Collaboration diagram for Matrix Mathematics:

Functions
static int	inv2 (const double a, double b)
static int	inv3 (const double a, double b)
static int	inv4 (const double a, double b)
void	matrix_add_sc (u32 n, u32 m, const double a, const double b, double gamma, double *c)
int	matrix_ataati (u32 n, u32 m, const double a, double b)
int	matrix_ataiat (u32 n, u32 m, const double a, double b)
int	matrix_atawati (u32 n, u32 m, const double a, const double w, double *b)
int	matrix_atwaiat (u32 n, u32 m, const double a, const double w, double *b)
void	matrix_copy (u32 n, u32 m, const double a, double b)
int	matrix_inverse (u32 n, const double const a, double b)
void	matrix_multiply (u32 n, u32 m, u32 p, const double a, const double b, double *c)
int	matrix_pseudoinverse (u32 n, u32 m, const double a, double b)
void	matrix_transpose (u32 n, u32 m, const double a, double b)
s32	qrdecomp (const double a, u32 rows, u32 cols, double qt, double *r)
s32	qrdecomp_square (const double a, u32 rows, double qt, double *r)
s32	qrsolve (const double a, u32 rows, u32 cols, const double b, double *x)
void	qtmult (const double qt, u32 n, const double b, double *x)
static void	row_swap (double a, double b, u32 size)
static int	rref (u32 order, u32 cols, double *m)
void	rsolve (const double r, u32 rows, u32 cols, const double b, double *x)

Detailed Description

Routines for working with matrices.

Function Documentation

static int inv2	(	const double *	a,
		double *	b
	)		`[inline, static]`

Invert a 2x2 matrix. Calculate the inverse of a 2x2 matrix: $b := a^{-1}$

Parameters:

a	The matrix to invert (input)
b	Where to put the inverse (output)

Returns:: -1 if a is singular; 0 otherwise.

Definition at line 384 of file linear_algebra.c.

static int inv3	(	const double *	a,
		double *	b
	)		`[inline, static]`

Invert a 3x3 matrix. Calculate the inverse of a 3x3 matrix: $b := a^{-1}$

Parameters:

a	The matrix to invert (input)
b	Where to put the inverse (output)

Returns:: -1 if a is singular; 0 otherwise.

Definition at line 403 of file linear_algebra.c.

static int inv4	(	const double *	a,
		double *	b
	)		`[inline, static]`

Invert a 4x4 matrix. Calculate the inverse of a 4x4 matrix: $b := a^{-1}$

Parameters:

a	The matrix to invert (input)
b	Where to put the inverse (output)

Returns:: -1 if a is singular; 0 otherwise.

Definition at line 434 of file linear_algebra.c.

void matrix_add_sc	(	u32	n,
		u32	m,
		const double *	a,
		const double *	b,
		double	gamma,
		double *	c
	)

Add a matrix to a scaled matrix. Add two matrices: $C := A + \gamma B$ , where $ A $ , $ B $ and $C$ are matrices on $\mathbb{R}^{n \times m}$ and $\gamma$ is a scalar coefficient.

Parameters:

n	Number of rows in a, b and c
m	Number of columns in a, b and c
a	First matrix (unscaled)
b	Second matrix (will be scaled)
gamma	Coefficient for second matrix
c	Output (sum) matrix

Definition at line 805 of file linear_algebra.c.

int matrix_ataati	(	u32	n,
		u32	m,
		const double *	a,
		double *	b
	)		`[inline]`

Compute $B := A^{T} (A A^{T})^{-1}$ . Compute $B := A^{T} (A A^{T})^{-1}$ , where $ A $ is a matrix on $\mathbb{R}^{n \times m}$ and $B$ is (therefore) a matrix on $\mathbb{R}^{n \times n}$ , for $ n < m $ .

Parameters:

n	Number of rows in a and rows and columns in b
m	Number of columns in a
a	Input matrix
b	Output matrix

Returns:: -1 if n >= m or singular; 0 otherwise

Definition at line 762 of file linear_algebra.c.

int matrix_ataiat	(	u32	n,
		u32	m,
		const double *	a,
		double *	b
	)		`[inline]`

Compute $B := (A^{T} A)^{-1} A^{T}$ . Compute $B := (A^{T} A)^{-1} A^{T}$ , where $ A $ is a matrix on $\mathbb{R}^{n \times m}$ and $B$ is (therefore) a matrix on $\mathbb{R}^{m \times m}$ , for $ n > m $ .

Parameters:

n	Number of rows in a
m	Number of columns in a and rows and columns in b
a	Input matrix
b	Output matrix

Returns:: -1 if n < m; 0 otherwise

Definition at line 743 of file linear_algebra.c.

int matrix_atawati	(	u32	n,
		u32	m,
		const double *	a,
		const double *	w,
		double *	b
	)		`[inline]`

Compute $B := A^T (A W A^{T})^{-1}$ . Compute $B := A^T (A W A^{T})^{-1}$ , where $ A $ is a matrix on $\mathbb{R}^{n \times m}$ , $ W $ is a diagonal weighting matrix on $\mathbb{R}^{m \times m}$ and $B$ is (therefore) a matrix on $\mathbb{R}^{n \times n}$ , for $ n < m $ .

Parameters:

n	Number of rows in a and rows and columns in b
m	Number of columns in a
a	Input matrix
w	Diagonal vector of weighting matrix
b	Output matrix

Returns:: -1 if n <= m or singular; 0 otherwise

Definition at line 696 of file linear_algebra.c.

int matrix_atwaiat	(	u32	n,
		u32	m,
		const double *	a,
		const double *	w,
		double *	b
	)		`[inline]`

Compute $B := (A^{T} W A)^{-1} A^{T}$ . Compute $B := (A^{T} W A)^{-1} A^{T}$ , where $ A $ is a matrix on $\mathbb{R}^{n \times m}$ , $ W $ is a diagonal weighting matrix on $\mathbb{R}^{n \times n}$ and $B$ is (therefore) a matrix on $\mathbb{R}^{m \times m}$ , for $ n > m $ .

Parameters:

n	Number of rows in a
m	Number of columns in a and rows and columns in b
a	Input matrix
w	Diagonal vector of weighting matrix
b	Output matrix

Returns:: -1 if n <= m or singular; 0 otherwise

Definition at line 647 of file linear_algebra.c.

void matrix_copy	(	u32	n,
		u32	m,
		const double *	a,
		double *	b
	)

Copy a matrix. Copy a matrix: $ B := A $ , where $A$ and $B$ are matrices on $\mathbb{R}^{n \times m}$ .

Parameters:

n	Number of rows in and
m	Number of columns in and
a	Matrix to copy
b	Copied (output) matrix

Definition at line 840 of file linear_algebra.c.

int matrix_inverse	(	u32	n,
		const double const *	a,
		double *	b
	)		`[inline]`

Invert a square matrix. Calculate the inverse of a square matrix: $B := A^{-1}$ , where $A$ and $B$ are matrices on $\mathbb{R}^{n \times n}$ . For matrices size 4x4 and smaller, this is done by autogenerated hard-coded routines. For larger matrices, this is done by Gauss-Jordan elimination (which is $O(n^{3})$ ).

Parameters:

n	The rank of a and b
a	The matrix to invert (input)
b	Where to put the inverse (output)

Returns:: -1 if a is singular; 0 otherwise.

Definition at line 534 of file linear_algebra.c.

void matrix_multiply	(	u32	n,
		u32	m,
		u32	p,
		const double *	a,
		const double *	b,
		double *	c
	)		`[inline]`

Multiply two matrices. Multiply two matrices: $ C := AB $ , where $ A $ is a matrix on $\mathbb{R}^{n \times m}$ , $B$ is a matrix on $\mathbb{R}^{m \times p}$ and $C$ is (therefore) a matrix in $\mathbb{R}^{n \times p}$ .

Parameters:

n	Number of rows in a and c
m	Number of columns in a and rows in b
p	Number of columns in b and c
a	First matrix to multiply
b	Second matrix to multiply
c	Output matrix

Definition at line 782 of file linear_algebra.c.

int matrix_pseudoinverse	(	u32	n,
		u32	m,
		const double *	a,
		double *	b
	)

Invert a non-square matrix (least-squares or least-norm solution). If $ A $ is of full rank, calculate the Moore-Penrose pseudoinverse $A^{+}$ of a square matrix:

$A \in \mathbb{R}^{n \times m}$

$B := A^{+} = \begin{cases} (A^{T} A)^{-1} A^{T} & \text{if } n > m \\ A^{T} (A A^{T})^{-1} & \text{if } m > n \\ A^{-1} & \text{if } n = m \end{cases}$

If $ n > m $ , then $ A $ must be of full column rank, and $A^{+}$ solves the linear least-squares (overconstrained) problem:

$x' = A^{+} b = \underset{x}{min} \|Ax - b\|_{2}$

If $ m > n $ , then $ A $ must be of full row rank, and $A^{+}$ solves the linear least-norm (underconstrained) problem:

$x' = A^{+} b = \underset{x}{min} \|x\|_{2} \text{s.t. } Ax = b$

If $ m = n $ , then $ A $ must be of full rank, and $A^{+} = A^{-1}$ .

Parameters:

n	The number of rows in a
m	The number of columns in a
a	The matrix to invert (input)
b	Where to put the inverse (output)

Returns:: -1 if a is singular; 0 otherwise.

Definition at line 622 of file linear_algebra.c.

void matrix_transpose	(	u32	n,
		u32	m,
		const double *	a,
		double *	b
	)

Transpose a matrix. Transpose a matrix: $B := A^{T}$ , where $A$ is a matrix on $\mathbb{R}^{n \times m}$ and $B$ is (therefore) a matrix on $\mathbb{R}^{m \times n}$ .

Parameters:

n	Number of rows in and columns in
m	Number of rows in and columns in
a	Matrix to transpose
b	Transposed (output) matrix

Definition at line 823 of file linear_algebra.c.

s32 qrdecomp	(	const double *	a,
		u32	rows,
		u32	cols,
		double *	qt,
		double *	r
	)

QR decomposition of a matrix. Compute the QR decomposition of an arbitrary matrix $A \in \mathbb{R}^{N \times M}$ : $A = Q \cdot R$ where $Q \in \mathbb{R}^{N \times N}$ is an orthogonal matrix and $R \in \mathbb{R}^{N \times M}$ is an upper-triangular matrix.

For an overdetermined (least-squares) problem, $ R $ will be of the form

$\begin{bmatrix} R' \\ 0 \end{bmatrix}$

where $ R' $ is an upper-triangular matrix on $\mathbb{R}^{M \times M}$ .

Parameters:

A	The matrix to decompose (input)
rows	How many rows in A
cols	How many columns in A
qt	$Q^{T}$ (output)
r	(output)

Returns:: -1 if A is singular; 0 otherwise;

Definition at line 103 of file linear_algebra.c.

s32 qrdecomp_square	(	const double *	a,
		u32	rows,
		double *	qt,
		double *	r
	)

QR decomposition of a square matrix. Compute the QR decomposition of a square matrix $A \in \mathbb{R}^{N \times N}$ : $A = Q \cdot R$ where $Q \in \mathbb{R}^{N \times N}$ is an orthogonal matrix and $R \in \mathbb{R}^{N \times N}$ is an upper-triangular matrix.

Parameters:

A	The matrix to decompose (input)
rows	How many rows in A
qt	$Q^{T}$ (output)
r	(output)

Returns:: -1 if A is singular; 0 otherwise;

Definition at line 257 of file linear_algebra.c.

s32 qrsolve	(	const double *	a,
		u32	rows,
		u32	cols,
		const double *	b,
		double *	x
	)

Solve a linear system using the QR decomposition. Solve the linear system $ Ax = b $ using the QR decomposition and backward substitution, where $ A $ is a matrix on $\mathbb{R}^{N \times N}$ and $ x $ and $ b $ are vectors on $\mathbb{R}^{N}$ .

Parameters:

a	Matrix (input)
rows	Number of rows in a
cols	Number of columns in a
b	Vector (input)
x	Vector (output)

Returns:: -1 if a is singular; 0 otherwise.

Definition at line 365 of file linear_algebra.c.

void qtmult	(	const double *	qt,
		u32	n,
		const double *	b,
		double *	x
	)

Solve Qx = b for x. Since $Q \in \mathbb{R}^{N \times N}$ is an orthogonal matrix, $Q^{T} Q = I$ and therefore $x = Q^{T} b$ . This function computes $x \in \mathbb{R}^{N}$ in this way.

Parameters:

qt	$Q^{T}$ to be used to solve for x (input)
n	size of qt (it is square)
b	to be used to solve for x (input)
x	result of the linear solve (output)

Definition at line 317 of file linear_algebra.c.

static void row_swap	(	double *	a,
		double *	b,
		u32	size
	)		`[static]`

Definition at line 468 of file linear_algebra.c.

static int rref	(	u32	order,
		u32	cols,
		double *	m
	)		`[static]`

Definition at line 479 of file linear_algebra.c.

void rsolve	(	const double *	r,
		u32	rows,
		u32	cols,
		const double *	b,
		double *	x
	)

Solve Rx = b for x. Solve $ Rx = b $ for $x \in \mathbb{R}^{N}$ . Since $R \in \mathbb{R}^{N \times M}$ is upper-triangular, this can be done efficiently by back-substitution. This function has two important properties: it must never be called with an $ R $ that results from the decomposition of a singular matrix, and it is safe to pass the same pointer for $ b $ and $ x $ .

Parameters:

r	Upper-triangular (input)
rows	Number of rows in r
cols	Number of columns in r
b	Vector to solve against, from qtmult()
x	Solution vector (output)

Definition at line 341 of file linear_algebra.c.

Functions

Detailed Description

Function Documentation