Matrix Decompositions


Functions
bool	SPUC::chol (const mat &X, mat &F)
	Cholesky factorisation of real symmetric and positive definite matrix.
mat	SPUC::chol (const mat &X)
	Cholesky factorisation of real symmetric and positive definite matrix.
bool	SPUC::chol (const mat &X, int p, mat &F)
	Cholesky factorisation of an n by n band-matrix X. Bandwidth p.
mat	SPUC::chol (const mat &X, int p)
	Cholesky factorisation of a band matrix.
bool	SPUC::lu (const mat &X, mat &L, mat &U, ivec &p)
	LU factorisation of real matrix.
void	SPUC::interchange_permutations (vec &b, const ivec &p)
	Makes swapping of vector b according to the inerchange permutation vector p.
bmat	SPUC::permutation_matrix (const ivec &p)
	Make permutation matrix P from the interchange permutation vector p.
bool	SPUC::svd (const mat &A, vec &S)
	Singular Value Decomposition (SVD).
bool	SPUC::svd (const cmat &A, vec &S)
	Singular Value Decomposition (SVD).
vec	SPUC::svd (const mat &A)
	Singular Value Decomposition (SVD).
vec	SPUC::svd (const cmat &A)
	Singular Value Decomposition (SVD).
bool	SPUC::svd (const mat &A, mat &U, vec &S, mat &V)
	Singular Value Decomposition (SVD).
bool	SPUC::svd (const cmat &A, cmat &U, vec &S, cmat &V)
	Singular Value Decomposition (SVD).

Function Documentation

mat SPUC::chol ( const mat & X,

int p

)

Cholesky factorisation of a band matrix.
Cholesky factorisation of an n by n band-matrix X. Bandwidth p. If X is positive definite, F=chol(X) produces an upper triangular F. If also X is symmetric then F'*F = X. If X is not positive definite, an error message is printed.
Uses n*(p^2+3*p) flops and n sqrt() (if n >> p). Uses Alg. 4.3.5 (outer product version) in Golub & van Loan "Matrix computations", 3rd ed., p. 156.

bool SPUC::chol ( const mat & X,

int p,

mat & F

)

Cholesky factorisation of an n by n band-matrix X. Bandwidth p.
If X is positive definite, true is returned and F=chol(X) produces an upper triangular F. If also X is symmetric then F'*F = X. If X is not positive definite, false is returned.
Uses n*(p^2+3*p) flops and n sqrt() (if n >> p). Uses Alg. 4.3.5 (outer product version) in Golub & van Loan "Matrix computations", 3rd ed., p. 156.

mat SPUC::chol ( const mat & X )

Cholesky factorisation of real symmetric and positive definite matrix.
The Cholesky factorisation of a real symmetric positive-definite matrix $\mathbf{X}$ of size $n \times n$ is given by
$\mathbf{X} = \mathbf{F}^T \mathbf{F}$
where $\mathbf{F}$ is an upper trangular $n \times n$ matrix.

bool SPUC::chol ( const mat & X,

mat & F

)

Cholesky factorisation of real symmetric and positive definite matrix.
The Cholesky factorisation of a real symmetric positive-definite matrix $\mathbf{X}$ of size $n \times n$ is given by
$\mathbf{X} = \mathbf{F}^T \mathbf{F}$
where $\mathbf{F}$ is an upper trangular $n \times n$ matrix.
Returns true if calcuation succeeded. False otherwise.

void SPUC::interchange_permutations ( vec & b,

const ivec & p

)

Makes swapping of vector b according to the inerchange permutation vector p.

bool SPUC::lu ( const mat & X,

mat & L,

mat & U,

ivec & p

)

LU factorisation of real matrix.
The LU factorization of the real matrix $\mathbf{X}$ of size $n \times n$ is given by
$\mathbf{X} = \mathbf{P}^T \mathbf{L} \mathbf{U} ,$
where $\mathbf{L}$ and $\mathbf{U}$ are lower and upper triangular matrices and $\mathbf{P}$ is a permutation matrix.
The interchange permutation vector p is such that k and p(k) should be changed for all k. Given this vector a permuation matrix can be constructed using the function
bmat permuation_matrix(const ivec &p)
If X is an n by n matrix lu(X,L,U,p) computes the LU decomposition. L is a lower trangular, U an upper triangular matrix. p is the interchange permutation vector such that k and p(k) should be changed for all k.
Returns true is calculation succeeds. False otherwise.

bmat SPUC::permutation_matrix ( const ivec & p )

Make permutation matrix P from the interchange permutation vector p.

bool SPUC::svd ( const cmat & A,

cmat & U,

vec & S,

cmat & V

)

Singular Value Decomposition (SVD).
The svd-algorithm computes the decomposition of a real $m \times n$ matrix $\mathbf{A}$ so that
$\mathbf{U}^T \mathbf{A} \mathbf{V} = \mathrm{diag}(\mathbf{s}) = \mathrm{diag}(\sigma_1, \ldots, \sigma_p)$
where the elements of $\mathbf{s}$ , $\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0$ are the singular values of $\mathbf{A}$ . Or put differently
$\mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H$

bool SPUC::svd ( const mat & A,

mat & U,

vec & S,

mat & V

)

Singular Value Decomposition (SVD).
The svd-algorithm computes the decomposition of a real $m \times n$ matrix $\mathbf{A}$ so that
$\mathbf{U}^T \mathbf{A} \mathbf{V} = \mathrm{diag}(\mathbf{s}) = \mathrm{diag}(\sigma_1, \ldots, \sigma_p)$
where the elements of $\mathbf{s}$ , $\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0$ are the singular values of $\mathbf{A}$ . Or put differently
$\mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H$

vec SPUC::svd ( const cmat & A )

Singular Value Decomposition (SVD).
The svd-algorithm computes the decomposition of a real $m \times n$ matrix $\mathbf{A}$ so that
$\mathbf{U}^T \mathbf{A} \mathbf{V} = \mathrm{diag}(\mathbf{s}) = \mathrm{diag}(\sigma_1, \ldots, \sigma_p)$
where the elements of $\mathbf{s}$ , $\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0$ are the singular values of $\mathbf{A}$ . Or put differently
$\mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H$

vec SPUC::svd ( const mat & A )

Singular Value Decomposition (SVD).
The svd-algorithm computes the decomposition of a real $m \times n$ matrix $\mathbf{A}$ so that
$\mathbf{U}^T \mathbf{A} \mathbf{V} = \mathrm{diag}(\mathbf{s}) = \mathrm{diag}(\sigma_1, \ldots, \sigma_p)$
where the elements of $\mathbf{s}$ , $\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0$ are the singular values of $\mathbf{A}$ . Or put differently
$\mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H$

bool SPUC::svd ( const cmat & A,

vec & S

)

Singular Value Decomposition (SVD).
The svd-algorithm computes the decomposition of a real $m \times n$ matrix $\mathbf{A}$ so that
$\mathbf{U}^T \mathbf{A} \mathbf{V} = \mathrm{diag}(\mathbf{s}) = \mathrm{diag}(\sigma_1, \ldots, \sigma_p)$
where the elements of $\mathbf{s}$ , $\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0$ are the singular values of $\mathbf{A}$ . Or put differently
$\mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H$

bool SPUC::svd ( const mat & A,

vec & S

)

Singular Value Decomposition (SVD).
The svd-algorithm computes the decomposition of a real $m \times n$ matrix $\mathbf{A}$ so that
$\mathbf{U}^T \mathbf{A} \mathbf{V} = \mathrm{diag}(\mathbf{s}) = \mathrm{diag}(\sigma_1, \ldots, \sigma_p)$
where the elements of $\mathbf{s}$ , $\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0$ are the singular values of $\mathbf{A}$ . Or put differently
$\mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H$

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1.4.4

Matrix Decompositions

Functions

Function Documentation