The routine may be called by the names f07cbf, nagf_lapacklin_dgtsvx or its LAPACK name dgtsvx.
3Description
f07cbf performs the following steps:
1.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.
2.If some , so that is exactly singular, then the routine 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, is returned as a warning, but the routine still goes on to solve for and compute error bounds as described below.
3.The system of equations is solved for using the factored form of .
4.Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.
4References
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
5Arguments
1: – Character(1)Input
On entry: specifies whether or not the factorized form of the matrix has been supplied.
The matrix will be copied to dlf, df and duf and factorized.
Constraint:
or .
2: – Character(1)Input
On entry: specifies the form of the system of equations.
(No transpose).
or
(Transpose).
Constraint:
, or .
3: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
4: – IntegerInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
Constraint:
.
5: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array dl
must be at least
.
On entry: the subdiagonal elements of .
6: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array d
must be at least
.
On entry: the diagonal elements of .
7: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array du
must be at least
.
On entry: the superdiagonal elements of .
8: – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array dlf
must be at least
.
On entry: if , dlf contains the multipliers that define the matrix from the factorization of .
On exit: if , dlf contains the multipliers that define the matrix from the factorization of .
9: – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array df
must be at least
.
On entry: if , df contains the diagonal elements of the upper triangular matrix from the factorization of .
On exit: if , df contains the diagonal elements of the upper triangular matrix from the factorization of .
10: – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array duf
must be at least
.
On entry: if , duf contains the elements of the first superdiagonal of .
On exit: if , duf contains the elements of the first superdiagonal of .
11: – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array du2
must be at least
.
On entry: if , du2 contains the () elements of the second superdiagonal of .
On exit: if , du2 contains the () elements of the second superdiagonal of .
12: – Integer arrayInput/Output
Note: the dimension of the array ipiv
must be at least
.
On entry: if , ipiv contains the pivot indices from the factorization of .
On exit: if , ipiv 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.
13: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array b
must be at least
.
On entry: the right-hand side matrix .
14: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07cbf is called.
Constraint:
.
15: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array x
must be at least
.
On exit: if or , the solution matrix .
16: – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07cbf is called.
Constraint:
.
17: – Real (Kind=nag_wp)Output
On exit: the estimate of the reciprocal condition number of the matrix . If , the matrix may be exactly singular. This condition is indicated by . Otherwise, if rcond is less than the machine precision, the matrix is singular to working precision. This condition is indicated by .
18: – Real (Kind=nag_wp) arrayOutput
On exit: if or , 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 x and is the corresponding column of the exact solution . The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
19: – Real (Kind=nag_wp) arrayOutput
On exit: if or , 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).
20: – Real (Kind=nag_wp) arrayWorkspace
21: – Integer arrayWorkspace
22: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
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.
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.
is nonsingular, but rcond 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 rcond would suggest.
7Accuracy
For each right-hand side vector , the computed solution is the exact solution of a perturbed system of equations , where
is a modest linear function of , and is the machine precision. See Section 9.3 of Higham (2002) for further details.
If is the true solution, then the computed solution satisfies a forward error bound of the form
where
.
If is the th column of , then is returned in and a bound on is returned in . See Section 4.4 of Anderson et al. (1999) for further details.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07cbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07cbf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The total number of floating-point operations required to solve the equations is proportional to .
The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization. The solution is then refined, and the errors estimated, using iterative refinement.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.