These are the variables
whose data was analyzed to produce the matrices.  @cmd{MATRIX DATA}
assigns continuous variables format F10.4.
@end enumerate

Case weights are ignored in matrix files. 

@subheading Row Types
@anchor{Matrix File Row Types}

Matrix files support a fixed set of types of matrix and vector data.
The @code{ROWTYPE_} variable in each case of a matrix file indicates
its row type.  

The supported matrix row types are listed below.  Each type is listed
with the keyword that identifies it in @code{ROWTYPE_}.  All supported
types of matrices are square, meaning that each matrix must include
one row per continuous variable, with the @code{VARNAME_} variable
indicating each continuous variable in turn in the same order as the
dictionary.

@table @code
@item CORR
Correlation coefficients.

@item COV
Covariance coefficients.

@item MAT
General-purpose matrix.

@item N_MATRIX
Counts.

@item PROX
Proximities matrix.
@end table

The supported vector row types are listed below, along with their
associated keyword.  Vector row types only require a single row, whose
@code{VARNAME_} is blank:

@table @code
@item COUNT
Unweighted counts.

@item DFE
Degrees of freedom.

@item MEAN
Means.

@item MSE
Mean squared errors.

@item N
Counts.

@item STDDEV
Standard deviations.
@end table

Only the row types listed above may appear in matrix files.  The
@cmd{MATRIX DATA} command, however, accepts the additional row types
listed below, which it changes into matrix file row types as part of
its conversion process:

@table @code
@item N_VECTOR
Synonym for @cmd{N}.

@item SD
Synonym for @code{STDDEV}.

@item N_SCALAR
Accepts a single number from the @code{MATRIX DATA} input and writes
it as an @code{N} row with the number replicated across all the
continuous variables.
@end table

@node MATRIX DATA
@section MATRIX DATA
@vindex MATRIX DATA

@display
MATRIX DATA
        VARIABLES=@var{variables}
        [FILE=@{'@var{file_name}' | INLINE@}
        [/FORMAT=[@{LIST | FREE@}]
                 [@{UPPER | LOWER | FULL@}]
                 [@{DIAGONAL | NODIAGONAL@}]]
        [/SPLIT=@var{split_vars}]
        [/FACTORS=@var{factor_vars}]
        [/N=@var{n}]

The following subcommands are only needed when ROWTYPE_ is not
specified on the VARIABLES subcommand:
        [/CONTENTS=@{CORR,COUNT,COV,DFE,MAT,MEAN,MSE,
                    N_MATRIX,N|N_VECTOR,N_SCALAR,PROX,SD|STDDEV@}]
        [/CELLS=@var{n_cells}]
@end display

The @cmd{MATRIX DATA} command convert matrices and vectors from text
format into the matrix file format (@xref{Matrix Files}) for use by
procedures that read matrices.  It reads a text file or inline data
and outputs to the active file, replacing any data already in the
active dataset.  The matrix file may then be used by other commands
directly from the active file, or it may be written to a @file{.sav}
file using the @cmd{SAVE} command.

The text data read by @cmd{MATRIX DATA} can be delimited by spaces or
commas.  A plus or minus sign, except immediately following a @samp{d}
or @samp{e}, also begins a new value.  Optionally, values may be
enclosed in single or double quotes.

@cmd{MATRIX DATA} can read the types of matrix and vector data
supported in matrix files (@pxref{Matrix File Row Types}).

The @subcmd{FILE} subcommand specifies the source of the command's
input.  To read input from a text file, specify its name in quotes.
To supply input inline, omit @subcmd{FILE} or specify @code{INLINE}.
Inline data must directly follow @code{MATRIX DATA}, inside @cmd{BEGIN
DATA} (@pxref{BEGIN DATA}).

@subcmd{VARIABLES} is the only required subcommand.  It names the
variables present in each input record in the order that they appear.
(@cmd{MATRIX DATA} reorders the variables in the matrix file it
produces, if needed to fit the matrix file format.)  The variable list
must include split variables and factor variables, if they are present
in the data, in addition to the continuous variables that form matrix
rows and columns.  It may also include a special variable named
@code{ROWTYPE_}.

Matrix data may include split variables or factor variables or both.
List split variables, if any, on the @subcmd{SPLIT} subcommand and
factor variables, if any, on the @subcmd{FACTORS} subcommand.  Split
and factor variables must be numeric.  Split and factor variables must
also be listed on @subcmd{VARIABLES}, with one exception: if
@subcmd{VARIABLES} does not include @code{ROWTYPE_}, then
@subcmd{SPLIT} may name a single variable that is not in
@subcmd{VARIABLES} (@pxref{MATRIX DATA Example 8}).

The @subcmd{FORMAT} subcommand accepts settings to describe the format
of the input data:

@table @asis
@item @code{LIST} (default)
@itemx @code{FREE}
LIST requires each row to begin at the start of a new input line.
FREE allows rows to begin in the middle of a line.  Either setting
allows a single row to continue across multiple input lines.

@item @code{LOWER} (default)
@itemx @code{UPPER}
@itemx @code{FULL}
With LOWER, only the lower triangle is read from the input data and
the upper triangle is mirrored across the main diagonal.  UPPER
behaves similarly for the upper triangle.  FULL reads the entire
matrix.

@item @code{DIAGONAL} (default)
@itemx @code{NODIAGONAL}
With DIAGONAL, the main diagonal is read from the input data.  With
NODIAGONAL, which is incompatible with FULL, the main diagonal is not
read from the input data but instead set to 1 for correlation matrices
and system-missing for others.
@end table

The @subcmd{N} subcommand is a way to specify the size of the
population.  It is equivalent to specifying an @code{N} vector with
the specified value for each split file.

@cmd{MATRIX DATA} supports two different ways to indicate the kinds of
matrices and vectors present in the data, depending on whether a
variable with the special name @code{ROWTYPE_} is present in
@code{VARIABLES}.  The following subsections explain @cmd{MATRIX DATA}
syntax and behavior in each case.

@node MATRIX DATA with ROWTYPE_
@subsection With @code{ROWTYPE_}

If @code{VARIABLES} includes @code{ROWTYPE_}, each case's
@code{ROWTYPE_} indicates the type of data contained in the row.
@xref{Matrix File Row Types}, for a list of supported row types.

@subsubheading Example 1: Defaults with @code{ROWTYPE_}
@anchor{MATRIX DATA Example 1}

This example shows a simple use of @cmd{MATRIX DATA} with
@code{ROWTYPE_} plus 8 variables named @code{var01} through
@code{var08}.

Because @code{ROWTYPE_} is the first variable in @subcmd{VARIABLES},
it appears first on each line. The first three lines in the example
data have @code{ROWTYPE_} values of @samp{MEAN}, @samp{SD}, and
@samp{N}.  These indicate that these lines contain vectors of means,
standard deviations, and counts, respectively, for @code{var01}
through @code{var08} in order.

The remaining 8 lines have a ROWTYPE_ of @samp{CORR} which indicates
that the values are correlation coefficients.  Each of the lines
corresponds to a row in the correlation matrix: the first line is for
@code{var01}, the next line for @code{var02}, and so on.  The input
only contains values for the lower triangle, including the diagonal,
since @code{FORMAT=LOWER DIAGONAL} is the default.

With @code{ROWTYPE_}, the @code{CONTENTS} subcommand is optional and
the @code{CELLS} subcommand may not be used.

@example
MATRIX DATA
    VARIABLES=ROWTYPE_ var01 TO var08.
BEGIN DATA.
MEAN  24.3   5.4  69.7  20.1  13.4   2.7  27.9   3.7
SD     5.7   1.5  23.5   5.8   2.8   4.5   5.4   1.5
N       92    92    92    92    92    92    92    92
CORR  1.00
CORR   .18  1.00
CORR  -.22  -.17  1.00
CORR   .36   .31  -.14  1.00
CORR   .27   .16  -.12   .22  1.00
CORR   .33   .15  -.17   .24   .21  1.00
CORR   .50   .29  -.20   .32   .12   .38  1.00
CORR   .17   .29  -.05   .20   .27   .20   .04  1.00
END DATA.
@end example

@subsubheading Example 2: @code{FORMAT=UPPER NODIAGONAL}

This syntax produces the same matrix file as example 1, but it uses
@code{FORMAT=UPPER NODIAGONAL} to specify the upper triangle and omit
the diagonal.  Because the matrix's @code{ROWTYPE_} is @code{CORR},
@pspp{} automatically fills in the diagonal with 1.

@example
MATRIX DATA
    VARIABLES=ROWTYPE_ var01 TO var08
    /FORMAT=UPPER NODIAGONAL.
BEGIN DATA.
MEAN  24.3   5.4  69.7  20.1  13.4   2.7  27.9   3.7
SD     5.7   1.5  23.5   5.8   2.8   4.5   5.4   1.5
N       92    92    92    92    92    92    92    92
CORR         .17   .50  -.33   .27   .36  -.22   .18
CORR               .29   .29  -.20   .32   .12   .38
CORR                     .05   .20  -.15   .16   .21
CORR                           .20   .32  -.17   .12
CORR                                 .27   .12  -.24
CORR                                      -.20  -.38
CORR                                             .04
END DATA.
@end example

@subsubheading Example 3: @subcmd{N} subcommand

This syntax uses the @subcmd{N} subcommand in place of an @code{N}
vector.  It produces the same matrix file as examples 1 and 2.

@example
MATRIX DATA
    VARIABLES=ROWTYPE_ var01 TO var08
    /FORMAT=UPPER NODIAGONAL
    /N 92.
BEGIN DATA.
MEAN  24.3   5.4  69.7  20.1  13.4   2.7  27.9   3.7
SD     5.7   1.5  23.5   5.8   2.8   4.5   5.4   1.5
CORR         .17   .50  -.33   .27   .36  -.22   .18
CORR               .29   .29  -.20   .32   .12   .38
CORR                     .05   .20  -.15   .16   .21
CORR                           .20   .32  -.17   .12
CORR                                 .27   .12  -.24
CORR                                      -.20  -.38
CORR                                             .04
END DATA.
@end example

@subsubheading Example 4: Split variables
@anchor{MATRIX DATA Example 4}

This syntax defines two matrices, using the variable @samp{s1} to
distinguish between them.  Notice how the order of variables in the
input matches their order on @subcmd{VARIABLES}.  This example also
uses @code{FORMAT=FULL}.

@example
MATRIX DATA
    VARIABLES=s1 ROWTYPE_  var01 TO var04
    /SPLIT=s1
    /FORMAT=FULL.
BEGIN DATA.
0 MEAN 34 35 36 37
0 SD   22 11 55 66
0 N    99 98 99 92
0 CORR  1 .9 .8 .7
0 CORR .9  1 .6 .5
0 CORR .8 .6  1 .4
0 CORR .7 .5 .4  1
1 MEAN 44 45 34 39
1 SD   23 15 51 46
1 N    98 34 87 23
1 CORR  1 .2 .3 .4
1 CORR .2  1 .5 .6
1 CORR .3 .5  1 .7
1 CORR .4 .6 .7  1
END DATA.
@end example

@subsubheading Example 5: Factor variables
@anchor{MATRIX DATA Example 5}

This syntax defines a matrix file that includes a factor variable
@samp{f1}.  The data includes mean, standard deviation, and count
vectors for two values of the factor variable, plus a correlation
matrix for pooled data.

@example
MATRIX DATA
    VARIABLES=ROWTYPE_ f1 var01 TO var04
    /FACTOR=f1.
BEGIN DATA.
MEAN 0 34 35 36 37
SD   0 22 11 55 66
N    0 99 98 99 92
MEAN 1 44 45 34 39
SD   1 23 15 51 46
N    1 98 34 87 23
CORR .  1
CORR . .9  1
CORR . .8 .6  1
CORR . .7 .5 .4  1
END DATA.
@end example

@node MATRIX DATA without ROWTYPE_
@subsection Without @code{ROWTYPE_}

If @code{VARIABLES} does not contain @code{ROWTYPE_}, the
@subcmd{CONTENTS} subcommand defines the row types that appear in the
file and their order.  If @subcmd{CONTENTS} is omitted,
@code{CONTENTS=CORR} is assumed.

Factor variables without @code{ROWTYPE_} introduce special
requirements, illustrated below in Examples 8 and 9.

@subsubheading Example 6: Defaults without @code{ROWTYPE_}

This example shows a simple use of @cmd{MATRIX DATA} with 8 variables
named @code{var01} through @code{var08}, without @code{ROWTYPE_}.
This yields the same matrix file as Example 1 (@pxref{MATRIX DATA
Example 1}).

@example
MATRIX DATA
    VARIABLES=var01 TO var08
   /CONTENTS=MEAN SD N CORR.
BEGIN DATA.
24.3   5.4  69.7  20.1  13.4   2.7  27.9   3.7
 5.7   1.5  23.5   5.8   2.8   4.5   5.4   1.5
  92    92    92    92    92    92    92    92
1.00
 .18  1.00
-.22  -.17  1.00
 .36   .31  -.14  1.00
 .27   .16  -.12   .22  1.00
 .33   .15  -.17   .24   .21  1.00
 .50   .29  -.20   .32   .12   .38  1.00
 .17   .29  -.05   .20   .27   .20   .04  1.00
END DATA.
@end example

@subsubheading Example 7: Split variables with explicit values

This syntax defines two matrices, using the variable @code{s1} to
distinguish between them.  Each line of data begins with @code{s1}.
This yields the same matrix file as Example 4 (@pxref{MATRIX DATA
Example 4}).

@example
MATRIX DATA
    VARIABLES=s1 var01 TO var04
    /SPLIT=s1
    /FORMAT=FULL
    /CONTENTS=MEAN SD N CORR.
BEGIN DATA.
0 34 35 36 37
0 22 11 55 66
0 99 98 99 92
0  1 .9 .8 .7
0 .9  1 .6 .5
0 .8 .6  1 .4
0 .7 .5 .4  1
1 44 45 34 39
1 23 15 51 46
1 98 34 87 23
1  1 .2 .3 .4
1 .2  1 .5 .6
1 .3 .5  1 .7
1 .4 .6 .7  1
END DATA.
@end example

@subsubheading Example 8: Split variable with sequential values
@anchor{MATRIX DATA Example 8}

Like this previous example, this syntax defines two matrices with
split variable @code{s1}.  In this case, though, @code{s1} is not
listed in @subcmd{VARIABLES}, which means that its value does not
appear in the data.  Instead, @cmd{MATRIX DATA} reads matrix data
until the input is exhausted, supplying 1 for the first split, 2 for
the second, and so on.

@example
MATRIX DATA
    VARIABLES=var01 TO var04
    /SPLIT=s1
    /FORMAT=FULL
    /CONTENTS=MEAN SD N CORR.
BEGIN DATA.
34 35 36 37
22 11 55 66
99 98 99 92
 1 .9 .8 .7
.9  1 .6 .5
.8 .6  1 .4
.7 .5 .4  1
44 45 34 39
23 15 51 46
98 34 87 23
 1 .2 .3 .4
.2  1 .5 .6
.3 .5  1 .7
.4 .6 .7  1
END DATA.
@end example

@subsubsection Factor variables without @code{ROWTYPE_}

Without @subcmd{ROWTYPE_}, factor variables introduce two new wrinkles
to @cmd{MATRIX DATA} syntax.  First, the @subcmd{CELLS} subcommand
must declare the number of combinations of factor variables present in
the data.  If there is, for example, one factor variable for which the
data contains three values, one would write @code{CELLS=3}; if there
are two (or more) factor variables for which the data contains five
combinations, one would use @code{CELLS=5}; and so on.

Second, the @subcmd{CONTENTS} subcommand must distinguish within-cell
data from pooled data by enclosing within-cell row types in
parentheses.  When different within-cell row types for a single factor
appear in subsequent lines, enclose the row types in a single set of
parentheses; when different factors' values for a given within-cell
row type appear in subsequent lines, enclose each row type in
individual parentheses.

Without @subcmd{ROWTYPE_}, input lines for pooled data do not include
factor values, not even as missing values, but input lines for
within-cell data do.

The following examples aim to clarify this syntax.

@subsubheading Example 9: Factor variables, grouping within-cell records by factor

This syntax defines the same matrix file as Example 5 (@pxref{MATRIX
DATA Example 5}), without using @code{ROWTYPE_}.  It declares
@code{CELLS=2} because the data contains two values (0 and 1) for
factor variable @code{f1}.  Within-cell vector row types @code{MEAN},
@code{SD}, and @code{N} are in a single set of parentheses on
@subcmd{CONTENTS} because they are grouped together in subsequent
lines for a single factor value.  The data lines with the pooled
correlation matrix do not have any factor values.

@example
MATRIX DATA
    VARIABLES=f1 var01 TO var04
    /FACTOR=f1
    /CELLS=2
    /CONTENTS=(MEAN SD N) CORR.
BEGIN DATA.
0 34 35 36 37
0 22 11 55 66
0 99 98 99 92
1 44 45 34 39
1 23 15 51 46
1 98 34 87 23
   1
  .9  1
  .8 .6  1
  .7 .5 .4  1
END DATA.
@end example

@subsubheading Example 10: Factor variables, grouping within-cell records by row type

This syntax defines the same matrix file as the previous example.  The
only difference is that the within-cell vector rows are grouped
differently: two rows of means (one for each factor), followed by two
rows of standard deviations, followed by two rows of counts.

@example
MATRIX DATA
    VARIABLES=f1 var01 TO var04
    /FACTOR=f1
    /CELLS=2
    /CONTENTS=(MEAN) (SD) (N) CORR.
BEGIN DATA.
0 34 35 36 37
1 44 45 34 39
0 22 11 55 66
1 23 15 51 46
0 99 98 99 92
1 98 34 87 23
   1
  .9  1
  .8 .6  1
  .7 .5 .4  1
END DATA.
@end example

@node MCONVERT
@section MCONVERT
@vindex MCONVERT

@display
MCONVERT
    [[MATRIX=]
     [IN(@{@samp{*}|'@var{file}'@})]
     [OUT(@{@samp{*}|'@var{file}'@})]]
    [/@{REPLACE,APPEND@}].
@end display

The @cmd{MCONVERT} command converts matrix data from a correlation
matrix and a vector of standard deviations into a covariance matrix,
or vice versa.

By default, @cmd{MCONVERT} both reads and writes the active file.  Use
the @cmd{MATRIX} subcommand to specify other files.  To read a matrix
file, specify its name inside parentheses following @code{IN}.  To
write a matrix file, specify its name inside parentheses following
@code{OUT}.  Use @samp{*} to explicitly specify the active file for
input or output.

When @cmd{MCONVERT} reads the input, by default it substitutes a
correlation matrix and a vector of standard deviations each time it
encounters a covariance matrix, and vice versa.  Specify
@code{/APPEND} to instead have @cmd{MCONVERT} add the other form of
data without removing the existing data.  Use @code{/REPLACE} to
explicitly request removing the existing data.

The @cmd{MCONVERT} command requires its input to be a matrix file.
Use @cmd{MATRIX DATA} to convert text input into matrix file format.
@xref{MATRIX DATA}, for details.

@node MATRIX
@section MATRIX
@vindex MATRIX
@vindex END MATRIX

@display
@t{MATRIX.}
@dots{}@i{matrix commands}@dots{}
@t{END MATRIX.}
@end display

@noindent
The following basic matrix commands are supported:

@display
@t{COMPUTE} @i{variable}[@t{(}@i{index}[@t{,}@i{index}]@t{)}]@t{=}@i{expression}@t{.}
@t{CALL} @i{procedure}@t{(}@i{argument}@t{,} @dots{}).
@t{PRINT} [@i{expression}]
      [@t{/FORMAT}@t{=}@i{format}]
      [@t{/TITLE}@t{=}@i{title}]
      [@t{/SPACE}@t{=}@{@t{NEWPAGE} @math{|} @i{n}@}]
      [@{@t{/RLABELS}@t{=}@i{string}@dots{} @math{|} @t{/RNAMES}@t{=}@i{expression}@}]
      [@{@t{/CLABELS}@t{=}@i{string}@dots{} @math{|} @t{/CNAMES}@t{=}@i{expression}@}]@t{.}
@end display

@noindent
The following matrix commands offer support for flow control:

@display
@t{DO IF} @i{expression}@t{.}
  @dots{}@i{matrix commands}@dots{}
[@t{ELSE IF} @i{expression}@t{.}
  @dots{}@i{matrix commands}@dots{}]@dots{}
[@t{ELSE}
  @dots{}@i{matrix commands}@dots{}]
@t{END IF}@t{.}

@t{LOOP} [@i{var}@t{=}@i{first} @t{TO} @i{last} [@t{BY} @i{step}]] [@t{IF} @i{expression}]@t{.}
  @dots{}@i{matrix commands}@dots{}
@t{END LOOP} [@t{IF} @i{expression}]@t{.}

@t{BREAK}@t{.}
@end display

@noindent
The following matrix commands support matrix input and output:

@display
@t{READ} @i{variable}[@t{(}@i{index}[@t{,}@i{index}]@t{)}]
     [@t{/FILE}@t{=}@i{file}]
     @t{/FIELD}@t{=}@i{first} @t{TO} @i{last} [@t{BY} @i{width}]
     [@t{/FORMAT}@t{=}@i{format}]
     [@t{/SIZE}@t{=}@i{expression}]
     [@t{/MODE}@t{=}@{@t{RECTANGULAR} @math{|} @t{SYMMETRIC}@}]
     [@t{/REREAD}]@t{.}
@t{WRITE} @i{expression}
      [@t{/OUTFILE}@t{=}@i{file}]
      @t{/FIELD}@t{=}@i{first} @t{TO} @i{last} [@t{BY} @i{width}]
      [@t{/MODE}@t{=}@{@t{RECTANGULAR} @math{|} @t{TRIANGULAR}@}]
      [@t{/HOLD}]
      [@t{/FORMAT}@t{=}@i{format}]@t{.}
@t{GET} @i{variable}[@t{(}@i{index}[@t{,}@i{index}]@t{)}]
    [@t{/FILE}@t{=}@{@i{file} @math{|} @t{*}@}]
    [@t{/VARIABLES}@t{=}@i{variable}@dots{}]
    [@t{/NAMES}@t{=}@i{expression}]
    [@t{/MISSING}@t{=}@{@t{ACCEPT} @math{|} @t{OMIT} @math{|} @i{number}@}]
    [@t{/SYSMIS}@t{=}@{@t{OMIT} @math{|} @i{number}@}]@t{.}
@t{SAVE} @i{expression}
     [@t{/OUTFILE}@t{=}@{@i{file} @math{|} @t{*}@}]
     [@t{/VARIABLES}@t{=}@i{variable}@dots{}]
     [@t{/NAMES}@t{=}@i{expression}]
     [@t{/STRINGS}@t{=}@i{variable}@dots{}]@t{.}
@t{MGET} [@t{/FILE}@t{=}@i{file}]
     [@t{/TYPE}@t{=}@{@t{COV} @math{|} @t{CORR} @math{|} @t{MEAN} @math{|} @t{STDDEV} @math{|} @t{N} @math{|} @t{COUNT}@}]@t{.}
@t{MSAVE} @i{expression}
      @t{/TYPE}@t{=}@{@t{COV} @math{|} @t{CORR} @math{|} @t{MEAN} @math{|} @t{STDDEV} @math{|} @t{N} @math{|} @t{COUNT}@}
      [@t{/OUTFILE}@t{=}@i{file}]
      [@t{/VARIABLES}@t{=}@i{variable}@dots{}]
      [@t{/SNAMES}@t{=}@i{variable}@dots{}]
      [@t{/SPLIT}@t{=}@i{expression}]
      [@t{/FNAMES}@t{=}@i{variable}@dots{}]
      [@t{/FACTOR}@t{=}@i{expression}]@t{.}
@end display

@noindent
The following matrix commands provide additional support:

@display
@t{DISPLAY} [@{@t{DICTIONARY} @math{|} @t{STATUS}@}]@t{.}
@t{RELEASE} @i{variable}@dots{}@t{.}
@end display

@code{MATRIX} and @code{END MATRIX} enclose a special @pspp{}
sub-language, called the matrix language.  The matrix language does
not require an active dataset to be defined and only a few of the
matrix language commands work with any datasets that are defined.
Each instance of @code{MATRIX}@dots{}@code{END MATRIX} is a separate
program whose state is independent of any instance, so that variables
declared within a matrix program are forgotten at its end.

The matrix language works with matrices, where a @dfn{matrix} is a
rectangular array of real numbers.  An @math{@var{n}@times{}@var{m}}
matrix has @var{n} rows and @var{m} columns.  Some special cases are
important: a @math{@var{n}@times{}1} matrix is a @dfn{column vector},
a @math{1@times{}@var{n}} is a @dfn{row vector}, and a
@math{1@times{}1} matrix is a @dfn{scalar}.

The matrix language also has limited support for matrices that contain
8-byte strings instead of numbers.  Strings longer than 8 bytes are
truncated, and shorter strings are padded with spaces.  String
matrices are mainly useful for labeling rows and columns when printing
numerical matrices with the @code{MATRIX PRINT} command.  Arithmetic
operations on string matrices will not produce useful results.  The
user should not mix strings and numbers within a matrix.

The matrix language does not work with cases.  A variable in the
matrix language represents a single matrix.

The matrix language does not support missing values.

@code{MATRIX} is a procedure, so it cannot be enclosed inside @code{DO
IF}, @code{LOOP}, etc.

Macros may be used within a matrix program, and macros may expand to
include entire matrix programs.  The @code{DEFINE} command may not
appear within a matrix program.  @xref{DEFINE}, for more information
about macros.

The following sections describe the details of the matrix language:
first, the syntax of matrix expressions, then each of the supported
commands.  The @code{COMMENT} command (@pxref{COMMENT}) is also
supported.

@node Matrix Expressions
@subsection Matrix Expressions

Many matrix commands use expressions.  A matrix expression may use the
following operators, listed in descending order of operator
precedence.  Within a single level, operators associate from left to
right.

@itemize @bullet
@item
Function call @t{()} and matrix construction @t{@{@}}

@item
Indexing @t{()}

@item
Unary @t{+} and @t{-}

@item
Integer sequence @t{:}

@item
Exponentiation @t{**} and @t{&**}

@item
Multiplication @t{*} and @t{&*}, and division @t{/} and @t{&/}

@item
Addition @t{+} and subtraction @t{-}

@item
Relational @t{< <= = >= > <>}

@item
Logical @t{NOT}

@item
Logical @t{AND}

@item
Logical @t{OR} and @t{XOR}
@end itemize

@xref{Matrix Functions}, for the available matrix functions.  The
remaining operators are described in more detail below.

@cindex restricted expressions
Expressions appear in the matrix language in some contexts where there
would be ambiguity whether @samp{/} is an operator or a separator
between subcommands.  In these contexts, only the operators with
higher precedence than @samp{/} are allowed outside parentheses.
Later sections call these @dfn{restricted expressions}.

@node Matrix Construction Operator
@subsubsection Matrix Construction Operator @t{@{@}}

Use the @t{@{}@t{@}} operator to construct matrices.  Within
the curly braces, commas separate elements within a row and semicolons
separate rows.  The following examples show a @math{2@times{}3}
matrix, a @math{1@times{}4} row vector, a @math{3@times{}1} column
vector, and a scalar.

@multitable @columnfractions .4 .05 .4
@item @t{@{1, 2, 3; 4, 5, 6@}}
@tab @result{}
@tab
@ifnottex
@t{[1  2  3] @* [4  5  6]}
@end ifnottex
@iftex
@math{\left(\matrix{1 & 2 & 3 \cr 4 & 5 & 6}\right)}
@end iftex
@
@item @t{@{3.14, 6.28, 9.24, 12.57@}}
@tab @result{}
@tab
@ifnottex
[3.14  6.28  9.42  12.57]
@end ifnottex
@iftex
@math{(\matrix{3.14 & 6.28 & 9.42 & 12.57})}
@end iftex
@
@item @t{@{1.41; 1.73; 2@}}
@tab @result{}
@tab
@ifnottex
@t{[1.41] @* [1.73] @* [2.00]}
@end ifnottex
@iftex
@math{(\matrix{1.41 & 1.73 & 2.00})}
@end iftex
@
@item @t{@{5@}}
@tab @result{}
@tab 5
@end multitable

Curly braces are not limited to holding numeric literals.  They can
contain calculations, and they can paste together matrices and vectors
in any way as long as the result is rectangular.  For example, if
@samp{m} is matrix @code{@{1, 2; 3, 4@}}, @samp{r} is row vector
@code{@{5, 6@}}, and @samp{c} is column vector @code{@{7, 8@}}, then
curly braces can be used as follows:

@multitable @columnfractions .4 .05 .4
@item @t{@{m, c; r, 10@}}
@tab @result{}
@tab
@ifnottex
@t{[1 2  7] @* [3 4  8] @* [5 6 10]}
@end ifnottex
@iftex
@math{\left(\matrix{1 & 2 & 7 \cr 3 & 4 & 8 \cr 5 & 6 & 10}\right)}
@end iftex
@
@item @t{@{c, 2 * c, T(r)@}}
@tab @result{}
@tab
@ifnottex
@t{[7 14 5] @* [8 16 6]}
@end ifnottex
@iftex
@math{\left(\matrix{7 & 14 & 5 \cr 8 & 16 & 6}\right)}
@end iftex
@end multitable

The final example above uses the transposition function @code{T}.

@node Matrix Sequence Operator
@subsubsection Integer Sequence Operator @samp{:}

The syntax @code{@var{first}:@var{last}:@var{step}} yields a row
vector of consecutive integers from @var{first} to @var{last} counting
by @var{step}.  The final @code{:@var{step}} is optional and
defaults to 1 when omitted.

Each of @var{first}, @var{last}, and @var{step} must be a scalar and
should be an integer (any fractional part is discarded).  Because
@samp{:} has a high precedence, operands other than numeric literals
must usually be parenthesized.

When @var{step} is positive (or omitted) and @math{@var{end} <
@var{start}}, or if @var{step} is negative and @math{@var{end} >
@var{start}}, then the result is an empty matrix.  If @var{step} is 0,
then @pspp{} reports an error.

Here are some examples:

@multitable @columnfractions .4 .05 .4
@item @t{1:6}      @tab @result{} @tab @t{@{1, 2, 3, 4, 5, 6@}}
@item @t{1:6:2}    @tab @result{} @tab @t{@{1, 3, 5@}}
@item @t{-1:-5:-1} @tab @result{} @tab @t{@{-1, -2, -3, -4, -5@}}
@item @t{-1:-5}    @tab @result{} @tab @t{@{@}}
@item @t{2:1:0}    @tab @result{} @tab (error)
@end multitable

@node Matrix Index Operator
@subsubsection Index Operator @code{()}

The result of the submatrix or indexing operator, written
@code{@var{m}(@var{rindex}, @var{cindex})}, contains the rows of
@var{m} whose indexes are given in vector @var{rindex} and the columns
whose indexes are given in vector @var{cindex}.

In the simplest case, if @var{rindex} and @var{cindex} are both
scalars, the result is also a scalar:

@multitable @columnfractions .4 .05 .4
@item @t{@{10, 20; 30, 40@}(1, 1)} @tab @result{} @tab @t{10}
@item @t{@{10, 20; 30, 40@}(1, 2)} @tab @result{} @tab @t{20}
@item @t{@{10, 20; 30, 40@}(2, 1)} @tab @result{} @tab @t{30}
@item @t{@{10, 20; 30, 40@}(2, 2)} @tab @result{} @tab @t{40}
@end multitable

If the index arguments have multiple elements, then the result
includes multiple rows or columns:

@multitable @columnfractions .4 .05 .4
@item @t{@{10, 20; 30, 40@}(1:2, 1)} @tab @result{} @tab @t{@{10; 30@}}
@item @t{@{10, 20; 30, 40@}(2, 1:2)} @tab @result{} @tab @t{@{30, 40@}}
@item @t{@{10, 20; 30, 40@}(1:2, 1:2)} @tab @result{} @tab @t{@{10, 20; 30, 40@}}
@end multitable

The special argument @samp{:} may stand in for all the rows or columns
in the matrix being indexed, like this:

@multitable @columnfractions .4 .05 .4
@item @t{@{10, 20; 30, 40@}(:, 1)} @tab @result{} @tab @t{@{10; 30@}}
@item @t{@{10, 20; 30, 40@}(2, :)} @tab @result{} @tab @t{@{30, 40@}}
@item @t{@{10, 20; 30, 40@}(:, :)} @tab @result{} @tab @t{@{10, 20; 30, 40@}}
@end multitable

The index arguments do not have to be in order, and they may contain
repeated values, like this:

@multitable @columnfractions .4 .05 .4
@item @t{@{10, 20; 30, 40@}(@{2, 1@}, 1)} @tab @result{} @tab @t{@{30; 10@}}
@item @t{@{10, 20; 30, 40@}(2, @{2; 2; 1@})} @tab @result{} @tab @t{@{40, 40, 30@}}
@item @t{@{10, 20; 30, 40@}(2:1:-1, :)} @tab @result{} @tab @t{@{30, 40; 10, 20@}}
@end multitable

When the matrix being indexed is a row or column vector, only a single
index argument is needed, like this:

@multitable @columnfractions .4 .05 .4
@item @t{@{11, 12, 13, 14, 15@}(2:4)} @tab @result{} @tab @t{@{12, 13, 14@}}
@item @t{@{11; 12; 13; 14; 15@}(2:4)} @tab @result{} @tab @t{@{12; 13; 14@}}
@end multitable

When an index is not an integer, @pspp{} discards the fractional part.
It is an error for an index to be less than 1 or greater than the
number of rows or columns:

@multitable @columnfractions .4 .05 .4
@item @t{@{11, 12, 13, 14@}(@{2.5, 4.6@})} @tab @result{} @tab @t{@{12, 14@}}
@item @t{@{11; 12; 13; 14@}(0)} @tab @result{} @tab (error)
@end multitable

@node Matrix Unary Operators
@subsubsection Unary Operators

The unary operators take a single operand of any dimensions and
operate on each of its elements independently.  The unary operators
are:

@table @code
@item -
Inverts the sign of each element.

@item +
No change.

@item NOT
Logical inversion: each positive value becomes 0 and each zero or
negative value becomes 1.
@end table

Examples:

@multitable @columnfractions .4 .05 .4
@item @t{-@{1, -2; 3, -4@}} @tab @result{} @tab @t{@{-1, 2; -3, 4@}}
@item @t{+@{1, -2; 3, -4@}} @tab @result{} @tab @t{@{1, -2; 3, -4@}}
@item @t{NOT @{1, 0; -1, 1@}} @tab @result{} @tab @t{@{0, 1; 1, 0@}}
@end multitable

@node Matrix Elementwise Binary Operators
@subsubsection Elementwise Binary Operators

The elementwise binary operators require their operands to be matrices
with the same dimensions.  Alternatively, if one operand is a scalar,
then its value is treated as if it were duplicated to the dimensions
of the other operand.  The result is a matrix of the same size as the
operands, in which each element is the result of the applying the
operator to the corresponding elements of the operands.

The elementwise binary operators are listed below.

@itemize @bullet
@item
The arithmetic operators, for familiar arithmetic operations:

@table @asis
@item @code{+}
Addition.

@item @code{-}
Subtraction.

@item @code{*}
Multiplication, if one operand is a scalar.  (Otherwise this is matrix
multiplication, described below.)

@item @code{/} or @code{&/}
Division.

@item @code{&*}
Multiplication.

@item @code{&**}
Exponentiation.
@end table

@item
The relational operators, whose results are 1 when a comparison is
true and 0 when it is false:

@table @asis
@item @code{<} or @code{LT}
Less than.

@item @code{<=} or @code{LE}
Less than or equal.

@item @code{=} or @code{EQ}
Equal.

@item @code{>} or @code{GT}
Greater than.

@item @code{>=} or @code{GE}
Greater than or equal.

@item @code{<>} or @code{~=} or @code{NE}
Not equal.
@end table

@item
The logical operators, which treat positive operands as true and
nonpositive operands as false.  They yield 0 for false and 1 for true:

@table @code
@item AND
True if both operands are true.

@item OR
True if at least one operand is true.

@item XOR
True if exactly one operand is true.
@end table
@end itemize

Examples:

@multitable @columnfractions .4 .05 .4
@item @t{1 + 2} @tab @result{} @tab @t{3}
@item @t{1 + @{3; 4@}} @tab @result{} @tab @t{@{4; 5@}}
@item @t{@{66, 77; 88, 99@} + 5} @tab @result{} @tab @t{@{71, 82; 93, 104@}}
@item @t{@{4, 8; 3, 7@} + @{1, 0; 5, 2@}} @tab @result{} @tab @t{@{5, 8; 8, 9@}}
@item @t{@{1, 2; 3, 4@} < @{4, 3; 2, 1@}} @tab @result{} @tab @t{@{1, 1; 0, 0@}}
@item @t{@{1, 3; 2, 4@} >= 3} @tab @result{} @tab @t{@{0, 1; 0, 1@}}
@item @t{@{0, 0; 1, 1@} AND @{0, 1; 0, 1@}} @tab @result{} @tab @t{@{0, 0; 0, 1@}}
@end multitable

@node Matrix Multiplication Operator
@subsubsection Matrix Multiplication Operator @samp{*}

If @code{A} is an @math{@var{m}@times{}@var{n}} matrix and @code{B} is
an @math{@var{n}@times{}@var{p}} matrix, then @code{A*B} is the
@math{@var{m}@times{}@var{p}} matrix multiplication product @code{C}.
@pspp{} reports an error if the number of columns in @code{A} differs
from the number of rows in @code{B}.

The @code{*} operator performs elementwise multiplication (see above)
if one of its operands is a scalar.

No built-in operator yields the inverse of matrix multiplication.
Instead, multiply by the result of @code{INV} or @code{GINV}.

Some examples:

@multitable @columnfractions .4 .05 .4
@item @t{@{1, 2, 3@} * @{4; 5; 6@}} @tab @result{} @tab @t{32}
@item @t{@{4; 5; 6@} * @{1, 2, 3@}} @tab @result{} @tab @t{@{4,@w{ } 8, 12; @*@w{ }5, 10, 15; @*@w{ }6, 12, 18@}}
@end multitable

@node Matrix Exponentiation Operator
@subsubsection Matrix Exponentiation Operator @code{**}

The result of @code{A**B} is defined as follows when @code{A} is a
square matrix and @code{B} is an integer scalar:

@itemize @bullet
@item
For @code{B > 0}, @code{A**B} is @code{A*@dots{}*A}, where there are
@code{B} @samp{A}s.  (@pspp{} implements this efficiently for large
@code{B}, using exponentiation by squaring.)

@item
For @code{B < 0}, @code{A**B} is @code{INV(A**(-B))}.

@item
For @code{B = 0}, @code{A**B} is the identity matrix.
@end itemize

@noindent
@pspp{} reports an error if @code{A} is not square or @code{B} is not
an integer.

Examples:

@multitable @columnfractions .4 .05 .4
@item @t{@{2, 5; 1, 4@}**3} @tab @result{} @tab @t{@{48, 165; 33, 114@}}
@item @t{@{2, 5; 1, 4@}**0} @tab @result{} @tab @t{@{1, 0; 0, 1@}}
@item @t{10*@{4, 7; 2, 6@}**-1} @tab @result{} @tab @t{@{6, -7; -2, 4@}}
@end multitable

@node Matrix Functions
@subsection Matrix Functions

The matrix language support numerous functions in multiple categories.
The following subsections document each of the currently supported
functions.  The first letter of each parameter's name indicate the
required argument type:

@table @var
@item s
A scalar.

@item n
A nonnegative integer scalar.  (Non-integers are accepted and silently
rounded down to the nearest integer.)

@item V
A row or column vector.

@item M
A matrix.
@end table

@node Matrix Elementwise Functions
@subsubsection Elementwise Functions

These functions act on each element of their argument independently,
like the elementwise operators (@pxref{Matrix Elementwise Binary
Operators}).

@deffn {Matrix Function} ABS (@var{M})
Takes the absolute value of each element of @var{M}.

@t{ABS(@{-1, 2; -3, 0@}) @result{} @{1, 2; 3, 0@}}
@end deffn

@deffn {Matrix Function} ARSIN (@var{M})
@deffnx {Matrix Function} ARTAN (@var{M})
Computes the inverse sine or tangent, respectively, of each element in
@var{M}.  The results are in radians, between @math{-\pi/2} and
@math{+\pi/2}, inclusive.

The value of @math{\pi} can be computed as @code{4*ARTAN(1)}.

@t{ARSIN(@{-1, 0, 1@}) @result{} @{-1.57, 0, 1.57@}} (approximately)

@t{ARTAN(@{-5, -1, 1, 5@}) @result{} @{-1.37, -.79, .79, 1.37@}} (approximately)
@end deffn

@deffn {Matrix Function} COS (@var{M})
@deffnx {Matrix Function} SIN (@var{M})
Computes the cosine or sine, respectively, of each element in @var{M},
which must be in radians.

@t{COS(@{0.785, 1.57; 3.14, 1.57 + 3.14@}) @result{} @{.71, 0; -1, 0@}} (approximately)
@end deffn

@deffn {Matrix Function} EXP (@var{M})
Computes @math{e^x} for each element @var{x} in @var{M}.

@t{EXP(@{2, 3; 4, 5@}) @result{} @{7.39, 20.09; 54.6, 148.4@}} (approximately)
@end deffn

@deffn {Matrix Function} LG10 (@var{M})
@deffnx {Matrix Function} LN (@var{M})
Takes the logarithm with base 10 or base @math{e}, respectively, of
each element in @var{M}.

@t{LG10(@{1, 10, 100, 1000@}) @result{} @{0, 1, 2, 3@}} @*
@t{LG10(0) @result{}} (error)

@t{LN(@{EXP(1), 1, 2, 3, 4@}) @result{} @{1, 0, .69, 1.1, 1.39@}} (approximately) @*
@t{LN(0) @result{}} (error)
@end deffn

@deffn {Matrix Function} MOD (@var{M}, @var{s})
Takes each element in @var{M} modulo nonzero scalar value @var{s},
that is, the remainder of division by @var{s}.  The sign of the result
is the same as the sign of the dividend.

@t{MOD(@{5, 4, 3, 2, 1, 0@}, 3) @result{} @{2, 1, 0, 2, 1, 0@}} @*
@t{MOD(@{5, 4, 3, 2, 1, 0@}, -3) @result{} @{2, 1, 0, 2, 1, 0@}} @*
@t{MOD(@{-5, -4, -3, -2, -1, 0@}, 3) @result{} @{-2, -1, 0, -2, -1, 0@}} @*
@t{MOD(@{-5, -4, -3, -2, -1, 0@}, -3) @result{} @{-2, -1, 0, -2, -1, 0@}} @*
@t{MOD(@{5, 4, 3, 2, 1, 0@}, 1.5) @result{} @{.5, 1.0, .0, .5, 1.0, .0@}} @*
@t{MOD(@{5, 4, 3, 2, 1, 0@}, 0) @result{}} (error)
@end deffn

@deffn {Matrix Function} RND (@var{M})
@deffnx {Matrix Function} TRUNC (@var{M})
Rounds each element of @var{M} to an integer.  @code{RND} rounds to
the nearest integer, with halves rounded to even integers, and
@code{TRUNC} rounds toward zero.

@t{RND(@{-1.6, -1.5, -1.4@}) @result{} @{-2, -2, -1@}} @*
@t{RND(@{-.6, -.5, -.4@}) @result{} @{-1, 0, 0@}} @*
@t{RND(@{.4, .5, .6@} @result{} @{0, 0, 1@}} @*
@t{RND(@{1.4, 1.5, 1.6@}) @result{} @{1, 2, 2@}}

@t{TRUNC(@{-1.6, -1.5, -1.4@}) @result{} @{-1, -1, -1@}} @*
@t{TRUNC(@{-.6, -.5, -.4@}) @result{} @{0, 0, 0@}} @*
@t{TRUNC(@{.4, .5, .6@} @result{} @{0, 0, 0@}} @*
@t{TRUNC(@{1.4, 1.5, 1.6@}) @result{} @{1, 1, 1@}}
@end deffn

@deffn {Matrix Function} SQRT (@var{M})
Takes the square root of each element of @var{M}, which must not be
negative.

@t{SQRT(@{0, 1, 2, 4, 9, 81@}) @result{} @{0, 1, 1.41, 2, 3, 9@}} (approximately) @*
@t{SQRT(-1) @result{}} (error)
@end deffn

@node Matrix Logical Functions
@subsubsection Logical Functions

@deffn {Matrix Function} ALL (@var{M})
Returns a scalar with value 1 if all of the elements in @var{M} are
nonzero, or 0 if at least one element is zero.

@t{ALL(@{1, 2, 3@} < @{2, 3, 4@}) @result{} 1} @*
@t{ALL(@{2, 2, 3@} < @{2, 3, 4@}) @result{} 0} @*
@t{ALL(@{2, 3, 3@} < @{2, 3, 4@}) @result{} 0} @*
@t{ALL(@{2, 3, 4@} < @{2, 3, 4@}) @result{} 0}
@end deffn

@deffn {Matrix Function} ANY (@var{M})
Returns a scalar with value 1 if any of the elements in @var{M} is
nonzero, or 0 if all of them are zero.

@t{ANY(@{1, 2, 3@} < @{2, 3, 4@}) @result{} 1} @*
@t{ANY(@{2, 2, 3@} < @{2, 3, 4@}) @result{} 1} @*
@t{ANY(@{2, 3, 3@} < @{2, 3, 4@}) @result{} 1} @*
@t{ANY(@{2, 3, 4@} < @{2, 3, 4@}) @result{} 0}
@end deffn

@node Matrix Construction Functions
@subsubsection Matrix Construction Functions

@deffn {Matrix Function} BLOCK (@var{M1}, @dots{}, @var{Mn})
Returns a block diagonal matrix with as many rows as the sum of its
arguments' row counts and as many columns as the sum of their columns.
Each argument matrix is placed along the main diagonal of the result,
and all other elements are zero.

@format
@t{BLOCK(@{1, 2; 3, 4@}, 5, @{7; 8; 9@}, @{10, 11@}) @result{}
   1   2   0   0   0   0
   3   4   0   0   0   0
   0   0   5   0   0   0
   0   0   0   7   0   0
   0   0   0   8   0   0
   0   0   0   9   0   0
   0   0   0   0  10  11}
@end format
@end deffn

@deffn {Matrix Function} IDENT (@var{n})
@deffnx {Matrix Function} IDENT (@var{nr}, @var{nc})
Returns an identity matrix, whose main diagonal elements are one and
whose other elements are zero.  The returned matrix has @var{n} rows
and columns or @var{nr} rows and @var{nc} columns, respectively.

@format
@t{IDENT(1) @result{} 1
IDENT(2) @result{}
  1  0
  0  1
IDENT(3, 5) @result{}
  1  0  0  0  0
  0  1  0  0  0
  0  0  1  0  0
IDENT(5, 3) @result{}
  1  0  0
  0  1  0
  0  0  1
  0  0  0
  0  0  0}
@end format
@end deffn

@deffn {Matrix Function} MAGIC (@var{n})
Returns an @math{@var{n}@times{}@var{n}} matrix that contains each of
the integers @math{1@dots{}@var{n}} once, in which each column, each
row, and each diagonal sums to @math{n(n^2+1)/2}.  There are many
magic squares with given dimensions, but this function always returns
the same one for a given value of @var{n}.

@t{MAGIC(3) @result{} @{8, 1, 6; 3, 5, 7; 4, 9, 2@}} @*
@t{MAGIC(4) @result{} @{1, 5, 12, 16; 15, 11, 6, 2; 14, 8, 9, 3; 4, 10, 7, 13@}}
@end deffn

@deffn {Matrix Function} MAKE (@var{nr}, @var{nc}, @var{s})
Returns an @math{@var{nr}@times{}@var{nc}} matrix whose elements are
all @var{s}.

@t{MAKE(1, 2, 3) @result{} @{3, 3@}} @*
@t{MAKE(2, 1, 4) @result{} @{4; 4@}} @*
@t{MAKE(2, 3, 5) @result{} @{5, 5, 5; 5, 5, 5@}}
@end deffn

@deffn {Matrix Function} MDIAG (@var{V})
@anchor{MDIAG} Given @var{n}-element vector @var{V}, returns a
@math{@var{n}@times{}@var{n}} matrix whose main diagonal is copied
from @var{V}.  The other elements in the returned vector are zero.

Use @code{CALL SETDIAG} (@pxref{CALL SETDIAG}) to replace the main
diagonal of a matrix in-place.

@format
@t{MDIAG(@{1, 2, 3, 4@}) @result{}
  1  0  0  0
  0  2  0  0
  0  0  3  0
  0  0  0  4}
@end format
@end deffn

@deffn {Matrix Function} RESHAPE (@var{M}, @var{nr}, @var{nc})
Returns an @math{@var{nr}@times{}@var{nc}} matrix whose elements come
from @var{M}, which must have the same number of elements as the new
matrix, copying elements from @var{M} to the new matrix row by row.

@format
@t{RESHAPE(1:12, 1, 12) @result{}
   1   2   3   4   5   6   7   8   9  10  11  12
RESHAPE(1:12, 2, 6) @result{}
   1   2   3   4   5   6
   7   8   9  10  11  12
RESHAPE(1:12, 3, 4) @result{}
   1   2   3   4
   5   6   7   8
   9  10  11  12
RESHAPE(1:12, 4, 3) @result{}
   1   2   3
   4   5   6
   7   8   9
  10  11  12}
@end format
@end deffn

@deffn {Matrix Function} T (@var{M})
@deffnx {Matrix Function} TRANSPOS (@var{M})
Returns @var{M} with rows exchanged for columns.

@t{T(@{1, 2, 3@}) @result{} @{1; 2; 3@}} @*
@t{T(@{1; 2; 3@}) @result{} @{1, 2, 3@}}
@end deffn

@deffn {Matrix Function} UNIFORM (@var{nr}, @var{nc})
Returns a @math{@var{nr}@times{}@var{nc}} matrix in which each element
is randomly chosen from a uniform distribution of real numbers between
0 and 1.  Random number generation honors the current seed setting
(@pxref{SET SEED}).

The following example shows one possible output, but of course every
result will be different (given different seeds):

@format
@t{UNIFORM(4, 5)*10 @result{}
  7.71  2.99   .21  4.95  6.34
  4.43  7.49  8.32  4.99  5.83
  2.25   .25  1.98  7.09  7.61
  2.66  1.69  2.64   .88  1.50}
@end format
@end deffn

@node Matrix Minimum and Maximum and Sum Functions
@subsubsection Minimum, Maximum, and Sum Functions

@deffn {Matrix Function} CMIN (@var{M})
@deffnx {Matrix Function} CMAX (@var{M})
@deffnx {Matrix Function} CSUM (@var{M})
@deffnx {Matrix Function} CSSQ (@var{M})
Returns a row vector with the same number of columns as @var{M}, in
which each element is the minimum, maximum, sum, or sum of squares,
respectively, of the elements in the same column of @var{M}.

@t{CMIN(@{1, 2, 3; 4, 5, 6; 7, 8, 9@} @result{} @{1, 2, 3@}} @*
@t{CMAX(@{1, 2, 3; 4, 5, 6; 7, 8, 9@} @result{} @{7, 8, 9@}} @*
@t{CSUM(@{1, 2, 3; 4, 5, 6; 7, 8, 9@} @result{} @{12, 15, 18@}} @*
@t{CSSQ(@{1, 2, 3; 4, 5, 6; 7, 8, 9@} @result{} @{66, 93, 126@}}
@end deffn

@deffn {Matrix Function} MMIN (@var{M})
@deffnx {Matrix Function} MMAX (@var{M})
@deffnx {Matrix Function} MSUM (@var{M})
@deffnx {Matrix Function} MSSQ (@var{M})
Returns the minimum, maximum, sum, or sum of squares, respectively, of
the elements of @var{M}.

@t{MMIN(@{1, 2, 3; 4, 5, 6; 7, 8, 9@} @result{} 1} @*
@t{MMAX(@{1, 2, 3; 4, 5, 6; 7, 8, 9@} @result{} 9} @*
@t{MSUM(@{1, 2, 3; 4, 5, 6; 7, 8, 9@} @result{} 45} @*
@t{MSSQ(@{1, 2, 3; 4, 5, 6; 7, 8, 9@} @result{} 285}
@end deffn

@deffn {Matrix Function} RMIN (@var{M})
@deffnx {Matrix Function} RMAX (@var{M})
@deffnx {Matrix Function} RSUM (@var{M})
@deffnx {Matrix Function} RSSQ (@var{M})
Returns a column vector with the same number of rows as @var{M}, in
which each element is the minimum, maximum, sum, or sum of squares,
respectively, of the elements in the same row of @var{M}.

@t{RMIN(@{1, 2, 3; 4, 5, 6; 7, 8, 9@} @result{} @{1; 4; 7@}} @*
@t{RMAX(@{1, 2, 3; 4, 5, 6; 7, 8, 9@} @result{} @{3; 6; 9@}} @*
@t{RSUM(@{1, 2, 3; 4, 5, 6; 7, 8, 9@} @result{} @{6; 15; 24@}} @*
@t{RSSQ(@{1, 2, 3; 4, 5, 6; 7, 8, 9@} @result{} @{14; 77; 194@}}
@end deffn

@deffn {Matrix Function} SSCP (@var{M})
Returns @math{@var{M}^T @times{} @var{M}}.

@t{SSCP(@{1, 2, 3; 4, 5, 6@}) @result{} @{17, 22, 27; 22, 29, 36; 27, 36, 45@}}
@end deffn

@deffn {Matrix Function} TRACE (@var{M})
Returns the sum of the elements along @var{M}'s main diagonal,
equivalent to @code{MSUM(DIAG(@var{M}))}.

@t{TRACE(MDIAG(1:5)) @result{} 15}
@end deffn

@node Matrix Property Functions
@subsubsection Matrix Property Functions

@deffn {Matrix Function} NROW (@var{M})
@deffnx {Matrix Function} NCOL (@var{M})
Returns the number of row or columns, respectively, in @var{M}.

@format
@t{NROW(@{1, 0; -2, -3; 3, 3@}) @result{} 3
NROW(1:5) @result{} 1

NCOL(@{1, 0; -2, -3; 3, 3@}) @result{} 2
NCOL(1:5) @result{} 5}
@end format
@end deffn

@deffn {Matrix Function} DIAG (@var{M})
Returns a column vector containing a copy of @var{M}'s main diagonal.
The vector's length is the lesser of @code{NCOL(@var{M})} and
@code{NROW(@var{M})}.

@t{DIAG(@{1, 0; -2, -3; 3, 3@}) @result{} @{1; -3@}}
@end deffn

@node Matrix Rank Ordering Functions
@subsubsection Matrix Rank Ordering Functions

The @code{GRADE} and @code{RANK} functions each take a matrix @var{M}
and return a matrix @var{r} with the same dimensions.  Each element in
@var{r} ranges between 1 and the number of elements @var{n} in
@var{M}, inclusive.  When the elements in @var{M} all have unique
values, both of these functions yield the same results: the smallest
element in @var{M} corresponds to value 1 in @var{r}, the next
smallest to 2, and so on, up to the largest to @var{n}.  When multiple
elements in @var{M} have the same value, these functions use different
rules for handling the ties.

@deffn {Matrix Function} GRADE (@var{M})
Returns a ranking of @var{M}, turning duplicate values into sequential
ranks.  The returned matrix always contains each of the integers 1
through the number of elements in the matrix exactly once.

@t{GRADE(@{1, 0, 3; 3, 1, 2; 3, 0, 5@})} @result{} @t{@{3, 1, 6; 7, 4, 5; 8, 2, 9@}}
@end deffn

@deffn {Matrix Function} RNKORDER (@var{M})
Returns a ranking of @var{M}, turning duplicate values into the mean
of their sequential ranks.

@t{RNKORDER(@{1, 0, 3; 3, 1, 2; 3, 0, 5@})} @*@w{ }@result{} @t{@{3.5, 1.5, 7; 7, 3.5, 5; 7, 1.5, 9@}}
@end deffn

@noindent
One may use @code{GRADE} to sort a vector:

@example
COMPUTE v(GRADE(v))=v.   /* Sort v in ascending order.
COMPUTE v(GRADE(-v))=v.  /* Sort v in descending order.
@end example

@node Matrix Algebra Functions
@subsubsection Matrix Algebra Functions

@deffn {Matrix Function} CHOL (@var{M})
Matrix @var{M} must be an @math{@var{n}@times{}@var{n}} symmetric
positive-definite matrix.  Returns an @math{@var{n}@times{}@var{n}}
matrix @var{B} such that @math{@var{B}^T@times{}@var{B}=@var{M}}.

@format
@t{CHOL(@{4, 12, -16; 12, 37, -43; -16, -43, 98@}) @result{}
  2  6 -8
  0  1  5
  0  0  3}
@end format
@end deffn

@deffn {Matrix Function} DESIGN (@var{M})
Returns a design matrix for @var{M}.  The design matrix has the same
number of rows as @var{M}.  Each column @var{c} in @var{M}, from left
to right, yields a group of columns in the output.  For each unique
value @var{v} in @var{c}, from top to bottom, add a column to the
output in which @var{v} becomes 1 and other values become 0.

@pspp{} issues a warning if a column only contains a single unique value.

@format
@t{DESIGN(@{1; 2; 3@}) @result{} @{1, 0, 0; 0, 1, 0; 0, 0, 1@}}
@t{DESIGN(@{5; 8; 5@}) @result{} @{1, 0; 0, 1; 1, 0@}}
@t{DESIGN(@{1, 5; 2, 8; 3, 5@})}
 @result{} @t{@{1, 0, 0, 1, 0; 0, 1, 0, 0, 1; 0, 0, 1, 1, 0@}}
@t{DESIGN(@{5; 5; 5@})} @result{} (warning)
@end format
@end deffn

@deffn {Matrix Function} DET (@var{M})
Returns the determinant of square matrix @var{M}.

@t{DET(@{3, 7; 1, -4@}) @result{} -19}
@end deffn

@deffn {Matrix Function} EVAL (@var{M})
@anchor{EVAL}
Returns a column vector containing the eigenvalues of symmetric matrix
@var{M}, sorted in ascending order.

Use @code{CALL EIGEN} (@pxref{CALL EIGEN}) to compute eigenvalues and
eigenvectors of a matrix.

@t{EVAL(@{2, 0, 0; 0, 3, 4; 0, 4, 9@}) @result{} @{11; 2; 1@}}
@end deffn

@deffn {Matrix Function} GINV (@var{M})
Returns the @math{@var{k}@times{}@var{n}} matrix @var{A} that is the
@dfn{generalized inverse} of @math{@var{n}@times{}@var{k}} matrix
@var{M}, defined such that
@math{@var{M}@times{}@var{A}@times{}@var{M}=@var{M}} and
@math{@var{A}@times{}@var{M}@times{}@var{A}=@var{A}}.

@t{GINV(@{1, 2@}) @result{} @{.2; .4@}} (approximately) @*
@t{@{1:9@} * GINV(1:9) * @{1:9@} @result{} @{1:9@}} (approximately)
@end deffn

@deffn {Matrix Function} GSCH (@var{M})
@var{M} must be a @math{@var{n}@times{}@var{m}} matrix, @math{@var{m}
@geq{} @var{n}}, with rank @var{n}.  Returns an
@math{@var{n}@times{}@var{n}} orthonormal basis for @var{M}, obtained
using the Gram-Schmidt process.

@t{GSCH(@{3, 2; 1, 2@}) * SQRT(10) @result{} @{3, -1; 1, 3@}} (approximately)
@end deffn

@deffn {Matrix Function} INV (@var{M})
Returns the @math{@var{n}@times{}@var{n}} matrix @var{A} that is the
inverse of @math{@var{n}@times{}@var{n}} matrix @var{M}, defined such
that @math{@var{M}@times{}@var{A} = @var{A}@times{}@var{M} = I}, where
@var{I} is the identity matrix.  @var{M} must not be singular, that
is, @math{\det(@var{M}) @ne{} 0}.

@t{INV(@{4, 7; 2, 6@}) @result{} @{.6, -.7; -.2, .4@}} (approximately)
@end deffn

@deffn {Matrix Function} KRONEKER (@var{Ma}, @var{Mb})
Returns the @math{@var{pm}@times{}@var{qn}} matrix @var{P} that is the
@dfn{Kroneker product} of @math{@var{m}@times{}@var{n}} matrix
@var{Ma} and @math{@var{p}@times{}@var{q}} matrix @var{Mb}.  One may
view @var{P} as the concatenation of multiple
@math{@var{p}@times{}@var{q}} blocks, each of which is the scalar
product of @var{Mb} by a different element of @var{Ma}.  For example,
when @code{A} is a @math{2@times{}2} matrix, @code{KRONEKER(A, B)} is
equivalent to @code{@{A(1,1)*B, A(1,2)*B; A(2,1)*B, A(2,2)*B@}}.

@format
@t{KRONEKER(@{1, 2; 3, 4@}, @{0, 5; 6, 7@}) @result{}
   0   5   0  10
   6   7  12  14
   0  15   0  20
  18  21  24  28}
@end format
@end deffn

@deffn {Matrix Function} RANK (@var{M})
Returns the rank of matrix @var{M}, a integer scalar whose value is
the dimension of the vector space spanned by its columns or,
equivalently, by its rows.

@format
@t{RANK(@{1, 0, 1; -2, -3, 1; 3, 3, 0@}) @result{} 2
RANK(@{1, 1, 0, 2; -1, -1, 0, -2@}) @result{} 1
RANK(@{1, -1; 1, -1; 0, 0; 2, -2@}) @result{} 1
RANK(@{1, 2, 1; -2, -3, 1; 3, 5, 0@}) @result{} 2
RANK(@{1, 0, 2; 2, 1, 0; 3, 2, 1@}) @result{} 3}
@end format
@end deffn

@deffn {Matrix Function} SOLVE (@var{Ma}, @var{Mb})
@var{Ma} must be an @math{@var{n}@times{}@var{n}} matrix, with
@math{\det(@var{Ma}) @ne{} 0}, and @var{Mb} an
@math{@var{n}@times{}@var{k}} matrix.  Returns an
@math{@var{n}@times{}@var{k}} matrix @var{X} such that @math{@var{Ma}
@times{} @var{X} = @var{Mb}}.

All of the following examples show approximate results:

@format
@t{SOLVE(@{2, 3; 4, 9@}, @{6, 2; 15, 5@}) @result{}
   1.50    .50
   1.00    .33
SOLVE(@{1, 3, -2; 3, 5, 6; 2, 4, 3@}, @{5; 7; 8@}) @result{}
 -15.00
   8.00
   2.00
SOLVE(@{2, 1, -1; -3, -1, 2; -2, 1, 2@}, @{8; -11; -3@}) @result{}
   2.00
   3.00
  -1.00}
@end format
@end deffn

@deffn {Matrix Function} SVAL (@var{M})
@anchor{SVAL}

Given @math{@var{n}@times{}@var{k}} matrix @var{M}, returns a
@math{\min(@var{n},@var{k})}-element column vector containing the
singular values of @var{M} in descending order.

Use @code{CALL SVD} (@pxref{CALL SVD}) to compute the full singular
value decomposition of a matrix.

@format
@t{SVAL(@{1, 1; 0, 0@}) @result{} @{1.41; .00@}
SVAL(@{1, 0, 1; 0, 1, 1; 0, 0, 0@}) @result{} @{1.73; 1.00; .00@}
SVAL(@{2, 4; 1, 3; 0, 0; 0, 0@}) @result{} @{5.46; .37@}}
@end format
@end deffn

@deffn {Matrix Function} SWEEP (@var{M}, @var{nk})
Given @math{@var{r}@times{}@var{c}} matrix @var{M} and integer scalar
@math{k = @var{nk}} such that @math{1 @leq{} k @leq{}
\min(@var{r},@var{c})}, returns the @math{@var{r}@times{}@var{c}}
sweep matrix @var{A}.

If @math{@var{M}_{kk} @ne{} 0}, then:

@display
@math{@var{A}_{kk} = 1/@var{M}_{kk}},
@math{@var{A}_{ik} = -@var{M}_{ik}/@var{M}_{kk} @r{for} i @ne{} k},
@math{@var{A}_{kj} = @var{M}_{kj}/@var{M}_{kk} @r{for} j @ne{} k, @r{and}}
@math{@var{A}_{ij} = @var{M}_{ij} - @var{M}_{ik}@var{M}_{kj}/@var{M}_{kk} @r{for} i @ne{} k @r{and} j @ne{} k}.
@end display

If @math{@var{M}_{kk} = 0}, then:

@display
@math{@var{A}_{ik} = @var{A}_{ki} = 0 @r{and}}
@math{@var{A}_{ij} = @var{M}_{ij}, @r{for} i @ne{} k @r{and} j @ne{} k}.
@end display

Given @t{M = @{0, 1, 2; 3, 4, 5; 6, 7, 8@}}, then (approximately):

@format
@t{SWEEP(M, 1) @result{}
   .00   .00   .00
   .00  4.00  5.00
   .00  7.00  8.00
SWEEP(M, 2) @result{}
  -.75  -.25   .75
   .75   .25  1.25
   .75 -1.75  -.75
SWEEP(M, 3) @result{}
 -1.50  -.75  -.25
  -.75  -.38  -.63
   .75   .88   .13}
@end format
@end deffn

@node Matrix IO
@subsubsection I/O Function

This function works with files being used on the @code{READ} statement.

@deffn {Matrix Function} EOF (@var{file})
@anchor{EOF Matrix Function}

Given a file handle or file name @var{file}, returns an integer scalar
1 if the last line in the file has been read or 0 if more lines are
available.  Determining this requires attempting to read another line,
which means that @code{REREAD} on the next @code{READ} command
following @code{EOF} on the same file will be ineffective.
@end deffn

The @code{EOF} function gives a matrix program the flexibility to read
a file with text data without knowing the length of the file in
advance.  For example, the following program will read all the lines
of data in @file{data.txt}, each consisting of three numbers, as rows
in matrix @code{data}:

@verbatim
MATRIX.
COMPUTE data={}.
LOOP IF NOT EOF('data.txt').
  READ row/FILE='data.txt'/FIELD=1 TO 1000/SIZE={1,3}.
  COMPUTE data={data; row}.
END LOOP.
PRINT data.
END MATRIX.

@end verbatim

@node Matrix COMPUTE Command
@subsection The @code{COMPUTE} Command

@display
@t{COMPUTE} @i{variable}[@t{(}@i{index}[@t{,}@i{index}]@t{)}]@t{=}@i{expression}@t{.}
@end display

The @code{COMPUTE} command evaluates an expression and assigns the
result to a variable or a submatrix of a variable.  Assigning to a
submatrix uses the same syntax as the index operator (@pxref{Matrix
Index Operator}).

@node Matrix CALL command
@subsection The @code{CALL} Command

A matrix function returns a single result.  The @code{CALL} command
implements procedures, which take a similar syntactic form to
functions but yield results by modifying their arguments rather than
returning a value.

Output arguments to a @code{CALL} procedure must be a single variable
name.

The following procedures are implemented via @code{CALL} to allow them
to return multiple results.  For these procedures, the output
arguments need not name existing variables; if they do, then their
previous values are replaced:

@table @asis
@item @t{CALL EIGEN(@var{M}, @var{evec}, @var{eval})}
@anchor{CALL EIGEN}

Computes the eigenvalues and eigenvector of symmetric
@math{@var{n}@times{}@var{n}} matrix @var{M}.  Assigns the
eigenvectors of @var{M} to the columns of
@math{@var{n}@times{}@var{n}} matrix @var{evec} and the eigenvalues in
descending order to @var{n}-element column vector @var{eval}.

Use the @code{EVAL} function (@pxref{EVAL}) to compute just the
eigenvalues of a symmetric matrix.

For example, the following matrix language commands:
@example
CALL EIGEN(@{1, 0; 0, 1@}, evec, eval).
PRINT evec.
PRINT eval.

CALL EIGEN(@{3, 2, 4; 2, 0, 2; 4, 2, 3@}, evec2, eval2).
PRINT evec2.
PRINT eval2.
@end example

@noindent
yield this output:

@example
evec
  1  0
  0  1

eval
  1
  1

evec2
  -.6666666667   .0000000000   .7453559925
  -.3333333333  -.8944271910  -.2981423970
  -.6666666667   .4472135955  -.5962847940

eval2
  8.0000000000
 -1.0000000000
 -1.0000000000
@end example
  
@item @t{CALL SVD(@var{M}, @var{U}, @var{S}, @var{V})}
@anchor{CALL SVD}

Computes the singular value decomposition of
@math{@var{n}@times{}@var{k}} matrix @var{M}, assigning @var{S} a
@math{@var{n}@times{}@var{k}} diagonal matrix and to @var{U} and
@var{V} unitary @math{@var{k}@times{}@var{k}} matrices such that
@math{@var{M} = @var{U}@times{}@var{S}@times{}@var{V}^T}.  The main
diagonal of @var{Q} contains the singular values of @var{M}.

Use the @code{SVAL} function (@pxref{SVAL}) to compute just the
singular values of a matrix.

For example, the following matrix program:

@example
CALL SVD(@{3, 2, 2; 2, 3, -2@}, u, s, v).
PRINT (u * s * T(v))/FORMAT F5.1.
@end example

@noindent
yields this output:

@example
(u * s * T(v))
   3.0   2.0   2.0
   2.0   3.0  -2.0
@end example
@end table

The final procedure is implemented via @code{CALL} to allow it to
modify a matrix instead of returning a modified version.  For this
procedure, the output argument must name an existing variable.

@table @asis
@item @t{CALL SETDIAG(@var{M}, @var{V})}
@anchor{CALL SETDIAG}

Replaces the main diagonal of @math{@var{n}@times{}@var{p}} matrix
@var{M} by the contents of @var{k}-element vector @var{V}.  If
@math{@var{k} = 1}, so that @var{V} is a scalar, replaces all of the
diagonal elements of @var{M} by @var{V}.  If @math{@var{k} <
\min(@var{n},@var{p})}, only the upper @var{k} diagonal elements are
replaced; if @math{@var{k} > \min(@var{n},@var{p})}, then the 
extra elements of @var{V} are ignored.

Use the @code{MDIAG} function (@pxref{MDIAG}) to construct a new
matrix with a specified main diagonal.

For example, this matrix program:

@example
COMPUTE x=@{1, 2, 3; 4, 5, 6; 7, 8, 9@}.
CALL SETDIAG(x, 10).
PRINT x.
@end example

@noindent
outputs the following:

@example
x
  10   2   3
   4  10   6
   7   8  10
@end example
@end table

@node Matrix PRINT Command
@subsection The @code{PRINT} Command

@display
@t{PRINT} [@i{expression}]
      [@t{/FORMAT}@t{=}@i{format}]
      [@t{/TITLE}@t{=}@i{title}]
      [@t{/SPACE}@t{=}@{@t{NEWPAGE} @math{|} @i{n}@}]
      [@{@t{/RLABELS}@t{=}@i{string}@dots{} @math{|} @t{/RNAMES}@t{=}@i{expression}@}]
      [@{@t{/CLABELS}@t{=}@i{string}@dots{} @math{|} @t{/CNAMES}@t{=}@i{expression}@}]@t{.}
@end display

The @code{PRINT} command is commonly used to display a matrix.  It
evaluates the restricted @var{expression}, if present, and outputs it
either as text or a pivot table, depending on the setting of
@code{MDISPLAY} (@pxref{SET MDISPLAY}).

Use the @code{FORMAT} subcommand to specify a format, such as
@code{F8.2}, for displaying the matrix elements.  @code{FORMAT} is
optional for numerical matrices.  When it is omitted, @pspp{} chooses
how to format entries automatically using @var{m}, the magnitude of
the largest-magnitude element in the matrix to be displayed:

@enumerate
@item
If @math{@var{m} < 10^{11}} and the matrix's elements are all
integers, @pspp{} chooses the narrowest @code{F} format that fits
@var{m} plus a sign.  For example, if the matrix is @t{@{1:10@}}, then
@math{m = 10}, which fits in 3 columns with room for a sign, the
format is @code{F3.0}.

@item
Otherwise, if @math{@var{m} @geq{} 10^9} or @math{@var{m} @leq{}
10^{-4}}, @pspp{} scales all of the numbers in the matrix by
@math{10^x}, where @var{x} is the exponent that would be used to
display @var{m} in scientific notation.  For example, for
@math{@var{m} = 5.123@times{}10^{20}}, the scale factor is
@math{10^{20}}.  @pspp{} displays the scaled values in format
@code{F13.10} and notes the scale factor in the output.

@item
Otherwise, @pspp{} displays the matrix values, without scaling, in
format @code{F13.10}.
@end enumerate

The optional @code{TITLE} subcommand specifies a title for the output
text or table, as a quoted string.  When it is omitted, the syntax of
the matrix expression is used as the title.

Use the @code{SPACE} subcommand to request extra space above the
matrix output.  With a numerical argument, it adds the specified
number of lines of blank space above the matrix.  With @code{NEWPAGE}
as an argument, it prints the matrix at the top of a new page.  The
@code{SPACE} subcommand has no effect when a matrix is output as a
pivot table.

The @code{RLABELS} and @code{RNAMES} subcommands, which are mutually
exclusive, can supply a label to accompany each row in the output.
With @code{RLABELS}, specify the labels as comma-separated strings or
other tokens.  With @code{RNAMES}, specify a single expression that
evaluates to a vector of strings.  Either way, if there are more
labels than rows, the extra labels are ignored, and if there are more
rows than labels, the extra rows are unlabeled.  For output to a pivot
table with @code{RLABELS}, the labels can be any length; otherwise,
the labels are truncated to 8 bytes.

The @code{CLABELS} and @code{CNAMES} subcommands work for labeling
columns as @code{RLABELS} and @code{RNAMES} do for labeling rows.

When the @var{expression} is omitted, @code{PRINT} does not output a
matrix.  Instead, it outputs only the text specified on @code{TITLE},
if any, preceded by any space specified on the @code{SPACE}
subcommand, if any.  Any other subcommands are ignored, and the
command acts as if @code{MDISPLAY} is set to @code{TEXT} regardless of
its actual setting.

The following syntax demonstrates two different ways to label the rows
and columns of a matrix with @code{PRINT}:

@example
MATRIX.
COMPUTE m=@{1, 2, 3; 4, 5, 6; 7, 8, 9@}.
PRINT m/RLABELS=a, b, c/CLABELS=x, y, z.

COMPUTE rlabels=@{"a", "b", "c"@}.
COMPUTE clabels=@{"x", "y", "z"@}.
PRINT m/RNAMES=rlabels/CNAMES=clabels.
END MATRIX.
@end example

@noindent
With @code{MDISPLAY=TEXT} (the default), this program outputs the
following (twice):

@example
m
                x        y        z
a               1        2        3
b               4        5        6
c               7        8        9
@end example

@noindent
With @samp{SET MDISPLAY=TABLES.} added above @samp{MATRIX.}, the
output becomes the following (twice):

@psppoutput {matrix-print}

@node Matrix DO IF Command
@subsection The @code{DO IF} Command

@display
@t{DO IF} @i{expression}@t{.}
  @dots{}@i{matrix commands}@dots{}
[@t{ELSE IF} @i{expression}@t{.}
  @dots{}@i{matrix commands}@dots{}]@dots{}
[@t{ELSE}
  @dots{}@i{matrix commands}@dots{}]
@t{END IF}@t{.}
@end display

A @code{DO IF} command evaluates its expression argument.  If the
@code{DO IF} expression evaluates to true, then @pspp{} executes the
associated commands.  Otherwise, @pspp{} evaluates the expression on
each @code{ELSE IF} clause (if any) in order, and executes the
commands associated with the first one that yields a true value.
Finally, if the @code{DO IF} and all the @code{ELSE IF} expressions
all evaluate to false, @pspp{} executes the commands following the
@code{ELSE} clause (if any).

Each expression on @code{DO IF} and @code{ELSE IF} must evaluate to a
scalar.  Positive scalars are considered to be true, and scalars that
are zero or negative are considered to be false.

The following matrix language fragment sets @samp{b} to the term
following @samp{a} in the
@url{https://en.wikipedia.org/wiki/Juggler_sequence, Juggler
sequence}:

@example
DO IF MOD(a, 2) = 0.
  COMPUTE b = TRUNC(a &** (1/2)).
ELSE.
  COMPUTE b = TRUNC(a &** (3/2)).
END IF.
@end example

@node Matrix LOOP and BREAK Commands
@subsection The @code{LOOP} and @code{BREAK} Commands

@display
@t{LOOP} [@i{var}@t{=}@i{first} @t{TO} @i{last} [@t{BY} @i{step}]] [@t{IF} @i{expression}]@t{.}
  @dots{}@i{matrix commands}@dots{}
@t{END LOOP} [@t{IF} @i{expression}]@t{.}

@t{BREAK}@t{.}
@end display

The @code{LOOP} command executes a nested group of matrix commands,
called the loop's @dfn{body}, repeatedly.  It has three optional
clauses that control how many times the loop body executes.
Regardless of these clauses, the global @code{MXLOOPS} setting, which
defaults to 40, also limits the number of iterations of a loop.  To
iterate more times, raise the maximum with @code{SET MXLOOPS} outside
of the @code{MATRIX} command (@pxref{SET MXLOOPS}).

The optional index clause causes @var{var} to be assigned successive
values on each trip through the loop: first @var{first}, then
@math{@var{first} + @var{step}}, then @math{@var{first} + 2 @times{}
@var{step}}, and so on.  The loop ends when @math{@var{var} >
@var{last}}, for positive @var{step}, or @math{@var{var} <
@var{last}}, for negative @var{step}.  If @var{step} is not specified,
it defaults to 1.  All the index clause expressions must evaluate to
scalars, and non-integers are rounded toward zero.  If @var{step}
evaluates as zero (or rounds to zero), then the loop body never
executes.

The optional @code{IF} on @code{LOOP} is evaluated before each
iteration through the loop body.  If its expression, which must
evaluate to a scalar, is zero or negative, then the loop terminates
without executing the loop body.

The optional @code{IF} on @code{END LOOP} is evaluated after each
iteration through the loop body.  If its expression, which must
evaluate to a scalar, is zero or negative, then the loop terminates.

The following computes and prints @math{l(n)}, whose value is the
number of steps in the
@url{https://en.wikipedia.org/wiki/Juggler_sequence, Juggler sequence}
for @math{n}, for @math{n} from 2 to 10 inclusive:

@example
COMPUTE l = @{@}.
LOOP n = 2 TO 10.
  COMPUTE a = n.
  LOOP i = 1 TO 100.
    DO IF MOD(a, 2) = 0.
      COMPUTE a = TRUNC(a &** (1/2)).
    ELSE.
      COMPUTE a = TRUNC(a &** (3/2)).
    END IF.
  END LOOP IF a = 1.
  COMPUTE l = @{l; i@}.
END LOOP.
PRINT l.
@end example

@node Matrix BREAK Command
@subsubsection The @code{BREAK} Command

The @code{BREAK} command may be used inside a loop body, ordinarily
within a @code{DO IF} command.  If it is executed, then the loop
terminates immediately, jumping to the command just following
@code{END LOOP}.  When multiple @code{LOOP} commands nest,
@code{BREAK} terminates the innermost loop.

The following example is a revision of the one above that shows how
@code{BREAK} could substitute for the index and @code{IF} clauses on
@code{LOOP} and @code{END LOOP}:

@example
COMPUTE l = @{@}.
LOOP n = 2 TO 10.
  COMPUTE a = n.
  COMPUTE i = 1.
  LOOP.
    DO IF MOD(a, 2) = 0.
      COMPUTE a = TRUNC(a &** (1/2)).
    ELSE.
      COMPUTE a = TRUNC(a &** (3/2)).
    END IF.
    DO IF a = 1.
      BREAK.
    END IF.
    COMPUTE i = i + 1.
  END LOOP.
  COMPUTE l = @{l; i@}.
END LOOP.
PRINT l.
@end example

@node Matrix READ and WRITE Commands
@subsection The @code{READ} and @code{WRITE} Commands

The @code{READ} and @code{WRITE} commands perform matrix input and
output with text files.  They share the following syntax for
specifying how data is divided among input lines:

@display
@t{/FIELD}@t{=}@i{first} @t{TO} @i{last} [@t{BY} @i{width}]
[@t{/FORMAT}@t{=}@i{format}]
@end display

Both commands require the @code{FIELD} subcommand.  It specifies the
range of columns, from @var{first} to @var{last}, inclusive, that the
data occupies on each line of the file.  The leftmost column is column
1.  The columns must be literal numbers, not expressions.  To use
entire lines, even if they might be very long, specify a column range
such as @code{1 TO 100000}.

The @code{FORMAT} subcommand is optional for numerical matrices.  For
string matrix input and output, specify an @code{A} format.  In
addition to @code{FORMAT}, the optional @code{BY} specification on
@code{FIELD} determine the meaning of each text line:

@itemize @bullet
@item
Without @code{BY} and @code{FORMAT}, the numbers in the text file are
in @code{F} format separated by spaces or commas.  For @code{WRITE},
@pspp{} uses as many digits of precision needed to represent the
numbers in the matrix

@item
With @code{BY @i{width}}, the input area is divided into fixed-width
fields with the given @i{width}.  The input area must be a multiple of
@i{width} columns wide.  Numbers are read or written as
@code{F@i{width}.0} format.

@item
With @code{FORMAT=@i{count}F}, the input area is divided into
@i{count} equal-width fields per line.  The input area must be a
multiple of @i{count} columns wide.  Another format type may be
substituted for @code{F}.

@item
@code{FORMAT=F@i{w}.@i{d}} divides the input area into fixed-width
fields with width @i{w}.  The input area must be a multiple of @i{w}
columns wide.  Another format type may be substituted for @code{F}.

@item
If @code{BY} and @code{FORMAT} are both used, then they must agree on
the field width.
@end itemize

@node Matrix READ Command
@subsubsection The @code{READ} Command

@display
@t{READ} @i{variable}[@t{(}@i{index}[@t{,}@i{index}]@t{)}]
     [@t{/FILE}@t{=}@i{file}]
     @t{/FIELD}@t{=}@i{first} @t{TO} @i{last} [@t{BY} @i{width}]
     [@t{/FORMAT}@t{=}@i{format}]
     [@t{/SIZE}@t{=}@i{expression}]
     [@t{/MODE}@t{=}@{@t{RECTANGULAR} @math{|} @t{SYMMETRIC}@}]
     [@t{/REREAD}]@t{.}
@end display

The @code{READ} command reads from a text file into a matrix variable.
Specify the target variable just after the command name, either just a
variable name to create or replace an entire variable, or a variable
name followed by an indexing expression to replace a submatrix of an
existing variable.

The @code{FILE} subcommand is required in the first @code{READ}
command that appears within @code{MATRIX}.  It specifies the text file
to be read, either as a file name in quotes or a file handle
previously declared on @code{FILE HANDLE} (@pxref{FILE HANDLE}).
Later @code{READ} commands (in syntax order) use the previous
referenced file if @code{FILE} is omitted.

The @code{FIELD} and @code{FORMAT} subcommands specify how input lines
are interpreted.  @xref{Matrix READ and WRITE Commands}, for details.

The @code{SIZE} subcommand is required for reading into an entire
variable.  Its restricted expression argument should evaluate to a
2-element vector @code{@{@var{n},@w{ }@var{m}@}} or
@code{@{@var{n};@w{ }@var{m}@}}, which indicates a
@math{@var{n}@times{}@var{m}} matrix destination.  A scalar @var{n} is
also allowed and indicates a @math{@var{n}@times{}1} column vector
destination.  When the destination is a submatrix, @code{SIZE} is
optional, and if it is present then it must match the size of the
submatrix.

By default, or with @code{MODE=RECTANGULAR}, the command reads an
entry for every row and column.  With @code{MODE=SYMMETRIC}, the
command reads only the entries on and below the matrix's main
diagonal, and copies the entries above the main diagonal from the
corresponding symmetric entries below it.  Only square matrices
may use @code{MODE=SYMMETRIC}.

Ordinarily, each @code{READ} command starts from a new line in the
text file.  Specify the @code{REREAD} subcommand to instead start from
the last line read by the previous @code{READ} command.  This has no
effect for the first @code{READ} command to read from a particular
file.  It is also ineffective just after a command that uses the
@code{EOF} matrix function (@pxref{EOF Matrix Function}) on a
particular file, because @code{EOF} has to try to read the next line
from the file to determine whether the file contains more input.

@subsubheading Example 1: Basic Use

The following matrix program reads the same matrix @code{@{1, 2, 4; 2,
3, 5; 4, 5, 6@}} into matrix variables @code{v}, @code{w}, and
@code{x}:

@example
READ v /FILE='input.txt' /FIELD=1 TO 100 /SIZE=@{3, 3@}.
READ w /FIELD=1 TO 100 /SIZE=@{3; 3@} /MODE=SYMMETRIC.
READ x /FIELD=1 TO 100 BY 1/SIZE=@{3, 3@} /MODE=SYMMETRIC.
@end example

@noindent
given that @file{input.txt} contains the following:

@example
1, 2, 4
2, 3, 5
4, 5, 6
1
2 3
4 5 6
1
23
456
@end example

The @code{READ} command will read as many lines of input as needed for
a particular row, so it's also acceptable to break any of the lines
above into multiple lines.  For example, the first line @code{1, 2, 4}
could be written with a line break following either or both commas.

@subsubheading Example 2: Reading into a Submatrix

The following reads a @math{5@times{}5} matrix from @file{input2.txt},
reversing the order of the rows:

@example
COMPUTE m = MAKE(5, 5, 0).
LOOP r = 5 TO 1 BY -1.
  READ m(r, :) /FILE='input2.txt' /FIELD=1 TO 100.
END LOOP.
@end example

@subsubheading Example 3: Using @code{REREAD}

Suppose each of the 5 lines in a file @file{input3.txt} starts with an
integer @var{count} followed by @var{count} numbers, e.g.:

@example
1 5
3 1 2 3
5 6 -1 2 5 1
2 8 9
3 1 3 2
@end example

@noindent
Then, the following reads this file into a matrix @code{m}:

@example
COMPUTE m = MAKE(5, 5, 0).
LOOP i = 1 TO 5.
  READ count /FILE='input3.txt' /FIELD=1 TO 1 /SIZE=1.
  READ m(i, 1:count) /FIELD=3 TO 100 /REREAD.
END LOOP.
@end example

@node Matrix WRITE Command
@subsubsection The @code{WRITE} Command

@display
@t{WRITE} @i{expression}
      [@t{/OUTFILE}@t{=}@i{file}]
      @t{/FIELD}@t{=}@i{first} @t{TO} @i{last} [@t{BY} @i{width}]
      [@t{/FORMAT}@t{=}@i{format}]
      [@t{/MODE}@t{=}@{@t{RECTANGULAR} @math{|} @t{TRIANGULAR}@}]
      [@t{/HOLD}]@t{.}
@end display

The @code{WRITE} command evaluates @i{expression} and writes it to a
text file in a specified format.  Write the expression to evaluate
just after the command name.

The @code{OUTFILE} subcommand is required in the first @code{WRITE}
command that appears within @code{MATRIX}.  It specifies the text file
to be written, either as a file name in quotes or a file handle
previously declared on @code{FILE HANDLE} (@pxref{FILE HANDLE}).
Later @code{WRITE} commands (in syntax order) use the previous
referenced file if @code{FILE} is omitted.

The @code{FIELD} and @code{FORMAT} subcommands specify how output
lines are formed.  @xref{Matrix READ and WRITE Commands}, for details.

By default, or with @code{MODE=RECTANGULAR}, the command writes an
entry for every row and column.  With @code{MODE=TRIANGULAR}, the
command writes only the entries on and below the matrix's main
diagonal.  Entries above the diagonal are not written.  Only square
matrices may be written with @code{MODE=TRIANGULAR}.

Ordinarily, each @code{WRITE} command starts a new line in the output
file.  With @code{HOLD}, the next @code{WRITE} command will write to
the same line as the current one.  This can be useful to write more
than one matrix on a single output line.

@node Matrix GET Command
@subsection The @code{GET} Command

@display
@t{GET} @i{variable}[@t{(}@i{index}[@t{,}@i{index}]@t{)}]
    [@t{/FILE}@t{=}@{@i{file} @math{|} @t{*}@}]
    [@t{/VARIABLES}@t{=}@i{variable}@dots{}]
    [@t{/NAMES}@t{=}@i{variable}]
    [@t{/MISSING}@t{=}@{@t{ACCEPT} @math{|} @t{OMIT} @math{|} @i{number}@}]
    [@t{/SYSMIS}@t{=}@{@t{OMIT} @math{|} @i{number}@}]@t{.}
@end display

The @code{READ} command reads numeric data from an SPSS system file,
SPSS/PC+ system file, or SPSS portable file into a matrix variable or
submatrix:

@itemize @bullet
@item
To read data into a variable, specify just its name following
@code{GET}.  The variable need not already exist; if it does, it is
replaced.  The variable will have as many columns as there are
variables specified on the @code{VARIABLES} subcommand and as many
rows as there are cases in the input file.

@item
To read data into a submatrix, specify the name of an existing
variable, followed by an indexing expression, just after @code{GET}.
The submatrix must have as many columns as variables specified on
@code{VARIABLES} and as many rows as cases in the input file.
@end itemize

Specify the name or handle of the file to be read on @code{FILE}.  Use
@samp{*}, or simply omit the @code{FILE} subcommand, to read from the
active file.  Reading from the active file is only permitted if it was
already defined outside @code{MATRIX}.

List the variables to be read as columns in the matrix on the
@code{VARIABLES} subcommand.  The list can use @code{TO} for
collections of variables or @code{ALL} for all variables.  If
@code{VARIABLES} is omitted, all variables are read.  Only numeric
variables may be read.

If a variable is named on @code{NAMES}, then the names of the
variables read as data columns are stored in a string vector within
the given name, replacing any existing matrix variable with that name.
Variable names are truncated to 8 bytes.

The @code{MISSING} and @code{SYSMIS} subcommands control the treatment
of missing values in the input file.  By default, any user- or
system-missing data in the variables being read from the input causes
an error that prevents @code{GET} from executing.  To accept missing
values, specify one of the following settings on @code{MISSING}:

@table @asis
@item @code{ACCEPT}
Accept user-missing values with no change.  By default, system-missing
values still yield an error.  Use the @code{SYSMIS} subcommand to
change this treatment:

@table @asis
@item @code{OMIT}
Skip any case that contains a system-missing value.

@item @i{number}
Recode the system-missing value to @i{number}.
@end table

@item @code{OMIT}
Skip any case that contains any user- or system-missing value.

@item @i{number}
Recode all user- and system-missing values to @i{number}.
@end table

The @code{SYSMIS} subcommand has an effect only with
@code{MISSING=ACCEPT}.

@node Matrix SAVE Command
@subsection The @code{SAVE} Command

@display
@t{SAVE} @i{expression}
     [@t{/OUTFILE}@t{=}@{@i{file} @math{|} @t{*}@}]
     [@t{/VARIABLES}@t{=}@i{variable}@dots{}]
     [@t{/NAMES}@t{=}@i{expression}]
     [@t{/STRINGS}@t{=}@i{variable}@dots{}]@t{.}
@end display

The @code{SAVE} matrix command evaluates @i{expression} and writes the
resulting matrix to an SPSS system file.  In the system file, each
matrix row becomes a case and each column becomes a variable.

Specify the name or handle of the SPSS system file on the
@code{OUTFILE} subcommand, or @samp{*} to write the output as the new
active file.  The @code{OUTFILE} subcommand is required on the first
@code{SAVE} command, in syntax order, within @code{MATRIX}.  For
@code{SAVE} commands after the first, the default output file is the
same as the previous.

When multiple @code{SAVE} commands write to one destination within a
single @code{MATRIX}, the later commands append to the same output
file.  All the matrices written to the file must have the same number
of columns.  The @code{VARIABLES}, @code{NAMES}, and @code{STRINGS}
subcommands are honored only for the first @code{SAVE} command that
writes to a given file.

By default, @code{SAVE} names the variables in the output file
@code{COL1} through @code{COL@i{n}}.  Use @code{VARIABLES} or
@code{NAMES} to give the variables meaningful names.  The
@code{VARIABLES} subcommand accepts a comma-separated list of variable
names.  Its alternative, @code{NAMES}, instead accepts an expression
that must evaluate to a row or column string vector of names.  The
number of names need not exactly match the number of columns in the
matrix to be written: extra names are ignored; extra columns use
default names.

By default, @code{SAVE} assumes that the matrix to be written is all
numeric.  To write string columns, specify a comma-separated list of
the string columns' variable names on @code{STRINGS}.

@node Matrix MGET Command
@subsection The @code{MGET} Command

@display
@t{MGET} [@t{/FILE}@t{=}@i{file}]
     [@t{/TYPE}@t{=}@{@t{COV} @math{|} @t{CORR} @math{|} @t{MEAN} @math{|} @t{STDDEV} @math{|} @t{N} @math{|} @t{COUNT}@}]@t{.}
@end display

The @code{MGET} command reads the data from a matrix file
(@pxref{Matrix Files}) into matrix variables.  Specify the name or
handle of the matrix file to be read on the @code{FILE} subcommand; if
it is omitted, then the command reads the active file.

By default, @code{MGET} reads all of the data from the matrix file.
Specify a space-delimited list of matrix types @code{TYPE} to limit the
kinds of data to the one specified:

@table @code
@item COV
Covariance matrix.

@item CORR
Correlation coefficient matrix.

@item MEAN
Vector of means.

@item STDDEV
Vector of standard deviations.

@item N
Vector of case counts.

@item COUNT
Vector of counts.
@end table

@code{MGET} reads the entire matrix file and automatically names,
creates, and populates matrix variables using its contents.  It
constructs the name of each variable by concatenating the following:

@itemize @bullet
@item
A 2-character prefix that identifies the type of the matrix:

@table @code
@item CV
Covariance matrix.

@item CR
Correlation coefficient matrix.

@item MN
Vector of means.

@item SD
Vector of standard deviations.

@item NC
Vector of case counts.

@item CN
Vector of counts.
@end table

@item
If the matrix file has factor variables, @code{F@i{n}}, where @i{n}
is a number identifying a group of factors: @code{F1} for the first
group, @code{F2} for the second, and so on.

@item
If the matrix file has split file variables, @code{S@i{n}}, where
@i{n} is a number identifying a split group: @code{S1} for the first
group, @code{S2} for the second, and so on.
@end itemize

If @code{MGET} chooses the name of an existing variable, it issues a
warning and does not change the variable.

@node Matrix MSAVE Command
@subsection The @code{MSAVE} Command

@display
@t{MSAVE} @i{expression}
      @t{/TYPE}@t{=}@{@t{COV} @math{|} @t{CORR} @math{|} @t{MEAN} @math{|} @t{STDDEV} @math{|} @t{N} @math{|} @t{COUNT}@}
      [@t{/FACTOR}@t{=}@i{expression}]
      [@t{/SPLIT}@t{=}@i{expression}]
      [@t{/OUTFILE}@t{=}@i{file}]
      [@t{/VARIABLES}@t{=}@i{variable}@dots{}]
      [@t{/SNAMES}@t{=}@i{variable}@dots{}]
      [@t{/FNAMES}@t{=}@i{variable}@dots{}]@t{.}
@end display

The @code{MSAVE} command evaluates the @i{expression} specifies just
after the command name, and writes the resulting matrix to a matrix
file (@pxref{Matrix Files}).

The @code{TYPE} subcommand is required.  It specifies the
@code{ROWTYPE_} to write along with this matrix.

The @code{FACTOR} and @code{SPLIT} subcommands are required if and
only if the matrix file has factor or split variables, respectively.
Each one takes an expression that must evaluate to a vector with the
same number of entries as the matrix has factor or split variables,
respectively.  Each @code{MSAVE} only writes data for a single
combination of factor and split variables, so many @code{MSAVE}
commands (or one inside a loop) may be needed to write a complete set.

The remaining @code{MSAVE} subcommands define the format of the matrix
file.  All of the @code{MSAVE} commands within a given matrix program
write to the same matrix file, so these subcommands are only
meaningful on the first @code{MSAVE} command within a matrix program.
(If they are given again on later @code{MSAVE} commands, then they
must have the same values as on the first.)

The @code{OUTFILE} subcommand specifies the name or handle of the
matrix file to be written.  Output must go to an external file, not a
data set or the active file.

The @code{VARIABLES} subcommand specifies a comma-separated list of
the names of the continuous variables to be written to the matrix
file.  The @code{TO} keyword can be used to define variables named
with consecutive integer suffixes.  These names become column names
and names that appear in @code{VARNAME_} in the matrix file.
@code{ROWTYPE_} and @code{VARNAME_} are not allowed on
@code{VARIABLES}.  If @code{VARIABLES} is omitted, then @pspp{} uses
the names @code{COL1}, @code{COL2}, and so on.

The @code{FNAMES} subcommand may be used to supply a comma-separated
list of factor variable names.  The default names are @code{FAC1},
@code{FAC2}, and so on.

The @code{SNAMES} subcommand can supply a comma-separated list of
split variable names.  The default names are @code{SPL1}, @code{SPL2},
and so on.

@node Matrix DISPLAY Command
@subsection The @code{DISPLAY} Command

@display
@t{DISPLAY} [@{@t{DICTIONARY} @math{|} @t{STATUS}@}]@t{.}
@end display

The @code{DISPLAY} command makes @pspp{} display a table with the name
and dimensions of each matrix variable.  The @code{DICTIONARY} and
@code{STATUS} keywords are accepted but have no effect.

@node Matrix RELEASE Command
@subsection The @code{RELEASE} Command

@display
@t{RELEASE} @i{variable}@dots{}@t{.}
@end display

The @code{RELEASE} command accepts a comma-separated list of matrix
variable names.  It deletes each variable and releases the memory
associated with it.

The @code{END MATRIX} command releases all matrix variables.