Matrix Decompositions
[Linear Algebra]


Functions

bool itpp::chol (const mat &X, mat &F)
 Cholesky factorisation of real symmetric and positive definite matrix.
mat itpp::chol (const mat &X)
 Cholesky factorisation of real symmetric and positive definite matrix.
bool itpp::chol (const cmat &X, cmat &F)
 Cholesky factorisation of complex hermitian and positive-definite matrix.
cmat itpp::chol (const cmat &X)
 Cholesky factorisation of complex hermitian and positive-definite matrix.
bool itpp::eig_sym (const mat &A, vec &d, mat &V)
 Calculates the eigenvalues and eigenvectors of a symmetric real matrix.
bool itpp::eig_sym (const mat &A, vec &d)
 Calculates the eigenvalues of a symmetric real matrix.
vec itpp::eig_sym (const mat &A)
 Calculates the eigenvalues of a symmetric real matrix.
bool itpp::eig_sym (const cmat &A, vec &d, cmat &V)
 Calculates the eigenvalues and eigenvectors of a hermitian complex matrix.
bool itpp::eig_sym (const cmat &A, vec &d)
 Calculates the eigenvalues of a hermitian complex matrix.
vec itpp::eig_sym (const cmat &A)
 Calculates the eigenvalues of a hermitian complex matrix.
bool itpp::eig (const mat &A, cvec &d, cmat &V)
 Calculates the eigenvalues and eigenvectors of a real non-symmetric matrix.
bool itpp::eig (const mat &A, cvec &d)
 Calculates the eigenvalues of a real non-symmetric matrix.
cvec itpp::eig (const mat &A)
 Calculates the eigenvalues of a real non-symmetric matrix.
bool itpp::eig (const cmat &A, cvec &d, cmat &V)
 Calculates the eigenvalues and eigenvectors of a complex non-hermitian matrix.
bool itpp::eig (const cmat &A, cvec &d)
 Calculates the eigenvalues of a complex non-hermitian matrix.
cvec itpp::eig (const cmat &A)
 Calculates the eigenvalues of a complex non-hermitian matrix.
bool itpp::lu (const mat &X, mat &L, mat &U, ivec &p)
 LU factorisation of real matrix.
bool itpp::lu (const cmat &X, cmat &L, cmat &U, ivec &p)
 LU factorisation of real matrix.
void itpp::interchange_permutations (vec &b, const ivec &p)
 Makes swapping of vector b according to the interchange permutation vector p.
bmat itpp::permutation_matrix (const ivec &p)
 Make permutation matrix P from the interchange permutation vector p.
bool itpp::qr (const mat &A, mat &Q, mat &R)
 QR factorisation of real matrix.
bool itpp::qr (const mat &A, mat &R)
 QR factorisation of real matrix with suppressed evaluation of Q.
bool itpp::qr (const mat &A, mat &Q, mat &R, bmat &P)
 QR factorisation of real matrix with pivoting.
bool itpp::qr (const cmat &A, cmat &Q, cmat &R)
 QR factorisation of a complex matrix.
bool itpp::qr (const cmat &A, cmat &R)
 QR factorisation of complex matrix with suppressed evaluation of Q.
bool itpp::qr (const cmat &A, cmat &Q, cmat &R, bmat &P)
 QR factorisation of a complex matrix with pivoting.
bool itpp::schur (const mat &A, mat &U, mat &T)
 Schur decomposition of a real matrix.
mat itpp::schur (const mat &A)
 Schur decomposition of a real matrix.
bool itpp::schur (const cmat &A, cmat &U, cmat &T)
 Schur decomposition of a complex matrix.
cmat itpp::schur (const cmat &A)
 Schur decomposition of a complex matrix.
bool itpp::svd (const mat &A, vec &s)
 Get singular values s of a real matrix A using SVD.
bool itpp::svd (const cmat &A, vec &s)
 Get singular values s of a complex matrix A using SVD.
vec itpp::svd (const mat &A)
 Return singular values of a real matrix A using SVD.
vec itpp::svd (const cmat &A)
 Return singular values of a complex matrix A using SVD.
bool itpp::svd (const mat &A, mat &U, vec &s, mat &V)
 Perform Singular Value Decomposition (SVD) of a real matrix A.
bool itpp::svd (const cmat &A, cmat &U, vec &s, cmat &V)
 Perform Singular Value Decomposition (SVD) of a complex matrix A.

Function Documentation

cmat itpp::chol ( const cmat &  X  ) 

Cholesky factorisation of complex hermitian and positive-definite matrix.

The Cholesky factorisation of a hermitian positive-definite matrix $\mathbf{X}$ of size $n \times n$ is given by

\[ \mathbf{X} = \mathbf{F}^H \mathbf{F} \]

where $\mathbf{F}$ is an upper triangular $n \times n$ matrix.

References itpp::chol(), and it_warning.

bool itpp::chol ( const cmat &  X,
cmat &  F 
)

Cholesky factorisation of complex hermitian and positive-definite matrix.

The Cholesky factorisation of a hermitian positive-definite matrix $\mathbf{X}$ of size $n \times n$ is given by

\[ \mathbf{X} = \mathbf{F}^H \mathbf{F} \]

where $\mathbf{F}$ is an upper triangular $n \times n$ matrix.

Returns true if calculation succeeded. False otherwise.

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.

References it_error.

mat itpp::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 triangular $n \times n$ matrix.

References it_warning.

Referenced by chmat::chmat(), itpp::chol(), ldmat::ldmat(), and fsqmat::sqrt_mult().

bool itpp::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 triangular $n \times n$ matrix.

Returns true if calculation succeeded. False otherwise.

References it_error.

cvec itpp::eig ( const cmat &  A  ) 

Calculates the eigenvalues of a complex non-hermitian matrix.

The Eigenvalues $\mathbf{d}(d_0, d_1, \ldots, d_{n-1})$ and the eigenvectors $\mathbf{v}_i, \: i=0, \ldots, n-1$ of the complex $n \times n$ matrix $\mathbf{A}$ satisfies

\[ \mathbf{A} \mathbf{v}_i = d_i \mathbf{v}_i\: i=0, \ldots, n-1. \]

Uses the LAPACK routine ZGEEV.

Referenced by itpp::eig(), and itpp::roots().

bool itpp::eig ( const cmat &  A,
cvec &  d 
)

Calculates the eigenvalues of a complex non-hermitian matrix.

The Eigenvalues $\mathbf{d}(d_0, d_1, \ldots, d_{n-1})$ and the eigenvectors $\mathbf{v}_i, \: i=0, \ldots, n-1$ of the complex $n \times n$ matrix $\mathbf{A}$ satisfies

\[ \mathbf{A} \mathbf{v}_i = d_i \mathbf{v}_i\: i=0, \ldots, n-1. \]

True is returned if the calculation was successful. Otherwise false.

Uses the LAPACK routine ZGEEV.

References it_error.

bool itpp::eig ( const cmat &  A,
cvec &  d,
cmat &  V 
)

Calculates the eigenvalues and eigenvectors of a complex non-hermitian matrix.

The Eigenvalues $\mathbf{d}(d_0, d_1, \ldots, d_{n-1})$ and the eigenvectors $\mathbf{v}_i, \: i=0, \ldots, n-1$ of the complex $n \times n$ matrix $\mathbf{A}$ satisfies

\[ \mathbf{A} \mathbf{v}_i = d_i \mathbf{v}_i\: i=0, \ldots, n-1. \]

The eigenvectors are the columns of the matrix V. True is returned if the calculation was successful. Otherwise false.

Uses the LAPACK routine ZGEEV.

References it_error.

cvec itpp::eig ( const mat &  A  ) 

Calculates the eigenvalues of a real non-symmetric matrix.

The Eigenvalues $\mathbf{d}(d_0, d_1, \ldots, d_{n-1})$ and the eigenvectors $\mathbf{v}_i, \: i=0, \ldots, n-1$ of the real $n \times n$ matrix $\mathbf{A}$ satisfies

\[ \mathbf{A} \mathbf{v}_i = d_i \mathbf{v}_i\: i=0, \ldots, n-1. \]

Uses the LAPACK routine DGEEV.

References itpp::eig().

bool itpp::eig ( const mat &  A,
cvec &  d 
)

Calculates the eigenvalues of a real non-symmetric matrix.

The Eigenvalues $\mathbf{d}(d_0, d_1, \ldots, d_{n-1})$ and the eigenvectors $\mathbf{v}_i, \: i=0, \ldots, n-1$ of the real $n \times n$ matrix $\mathbf{A}$ satisfies

\[ \mathbf{A} \mathbf{v}_i = d_i \mathbf{v}_i\: i=0, \ldots, n-1. \]

True is returned if the calculation was successful. Otherwise false.

Uses the LAPACK routine DGEEV.

References it_error.

bool itpp::eig ( const mat &  A,
cvec &  d,
cmat &  V 
)

Calculates the eigenvalues and eigenvectors of a real non-symmetric matrix.

The Eigenvalues $\mathbf{d}(d_0, d_1, \ldots, d_{n-1})$ and the eigenvectors $\mathbf{v}_i, \: i=0, \ldots, n-1$ of the real $n \times n$ matrix $\mathbf{A}$ satisfies

\[ \mathbf{A} \mathbf{v}_i = d_i \mathbf{v}_i\: i=0, \ldots, n-1. \]

The eigenvectors are the columns of the matrix V. True is returned if the calculation was successful. Otherwise false.

Uses the LAPACK routine DGEEV.

References it_error.

vec itpp::eig_sym ( const cmat &  A  ) 

Calculates the eigenvalues of a hermitian complex matrix.

The Eigenvalues $\mathbf{d}(d_0, d_1, \ldots, d_{n-1})$ and the eigenvectors $\mathbf{v}_i, \: i=0, \ldots, n-1$ of the complex and hermitian $n \times n$ matrix $\mathbf{A}$ satisfies

\[ \mathbf{A} \mathbf{v}_i = d_i \mathbf{v}_i\: i=0, \ldots, n-1. \]

Uses the LAPACK routine ZHEEV.

Referenced by itpp::eig_sym().

bool itpp::eig_sym ( const cmat &  A,
vec &  d 
)

Calculates the eigenvalues of a hermitian complex matrix.

The Eigenvalues $\mathbf{d}(d_0, d_1, \ldots, d_{n-1})$ and the eigenvectors $\mathbf{v}_i, \: i=0, \ldots, n-1$ of the complex and hermitian $n \times n$ matrix $\mathbf{A}$ satisfies

\[ \mathbf{A} \mathbf{v}_i = d_i \mathbf{v}_i\: i=0, \ldots, n-1. \]

True is returned if the calculation was successful. Otherwise false.

Uses the LAPACK routine ZHEEV.

References it_error.

bool itpp::eig_sym ( const cmat &  A,
vec &  d,
cmat &  V 
)

Calculates the eigenvalues and eigenvectors of a hermitian complex matrix.

The Eigenvalues $\mathbf{d}(d_0, d_1, \ldots, d_{n-1})$ and the eigenvectors $\mathbf{v}_i, \: i=0, \ldots, n-1$ of the complex and hermitian $n \times n$ matrix $\mathbf{A}$ satisfies

\[ \mathbf{A} \mathbf{v}_i = d_i \mathbf{v}_i\: i=0, \ldots, n-1. \]

The eigenvectors are the columns of the matrix V. True is returned if the calculation was successful. Otherwise false.

Uses the LAPACK routine ZHEEV.

References it_error.

vec itpp::eig_sym ( const mat &  A  ) 

Calculates the eigenvalues of a symmetric real matrix.

The Eigenvalues $\mathbf{d}(d_0, d_1, \ldots, d_{n-1})$ and the eigenvectors $\mathbf{v}_i, \: i=0, \ldots, n-1$ of the real and symmetric $n \times n$ matrix $\mathbf{A}$ satisfies

\[ \mathbf{A} \mathbf{v}_i = d_i \mathbf{v}_i\: i=0, \ldots, n-1. \]

Uses the LAPACK routine DSYEV.

References itpp::eig_sym().

bool itpp::eig_sym ( const mat &  A,
vec &  d 
)

Calculates the eigenvalues of a symmetric real matrix.

The Eigenvalues $\mathbf{d}(d_0, d_1, \ldots, d_{n-1})$ and the eigenvectors $\mathbf{v}_i, \: i=0, \ldots, n-1$ of the real and symmetric $n \times n$ matrix $\mathbf{A}$ satisfies

\[ \mathbf{A} \mathbf{v}_i = d_i \mathbf{v}_i\: i=0, \ldots, n-1. \]

True is returned if the calculation was successful. Otherwise false.

Uses the LAPACK routine DSYEV.

References it_error.

bool itpp::eig_sym ( const mat &  A,
vec &  d,
mat &  V 
)

Calculates the eigenvalues and eigenvectors of a symmetric real matrix.

The Eigenvalues $\mathbf{d}(d_0, d_1, \ldots, d_{n-1})$ and the eigenvectors $\mathbf{v}_i, \: i=0, \ldots, n-1$ of the real and symmetric $n \times n$ matrix $\mathbf{A}$ satisfies

\[ \mathbf{A} \mathbf{v}_i = d_i \mathbf{v}_i\: i=0, \ldots, n-1. \]

The eigenvectors are the columns of the matrix V. True is returned if the calculation was successful. Otherwise false.

Uses the LAPACK routine DSYEV.

References it_error.

bool itpp::lu ( const cmat &  X,
cmat &  L,
cmat &  U,
ivec &  p 
)

LU factorisation of real matrix.

The LU factorization of the complex 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 permutation matrix can be constructed using the function 

  bmat permutation_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 triangular, U an upper triangular matrix. p is the interchange permutation vector such that elements k and row p(k) should be interchanged.

Returns true is calculation succeeds. False otherwise.

References it_error.

Referenced by itpp::det().

bool itpp::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 permutation matrix can be constructed using the function 

  bmat permutation_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 triangular, 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.

References it_error.

bool itpp::qr ( const cmat &  A,
cmat &  Q,
cmat &  R,
bmat &  P 
)

QR factorisation of a complex matrix with pivoting.

The QR factorization of the complex matrix $\mathbf{A}$ of size $m \times n$ is given by

\[ \mathbf{A} \mathbf{P} = \mathbf{Q} \mathbf{R} , \]

where $\mathbf{Q}$ is an $m \times m$ unitary matrix, $\mathbf{R}$ is an $m \times n$ upper triangular matrix and $\mathbf{P}$ is an $n \times n$ permutation matrix.

Returns true is calculation succeeds. False otherwise. Uses the LAPACK routines ZGEQP3 and ZUNGQR.

References it_error.

Referenced by bdm::EKFCh::bayes(), bdm::KalmanCh::bayes(), and chmat::opupdt().

bool itpp::qr ( const cmat &  A,
cmat &  R 
)

QR factorisation of complex matrix with suppressed evaluation of Q.

For certain type of applications only the $\mathbf{R}$ matrix of full QR factorization of the complex matrix $\mathbf{A}=\mathbf{Q}\mathbf{R}$ is needed. These situations arise typically in designs of square-root algorithms where it is required that $\mathbf{A}^{H}\mathbf{A}=\mathbf{R}^{H}\mathbf{R}$. In such cases, evaluation of $\mathbf{Q}$ can be skipped.

Modification of qr(A,Q,R).

Author:
Vasek Smidl

References it_error.

bool itpp::qr ( const cmat &  A,
cmat &  Q,
cmat &  R 
)

QR factorisation of a complex matrix.

The QR factorization of the complex matrix $\mathbf{A}$ of size $m \times n$ is given by

\[ \mathbf{A} = \mathbf{Q} \mathbf{R} , \]

where $\mathbf{Q}$ is an $m \times m$ unitary matrix and $\mathbf{R}$ is an $m \times n$ upper triangular matrix.

Returns true is calculation succeeds. False otherwise. Uses the LAPACK routines ZGEQRF and ZUNGQR.

References it_error.

bool itpp::qr ( const mat &  A,
mat &  Q,
mat &  R,
bmat &  P 
)

QR factorisation of real matrix with pivoting.

The QR factorization of the real matrix $\mathbf{A}$ of size $m \times n$ is given by

\[ \mathbf{A} \mathbf{P} = \mathbf{Q} \mathbf{R} , \]

where $\mathbf{Q}$ is an $m \times m$ orthogonal matrix, $\mathbf{R}$ is an $m \times n$ upper triangular matrix and $\mathbf{P}$ is an $n \times n$ permutation matrix.

Returns true is calculation succeeds. False otherwise. Uses the LAPACK routines DGEQP3 and DORGQR.

References it_error.

bool itpp::qr ( const mat &  A,
mat &  R 
)

QR factorisation of real matrix with suppressed evaluation of Q.

For certain type of applications only the $\mathbf{R}$ matrix of full QR factorization of the real matrix $\mathbf{A}=\mathbf{Q}\mathbf{R}$ is needed. These situations arise typically in designs of square-root algorithms where it is required that $\mathbf{A}^{T}\mathbf{A}=\mathbf{R}^{T}\mathbf{R}$. In such cases, evaluation of $\mathbf{Q}$ can be skipped.

Modification of qr(A,Q,R).

Author:
Vasek Smidl

References it_error.

bool itpp::qr ( const mat &  A,
mat &  Q,
mat &  R 
)

QR factorisation of real matrix.

The QR factorization of the real matrix $\mathbf{A}$ of size $m \times n$ is given by

\[ \mathbf{A} = \mathbf{Q} \mathbf{R} , \]

where $\mathbf{Q}$ is an $m \times m$ orthogonal matrix and $\mathbf{R}$ is an $m \times n$ upper triangular matrix.

Returns true is calculation succeeds. False otherwise. Uses the LAPACK routine DGEQRF and DORGQR.

References it_error.

cmat itpp::schur ( const cmat &  A  ) 

Schur decomposition of a complex matrix.

This function computes the Schur form of a square complex matrix $ \mathbf{A} $. The Schur decomposition satisfies the following equation:

\[ \mathbf{U} \mathbf{T} \mathbf{U}^{H} = \mathbf{A} \]

where: $ \mathbf{U} $ is a unitary, $ \mathbf{T} $ is upper triangular, and $ \mathbf{U}^{H} $ is the Hermitian transposition of the $ \mathbf{U} $ matrix.

Returns:
Complex Schur matrix $ \mathbf{T} $
Uses the LAPACK routine ZGEES.

Referenced by itpp::schur().

bool itpp::schur ( const cmat &  A,
cmat &  U,
cmat &  T 
)

Schur decomposition of a complex matrix.

This function computes the Schur form of a square complex matrix $ \mathbf{A} $. The Schur decomposition satisfies the following equation:

\[ \mathbf{U} \mathbf{T} \mathbf{U}^{H} = \mathbf{A} \]

where: $ \mathbf{U} $ is a unitary, $ \mathbf{T} $ is upper triangular, and $ \mathbf{U}^{H} $ is the Hermitian transposition of the $ \mathbf{U} $ matrix.

Uses the LAPACK routine ZGEES.

References it_error.

mat itpp::schur ( const mat &  A  ) 

Schur decomposition of a real matrix.

This function computes the Schur form of a square real matrix $ \mathbf{A} $. The Schur decomposition satisfies the following equation:

\[ \mathbf{U} \mathbf{T} \mathbf{U}^{T} = \mathbf{A} \]

where: $ \mathbf{U} $ is a unitary, $ \mathbf{T} $ is upper quasi-triangular, and $ \mathbf{U}^{T} $ is the transposed $ \mathbf{U} $ matrix.

The upper quasi-triangular matrix may have $ 2 \times 2 $ blocks on its diagonal.

Returns:
Real Schur matrix $ \mathbf{T} $
uses the LAPACK routine DGEES.

References itpp::schur().

bool itpp::schur ( const mat &  A,
mat &  U,
mat &  T 
)

Schur decomposition of a real matrix.

This function computes the Schur form of a square real matrix $ \mathbf{A} $. The Schur decomposition satisfies the following equation:

\[ \mathbf{U} \mathbf{T} \mathbf{U}^{T} = \mathbf{A} \]

where: $ \mathbf{U} $ is a unitary, $ \mathbf{T} $ is upper quasi-triangular, and $ \mathbf{U}^{T} $ is the transposed $ \mathbf{U} $ matrix.

The upper quasi-triangular matrix may have $ 2 \times 2 $ blocks on its diagonal.

Uses the LAPACK routine DGEES.

References it_error.

bool itpp::svd ( const cmat &  A,
cmat &  U,
vec &  s,
cmat &  V 
)

Perform Singular Value Decomposition (SVD) of a complex matrix A.

This function returns two orthonormal matrices $U$ and $V$ and a vector of singular values $s$. The SVD algorithm computes the decomposition of a complex $m \times n$ matrix $\mathbf{A}$ so that

\[ \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s} = \sigma_1, \ldots, \sigma_p \]

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:

\[ \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H \]

where $ \mathrm{diag}(\mathbf{S}) = \mathbf{s} $

Note:
An external LAPACK library is required by this function.

References it_error.

bool itpp::svd ( const mat &  A,
mat &  U,
vec &  s,
mat &  V 
)

Perform Singular Value Decomposition (SVD) of a real matrix A.

This function returns two orthonormal matrices $U$ and $V$ and a vector of singular values $s$. The SVD algorithm computes the decomposition of a real $m \times n$ matrix $\mathbf{A}$ so that

\[ \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s} = \sigma_1, \ldots, \sigma_p \]

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:

\[ \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T \]

where $ \mathrm{diag}(\mathbf{S}) = \mathbf{s} $

Note:
An external LAPACK library is required by this function.

References it_error.

vec itpp::svd ( const cmat &  A  ) 

Return singular values of a complex matrix A using SVD.

This function returns singular values from the SVD decomposition of a complex matrix $A$. The SVD algorithm computes the decomposition of a complex $m \times n$ matrix $\mathbf{A}$ so that

\[ \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s} = \sigma_1, \ldots, \sigma_p \]

where $\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0$ are the singular values of $\mathbf{A}$. Or put differently:

\[ \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H \]

where $ \mathrm{diag}(\mathbf{S}) = \mathbf{s} $

Note:
An external LAPACK library is required by this function.

Referenced by itpp::norm(), itpp::rank(), and itpp::svd().

vec itpp::svd ( const mat &  A  ) 

Return singular values of a real matrix A using SVD.

This function returns singular values from the SVD decomposition of a real matrix $A$. The SVD algorithm computes the decomposition of a real $m \times n$ matrix $\mathbf{A}$ so that

\[ \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s} = \sigma_1, \ldots, \sigma_p \]

where $\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0$ are the singular values of $\mathbf{A}$. Or put differently:

\[ \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T \]

where $ \mathrm{diag}(\mathbf{S}) = \mathbf{s} $

Note:
An external LAPACK library is required by this function.

References itpp::svd().

bool itpp::svd ( const cmat &  A,
vec &  s 
)

Get singular values s of a complex matrix A using SVD.

This function calculates singular values $s$ from the SVD decomposition of a complex matrix $A$. The SVD algorithm computes the decomposition of a complex $m \times n$ matrix $\mathbf{A}$ so that

\[ \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s} = \sigma_1, \ldots, \sigma_p \]

where $\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0$ are the singular values of $\mathbf{A}$. Or put differently:

\[ \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H \]

where $ \mathrm{diag}(\mathbf{S}) = \mathbf{s} $

Note:
An external LAPACK library is required by this function.

References it_error.

bool itpp::svd ( const mat &  A,
vec &  s 
)

Get singular values s of a real matrix A using SVD.

This function calculates singular values $s$ from the SVD decomposition of a real matrix $A$. The SVD algorithm computes the decomposition of a real $m \times n$ matrix $\mathbf{A}$ so that

\[ \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s} = \sigma_1, \ldots, \sigma_p \]

where $\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0$ are the singular values of $\mathbf{A}$. Or put differently:

\[ \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T \]

where $ \mathrm{diag}(\mathbf{S}) = \mathbf{s} $

Note:
An external LAPACK library is required by this function.

References it_error.

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