General matrix: LU: tridiagonal

Functions
int64_t	lapack::gtcon (lapack::Norm norm, int64_t n, double const *DL, double const *D, double const *DU, double const *DU2, int64_t const *ipiv, double anorm, double *rcond)
int64_t	lapack::gtcon (lapack::Norm norm, int64_t n, float const *DL, float const *D, float const *DU, float const *DU2, int64_t const *ipiv, float anorm, float *rcond)
int64_t	lapack::gtcon (lapack::Norm norm, int64_t n, std::complex< double > const *DL, std::complex< double > const *D, std::complex< double > const *DU, std::complex< double > const *DU2, int64_t const *ipiv, double anorm, double *rcond)
	Estimates the reciprocal of the condition number of a complex tridiagonal matrix A using the LU factorization as computed by lapack::gttrf.
int64_t	lapack::gtcon (lapack::Norm norm, int64_t n, std::complex< float > const *DL, std::complex< float > const *D, std::complex< float > const *DU, std::complex< float > const *DU2, int64_t const *ipiv, float anorm, float *rcond)
int64_t	lapack::gtrfs (lapack::Op trans, int64_t n, int64_t nrhs, double const *DL, double const *D, double const *DU, double const *DLF, double const *DF, double const *DUF, double const *DU2, int64_t const *ipiv, double const *B, int64_t ldb, double *X, int64_t ldx, double *ferr, double *berr)
int64_t	lapack::gtrfs (lapack::Op trans, int64_t n, int64_t nrhs, float const *DL, float const *D, float const *DU, float const *DLF, float const *DF, float const *DUF, float const *DU2, int64_t const *ipiv, float const *B, int64_t ldb, float *X, int64_t ldx, float *ferr, float *berr)
int64_t	lapack::gtrfs (lapack::Op trans, int64_t n, int64_t nrhs, std::complex< double > const *DL, std::complex< double > const *D, std::complex< double > const *DU, std::complex< double > const *DLF, std::complex< double > const *DF, std::complex< double > const *DUF, std::complex< double > const *DU2, int64_t const *ipiv, std::complex< double > const *B, int64_t ldb, std::complex< double > *X, int64_t ldx, double *ferr, double *berr)
	Improves the computed solution to a system of linear equations when the coefficient matrix is tridiagonal, and provides error bounds and backward error estimates for the solution.
int64_t	lapack::gtrfs (lapack::Op trans, int64_t n, int64_t nrhs, std::complex< float > const *DL, std::complex< float > const *D, std::complex< float > const *DU, std::complex< float > const *DLF, std::complex< float > const *DF, std::complex< float > const *DUF, std::complex< float > const *DU2, int64_t const *ipiv, std::complex< float > const *B, int64_t ldb, std::complex< float > *X, int64_t ldx, float *ferr, float *berr)
int64_t	lapack::gttrf (int64_t n, double *DL, double *D, double *DU, double *DU2, int64_t *ipiv)
int64_t	lapack::gttrf (int64_t n, float *DL, float *D, float *DU, float *DU2, int64_t *ipiv)
int64_t	lapack::gttrf (int64_t n, std::complex< double > *DL, std::complex< double > *D, std::complex< double > *DU, std::complex< double > *DU2, int64_t *ipiv)
	Computes an LU factorization of a complex tridiagonal matrix A using elimination with partial pivoting and row interchanges.
int64_t	lapack::gttrf (int64_t n, std::complex< float > *DL, std::complex< float > *D, std::complex< float > *DU, std::complex< float > *DU2, int64_t *ipiv)
int64_t	lapack::gttrs (lapack::Op trans, int64_t n, int64_t nrhs, double const *DL, double const *D, double const *DU, double const *DU2, int64_t const *ipiv, double *B, int64_t ldb)
int64_t	lapack::gttrs (lapack::Op trans, int64_t n, int64_t nrhs, float const *DL, float const *D, float const *DU, float const *DU2, int64_t const *ipiv, float *B, int64_t ldb)
int64_t	lapack::gttrs (lapack::Op trans, int64_t n, int64_t nrhs, std::complex< double > const *DL, std::complex< double > const *D, std::complex< double > const *DU, std::complex< double > const *DU2, int64_t const *ipiv, std::complex< double > *B, int64_t ldb)
	Solves one of the systems of equations.
int64_t	lapack::gttrs (lapack::Op trans, int64_t n, int64_t nrhs, std::complex< float > const *DL, std::complex< float > const *D, std::complex< float > const *DU, std::complex< float > const *DU2, int64_t const *ipiv, std::complex< float > *B, int64_t ldb)

Detailed Description

Function Documentation

gtcon()

int64_t lapack::gtcon	(	lapack::Norm	norm,
		int64_t	n,
		std::complex< double > const *	DL,
		std::complex< double > const *	D,
		std::complex< double > const *	DU,
		std::complex< double > const *	DU2,
		int64_t const *	ipiv,
		double	anorm,
		double *	rcond
	)

Estimates the reciprocal of the condition number of a complex tridiagonal matrix A using the LU factorization as computed by lapack::gttrf.

An estimate is obtained for \(|| A^{-1} ||,\) and the reciprocal of the condition number is computed as \(\text{rcond} = 1 / (|| A || \cdot || A^{-1} ||).\)

Overloaded versions are available for float, double, std::complex<float>, and std::complex<double>.

Parameters

[in]	norm	Specifies whether the 1-norm condition number or the infinity-norm condition number is required: lapack::Norm::One: 1-norm; lapack::Norm::Inf: Infinity-norm.
[in]	n	The order of the matrix A. n >= 0.
[in]	DL	The vector DL of length n-1. The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by lapack::gttrf.
[in]	D	The vector D of length n. The n diagonal elements of the upper triangular matrix U from the LU factorization of A.
[in]	DU	The vector DU of length n-1. The (n-1) elements of the first superdiagonal of U.
[in]	DU2	The vector DU2 of length n-2. The (n-2) elements of the second superdiagonal of U.
[in]	ipiv	The vector ipiv of length n. The pivot indices; for 1 <= i <= n, row i of the matrix was interchanged with row ipiv(i). ipiv(i) will always be either i or i+1; ipiv(i) = i indicates a row interchange was not required.
[in]	anorm	If norm = One, the 1-norm of the original matrix A. If norm = Inf, the infinity-norm of the original matrix A.
[out]	rcond	The reciprocal of the condition number of the matrix A, computed as rcond = 1/(anorm * ainv_norm), where ainv_norm is an estimate of the 1-norm or infinity-norm of \(A^{-1}\) computed in this routine.

Returns

= 0: successful exit

gtrfs()

int64_t lapack::gtrfs	(	lapack::Op	trans,
		int64_t	n,
		int64_t	nrhs,
		std::complex< double > const *	DL,
		std::complex< double > const *	D,
		std::complex< double > const *	DU,
		std::complex< double > const *	DLF,
		std::complex< double > const *	DF,
		std::complex< double > const *	DUF,
		std::complex< double > const *	DU2,
		int64_t const *	ipiv,
		std::complex< double > const *	B,
		int64_t	ldb,
		std::complex< double > *	X,
		int64_t	ldx,
		double *	ferr,
		double *	berr
	)

Improves the computed solution to a system of linear equations when the coefficient matrix is tridiagonal, and provides error bounds and backward error estimates for the solution.

Overloaded versions are available for float, double, std::complex<float>, and std::complex<double>.

Parameters

[in]	trans	Specifies the form of the system of equations: lapack::Op::NoTrans: \(A X = B\) (No transpose) lapack::Op::Trans: \(A^T X = B\) (Transpose) lapack::Op::ConjTrans: \(A^H X = B\) (Conjugate transpose)
[in]	n	The order of the matrix A. n >= 0.
[in]	nrhs	The number of right hand sides, i.e., the number of columns of the matrix B. nrhs >= 0.
[in]	DL	The vector DL of length n-1. The (n-1) subdiagonal elements of A.
[in]	D	The vector D of length n. The diagonal elements of A.
[in]	DU	The vector DU of length n-1. The (n-1) superdiagonal elements of A.
[in]	DLF	The vector DLF of length n-1. The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by lapack::gttrf.
[in]	DF	The vector DF of length n. The n diagonal elements of the upper triangular matrix U from the LU factorization of A.
[in]	DUF	The vector DUF of length n-1. The (n-1) elements of the first superdiagonal of U.
[in]	DU2	The vector DU2 of length n-2. The (n-2) elements of the second superdiagonal of U.
[in]	ipiv	The vector ipiv of length n. The pivot indices; for 1 <= i <= n, row i of the matrix was interchanged with row ipiv(i). ipiv(i) will always be either i or i+1; ipiv(i) = i indicates a row interchange was not required.
[in]	B	The n-by-nrhs matrix B, stored in an ldb-by-nrhs array. The right hand side matrix B.
[in]	ldb	The leading dimension of the array B. ldb >= max(1,n).
[in,out]	X	The n-by-nrhs matrix X, stored in an ldx-by-nrhs array. On entry, the solution matrix X, as computed by lapack::gttrs. On exit, the improved solution matrix X.
[in]	ldx	The leading dimension of the array X. ldx >= max(1,n).
[out]	ferr	The vector ferr of length nrhs. The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), ferr(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error.
[out]	berr	The vector berr of length nrhs. The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution).

Returns

= 0: successful exit

gttrf()

int64_t lapack::gttrf	(	int64_t	n,
		std::complex< double > *	DL,
		std::complex< double > *	D,
		std::complex< double > *	DU,
		std::complex< double > *	DU2,
		int64_t *	ipiv
	)

Computes an LU factorization of a complex tridiagonal matrix A using elimination with partial pivoting and row interchanges.

The factorization has the form

\[ A = L U \]

where L is a product of permutation and unit lower bidiagonal matrices and U is upper triangular with nonzeros in only the main diagonal and first two superdiagonals.

Overloaded versions are available for float, double, std::complex<float>, and std::complex<double>.

Parameters

[in]	n	The order of the matrix A.
[in,out]	DL	The vector DL of length n-1. On entry, DL must contain the (n-1) sub-diagonal elements of A. On exit, DL is overwritten by the (n-1) multipliers that define the matrix L from the LU factorization of A.
[in,out]	D	The vector D of length n. On entry, D must contain the diagonal elements of A. On exit, D is overwritten by the n diagonal elements of the upper triangular matrix U from the LU factorization of A.
[in,out]	DU	The vector DU of length n-1. On entry, DU must contain the (n-1) super-diagonal elements of A. On exit, DU is overwritten by the (n-1) elements of the first super-diagonal of U.
[out]	DU2	The vector DU2 of length n-2. On exit, DU2 is overwritten by the (n-2) elements of the second super-diagonal of U.
[out]	ipiv	The vector ipiv of length n. The pivot indices; for 1 <= i <= n, row i of the matrix was interchanged with row ipiv(i). ipiv(i) will always be either i or i+1; ipiv(i) = i indicates a row interchange was not required.

Returns

= 0: successful exit

> 0: if return value = i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations.

gttrs()

int64_t lapack::gttrs	(	lapack::Op	trans,
		int64_t	n,
		int64_t	nrhs,
		std::complex< double > const *	DL,
		std::complex< double > const *	D,
		std::complex< double > const *	DU,
		std::complex< double > const *	DU2,
		int64_t const *	ipiv,
		std::complex< double > *	B,
		int64_t	ldb
	)

Solves one of the systems of equations.

\[ A X = B, \]

\[ A^T X = B, \]

or

\[ A^H X = B, \]

with a tridiagonal matrix A using the LU factorization computed by lapack::gttrf.

Overloaded versions are available for float, double, std::complex<float>, and std::complex<double>.

Parameters

[in]	trans	Specifies the form of the system of equations. lapack::Op::NoTrans: \(A X = B\) (No transpose) lapack::Op::Trans: \(A^T X = B\) (Transpose) lapack::Op::ConjTrans: \(A^H X = B\) (Conjugate transpose)
[in]	n	The order of the matrix A.
[in]	nrhs	The number of right hand sides, i.e., the number of columns of the matrix B. nrhs >= 0.
[in]	DL	The vector DL of length n-1. The (n-1) multipliers that define the matrix L from the LU factorization of A.
[in]	D	The vector D of length n. The n diagonal elements of the upper triangular matrix U from the LU factorization of A.
[in]	DU	The vector DU of length n-1. The (n-1) elements of the first super-diagonal of U.
[in]	DU2	The vector DU2 of length n-2. The (n-2) elements of the second super-diagonal of U.
[in]	ipiv	The vector ipiv of length n. The pivot indices; for 1 <= i <= n, row i of the matrix was interchanged with row ipiv(i). ipiv(i) will always be either i or i+1; ipiv(i) = i indicates a row interchange was not required.
[in,out]	B	The n-by-nrhs matrix B, stored in an ldb-by-nrhs array. On entry, the matrix of right hand side vectors B. On exit, B is overwritten by the solution vectors X.
[in]	ldb	The leading dimension of the array B. ldb >= max(1,n).

Returns

= 0: successful exit