to 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.
The routine may be called by the names f07jbf, nagf_lapacklin_dptsvx or its LAPACK name dptsvx.
3Description
f07jbf performs the following steps:
1.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 .
2.If the leading principal minor is not positive definite, 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.
df and ef contain the factorized form of the matrix . df and ef will not be modified.
The matrix will be copied to df and ef and factorized.
Constraint:
or .
2: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
3: – IntegerInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
Constraint:
.
4: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array d
must be at least
.
On entry: the diagonal elements of the tridiagonal matrix .
5: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array e
must be at least
.
On entry: the subdiagonal elements of the tridiagonal matrix .
6: – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array df
must be at least
.
On entry: if , df must contain the diagonal elements of the diagonal matrix from the factorization of .
On exit: if , df contains the diagonal elements of the diagonal matrix from the factorization of .
7: – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array ef
must be at least
.
On entry: if , ef must contain the subdiagonal elements of the unit bidiagonal factor from the factorization of .
On exit: if , ef contains the subdiagonal elements of the unit bidiagonal factor from the factorization of .
8: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array b
must be at least
.
On entry: the right-hand side matrix .
9: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07jbf is called.
Constraint:
.
10: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array x
must be at least
.
On exit: if or , the solution matrix .
11: – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07jbf is called.
Constraint:
.
12: – Real (Kind=nag_wp)Output
On exit: the reciprocal condition number of the matrix . If rcond 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 .
13: – Real (Kind=nag_wp) arrayOutput
On exit: 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 .
14: – Real (Kind=nag_wp) arrayOutput
On exit: 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).
15: – Real (Kind=nag_wp) arrayWorkspace
16: – 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.
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.
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 10.1 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.
f07jbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07jbf 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 number of floating-point operations required for the factorization, and for the estimation of the condition number of is proportional to . The number of floating-point operations required for the solution of the equations, and for the estimation of the forward and backward error is proportional to , where is the number of right-hand sides.
The condition estimation is based upon Equation (15.11) of Higham (2002). For further details of the error estimation, see Section 4.4 of Anderson et al. (1999).