naginterfaces.library.lapacklin.dgbsvx¶
- naginterfaces.library.lapacklin.dgbsvx(fact, trans, n, kl, ku, nrhs, ab, afb, ipiv, equed, r, c, b)[source]¶
dgbsvx
uses the factorization to compute the solution to a real system of linear equationswhere is an band matrix with subdiagonals and superdiagonals, 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 f07bb
https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f07/f07bbf.html
- Parameters
- factstr, length 1
Specifies whether or not the factorized form of the matrix is supplied on entry, and if not, whether the matrix should be equilibrated before it is factorized.
and contain the factorized form of . If , the matrix has been equilibrated with scaling factors given by and . , and are not modified.
The matrix will be copied to and factorized.
The matrix will be equilibrated if necessary, then copied to and factorized.
- transstr, length 1
Specifies the form of the system of equations.
(No transpose).
or
(Transpose).
- nint
, the number of linear equations, i.e., the order of the matrix .
- klint
, the number of subdiagonals within the band of the matrix .
- kuint
, the number of superdiagonals within the band of the matrix .
- nrhsint
, the number of right-hand sides, i.e., the number of columns of the matrix .
- abfloat, array-like, shape
The coefficient matrix .
See Further Comments for further details.
If and , must have been equilibrated by the scaling factors in and/or .
- afbfloat, array-like, shape
If or , need not be set.
If , details of the factorization of the band matrix , as computed by
dgbtrf()
.If , is the factorized form of the equilibrated matrix .
- ipivint, array-like, shape
If or , need not be set.
If , contains the pivot indices from the factorization , as computed by
dgbtrf()
; row of the matrix was interchanged with row .- equedstr, length 1
If or , need not be set.
If , must specify the form of the equilibration that was performed as follows:
if , no equilibration;
if , row equilibration, i.e., has been premultiplied by ;
if , column equilibration, i.e., has been postmultiplied by ;
if , both row and column equilibration, i.e., has been replaced by .
- rfloat, array-like, shape
If or , need not be set.
If and or , must contain the row scale factors for , ; each element of must be positive.
- cfloat, array-like, shape
If or , need not be set.
If and or , must contain the column scale factors for , ; each element of must be positive.
- bfloat, array-like, shape
The right-hand side matrix .
- Returns
- abfloat, ndarray, shape
If or , or if and , is not modified.
If then, if no constraints are violated, is scaled as follows:
if , ;
if , ;
if , .
- afbfloat, ndarray, shape
If , is unchanged from entry.
Otherwise, if no constraints are violated, then if , returns details of the factorization of the band matrix , and if , returns details of the factorization of the equilibrated band matrix (see the description of for the form of the equilibrated matrix).
- ipivint, ndarray, shape
If , is unchanged from entry.
Otherwise, if no constraints are violated, contains the pivot indices that define the permutation matrix ; at the th step row of the matrix was interchanged with row . indicates a row interchange was not required.
If , the pivot indices are those corresponding to the factorization of the original matrix .
If , the pivot indices are those corresponding to the factorization of of the equilibrated matrix .
- equedstr, length 1
If , is unchanged from entry.
Otherwise, if no constraints are violated, specifies the form of equilibration that was performed as specified above.
- rfloat, ndarray, shape
If , is unchanged from entry.
Otherwise, if no constraints are violated and or , contains the row scale factors for , , such that is multiplied on the left by ; each element of is positive.
- cfloat, ndarray, shape
If , is unchanged from entry.
Otherwise, if no constraints are violated and or , contains the row scale factors for , ; each element of is positive.
- bfloat, ndarray, shape
If , is not modified.
If and or , is overwritten by .
If or and or , is overwritten by .
- xfloat, ndarray, shape
If the function exits successfully or = + 1, the solution matrix to the original system of equations. Note that the arrays and are modified on exit if , and the solution to the equilibrated system is if and or , or if or and or .
- rcondfloat
If no constraints are violated, an estimate of the reciprocal condition number of the matrix (after equilibration if that is performed), computed as .
- 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).
- recip_growth_factorfloat
If no exception or warning is raised, the reciprocal pivot growth factor , where denotes the maximum absolute element norm. If , the stability of the factorization of (equilibrated) could be poor. This also means that the solution , condition estimate , and forward error bound could be unreliable. If the factorization fails with in 1 … , then contains the reciprocal pivot growth factor for the leading columns of .
- 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: .
- (errno )
On entry, error in parameter .
Constraint: .
- (errno )
On entry, error in parameter .
Constraint: .
- (errno )
On entry, error in parameter .
Constraint: , , or .
- (errno )
On entry, error in parameter .
- (errno )
On entry, error in parameter .
- Warns
- NagAlgorithmicWarning
- (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
dgbsvx
performs the following steps:Equilibration
The linear system to be solved may be badly scaled. However, the system can be equilibrated as a first stage by setting . In this case, real scaling factors are computed and these factors then determine whether the system is to be equilibrated. Equilibrated forms of the systems and are
and
respectively, where and are diagonal matrices, with positive diagonal elements, formed from the computed scaling factors.
When equilibration is used, will be overwritten by and will be overwritten by (or when the solution of is sought).
Factorization
The matrix , or its scaled form, is copied and factored using the decomposition
where is a permutation matrix, is a unit lower triangular matrix, and is upper triangular.
This stage can be by-passed when a factored matrix (with scaled matrices and scaling factors) are supplied; for example, as provided by a previous call to
dgbsvx
with the same matrix .Condition Number Estimation
The factorization of determines whether a solution to the linear system exists. If some diagonal element of is zero, then is exactly singular, no solution exists and the function returns with a failure. Otherwise the factorized form of is used to estimate the condition number of the matrix . If the reciprocal of the condition number is less than machine precision then a warning code is returned on final exit.
Solution
The (equilibrated) system is solved for ( or ) using the factored form of ().
Iterative Refinement
Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for the computed solution.
Construct Solution Matrix
If equilibration was used, the matrix is premultiplied by (if ) or (if or ) so that it solves the original system before equilibration.
- 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