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<title>MADlib: Matrix Operations</title>
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<div class="title">Matrix Operations<div class="ingroups"><a class="el" href="group__grp__datatrans.html">Data Types and Transforms</a> &raquo; <a class="el" href="group__grp__arraysmatrix.html">Arrays and Matrices</a></div></div> </div>
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<div class="contents">
<div class="toc"><b>Contents</b> </p><ul>
<li class="level1">
<a href="#description">Description</a> </li>
<li class="level1">
<a href="#operations">Matrix Operations</a> </li>
<li class="level1">
<a href="#glossary">Glossary of arguments</a> </li>
<li class="level1">
<a href="#examples">Examples</a> </li>
<li class="level1">
<a href="#related">Related Topics</a> </li>
</ul>
</div><p><a class="anchor" id="description"></a>This module provides a set of basic matrix operations for matrices that are too big to fit in memory. We provide two storage formats for a matrix:</p>
<ul>
<li>Dense: The matrix is represented as a distributed collection of 1-D arrays. An example 3x10 matrix would be the below table: <pre>
row_id | row_vec
--------+-------------------------
1 | {9,6,5,8,5,6,6,3,10,8}
2 | {8,2,2,6,6,10,2,1,9,9}
3 | {3,9,9,9,8,6,3,9,5,6}
</pre></li>
</ul>
<p>A '<em>row</em>' column (called as <em>row_id</em> above) provides the row number of each row and a '<em>val</em>' column (called as <em>row_vec</em> above) provides each row as an array. <b>The <em>row</em> column should contain a series of integers from 1 to <em>N</em> with no duplicates, where <em>N</em> is the row dimensionality</b>.</p>
<ul>
<li>Sparse: The matrix is represented using the row and column indices for each non-zero entry of the matrix. This representation is useful for sparse matrices, containing multiple zero elements. Given below is an example of a sparse 4x7 matrix with just 6 out of 28 entries being non-zero. The dimensionality of the matrix is inferred using the max value in <em>row</em> and <em>col</em> columns. Note the last entry is included (even though it is 0) to provide the dimensionality of the matrix (indicating that the 4th row and 7th column contain all zeros). <pre>
row_id | col_id | value
--------+--------+-------
1 | 1 | 9
1 | 5 | 6
1 | 6 | 6
2 | 1 | 8
3 | 1 | 3
3 | 2 | 9
4 | 7 | 0
(6 rows)
</pre></li>
</ul>
<p><b>For sparse matrices, the <em>row</em> and <em>col</em> columns together should not contain a duplicate entry and the <em>val</em> column should be of scalar (non-array) data type</b>. <br />
For comparison, the dense representation of this matrix is shown below. Note the dimensionality of the dense matrix is 4 x 7 since the max value of <em>row</em> and <em>col</em> is 4 and 7 respectively, leading to all zeros in the last row and last column. &#160; </p><pre>
row_id | row_vec
--------+-------------------------
1 | {9,0,0,0,6,6,0}
2 | {8,0,0,0,0,0,0}
3 | {3,9,0,0,0,0,0}
4 | {0,0,0,0,0,0,0}
</pre><dl class="section note"><dt>Note</dt><dd>The functions below support several numeric types (unless otherwise noted) including SMALLINT, INTEGER, BIGINT, DOUBLE PRECISION (FLOAT8), NUMERIC (internally casted into FLOAT8, loss of precision can happen).</dd></dl>
<p><a class="anchor" id="operations"></a></p><dl class="section user"><dt>Matrix Operations</dt><dd></dd></dl>
<p>Given below are the supported matrix operations. The meaning of the arguments and other terms are common to all functions and provided at the end of the list as a glossary.</p>
<ul>
<li><b>Representation</b> <pre class="syntax">
-- Convert to sparse representation
&#160; <b>matrix_sparsify</b>( matrix_in, in_args, matrix_out, out_args)
&#160;
-- Convert to dense representation
&#160; <b>matrix_densify</b>( matrix_in, in_args, matrix_out, out_args)
</pre></li>
<li><b>Mathematical operations</b> <pre class="syntax">
-- Matrix transposition
&#160; <b>matrix_trans</b>( matrix_in, in_args, matrix_out, out_args)
&#160;
-- Matrix addition
&#160; <b>matrix_add</b>( matrix_a, a_args, matrix_b, b_args, matrix_out, out_args)
&#160;
-- Matrix subtraction
&#160; <b>matrix_sub</b>( matrix_a, a_args, matrix_b, b_args, matrix_out, out_args)
&#160;
-- Matrix multiplication
&#160; <b>matrix_mult</b>( matrix_a, a_args, matrix_b, b_args, matrix_out, out_args)
&#160;
-- Element-wise matrix multiplication
&#160; <b>matrix_elem_mult</b>( matrix_a, a_args, matrix_b, b_args, matrix_out, out_args)
&#160;
-- Multiply matrix with scalar.
&#160; <b>matrix_scalar_mult</b>( matrix_in, in_args, scalar, matrix_out, out_args)
&#160;
-- Multiply matrix with vector.
&#160; <b>matrix_vec_mult</b>( matrix_in, in_args, vector)
</pre></li>
<li><b>Extraction/visitor methods</b> <pre class="syntax">
-- Extract row from matrix given row index
&#160; <b>matrix_extract_row</b>( matrix_in, in_args, index)
&#160;
-- Extract column from matrix given column index
&#160; <b>matrix_extract_col</b>( matrix_in, in_args, index)
</pre></li>
<li><b>Reduction operations (aggregate across specific dimension)</b> <pre class="syntax">
-- Get max value along dim. Also returns corresponding index if <em>fetch_index</em> = True
&#160; <b>matrix_max</b>( matrix_in, in_args, dim, matrix_out, fetch_index)
&#160;
-- Get min value along dim. Also returns corresponding index if <em>fetch_index</em> = True
&#160; <b>matrix_min</b>( matrix_in, in_args, dim, matrix_out, fetch_index)
&#160;
-- Get sum value along dimension from matrix.
&#160; <b>matrix_sum</b>( matrix_in, in_args, dim)
&#160;
-- Get mean value along dimension from matrix.
&#160; <b>matrix_mean</b>( matrix_in, in_args, dim)
</pre></li>
</ul>
<p><a class="anchor" id="glossary"></a><b>Glossary</b> </p>
<p>The table below provides a glossary of the terms used in the matrix operations.</p>
<dl class="arglist">
<dt>matrix_in, matrix_a, matrix_b </dt>
<dd><p class="startdd">TEXT. Name of the table containing the input matrix.</p><ul>
<li>For functions accepting one matrix, <em>matrix_in</em> denotes the input matrix.</li>
<li>For functions accepting two matrices, <em>matrix_a</em> denotes the first matrix and <em>matrix_b</em> denotes the second matrix. These two matrices can <b>independently</b> be in either dense or sparse format. </li>
</ul>
<p class="enddd"></p>
</dd>
<dt>in_args, a_args, b_args </dt>
<dd><p class="startdd">TEXT. A comma-delimited string containing multiple named arguments of the form "name=value". This argument is used as a container for multiple parameters related to a single matrix.</p>
<p>The following parameters are supported for this string argument: </p><table class="output">
<tr>
<th>row </th><td>(Default: 'row_num') Name of the column containing row index of the matrix. </td></tr>
<tr>
<th>col </th><td>(Default: 'col_num') Name of the column containing column index of the matrix. </td></tr>
<tr>
<th>val </th><td>(Default: 'val') Name of the column containing the entries of the matrix. </td></tr>
<tr>
<th>trans </th><td>(Default: False) Boolean flag to indicate if the matrix should be transposed before the operation. This is currently functional only for <em>matrix_mult</em>. </td></tr>
</table>
<p>For example, the string argument with default values will be 'row=row_num, col=col_num, val=val, trans=False'. Alternatively, the string argument can be set to <em>NULL</em> or be blank ('') if default values are to be used. </p>
<p class="enddd"></p>
</dd>
<dt>matrix_out </dt>
<dd><p class="startdd">TEXT. Name of the table to store the result matrix. </p>
<p class="enddd"></p>
</dd>
<dt>out_args </dt>
<dd><p class="startdd">TEXT. A comma-delimited string containing named arguments of the form "name=value". This is an <b>optional parameter</b> and the default value is set as follows:</p><ul>
<li>For functions with one input matrix, default = <em>in_args</em></li>
<li>For functions with two input matrices, default = <em>a_args</em>.</li>
</ul>
<p>The following parameters are supported for this string argument: </p><table class="output">
<tr>
<th>row </th><td>Name of the column containing row index of the matrix. </td></tr>
<tr>
<th>col </th><td>Name of the column containing column index of the matrix. </td></tr>
<tr>
<th>val </th><td>Name of the column containing the entries of the matrix. </td></tr>
</table>
<p class="enddd"></p>
</dd>
<dt>index </dt>
<dd><p class="startdd">INTEGER. An integer representing a row or column index of the matrix. Should be a number from 1 to <em>N</em>, where <em>N</em> is the maximum size of the dimension.</p>
<p class="enddd"></p>
</dd>
<dt>dim </dt>
<dd><p class="startdd">INTEGER. Should either be 1 or 2. This value indicates the dimension to operate along for the reduction/aggregation operations. <b>The value of <em>dim</em> should be interpreted as the dimension to be flattened i.e. whose length reduces to 1 in the result.</b> <br />
For <em>dim=1</em>, a reduction function on an <em>NxM</em> matrix operates on successive elements in each column and returns a single vector with <em>M</em> elements (i.e. matrix with just 1 row and <em>M</em> columns). <br />
For <em>dim=2</em>, a single vector is returned with <em>N</em> elements (i.e. matrix with just 1 column and <em>N</em> rows). </p>
<p class="enddd"></p>
</dd>
</dl>
<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
<ul>
<li>Create some random data tables in dense format. <pre class="syntax">
CREATE TABLE "mat_A" (
row_id integer,
row_vec integer[]
);
INSERT INTO "mat_A" (row_id, row_vec) VALUES (1, '{9,6,5,8,5,6,6,3,10,8}');
INSERT INTO "mat_A" (row_id, row_vec) VALUES (2, '{8,2,2,6,6,10,2,1,9,9}');
INSERT INTO "mat_A" (row_id, row_vec) VALUES (3, '{3,9,9,9,8,6,3,9,5,6}');
INSERT INTO "mat_A" (row_id, row_vec) VALUES (4, '{6,4,2,2,2,7,8,8,0,7}');
INSERT INTO "mat_A" (row_id, row_vec) VALUES (5, '{6,8,9,9,4,6,9,5,7,7}');
INSERT INTO "mat_A" (row_id, row_vec) VALUES (6, '{4,10,7,3,9,5,9,2,3,4}');
INSERT INTO "mat_A" (row_id, row_vec) VALUES (7, '{8,10,7,10,1,9,7,9,8,7}');
INSERT INTO "mat_A" (row_id, row_vec) VALUES (8, '{7,4,5,6,2,8,1,1,4,8}');
INSERT INTO "mat_A" (row_id, row_vec) VALUES (9, '{8,8,8,5,2,6,9,1,8,3}');
INSERT INTO "mat_A" (row_id, row_vec) VALUES (10, '{4,6,3,2,6,4,1,2,3,8}');
CREATE TABLE "mat_B" (
row_id integer,
vector integer[]
);
INSERT INTO "mat_B" (row_id, vector) VALUES (1, '{9,10,2,4,6,5,3,7,5,6}');
INSERT INTO "mat_B" (row_id, vector) VALUES (2, '{5,3,5,2,8,6,9,7,7,6}');
INSERT INTO "mat_B" (row_id, vector) VALUES (3, '{0,1,2,3,2,7,7,3,10,1}');
INSERT INTO "mat_B" (row_id, vector) VALUES (4, '{2,9,0,4,3,6,8,6,3,4}');
INSERT INTO "mat_B" (row_id, vector) VALUES (5, '{3,8,7,7,0,5,3,9,2,10}');
INSERT INTO "mat_B" (row_id, vector) VALUES (6, '{5,3,1,7,6,3,5,3,6,4}');
INSERT INTO "mat_B" (row_id, vector) VALUES (7, '{4,8,4,4,2,7,10,0,3,3}');
INSERT INTO "mat_B" (row_id, vector) VALUES (8, '{4,6,0,1,3,1,6,6,9,8}');
INSERT INTO "mat_B" (row_id, vector) VALUES (9, '{6,5,1,7,2,7,10,6,0,6}');
INSERT INTO "mat_B" (row_id, vector) VALUES (10, '{1,4,4,4,8,5,2,8,5,5}');
</pre></li>
<li>Transpose a matrix <pre class="syntax">
SELECT madlib.matrix_trans('"mat_B"', 'row=row_id, val=vector',
'mat_r');
SELECT * FROM mat_r ORDER BY row_id;
</pre> <pre class="result">
-- Note the result matrix has inherited 'vector' as the name of value column
row_id | vector
--------+-------------------------
1 | {9,5,0,2,3,5,4,4,6,1}
2 | {10,3,1,9,8,3,8,6,5,4}
3 | {2,5,2,0,7,1,4,0,1,4}
4 | {4,2,3,4,7,7,4,1,7,4}
5 | {6,8,2,3,0,6,2,3,2,8}
6 | {5,6,7,6,5,3,7,1,7,5}
7 | {3,9,7,8,3,5,10,6,10,2}
8 | {7,7,3,6,9,3,0,6,6,8}
9 | {5,7,10,3,2,6,3,9,0,5}
10 | {6,6,1,4,10,4,3,8,6,5}
(10 rows)
</pre></li>
<li>Add the two matrices <pre class="syntax">
SELECT madlib.matrix_add('"mat_A"', 'row=row_id, val=row_vec',
'"mat_B"', 'row=row_id, val=vector',
'mat_r', 'val=vector');
SELECT * FROM mat_r ORDER BY row_id;
</pre> <pre class="result">
row_id | vector
--------+-------------------------------
1 | {18,16,7,12,11,11,9,10,15,14}
2 | {13,5,7,8,14,16,11,8,16,15}
3 | {3,10,11,12,10,13,10,12,15,7}
4 | {8,13,2,6,5,13,16,14,3,11}
5 | {9,16,16,16,4,11,12,14,9,17}
6 | {9,13,8,10,15,8,14,5,9,8}
7 | {12,18,11,14,3,16,17,9,11,10}
8 | {11,10,5,7,5,9,7,7,13,16}
9 | {14,13,9,12,4,13,19,7,8,9}
10 | {5,10,7,6,14,9,3,10,8,13}
(10 rows)
</pre></li>
<li>Multiply the two matrices <pre class="syntax">
DROP TABLE IF EXISTS mat_r;
SELECT madlib.matrix_mult('"mat_A"', 'row=row_id, val=row_vec',
'"mat_B"', 'row=row_id, val=vector, trans=true',
'mat_r');
SELECT * FROM mat_r ORDER BY row_id;
</pre> <pre class="result">
row_id | row_vec
--------+-------------------------------------------
1 | {380,373,251,283,341,303,302,309,323,281}
2 | {318,318,222,221,269,259,236,249,264,248}
3 | {382,366,216,300,397,276,277,270,313,338}
4 | {275,284,154,244,279,183,226,215,295,204}
5 | {381,392,258,319,394,298,342,302,360,300}
6 | {321,333,189,276,278,232,300,236,281,250}
7 | {443,411,282,365,456,318,360,338,406,330}
8 | {267,240,150,186,270,194,210,184,233,193}
9 | {322,328,234,264,291,245,317,253,291,219}
10 | {246,221,109,173,222,164,167,185,181,189}
(10 rows)
</pre></li>
<li>Extract row and column from matrix given index <pre class="syntax">
SELECT madlib.matrix_extract_row('"mat_A"', 'row=row_id, val=row_vec', 2) as row,
madlib.matrix_extract_col('"mat_A"', 'row=row_id, val=row_vec', 3) as col;
</pre> <pre class="result">
row | col
------------------------+-----------------------
{8,2,2,6,6,10,2,1,9,9} | {5,2,9,2,9,7,7,5,8,3}
(1 rows)
</pre></li>
<li>Get min and max values along a specific dimension as well as corresponding indicies. Note that <em>dim=2</em> implies that the min and max is computed on each row returning a column vector i.e. the column (dim=2) is flattened. <pre class="syntax">
SELECT madlib.matrix_max('"mat_A"', 'row=row_id, val=row_vec', 2, 'mat_max_r', true),
madlib.matrix_min('"mat_A"', 'row=row_id, val=row_vec', 2, 'mat_min_r', true);
SELECT * from mat_max_r;
SELECT * from mat_min_r;
</pre> <pre class="result">
index | max
-----------------------+---------------------------
{8,5,1,6,2,1,1,5,6,9} | {10,10,9,8,9,10,10,8,9,8}
(1 rows)
&#160;
index | min
-----------------------+-----------------------
{7,7,0,8,4,7,4,6,7,6} | {3,1,3,0,4,2,1,1,1,1}
(1 rows)
</pre></li>
<li>Element-wise multiplication between two matrices <pre class="syntax">
SELECT madlib.matrix_elem_mult('"mat_A"', 'row=row_id, val=row_vec',
'"mat_B"', 'val=vector',
'mat_r', 'val=vector');
SELECT * FROM mat_r ORDER BY row_id;
</pre> <pre class="result">
row_id | vector
--------+---------------------------------
1 | {81,60,10,32,30,30,18,21,50,48}
2 | {40,6,10,12,48,60,18,7,63,54}
3 | {0,9,18,27,16,42,21,27,50,6}
4 | {12,36,0,8,6,42,64,48,0,28}
5 | {18,64,63,63,0,30,27,45,14,70}
6 | {20,30,7,21,54,15,45,6,18,16}
7 | {32,80,28,40,2,63,70,0,24,21}
8 | {28,24,0,6,6,8,6,6,36,64}
9 | {48,40,8,35,4,42,90,6,0,18}
10 | {4,24,12,8,48,20,2,16,15,40}
</pre></li>
<li>Get sum values along dimension. Sum is computed for each row (i.e. column is flattened since dim=2) <pre class="syntax">
SELECT madlib.matrix_sum('"mat_A"', 'row=row_id, val=row_vec', 2);
</pre> <pre class="result">
matrix_sum
---------------------------------
{66,55,67,46,70,56,76,46,58,39}
(1 rows)
</pre></li>
<li>Multiply matrix with a scalar <pre class="syntax">
DROP TABLE IF EXISTS mat_r;
SELECT madlib.matrix_scalar_mult('"mat_A"', 'row=row_id, val=row_vec', 3, 'mat_r');
SELECT * FROM mat_r ORDER BY row_id;
</pre> <pre class="result">
row_id | row_vec
--------+---------------------------------
0 | {27,18,15,24,15,18,18,9,30,24}
1 | {24,6,6,18,18,30,6,3,27,27}
2 | {9,27,27,27,24,18,9,27,15,18}
3 | {18,12,6,6,6,21,24,24,0,21}
4 | {18,24,27,27,12,18,27,15,21,21}
5 | {12,30,21,9,27,15,27,6,9,12}
6 | {24,30,21,30,3,27,21,27,24,21}
7 | {21,12,15,18,6,24,3,3,12,24}
8 | {24,24,24,15,6,18,27,3,24,9}
9 | {12,18,9,6,18,12,3,6,9,24}
(10 rows)
</pre></li>
<li>Multiply matrix with a vector <pre class="syntax">
SELECT madlib.matrix_vec_mult('"mat_A"', 'row=row_id, val=row_vec',
array[1,2,3,4,5,6,7,8,9,10]);
</pre> <pre class="result">
matrix_vec_mult
-------------------------------------------
{365,325,358,270,377,278,411,243,287,217}
(10 rows)
</pre></li>
<li><b>Examples with sparse representation</b>. The function calls are the same as before. <pre class="syntax">
SELECT madlib.matrix_sparsify('"mat_B"', 'row=row_id, val=vector',
'"mat_B_sparse"', 'col=col_id, val=val');
</pre></li>
<li>Create a matrix in sparse format. <pre class="syntax">
CREATE TABLE "mat_A_sparse" (
"rowNum" integer,
col_num integer,
entry integer
);
INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (1, 1, 9);
INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (1, 2, 6);
INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (1, 7, 3);
INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (1, 8, 10);
INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (1, 9, 8);
INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (2, 1, 8);
INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (2, 2, 2);
INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (2, 3, 6);
INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (3, 5, 6);
INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (3, 6, 3);
INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (7, 1, 7);
INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (8, 2, 8);
INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (8, 3, 5);
INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (9, 1, 6);
INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (9, 2, 3);
INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (10, 10, 0);
</pre></li>
<li>Transpose a matrix in sparse format <pre class="syntax">
-- Note the double quotes for '"rowNum"' required per PostgreSQL rules
SELECT madlib.matrix_trans('"mat_A_sparse"', 'row="rowNum", val=entry',
'matrix_r_sparse');
SELECT "rowNum", col_num, entry FROM matrix_r_sparse ORDER BY col_num;
</pre> <pre class="result">
rowNum | col_num | entry
--------+---------+-------
1 | 1 | 9
2 | 1 | 6
7 | 1 | 3
8 | 1 | 10
9 | 1 | 8
1 | 2 | 8
2 | 2 | 2
3 | 2 | 6
5 | 3 | 6
6 | 3 | 3
1 | 7 | 7
2 | 8 | 8
3 | 8 | 5
1 | 9 | 6
2 | 9 | 3
10 | 10 | 0
(16 rows)
</pre></li>
<li>Add two sparse matrices <pre class="syntax">
SELECT madlib.matrix_add('"mat_A_sparse"', 'row="rowNum", val=entry',
'"mat_B_sparse"', 'row=row_id, col=col_id, val=val',
'matrix_r_sparse', 'col=col_out');
SELECT madlib.matrix_densify('matrix_r_sparse', 'row="rowNum", col=col_out, val=entry',
'matrix_r');
SELECT * FROM matrix_r ORDER BY "rowNum";
</pre> <pre class="result">
rowNum | entry
--------+---------------------------
1 | {18,16,2,4,6,5,6,17,13,6}
2 | {13,5,11,2,8,6,9,7,7,6}
3 | {0,1,2,3,8,10,7,3,10,1}
4 | {2,9,0,4,3,6,8,6,3,4}
5 | {3,8,7,7,0,5,3,9,2,10}
6 | {5,3,1,7,6,3,5,3,6,4}
7 | {11,8,4,4,2,7,10,0,3,3}
8 | {4,14,5,1,3,1,6,6,9,8}
9 | {12,8,1,7,2,7,10,6,0,6}
10 | {1,4,4,4,8,5,2,8,5,5}
(10 rows)
</pre></li>
<li>Multiply two sparse matrices <pre class="syntax">
SELECT madlib.matrix_mult('"mat_A_sparse"', 'row="rowNum", col=col_num, val=entry',
'"mat_B_sparse"', 'row=row_id, col=col_id, val=val, trans=true',
'matrix_r');
SELECT * FROM matrix_r ORDER BY "rowNum";
</pre> <pre class="result">
rowNum | entry
--------+-------------------------------------------
1 | {260,216,137,180,190,156,138,222,174,159}
2 | {104,76,14,34,82,52,72,44,64,40}
3 | {51,66,33,36,15,45,33,21,33,63}
4 | {0,0,0,0,0,0,0,0,0,0}
5 | {0,0,0,0,0,0,0,0,0,0}
6 | {0,0,0,0,0,0,0,0,0,0}
7 | {63,35,0,14,21,35,28,28,42,7}
8 | {90,49,18,72,99,29,84,48,45,52}
9 | {84,39,3,39,42,39,48,42,51,18}
10 | {0,0,0,0,0,0,0,0,0,0}
(10 rows)
</pre></li>
</ul>
<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd></dd></dl>
<p>File <a class="el" href="array__ops_8sql__in.html" title="implementation of array operations in SQL ">array_ops.sql_in</a> documenting the array operations <a class="el" href="group__grp__array.html">Array Operations</a></p>
<p>File <a class="el" href="matrix__ops_8sql__in.html" title="Implementation of matrix operations in SQL. ">matrix_ops.sql_in</a> for list of functions and usage. </p>
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