naginterfaces.library.lapacklin.dptsvx¶
- naginterfaces.library.lapacklin.dptsvx(fact, nrhs, d, e, df, ef, b)[source]¶
dptsvx
uses the factorizationto compute the solution to a real system of linear equations
where is an symmetric positive definite tridiagonal matrix 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 f07jb
https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/f07/f07jbf.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.
- nrhsint
, the number of right-hand sides, i.e., the number of columns of the matrix .
- dfloat, array-like, shape
The diagonal elements of the tridiagonal matrix .
- efloat, array-like, shape
The subdiagonal elements of the tridiagonal matrix .
- dffloat, array-like, shape
If , must contain the diagonal elements of the diagonal matrix from the factorization of .
- effloat, array-like, shape
If , must contain the subdiagonal elements of the unit bidiagonal factor from the factorization of .
- bfloat, array-like, shape
The right-hand side matrix .
- Returns
- dffloat, ndarray, shape
If , contains the diagonal elements of the diagonal matrix from the factorization of .
- effloat, ndarray, shape
If , contains the subdiagonal elements of the unit bidiagonal factor from the factorization of .
- xfloat, ndarray, shape
If the function exits successfully or = + 1, the solution matrix .
- rcondfloat
The reciprocal condition number of the matrix . If is less than the machine precision (in particular, if ), the matrix is singular to working precision. This condition is indicated by a return code of = + 1.
- ferrfloat, ndarray, shape
The forward error bound for each solution vector (the th column of the solution matrix ). If is the true solution corresponding to , is an estimated upper bound for the magnitude of the largest element in () divided by the magnitude of the largest element in .
- berrfloat, ndarray, shape
The component-wise relative backward error of each 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: .
- (errno )
On entry, error in parameter .
Constraint: .
- (errno )
The leading minor of order of is not positive definite, so the factorization could not be completed, and the solution has not been computed. is returned.
- Warns
- NagAlgorithmicWarning
- (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
dptsvx
performs the following steps:If , the matrix is factorized as , where is a unit lower bidiagonal matrix and is diagonal. The factorization can also be regarded as having the form .
If the leading principal minor is not positive definite, 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