naginterfaces.library.lapacklin.dgtsvx¶
- naginterfaces.library.lapacklin.dgtsvx(fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b)[source]¶
dgtsvx
uses the factorization to compute the solution to a real system of linear equationswhere is a tridiagonal matrix of order and and are matrices. Error bounds on the solution and a condition estimate are also provided.
For full information please refer to the NAG Library document for f07cb
https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f07/f07cbf.html
- Parameters
- factstr, length 1
Specifies whether or not the factorized form of the matrix has been supplied.
, , , and contain the factorized form of the matrix . , , , and will not be modified.
The matrix will be copied to , and and factorized.
- transstr, length 1
Specifies the form of the system of equations.
(No transpose).
or
(Transpose).
- nint
, the order of the matrix .
- nrhsint
, the number of right-hand sides, i.e., the number of columns of the matrix .
- dlfloat, array-like, shape
The subdiagonal elements of .
- dfloat, array-like, shape
The diagonal elements of .
- dufloat, array-like, shape
The superdiagonal elements of .
- dlffloat, array-like, shape
If , contains the multipliers that define the matrix from the factorization of .
- dffloat, array-like, shape
If , contains the diagonal elements of the upper triangular matrix from the factorization of .
- duffloat, array-like, shape
If , contains the elements of the first superdiagonal of .
- du2float, array-like, shape
If , contains the () elements of the second superdiagonal of .
- ipivint, array-like, shape
If , contains the pivot indices from the factorization of .
- bfloat, array-like, shape
The right-hand side matrix .
- Returns
- dlffloat, ndarray, shape
If , contains the multipliers that define the matrix from the factorization of .
- dffloat, ndarray, shape
If , contains the diagonal elements of the upper triangular matrix from the factorization of .
- duffloat, ndarray, shape
If , contains the elements of the first superdiagonal of .
- du2float, ndarray, shape
If , contains the () elements of the second superdiagonal of .
- ipivint, ndarray, shape
If , contains the pivot indices from the factorization of ; row of the matrix was interchanged with row . will always be either or ; indicates a row interchange was not required.
- xfloat, ndarray, shape
If the function exits successfully or = + 1, the solution matrix .
- rcondfloat
The estimate of the reciprocal condition number of the matrix . If , the matrix may be exactly singular. This condition is indicated by in 1 … . Otherwise, if is less than the machine precision, the matrix is singular to working precision. This condition is indicated by = + 1.
- ferrfloat, ndarray, shape
If the function exits successfully or = + 1, an estimate of the forward error bound for each computed solution vector, such that where is the th column of the computed solution returned in the array and is the corresponding column of the exact solution . The estimate is as reliable as the estimate for , and is almost always a slight overestimate of the true error.
- berrfloat, ndarray, shape
If the function exits successfully or = + 1, an estimate of the component-wise relative backward error of each computed solution vector (i.e., the smallest relative change in any element of or that makes an exact solution).
- Raises
- NagValueError
- (errno )
On entry, error in parameter .
Constraint: or .
- (errno )
On entry, error in parameter .
Constraint: , or .
- (errno )
On entry, error in parameter .
Constraint: .
- (errno )
On entry, error in parameter .
Constraint: .
- Warns
- NagAlgorithmicWarning
- (errno )
Element of the diagonal is exactly zero. The factorization has not been completed, but the factor is exactly singular, so the solution and error bounds could not be computed. is returned.
- (errno )
Element of the diagonal is exactly zero. The factorization has been completed, but the factor is exactly singular, so the solution and error bounds could not be computed. is returned.
- (errno )
is nonsingular, but is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of would suggest.
- Notes
dgtsvx
performs the following steps:If , the decomposition is used to factor the matrix as , where is a product of permutation and unit lower bidiagonal matrices and is upper triangular with nonzeros in only the main diagonal and first two superdiagonals.
If some , so that is exactly singular, then the function returns with . Otherwise, the factored form of is used to estimate the condition number of the matrix . If the reciprocal of the condition number is less than machine precision, = + 1 is returned as a warning, but the function still goes on to solve for and compute error bounds as described below.
The system of equations is solved for using the factored form of .
Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.
- References
Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug
Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore
Higham, N J, 2002, Accuracy and Stability of Numerical Algorithms, (2nd Edition), SIAM, Philadelphia