[pspp] / doc / matrices.texi

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@c A copy of the license is included in the section entitled "GNU
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@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{/SIZE}@t{=}@i{expression}]
     [@t{/MODE}@t{=}@{@t{RECTANGULAR} @math{|} @t{SYMMETRIC}@}]
     [@t{/REREAD}]
     [@t{/FORMAT}@t{=}@i{format}]@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 Operators
@subsection Matrix Operators

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.

@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)}.
@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.
@end deffn

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

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

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

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

@node Matrix IO
@subsubsection I/O

@deffn {Matrix Function} EOF (@var{file})
Given a file handle or file name @var{file}, returns an integer scalar
that indicates whether the last record in the file has been read.
Determining this requires attempting reading past the current record,
which means that @code{REREAD} on the next @code{READ} command
following @code{EOF} on the same file will be ineffective. 
@end deffn

@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.

@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.
@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.
@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 @i{expression}, if present, and outputs it either as
text or a pivot table, depending on the setting of @code{MDISPLAY}
(@pxref{SET MDISPLAY}).

Any matrix expression is allowed as @var{expression}, but an
expression with operators with lower precedence than exponentiation
(@pxref{Matrix Operators}) must be parenthesized.  (This avoids
ambiguity between @t{/} as an operator and @t{/} to separate
subcommands.)

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 the matrix's elements are all integers, then, if possible, @pspp{}
chooses the narrowest @code{F} format with width 12 or less that fits
@var{m} plus a sign.

@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 used when @var{m} is
displayed in scientific notation, and displays the scaled value in
format @code{F13.10}.  @pspp{} adds a note to the output to indicate
the scale factor.

@item
Otherwise, @pspp{} displays the value, 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.

@subsubheading Text Output

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.

@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.

@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 @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.

@node Matrix READ Command
@subsection 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{/SIZE}@t{=}@i{expression}]
     [@t{/MODE}@t{=}@{@t{RECTANGULAR} @math{|} @t{SYMMETRIC}@}]
     [@t{/REREAD}]
     [@t{/FORMAT}@t{=}@i{format}]@t{.}
@end display

@node Matrix WRITE Command
@subsection 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{/MODE}@t{=}@{@t{RECTANGULAR} @math{|} @t{TRIANGULAR}@}]
      [@t{/HOLD}]
      [@t{/FORMAT}@t{=}@i{format}]@t{.}
@end display

@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{expression}]
    [@t{/MISSING}@t{=}@{@t{ACCEPT} @math{|} @t{OMIT} @math{|} @i{number}@}]
    [@t{/SYSMIS}@t{=}@{@t{OMIT} @math{|} @i{number}@}]@t{.}
@end display

@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
@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

@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{/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

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

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

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

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