SQL functions for linear algebra. More...

Functions
float8	norm1 (float8[] x)
	1-norm of a vector
float8	norm2 (float8[] x)
	2-norm of a vector
float8	dist_norm1 (float8[] x, float8[] y)
	1-norm of the difference between two vectors
float8	dist_norm2 (float8[] x, float8[] y)
	2-norm of the difference between two vectors
float8	squared_dist_norm2 (float8[] x, float8[] y)
	Squared 2-norm of the difference between two vectors.
float8	dist_angle (float8[] x, float8[] y)
	Angle between two vectors.
float8	dist_tanimoto (float8[] x, float8[] y)
	Tanimoto distance between two vectors.
closest_column_result	closest_column (float8[][] m, float8[] x, regproc dist="squared_dist_norm2")
	Given matrix \( M \) and vector \( \vec x \) compute the column of \( M \) that is closest to \( \vec x \).
closest_columns_result	closest_columns (float8[][] m, float8[] x, integer num, regproc dist="squared_dist_norm2")
	Given matrix \( M \) and vector \( \vec x \) compute the columns of \( M \) that are closest to \( \vec x \).
aggregate float8[]	avg (float8[] x)
	Compute the average of vectors.
aggregate float8[]	normalized_avg (float8[] x)
	Compute the normalized average of vectors.
aggregate float8[]	matrix_agg (float8[] x)
	Combine vectors to a matrix.
float8[]	matrix_column (float8[][] matrix, integer col)
	Return the column of a matrix.

Detailed Description

See also:: For an overview of linear-algebra functions, see the module description Linear-Algebra Operations.

Definition in file linalg.sql_in.

Function Documentation

aggregate float8 [] avg ( float8[] x )

Given vectors \( x_1, \dots, x_n \), compute the average \( \frac 1n \sum_{i=1}^n x_i \).

Parameters:

x	Point \( x_i \)

Returns:: Average \( \frac 1n \sum_{i=1}^n x_i \)

Definition at line 291 of file linalg.sql_in.

closest_column_result closest_column	(	float8	m[][],
		float8[]	x,
		regproc	dist = `"squared_dist_norm2"`
	)

Parameters:

M	Matrix \( M = (\vec{m_0} \dots \vec{m_{l-1}}) \in \mathbb{R}^{k \times l} \)
x	Vector \( \vec x \in \mathbb R^k \)
dist	The metric \( \operatorname{dist} \). This needs to be a function with signature `DOUBLE PRECISION[] x DOUBLE PRECISION[] -> DOUBLE PRECISION`.

Returns:

A composite value:

columns_id INTEGER - The 0-based index of the column of \( M \) that is closest to \( x \). In case of ties, the first such index is returned. That is, columns_id is the minimum element in the set \( \arg\min_{i=0,\dots,l-1} \operatorname{dist}(\vec{m_i}, \vec x) \).
distance DOUBLE PRECISION - The minimum distance between any column of \( M \) and \( x \). That is, \( \min_{i=0,\dots,l-1} \operatorname{dist}(\vec{m_i}, \vec x) \).

Definition at line 186 of file linalg.sql_in.

closest_columns_result closest_columns	(	float8	m[][],
		float8[]	x,
		integer	num,
		regproc	dist = `"squared_dist_norm2"`
	)

This function does essentially the same as closest_column(), except that it allows to specify the number of closest columns to return. The return value is a composite value:

columns_ids INTEGER[] - The 0-based indices of the num columns of \( M \) that are closest to \( x \). In case of ties, the first such indices are returned.
distances DOUBLE PRECISION[] - The distances between the columns of \( M \) with indices in columns_ids and \( x \). That is, distances[i] contains \( \operatorname{dist}(\vec{m_j}, \vec x) \), where \( j = \) columns_ids[i].

Definition at line 232 of file linalg.sql_in.

float8 dist_angle	(	float8[]	x,
		float8[]	y
	)

Parameters:

x	Vector \( \vec x = (x_1, \dots, x_n) \)
y	Vector \( \vec y = (y_1, \dots, y_n) \)

Returns:: \( \arccos\left(\frac{\langle \vec x, \vec y \rangle} {\| \vec x \| \cdot \| \vec y \|}\right) \)

Definition at line 131 of file linalg.sql_in.

float8 dist_norm1	(	float8[]	x,
		float8[]	y
	)

Parameters:

x	Vector \( \vec x = (x_1, \dots, x_n) \)
y	Vector \( \vec y = (y_1, \dots, y_n) \)

Returns:: \( \| x - y \|_1 = \sum_{i=1}^n |x_i - y_i| \)

Definition at line 82 of file linalg.sql_in.

float8 dist_norm2	(	float8[]	x,
		float8[]	y
	)

Parameters:

x	Vector \( \vec x = (x_1, \dots, x_n) \)
y	Vector \( \vec y = (y_1, \dots, y_n) \)

Returns:: \( \| x - y \|_2 = \sqrt{\sum_{i=1}^n (x_i - y_i)^2} \)

Definition at line 98 of file linalg.sql_in.

float8 dist_tanimoto	(	float8[]	x,
		float8[]	y
	)

Parameters:

x	Vector \( \vec x = (x_1, \dots, x_n) \)
y	Vector \( \vec y = (y_1, \dots, y_n) \)

Returns:: \( 1 - \frac{\langle \vec x, \vec y \rangle} {\| \vec x \|^2 \cdot \| \vec y \|^2 - \langle \vec x, \vec y \rangle} \)

Definition at line 149 of file linalg.sql_in.

aggregate float8 [] matrix_agg ( float8[] x )

Given vectors \( \vec x_1, \dots, \vec x_n \in \mathbb R^m \), return matrix \( ( \vec x_1 \dots \vec x_n ) \in \mathbb R^{m \times n}\).

Parameters:

x	Vector \( x_i \)

Returns:: Matrix with columns \( x_1, \dots, x_n \)

Definition at line 365 of file linalg.sql_in.

float8 [] matrix_column	(	float8	matrix[][],
		integer	col
	)

Parameters:

matrix	Two-dimensional matrix
col	Column of the matrix to return (0-based index)

Definition at line 382 of file linalg.sql_in.

float8 norm1 ( float8[] x )

Parameters:

x	Vector \( \vec x = (x_1, \dots, x_n) \)

Returns:: \( \| x \|_1 = \sum_{i=1}^n |x_i| \)

Definition at line 53 of file linalg.sql_in.

float8 norm2 ( float8[] x )

Parameters:

x	Vector \( \vec x = (x_1, \dots, x_n) \)

Returns:: \( \| x \|_2 = \sqrt{\sum_{i=1}^n x_i^2} \)

Definition at line 67 of file linalg.sql_in.

aggregate float8 [] normalized_avg ( float8[] x )

Given vectors \( x_1, \dots, x_n \), define \( \widetilde{x} := \frac 1n \sum_{i=1}^n \frac{x_i}{\| x_i \|} \), and compute the normalized average \( \frac{\widetilde{x}}{\| \widetilde{x} \|} \).

Parameters:

x	Point \( x_i \)

Returns:: Normalized average \( \frac{\widetilde{x}}{\| \widetilde{x} \|} \)

Definition at line 329 of file linalg.sql_in.

float8 squared_dist_norm2	(	float8[]	x,
		float8[]	y
	)

Parameters:

x	Vector \( \vec x = (x_1, \dots, x_n) \)
y	Vector \( \vec y = (y_1, \dots, y_n) \)

Returns:: \( \| x - y \|_2^2 = \sum_{i=1}^n (x_i - y_i)^2 \)

Definition at line 114 of file linalg.sql_in.

Functions

Detailed Description

Function Documentation