Solving Linear Equation Systems
[Linear Algebra]


Functions

bool itpp::ls_solve (const mat &A, const vec &b, vec &x)
 Solve linear equation system by LU factorisation.
vec itpp::ls_solve (const mat &A, const vec &b)
 Solve linear equation system by LU factorisation.
bool itpp::ls_solve (const mat &A, const mat &B, mat &X)
 Solve multiple linear equations by LU factorisation.
mat itpp::ls_solve (const mat &A, const mat &B)
 Solve multiple linear equations by LU factorisation.
bool itpp::ls_solve (const cmat &A, const cvec &b, cvec &x)
 Solve linear equation system by LU factorisation.
cvec itpp::ls_solve (const cmat &A, const cvec &b)
 Solve linear equation system by LU factorisation.
bool itpp::ls_solve (const cmat &A, const cmat &B, cmat &X)
 Solve multiple linear equations by LU factorisation.
cmat itpp::ls_solve (const cmat &A, const cmat &B)
 Solve multiple linear equations by LU factorisation.
bool itpp::ls_solve_chol (const mat &A, const vec &b, vec &x)
 Solve linear equation system by Cholesky factorisation.
vec itpp::ls_solve_chol (const mat &A, const vec &b)
 Solve linear equation system by Cholesky factorisation.
bool itpp::ls_solve_chol (const mat &A, const mat &B, mat &X)
 Solve linear equation system by Cholesky factorisation.
mat itpp::ls_solve_chol (const mat &A, const mat &B)
 Solve linear equation system by Cholesky factorisation.
bool itpp::ls_solve_chol (const cmat &A, const cvec &b, cvec &x)
 Solve linear equation system by Cholesky factorisation.
cvec itpp::ls_solve_chol (const cmat &A, const cvec &b)
 Solve linear equation system by Cholesky factorisation.
bool itpp::ls_solve_chol (const cmat &A, const cmat &B, cmat &X)
 Solve linear equation system by Cholesky factorisation.
cmat itpp::ls_solve_chol (const cmat &A, const cmat &B)
 Solve linear equation system by Cholesky factorisation.
bool itpp::ls_solve_od (const mat &A, const vec &b, vec &x)
 Solves overdetermined linear equation systems.
vec itpp::ls_solve_od (const mat &A, const vec &b)
 Solves overdetermined linear equation systems.
bool itpp::ls_solve_od (const mat &A, const mat &B, mat &X)
 Solves overdetermined linear equation systems.
mat itpp::ls_solve_od (const mat &A, const mat &B)
 Solves overdetermined linear equation systems.
bool itpp::ls_solve_od (const cmat &A, const cvec &b, cvec &x)
 Solves overdetermined linear equation systems.
cvec itpp::ls_solve_od (const cmat &A, const cvec &b)
 Solves overdetermined linear equation systems.
bool itpp::ls_solve_od (const cmat &A, const cmat &B, cmat &X)
 Solves overdetermined linear equation systems.
cmat itpp::ls_solve_od (const cmat &A, const cmat &B)
 Solves overdetermined linear equation systems.
bool itpp::ls_solve_ud (const mat &A, const vec &b, vec &x)
 Solves underdetermined linear equation systems.
vec itpp::ls_solve_ud (const mat &A, const vec &b)
 Solves overdetermined linear equation systems.
bool itpp::ls_solve_ud (const mat &A, const mat &B, mat &X)
 Solves underdetermined linear equation systems.
mat itpp::ls_solve_ud (const mat &A, const mat &B)
 Solves underdetermined linear equation systems.
bool itpp::ls_solve_ud (const cmat &A, const cvec &b, cvec &x)
 Solves underdetermined linear equation systems.
cvec itpp::ls_solve_ud (const cmat &A, const cvec &b)
 Solves overdetermined linear equation systems.
bool itpp::ls_solve_ud (const cmat &A, const cmat &B, cmat &X)
 Solves underdetermined linear equation systems.
cmat itpp::ls_solve_ud (const cmat &A, const cmat &B)
 Solves underdetermined linear equation systems.
bool itpp::backslash (const mat &A, const vec &b, vec &x)
 A general linear equation system solver.
vec itpp::backslash (const mat &A, const vec &b)
 A general linear equation system solver.
bool itpp::backslash (const mat &A, const mat &B, mat &X)
 A general linear equation system solver.
mat itpp::backslash (const mat &A, const mat &B)
 A general linear equation system solver.
bool itpp::backslash (const cmat &A, const cvec &b, cvec &x)
 A general linear equation system solver.
cvec itpp::backslash (const cmat &A, const cvec &b)
 A general linear equation system solver.
bool itpp::backslash (const cmat &A, const cmat &B, cmat &X)
 A general linear equation system solver.
cmat itpp::backslash (const cmat &A, const cmat &B)
 A general linear equation system solver.
vec itpp::forward_substitution (const mat &L, const vec &b)
 Forward substitution of square matrix.
void itpp::forward_substitution (const mat &L, const vec &b, vec &x)
 Forward substitution of square matrix.
vec itpp::forward_substitution (const mat &L, int p, const vec &b)
 Forward substitution of band matrices.
void itpp::forward_substitution (const mat &L, int p, const vec &b, vec &x)
 Forward substitution of band matrices.
vec itpp::backward_substitution (const mat &U, const vec &b)
 Backward substitution of square matrix.
void itpp::backward_substitution (const mat &U, const vec &b, vec &x)
 Backward substitution of square matrix.
vec itpp::backward_substitution (const mat &U, int q, const vec &b)
 Backward substitution of band matrix.
void itpp::backward_substitution (const mat &U, int q, const vec &b, vec &x)
 Backward substitution of band matrix.

Function Documentation

cmat itpp::backslash ( const cmat &  A,
const cmat &  B 
)

A general linear equation system solver.

Tries to emulate the backslash operator in Matlab by calling ls_solve(A,B), ls_solve_od(A,B), or ls_solve_ud(A,B).

References it_assert_debug.

Referenced by itpp::arma_estimator(), itpp::backslash(), and itpp::modified_yule_walker().

bool itpp::backslash ( const cmat &  A,
const cmat &  B,
cmat &  X 
)

A general linear equation system solver.

Tries to emulate the backslash operator in Matlab by calling ls_solve(A,B,X), ls_solve_od(A,B,X), or ls_solve_ud(A,B,X).

References itpp::ls_solve(), itpp::ls_solve_od(), and itpp::ls_solve_ud().

cvec itpp::backslash ( const cmat &  A,
const cvec &  b 
)

A general linear equation system solver.

Tries to emulate the backslash operator in Matlab by calling ls_solve(A,b), ls_solve_od(A,b) or ls_solve_ud(A,b)

References itpp::backslash(), and it_assert_debug.

bool itpp::backslash ( const cmat &  A,
const cvec &  b,
cvec &  x 
)

A general linear equation system solver.

Tries to emulate the backslash operator in Matlab by calling ls_solve(A,b,x), ls_solve_od(A,b,x) or ls_solve_ud(A,b,x)

References itpp::ls_solve(), itpp::ls_solve_od(), and itpp::ls_solve_ud().

mat itpp::backslash ( const mat &  A,
const mat &  B 
)

A general linear equation system solver.

Tries to emulate the backslash operator in Matlab by calling ls_solve(A,B), ls_solve_od(A,B), or ls_solve_ud(A,B).

References itpp::backslash(), and it_assert_debug.

bool itpp::backslash ( const mat &  A,
const mat &  B,
mat &  X 
)

A general linear equation system solver.

Tries to emulate the backslash operator in Matlab by calling ls_solve(A,B,X), ls_solve_od(A,B,X), or ls_solve_ud(A,B,X).

References itpp::ls_solve(), itpp::ls_solve_od(), and itpp::ls_solve_ud().

vec itpp::backslash ( const mat &  A,
const vec &  b 
)

A general linear equation system solver.

Tries to emulate the backslash operator in Matlab by calling ls_solve(A,b), ls_solve_od(A,b) or ls_solve_ud(A,b)

References itpp::backslash(), and it_assert_debug.

bool itpp::backslash ( const mat &  A,
const vec &  b,
vec &  x 
)

A general linear equation system solver.

Tries to emulate the backslash operator in Matlab by calling ls_solve(A,b,x), ls_solve_od(A,b,x) or ls_solve_ud(A,b,x)

References itpp::ls_solve(), itpp::ls_solve_od(), and itpp::ls_solve_ud().

void itpp::backward_substitution ( const mat &  U,
int  q,
const vec &  b,
vec &  x 
)

Backward substitution of band matrix.

Solves Ux=b, where U is a upper triangular n by n matrix band-matrix with upper bandwidth q. Assumes that U is nonsingular. Requires about 2nq flops (if n >> q). Uses Alg. 4.3.3 in Golub & van Loan "Matrix computations", 3rd ed., p. 153.

References it_assert, and itpp::max().

Referenced by itpp::backward_substitution(), bdm::EKFCh::bayes(), bdm::KalmanCh::bayes(), and ldmat::invqform().

vec itpp::backward_substitution ( const mat &  U,
int  q,
const vec &  b 
)

Backward substitution of band matrix.

Solves Ux=b, where U is a upper triangular n by n matrix band-matrix with upper bandwidth q. Assumes that U is nonsingular. Requires about 2nq flops (if n >> q). Uses Alg. 4.3.3 in Golub & van Loan "Matrix computations", 3rd ed., p. 153.

References itpp::backward_substitution().

void itpp::backward_substitution ( const mat &  U,
const vec &  b,
vec &  x 
)

Backward substitution of square matrix.

Solves Ux=b, where U is a upper triangular n by n matrix. Assumes that U is nonsingular. Requires n^2 flops. Uses Alg. 3.1.2 in Golub & van Loan "Matrix computations", 3rd ed., p. 89.

References it_assert.

vec itpp::backward_substitution ( const mat &  U,
const vec &  b 
)

Backward substitution of square matrix.

Solves Ux=b, where U is a upper triangular n by n matrix. Assumes that U is nonsingular. Requires n^2 flops. Uses Alg. 3.1.2 in Golub & van Loan "Matrix computations", 3rd ed., p. 89.

References itpp::backward_substitution().

void itpp::forward_substitution ( const mat &  L,
int  p,
const vec &  b,
vec &  x 
)

Forward substitution of band matrices.

Solves Lx=b, where L is a lower triangular n by n band-matrix with lower bandwidth p. Assumes that L is nonsingular. Requires about 2np flops (if n >> p). Uses Alg. 4.3.2 in Golub & van Loan "Matrix computations", 3rd ed., p. 153.

References it_assert, and itpp::min().

Referenced by itpp::forward_substitution(), and chmat::invqform().

vec itpp::forward_substitution ( const mat &  L,
int  p,
const vec &  b 
)

Forward substitution of band matrices.

Solves Lx=b, where L is a lower triangular n by n band-matrix with lower bandwidth p. Assumes that L is nonsingular. Requires about 2np flops (if n >> p). Uses Alg. 4.3.2 in Golub & van Loan "Matrix computations", 3rd ed., p. 153.

References itpp::forward_substitution().

void itpp::forward_substitution ( const mat &  L,
const vec &  b,
vec &  x 
)

Forward substitution of square matrix.

Solves Lx=b, where L is a lower triangular n by n matrix. Assumes that L is nonsingular. Requires n^2 flops. Uses Alg. 3.1.1 in Golub & van Loan "Matrix computations", 3rd ed., p. 89.

References it_assert.

vec itpp::forward_substitution ( const mat &  L,
const vec &  b 
)

Forward substitution of square matrix.

Solves Lx=b, where L is a lower triangular n by n matrix. Assumes that L is nonsingular. Requires n^2 flops. Uses Alg. 3.1.1 in Golub & van Loan "Matrix computations", 3rd ed., p. 89.

References itpp::forward_substitution().

cmat itpp::ls_solve ( const cmat &  A,
const cmat &  B 
)

Solve multiple linear equations by LU factorisation.

Solves the linear system $AX=B$. Here $A$ is a nonsingular $n \times n$ matrix. Uses the LAPACK routine ZGESV.

References it_assert_debug.

Referenced by itpp::backslash(), and itpp::ls_solve().

bool itpp::ls_solve ( const cmat &  A,
const cmat &  B,
cmat &  X 
)

Solve multiple linear equations by LU factorisation.

Solves the linear system $AX=B$. Here $A$ is a nonsingular $n \times n$ matrix. Uses the LAPACK routine ZGESV.

References it_error.

cvec itpp::ls_solve ( const cmat &  A,
const cvec &  b 
)

Solve linear equation system by LU factorisation.

Solves the linear system $Ax=b$, where $A$ is a $n \times n$ matrix. Uses the LAPACK routine ZGESV.

References it_assert_debug, and itpp::ls_solve().

bool itpp::ls_solve ( const cmat &  A,
const cvec &  b,
cvec &  x 
)

Solve linear equation system by LU factorisation.

Solves the linear system $Ax=b$, where $A$ is a $n \times n$ matrix. Uses the LAPACK routine ZGESV.

References it_error.

mat itpp::ls_solve ( const mat &  A,
const mat &  B 
)

Solve multiple linear equations by LU factorisation.

Solves the linear system $AX=B$. Here $A$ is a nonsingular $n \times n$ matrix. Uses the LAPACK routine DGESV.

References it_assert_debug, and itpp::ls_solve().

bool itpp::ls_solve ( const mat &  A,
const mat &  B,
mat &  X 
)

Solve multiple linear equations by LU factorisation.

Solves the linear system $AX=B$. Here $A$ is a nonsingular $n \times n$ matrix. Uses the LAPACK routine DGESV.

References it_error.

vec itpp::ls_solve ( const mat &  A,
const vec &  b 
)

Solve linear equation system by LU factorisation.

Solves the linear system $Ax=b$, where $A$ is a $n \times n$ matrix. Uses the LAPACK routine DGESV.

References it_assert_debug, and itpp::ls_solve().

bool itpp::ls_solve ( const mat &  A,
const vec &  b,
vec &  x 
)

Solve linear equation system by LU factorisation.

Solves the linear system $Ax=b$, where $A$ is a $n \times n$ matrix. Uses the LAPACK routine DGESV.

References it_error.

cmat itpp::ls_solve_chol ( const cmat &  A,
const cmat &  B 
)

Solve linear equation system by Cholesky factorisation.

Solves the linear system $AX=B$, where $A$ is a Hermitian positive definite $n \times n$ matrix. Uses the LAPACK routine ZPOSV.

References it_assert_debug.

Referenced by itpp::ls_solve_chol().

bool itpp::ls_solve_chol ( const cmat &  A,
const cmat &  B,
cmat &  X 
)

Solve linear equation system by Cholesky factorisation.

Solves the linear system $AX=B$, where $A$ is a Hermitian positive definite $n \times n$ matrix. Uses the LAPACK routine ZPOSV.

References it_error.

cvec itpp::ls_solve_chol ( const cmat &  A,
const cvec &  b 
)

Solve linear equation system by Cholesky factorisation.

Solves the linear system $Ax=b$, where $A$ is a Hermitian positive definite $n \times n$ matrix. Uses the LAPACK routine ZPOSV.

References it_assert_debug, and itpp::ls_solve_chol().

bool itpp::ls_solve_chol ( const cmat &  A,
const cvec &  b,
cvec &  x 
)

Solve linear equation system by Cholesky factorisation.

Solves the linear system $Ax=b$, where $A$ is a Hermitian positive definite $n \times n$ matrix. Uses the LAPACK routine ZPOSV.

References it_error.

mat itpp::ls_solve_chol ( const mat &  A,
const mat &  B 
)

Solve linear equation system by Cholesky factorisation.

Solves the linear system $AX=B$, where $A$ is a symmetric positive definite $n \times n$ matrix. Uses the LAPACK routine DPOSV.

References it_assert_debug, and itpp::ls_solve_chol().

bool itpp::ls_solve_chol ( const mat &  A,
const mat &  B,
mat &  X 
)

Solve linear equation system by Cholesky factorisation.

Solves the linear system $AX=B$, where $A$ is a symmetric positive definite $n \times n$ matrix. Uses the LAPACK routine DPOSV.

References it_error.

vec itpp::ls_solve_chol ( const mat &  A,
const vec &  b 
)

Solve linear equation system by Cholesky factorisation.

Solves the linear system $Ax=b$, where $A$ is a symmetric positive definite $n \times n$ matrix. Uses the LAPACK routine DPOSV.

References it_assert_debug, and itpp::ls_solve_chol().

bool itpp::ls_solve_chol ( const mat &  A,
const vec &  b,
vec &  x 
)

Solve linear equation system by Cholesky factorisation.

Solves the linear system $Ax=b$, where $A$ is a symmetric positive definite $n \times n$ matrix. Uses the LAPACK routine DPOSV.

References it_error.

cmat itpp::ls_solve_od ( const cmat &  A,
const cmat &  B 
)

Solves overdetermined linear equation systems.

Solves the overdetermined linear system $AX=B$, where $A$ is a $m \times n$ matrix and $m \geq n$. Uses QR-factorization and assumes that $A$ is full rank. Based on the LAPACK routine ZGELS.

References it_assert_debug.

Referenced by itpp::backslash(), and itpp::ls_solve_od().

bool itpp::ls_solve_od ( const cmat &  A,
const cmat &  B,
cmat &  X 
)

Solves overdetermined linear equation systems.

Solves the overdetermined linear system $AX=B$, where $A$ is a $m \times n$ matrix and $m \geq n$. Uses QR-factorization and assumes that $A$ is full rank. Based on the LAPACK routine ZGELS.

References it_error.

cvec itpp::ls_solve_od ( const cmat &  A,
const cvec &  b 
)

Solves overdetermined linear equation systems.

Solves the overdetermined linear system $Ax=b$, where $A$ is a $m \times n$ matrix and $m \geq n$. Uses QR-factorization and assumes that $A$ is full rank. Based on the LAPACK routine ZGELS.

References it_assert_debug, and itpp::ls_solve_od().

bool itpp::ls_solve_od ( const cmat &  A,
const cvec &  b,
cvec &  x 
)

Solves overdetermined linear equation systems.

Solves the overdetermined linear system $Ax=b$, where $A$ is a $m \times n$ matrix and $m \geq n$. Uses QR-factorization and is built upon the LAPACK routine ZGELS.

References it_error.

mat itpp::ls_solve_od ( const mat &  A,
const mat &  B 
)

Solves overdetermined linear equation systems.

Solves the overdetermined linear system $AX=B$, where $A$ is a $m \times n$ matrix and $m \geq n$. Uses QR-factorization and assumes that $A$ is full rank. Based on the LAPACK routine DGELS.

References it_assert_debug, and itpp::ls_solve_od().

bool itpp::ls_solve_od ( const mat &  A,
const mat &  B,
mat &  X 
)

Solves overdetermined linear equation systems.

Solves the overdetermined linear system $AX=B$, where $A$ is a $m \times n$ matrix and $m \geq n$. Uses QR-factorization and assumes that $A$ is full rank. Based on the LAPACK routine DGELS.

References it_error.

vec itpp::ls_solve_od ( const mat &  A,
const vec &  b 
)

Solves overdetermined linear equation systems.

Solves the overdetermined linear system $Ax=b$, where $A$ is a $m \times n$ matrix and $m \geq n$. Uses QR-factorization and assumes that $A$ is full rank. Based on the LAPACK routine DGELS.

References it_assert_debug, and itpp::ls_solve_od().

bool itpp::ls_solve_od ( const mat &  A,
const vec &  b,
vec &  x 
)

Solves overdetermined linear equation systems.

Solves the overdetermined linear system $Ax=b$, where $A$ is a $m \times n$ matrix and $m \geq n$. Uses QR-factorization and is built upon the LAPACK routine DGELS.

References it_error.

cmat itpp::ls_solve_ud ( const cmat &  A,
const cmat &  B 
)

Solves underdetermined linear equation systems.

Solves the underdetermined linear system $AX=B$, where $A$ is a $m \times n$ matrix and $m \leq n$. Uses LQ-factorization and assumes that $A$ is full rank. Based on the LAPACK routine ZGELS.

References it_assert_debug.

Referenced by itpp::backslash(), and itpp::ls_solve_ud().

bool itpp::ls_solve_ud ( const cmat &  A,
const cmat &  B,
cmat &  X 
)

Solves underdetermined linear equation systems.

Solves the underdetermined linear system $AX=B$, where $A$ is a $m \times n$ matrix and $m \leq n$. Uses LQ-factorization and assumes that $A$ is full rank. Based on the LAPACK routine ZGELS.

References it_error.

cvec itpp::ls_solve_ud ( const cmat &  A,
const cvec &  b 
)

Solves overdetermined linear equation systems.

Solves the underdetermined linear system $Ax=b$, where $A$ is a $m \times n$ matrix and $m \leq n$. Uses LQ-factorization and assumes that $A$ is full rank. Based on the LAPACK routine ZGELS.

References it_assert_debug, and itpp::ls_solve_ud().

bool itpp::ls_solve_ud ( const cmat &  A,
const cvec &  b,
cvec &  x 
)

Solves underdetermined linear equation systems.

Solves the underdetermined linear system $Ax=b$, where $A$ is a $m \times n$ matrix and $m \leq n$. Uses LQ-factorization and is built upon the LAPACK routine ZGELS.

References it_error.

mat itpp::ls_solve_ud ( const mat &  A,
const mat &  B 
)

Solves underdetermined linear equation systems.

Solves the underdetermined linear system $AX=B$, where $A$ is a $m \times n$ matrix and $m \leq n$. Uses LQ-factorization and assumes that $A$ is full rank. Based on the LAPACK routine DGELS.

References it_assert_debug, and itpp::ls_solve_ud().

bool itpp::ls_solve_ud ( const mat &  A,
const mat &  B,
mat &  X 
)

Solves underdetermined linear equation systems.

Solves the underdetermined linear system $AX=B$, where $A$ is a $m \times n$ matrix and $m \leq n$. Uses LQ-factorization and assumes that $A$ is full rank. Based on the LAPACK routine DGELS.

References it_error.

vec itpp::ls_solve_ud ( const mat &  A,
const vec &  b 
)

Solves overdetermined linear equation systems.

Solves the underdetermined linear system $Ax=b$, where $A$ is a $m \times n$ matrix and $m \leq n$. Uses LQ-factorization and assumes that $A$ is full rank. Based on the LAPACK routine DGELS.

References it_assert_debug, and itpp::ls_solve_ud().

bool itpp::ls_solve_ud ( const mat &  A,
const vec &  b,
vec &  x 
)

Solves underdetermined linear equation systems.

Solves the underdetermined linear system $Ax=b$, where $A$ is a $m \times n$ matrix and $m \leq n$. Uses LQ-factorization and is built upon the LAPACK routine DGELS.

References it_error.

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