f04ccc computes the solution to a complex system of linear equations , where is an tridiagonal matrix and and are matrices. An estimate of the condition number of and an error bound for the computed solution are also returned.
The function may be called by the names: f04ccc, nag_linsys_complex_tridiag_solve or nag_complex_tridiag_lin_solve.
3Description
The decomposition with partial pivoting and row interchanges is used to factor as , where is a permutation matrix, is unit lower triangular with at most one nonzero subdiagonal element, and is an upper triangular band matrix with two superdiagonals. The factored form of is then used to solve the system of equations .
Note that the equations may be solved by interchanging the order of the arguments du and dl.
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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5Arguments
1: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
2: – IntegerInput
On entry: the number of linear equations , i.e., 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: – ComplexInput/Output
Note: the dimension, dim, of the array dl
must be at least
.
On entry: must contain the subdiagonal elements of the matrix .
On exit: if NE_NOERROR, dl is overwritten by the multipliers that define the matrix from the factorization of .
5: – ComplexInput/Output
Note: the dimension, dim, of the array d
must be at least
.
On entry: must contain the diagonal elements of the matrix .
On exit: if NE_NOERROR, d is overwritten by the diagonal elements of the upper triangular matrix from the factorization of .
6: – ComplexInput/Output
Note: the dimension, dim, of the array du
must be at least
.
On entry: must contain the superdiagonal elements of the matrix
On exit: if NE_NOERROR, du is overwritten by the elements of the first superdiagonal of .
7: – ComplexOutput
On exit: if NE_NOERROR, du2 returns the elements of the second superdiagonal of .
8: – IntegerOutput
On exit: if NE_NOERROR, the pivot indices that define the permutation matrix ; at the th step row of the matrix was interchanged with row . will always be either or ; indicates a row interchange was not required.
9: – ComplexInput/Output
Note: the dimension, dim, of the array b
must be at least
when
;
when
.
The th element of the matrix is stored in
when ;
when .
On entry: the matrix of right-hand sides .
On exit: if NE_NOERROR or NE_RCOND, the solution matrix .
10: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
if ,
;
if , .
11: – double *Output
On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix , computed as .
12: – double *Output
On exit: if NE_NOERROR or NE_RCOND, an estimate of the forward error bound for a computed solution , such that , where is a column of the computed solution returned in the array b and is the corresponding column of the exact solution . If rcond is less than machine precision, errbnd is returned as unity.
13: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
The Complex allocatable memory required is . In this case the factorization and the solution have been computed, but rcond and errbnd have not been computed. See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_INT
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, . Constraint: .
NE_INT_2
On entry, and .
Constraint: .
On entry, and .
Constraint: .
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_RCOND
A solution has been computed, but rcond is less than machine precision so that the matrix is numerically singular.
NE_SINGULAR
Diagonal element of the upper triangular factor is zero. The factorization has been completed, but the solution could not be computed.
7Accuracy
The computed solution for a single right-hand side, , satisfies an equation of the form
where
and is the machine precision. An approximate error bound for the computed solution is given by
where , the condition number of with respect to the solution of the linear equations. f04ccc uses the approximation to estimate errbnd. 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.
f04ccc 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 function. 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.
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.