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 <div class="header">
   <div class="headertitle">
 <div class="title">Norms and Distance Functions<div class="ingroups"><a class="el" href="group__grp__datatrans.html">Data Types and Transformations</a> &raquo; <a class="el" href="group__grp__arraysmatrix.html">Arrays and Matrices</a></div></div>  </div>
 </div><!--header-->
 <div class="contents">
 <div class="toc"><b>Contents</b> <ul>
 <li class="level1">
 <a href="#functions">Linear Algebra Utility Functions</a> </li>
 <li class="level1">
 <a href="#examples">Examples</a> </li>
 <li class="level1">
 <a href="#literature">Literature</a> </li>
 <li class="level1">
 <a href="#related">Related Functions</a> </li>
 </ul>
 </div><p>The linalg module consists of useful utility functions for basic linear algebra operations. Several of these functions can be used while implementing new algorithms. These functions operate on vectors (1-D FLOAT8 array) and matrices (2-D FLOAT8 array). Note that other array types may need to be casted into FLOAT8[] before calling the functions.</p>
 <p>Refer to the <a class="el" href="linalg_8sql__in.html" title="SQL functions for linear algebra. ">linalg.sql_in</a> file for documentation on each of the utility functions.</p>
 <p><a class="anchor" id="functions"></a></p><dl class="section user"><dt>Linear Algebra Utility Functions</dt><dd><table class="output">
 <tr>
 <th><a class="el" href="linalg_8sql__in.html#a300300fe4b8576ba0b97b95d8dea3057" title="1-norm of a vector ">norm1()</a> </th><td><p class="starttd">1-norm of a vector, \(\|\vec{a}\|_1\).</p>
 <p class="endtd"></p>
 </td></tr>
 <tr>
 <th><a class="el" href="linalg_8sql__in.html#a50fdfe30cc0edc6888a909dbb4b4c239" title="2-norm of a vector ">norm2()</a> </th><td><p class="starttd">2-norm of a vector, \(\|\vec{a}\|_2\). </p>
 <p class="endtd"></p>
 </td></tr>
 <tr>
 <th><a class="el" href="linalg_8sql__in.html#aad193850e79c4b9d811ca9bc53e13476" title="1-norm of the difference between two vectors ">dist_norm1()</a> </th><td><p class="starttd">1-norm of the difference between two vectors, \(\|\vec{a} - \vec{b}\|_1\). </p>
 <p class="endtd"></p>
 </td></tr>
 <tr>
 <th><a class="el" href="linalg_8sql__in.html#aa58e51526edea6ea98db30b6f250adb4" title="2-norm of the difference between two vectors ">dist_norm2()</a> </th><td><p class="starttd">2-norm of the difference between two vectors, \(\|\vec{a} - \vec{b}\|_2\). </p>
 <p class="endtd"></p>
 </td></tr>
 <tr>
 <th><a class="el" href="linalg_8sql__in.html#ad9cc156ae57bf7c0a2fe90798259105a" title="p-norm of the difference between two vectors ">dist_pnorm()</a> </th><td><p class="starttd">Generic p-norm of the difference between two vectors, \(\|\vec{a} - \vec{b}\|_p, p &gt; 0\). </p>
 <p class="endtd"></p>
 </td></tr>
 <tr>
 <th><a class="el" href="linalg_8sql__in.html#a31fa9f2f5b45507c09f136464fdad1db" title="Infinity-norm of the difference between two vectors. ">dist_inf_norm()</a> </th><td><p class="starttd">Infinity-norm of the difference between two vectors, \(\|\vec{a} - \vec{b}\|_\infty\). </p>
 <p class="endtd"></p>
 </td></tr>
 <tr>
 <th><a class="el" href="linalg_8sql__in.html#a00a08e69f27524f2096032214e15b668" title="Squared 2-norm of the difference between two vectors. ">squared_dist_norm2()</a> </th><td><p class="starttd">Squared 2-norm of the difference between two vectors, \(\|\vec{a} - \vec{b}\|_2^2\). </p>
 <p class="endtd"></p>
 </td></tr>
 <tr>
 <th><a class="el" href="linalg_8sql__in.html#a1782f2ba00d9f9fab20894a576079f87" title="cosine similarity score between two vectors ">cosine_similarity()</a> </th><td><p class="starttd">Cosine score between two vectors, \(\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|_2 \|\vec{b}\|_2}\). </p>
 <p class="endtd"></p>
 </td></tr>
 <tr>
 <th><a class="el" href="linalg_8sql__in.html#a8c7b9281a72ff22caf06161701b27e84" title="Angle between two vectors. ">dist_angle()</a> </th><td><p class="starttd">Angle between two vectors in an Euclidean space, \(\cos^{-1}(\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|_2 \|\vec{b}\|_2})\). </p>
 <p class="endtd"></p>
 </td></tr>
 <tr>
 <th><a class="el" href="linalg_8sql__in.html#afa13b4c6122b99422d666dedea136c18" title="Tanimoto distance between two vectors. ">dist_tanimoto()</a> </th><td><p class="starttd">Tanimoto distance between two vectors. [1] </p>
 <p class="endtd"></p>
 </td></tr>
 <tr>
 <th><a class="el" href="linalg_8sql__in.html#ac1397ac9f4a35b3b67c3be05b5e1a828" title="Jaccard distance between two vectors (treated as sets) ">dist_jaccard()</a> </th><td><p class="starttd">Jaccard distance between two varchar vectors treated as sets. </p>
 <p class="endtd"></p>
 </td></tr>
 <tr>
 <th><a class="el" href="linalg_8sql__in.html#af6b905fcac7746ef0ed0c36df4a1e070" title="Get an indexed row of the given matrix (2-D array) ">get_row()</a> </th><td><p class="starttd">Return the indexed row of a matrix (2-D array). </p>
 <p class="endtd"></p>
 </td></tr>
 <tr>
 <th><a class="el" href="linalg_8sql__in.html#a20f34c9e661191e5225cca7bc23252c5" title="Get an indexed col of the given matrix (2-D array) ">get_col()</a> </th><td><p class="starttd">Return the indexed col of a matrix (2-D array). </p>
 <p class="endtd"></p>
 </td></tr>
 <tr>
 <th><a class="el" href="linalg_8sql__in.html#a1aa37f73fb1cd8d7d106aa518dd8c0b4" title="Compute the average of vectors. ">avg()</a> </th><td><p class="starttd">Compute the average of vectors. </p>
 <p class="endtd"></p>
 </td></tr>
 <tr>
 <th><a class="el" href="linalg_8sql__in.html#a0b04663ca206f03e66aed5ea2b4cc461" title="Compute the normalized average of vectors. ">normalized_avg()</a> </th><td><p class="starttd">Compute the normalized average of vectors (unit vector in an Euclidean space). </p>
 <p class="endtd"></p>
 </td></tr>
 <tr>
 <th><a class="el" href="linalg_8sql__in.html#a9c439706f35d6cac89f151d553a5f111" title="Combine vectors to a matrix. ">matrix_agg()</a> </th><td><p class="starttd">Combine vectors to a matrix. </p>
 <p class="endtd"></p>
 </td></tr>
 </table>
 </dd></dl>
 <p><a class="anchor" id="examples"></a></p>
 <p><b>Vector Norms and Distances</b></p>
 <ol type="1">
 <li>Create a database table with two vector columns and add some data. <pre class="example">
 CREATE TABLE two_vectors(
     id  integer,
     a   float8[],
     b   float8[]);
 </pre> <pre class="example">
 INSERT INTO two_vectors VALUES
 (1, '{3,4}', '{4,5}'),
 (2, '{1,1,0,-4,5,3,4,106,14}', '{1,1,0,6,-3,1,2,92,2}');
 </pre></li>
 <li>Invoke norm functions. <pre class="example">
 SELECT
     id,
     madlib.norm1(a),
     madlib.norm2(a)
 FROM two_vectors;
 </pre> Result: <pre class="result">
  id | norm1 |      norm2
 ----+-------+------------------
   1 |     7 |                5
   2 |   138 | 107.238052947636
 (2 rows)
 </pre></li>
 <li>Invoke distance functions. <pre class="example">
 SELECT
     id,
     madlib.dist_norm1(a, b),
     madlib.dist_norm2(a, b),
     madlib.dist_pnorm(a, b, 5) AS norm5,
     madlib.dist_inf_norm(a, b),
     madlib.squared_dist_norm2(a, b) AS sq_dist_norm2,
     madlib.cosine_similarity(a, b),
     madlib.dist_angle(a, b),
     madlib.dist_tanimoto(a, b),
     madlib.dist_jaccard(a::text[], b::text[])
 FROM two_vectors;
 </pre> Result: <pre class="result">
  id | dist_norm1 |    dist_norm2    |      norm5       | dist_inf_norm | sq_dist_norm2 | cosine_similarity |     dist_angle     |   dist_tanimoto    |   dist_jaccard
 ----+------------+------------------+------------------+---------------+---------------+-------------------+--------------------+--------------------+-------------------
   1 |          2 |  1.4142135623731 | 1.14869835499704 |             1 |             2 | 0.999512076087079 | 0.0312398334302684 | 0.0588235294117647 | 0.666666666666667
   2 |         48 | 22.6274169979695 |  15.585086360695 |            14 |           512 | 0.985403348449008 |   0.17106899659286 | 0.0498733684005455 | 0.833333333333333
 (2 rows)
 </pre></li>
 </ol>
 <p><b>Matrix Functions</b></p>
 <ol type="1">
 <li>Create a database table with a matrix column. <pre class="example">
 CREATE TABLE matrix(
     id  integer,
     m   float8[]);
 </pre> <pre class="example">
 INSERT INTO matrix VALUES
 (1, '{{4,5},{3,5},{9,0}}');
 </pre></li>
 <li>Invoke matrix functions. <pre class="example">
 SELECT
     madlib.get_row(m, 1) AS row_1,
     madlib.get_row(m, 2) AS row_2,
     madlib.get_row(m, 3) AS row_3,
     madlib.get_col(m, 1) AS col_1,
     madlib.get_col(m, 2) AS col_2
 FROM matrix;
 </pre> Result: <pre class="result">
  row_1 | row_2 | row_3 |  col_1  |  col_2
 -------+-------+-------+---------+---------
  {4,5} | {3,5} | {9,0} | {4,3,9} | {5,5,0}
 (1 row)
 </pre></li>
 </ol>
 <p><b>Aggregate Functions</b></p>
 <ol type="1">
 <li>Create a database table with a vector column. <pre class="example">
 CREATE TABLE vector(
     id  integer,
     v   float8[]);
 </pre> <pre class="example">
 INSERT INTO vector VALUES
 (1, '{4,3}'),
 (2, '{8,6}'),
 (3, '{12,9}');
 </pre></li>
 <li>Invoke aggregate functions. <pre class="example">
 SELECT
     madlib.avg(v),
     madlib.normalized_avg(v),
     madlib.matrix_agg(v)
 FROM vector;
 </pre> Result: <pre class="result">
   avg  | normalized_avg |      matrix_agg
 -------+----------------+----------------------
  {8,6} | {0.8,0.6}      | {{4,3},{8,6},{12,9}}
 (1 row)
 </pre></li>
 </ol>
 <p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
 <p>[1] <a href="http://en.wikipedia.org/wiki/Jaccard_index#Tanimoto_similarity_and_distance">http://en.wikipedia.org/wiki/Jaccard_index#Tanimoto_similarity_and_distance</a></p>
 <p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="linalg_8sql__in.html" title="SQL functions for linear algebra. ">linalg.sql_in</a> documenting the SQL functions. </dd></dl>
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	<div class="header">
	<div class="headertitle">
	<div class="title">Norms and Distance Functions<div class="ingroups"><a class="el" href="group__grp__datatrans.html">Data Types and Transformations</a> » <a class="el" href="group__grp__arraysmatrix.html">Arrays and Matrices</a></div></div> </div>
	</div><!--header-->
	<div class="contents">
	<div class="toc"><b>Contents</b> <ul>
	<li class="level1">
	<a href="#functions">Linear Algebra Utility Functions</a> </li>
	<li class="level1">
	<a href="#examples">Examples</a> </li>
	<li class="level1">
	<a href="#literature">Literature</a> </li>
	<li class="level1">
	<a href="#related">Related Functions</a> </li>
	</ul>
	</div><p>The linalg module consists of useful utility functions for basic linear algebra operations. Several of these functions can be used while implementing new algorithms. These functions operate on vectors (1-D FLOAT8 array) and matrices (2-D FLOAT8 array). Note that other array types may need to be casted into FLOAT8[] before calling the functions.</p>
	<p>Refer to the <a class="el" href="linalg_8sql__in.html" title="SQL functions for linear algebra. ">linalg.sql_in</a> file for documentation on each of the utility functions.</p>
	<p><a class="anchor" id="functions"></a></p><dl class="section user"><dt>Linear Algebra Utility Functions</dt><dd><table class="output">
	<tr>
	<th><a class="el" href="linalg_8sql__in.html#a300300fe4b8576ba0b97b95d8dea3057" title="1-norm of a vector ">norm1()</a> </th><td><p class="starttd">1-norm of a vector, \(\\|\vec{a}\\|_1\).</p>
	<p class="endtd"></p>
	</td></tr>
	<tr>
	<th><a class="el" href="linalg_8sql__in.html#a50fdfe30cc0edc6888a909dbb4b4c239" title="2-norm of a vector ">norm2()</a> </th><td><p class="starttd">2-norm of a vector, \(\\|\vec{a}\\|_2\). </p>
	<p class="endtd"></p>
	</td></tr>
	<tr>
	<th><a class="el" href="linalg_8sql__in.html#aad193850e79c4b9d811ca9bc53e13476" title="1-norm of the difference between two vectors ">dist_norm1()</a> </th><td><p class="starttd">1-norm of the difference between two vectors, \(\\|\vec{a} - \vec{b}\\|_1\). </p>
	<p class="endtd"></p>
	</td></tr>
	<tr>
	<th><a class="el" href="linalg_8sql__in.html#aa58e51526edea6ea98db30b6f250adb4" title="2-norm of the difference between two vectors ">dist_norm2()</a> </th><td><p class="starttd">2-norm of the difference between two vectors, \(\\|\vec{a} - \vec{b}\\|_2\). </p>
	<p class="endtd"></p>
	</td></tr>
	<tr>
	<th><a class="el" href="linalg_8sql__in.html#ad9cc156ae57bf7c0a2fe90798259105a" title="p-norm of the difference between two vectors ">dist_pnorm()</a> </th><td><p class="starttd">Generic p-norm of the difference between two vectors, \(\\|\vec{a} - \vec{b}\\|_p, p > 0\). </p>
	<p class="endtd"></p>
	</td></tr>
	<tr>
	<th><a class="el" href="linalg_8sql__in.html#a31fa9f2f5b45507c09f136464fdad1db" title="Infinity-norm of the difference between two vectors. ">dist_inf_norm()</a> </th><td><p class="starttd">Infinity-norm of the difference between two vectors, \(\\|\vec{a} - \vec{b}\\|_\infty\). </p>
	<p class="endtd"></p>
	</td></tr>
	<tr>
	<th><a class="el" href="linalg_8sql__in.html#a00a08e69f27524f2096032214e15b668" title="Squared 2-norm of the difference between two vectors. ">squared_dist_norm2()</a> </th><td><p class="starttd">Squared 2-norm of the difference between two vectors, \(\\|\vec{a} - \vec{b}\\|_2^2\). </p>
	<p class="endtd"></p>
	</td></tr>
	<tr>
	<th><a class="el" href="linalg_8sql__in.html#a1782f2ba00d9f9fab20894a576079f87" title="cosine similarity score between two vectors ">cosine_similarity()</a> </th><td><p class="starttd">Cosine score between two vectors, \(\frac{\vec{a} \cdot \vec{b}}{\\|\vec{a}\\|_2 \\|\vec{b}\\|_2}\). </p>
	<p class="endtd"></p>
	</td></tr>
	<tr>
	<th><a class="el" href="linalg_8sql__in.html#a8c7b9281a72ff22caf06161701b27e84" title="Angle between two vectors. ">dist_angle()</a> </th><td><p class="starttd">Angle between two vectors in an Euclidean space, \(\cos^{-1}(\frac{\vec{a} \cdot \vec{b}}{\\|\vec{a}\\|_2 \\|\vec{b}\\|_2})\). </p>
	<p class="endtd"></p>
	</td></tr>
	<tr>
	<th><a class="el" href="linalg_8sql__in.html#afa13b4c6122b99422d666dedea136c18" title="Tanimoto distance between two vectors. ">dist_tanimoto()</a> </th><td><p class="starttd">Tanimoto distance between two vectors. [1] </p>
	<p class="endtd"></p>
	</td></tr>
	<tr>
	<th><a class="el" href="linalg_8sql__in.html#ac1397ac9f4a35b3b67c3be05b5e1a828" title="Jaccard distance between two vectors (treated as sets) ">dist_jaccard()</a> </th><td><p class="starttd">Jaccard distance between two varchar vectors treated as sets. </p>
	<p class="endtd"></p>
	</td></tr>
	<tr>
	<th><a class="el" href="linalg_8sql__in.html#af6b905fcac7746ef0ed0c36df4a1e070" title="Get an indexed row of the given matrix (2-D array) ">get_row()</a> </th><td><p class="starttd">Return the indexed row of a matrix (2-D array). </p>
	<p class="endtd"></p>
	</td></tr>
	<tr>
	<th><a class="el" href="linalg_8sql__in.html#a20f34c9e661191e5225cca7bc23252c5" title="Get an indexed col of the given matrix (2-D array) ">get_col()</a> </th><td><p class="starttd">Return the indexed col of a matrix (2-D array). </p>
	<p class="endtd"></p>
	</td></tr>
	<tr>
	<th><a class="el" href="linalg_8sql__in.html#a1aa37f73fb1cd8d7d106aa518dd8c0b4" title="Compute the average of vectors. ">avg()</a> </th><td><p class="starttd">Compute the average of vectors. </p>
	<p class="endtd"></p>
	</td></tr>
	<tr>
	<th><a class="el" href="linalg_8sql__in.html#a0b04663ca206f03e66aed5ea2b4cc461" title="Compute the normalized average of vectors. ">normalized_avg()</a> </th><td><p class="starttd">Compute the normalized average of vectors (unit vector in an Euclidean space). </p>
	<p class="endtd"></p>
	</td></tr>
	<tr>
	<th><a class="el" href="linalg_8sql__in.html#a9c439706f35d6cac89f151d553a5f111" title="Combine vectors to a matrix. ">matrix_agg()</a> </th><td><p class="starttd">Combine vectors to a matrix. </p>
	<p class="endtd"></p>
	</td></tr>
	</table>
	</dd></dl>
	<p><a class="anchor" id="examples"></a></p>
	<p><b>Vector Norms and Distances</b></p>
	<ol type="1">
	<li>Create a database table with two vector columns and add some data. <pre class="example">
	CREATE TABLE two_vectors(
	id integer,
	a float8[],
	b float8[]);
	</pre> <pre class="example">
	INSERT INTO two_vectors VALUES
	(1, '{3,4}', '{4,5}'),
	(2, '{1,1,0,-4,5,3,4,106,14}', '{1,1,0,6,-3,1,2,92,2}');
	</pre></li>
	<li>Invoke norm functions. <pre class="example">
	SELECT
	id,
	madlib.norm1(a),
	madlib.norm2(a)
	FROM two_vectors;
	</pre> Result: <pre class="result">
	id \| norm1 \| norm2
	----+-------+------------------
	1 \| 7 \| 5
	2 \| 138 \| 107.238052947636
	(2 rows)
	</pre></li>
	<li>Invoke distance functions. <pre class="example">
	SELECT
	id,
	madlib.dist_norm1(a, b),
	madlib.dist_norm2(a, b),
	madlib.dist_pnorm(a, b, 5) AS norm5,
	madlib.dist_inf_norm(a, b),
	madlib.squared_dist_norm2(a, b) AS sq_dist_norm2,
	madlib.cosine_similarity(a, b),
	madlib.dist_angle(a, b),
	madlib.dist_tanimoto(a, b),
	madlib.dist_jaccard(a::text[], b::text[])
	FROM two_vectors;
	</pre> Result: <pre class="result">
	id \| dist_norm1 \| dist_norm2 \| norm5 \| dist_inf_norm \| sq_dist_norm2 \| cosine_similarity \| dist_angle \| dist_tanimoto \| dist_jaccard
	----+------------+------------------+------------------+---------------+---------------+-------------------+--------------------+--------------------+-------------------
	1 \| 2 \| 1.4142135623731 \| 1.14869835499704 \| 1 \| 2 \| 0.999512076087079 \| 0.0312398334302684 \| 0.0588235294117647 \| 0.666666666666667
	2 \| 48 \| 22.6274169979695 \| 15.585086360695 \| 14 \| 512 \| 0.985403348449008 \| 0.17106899659286 \| 0.0498733684005455 \| 0.833333333333333
	(2 rows)
	</pre></li>
	</ol>
	<p><b>Matrix Functions</b></p>
	<ol type="1">
	<li>Create a database table with a matrix column. <pre class="example">
	CREATE TABLE matrix(
	id integer,
	m float8[]);
	</pre> <pre class="example">
	INSERT INTO matrix VALUES
	(1, '{{4,5},{3,5},{9,0}}');
	</pre></li>
	<li>Invoke matrix functions. <pre class="example">
	SELECT
	madlib.get_row(m, 1) AS row_1,
	madlib.get_row(m, 2) AS row_2,
	madlib.get_row(m, 3) AS row_3,
	madlib.get_col(m, 1) AS col_1,
	madlib.get_col(m, 2) AS col_2
	FROM matrix;
	</pre> Result: <pre class="result">
	row_1 \| row_2 \| row_3 \| col_1 \| col_2
	-------+-------+-------+---------+---------
	{4,5} \| {3,5} \| {9,0} \| {4,3,9} \| {5,5,0}
	(1 row)
	</pre></li>
	</ol>
	<p><b>Aggregate Functions</b></p>
	<ol type="1">
	<li>Create a database table with a vector column. <pre class="example">
	CREATE TABLE vector(
	id integer,
	v float8[]);
	</pre> <pre class="example">
	INSERT INTO vector VALUES
	(1, '{4,3}'),
	(2, '{8,6}'),
	(3, '{12,9}');
	</pre></li>
	<li>Invoke aggregate functions. <pre class="example">
	SELECT
	madlib.avg(v),
	madlib.normalized_avg(v),
	madlib.matrix_agg(v)
	FROM vector;
	</pre> Result: <pre class="result">
	avg \| normalized_avg \| matrix_agg
	-------+----------------+----------------------
	{8,6} \| {0.8,0.6} \| {{4,3},{8,6},{12,9}}
	(1 row)
	</pre></li>
	</ol>
	<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
	<p>[1] <a href="http://en.wikipedia.org/wiki/Jaccard_index#Tanimoto_similarity_and_distance">http://en.wikipedia.org/wiki/Jaccard_index#Tanimoto_similarity_and_distance</a></p>
	<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="linalg_8sql__in.html" title="SQL functions for linear algebra. ">linalg.sql_in</a> documenting the SQL functions. </dd></dl>
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