The function may be called by the names: f07usc, nag_lapacklin_ztptrs or nag_ztptrs.
3Description
f07usc solves a complex triangular system of linear equations , or , using packed storage.
4References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal.26 1252–1265
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: – Nag_UploTypeInput
On entry: specifies whether is upper or lower triangular.
is upper triangular.
is lower triangular.
Constraint:
or .
3: – Nag_TransTypeInput
On entry: indicates the form of the equations.
The equations are of the form .
The equations are of the form .
The equations are of the form .
Constraint:
, or .
4: – Nag_DiagTypeInput
On entry: indicates whether is a nonunit or unit triangular matrix.
is a nonunit triangular matrix.
is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be .
Constraint:
or .
5: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
6: – IntegerInput
On entry: , the number of right-hand sides.
Constraint:
.
7: – const ComplexInput
Note: the dimension, dim, of the array ap
must be at least
.
On entry: the triangular matrix , packed by rows or columns.
The storage of elements depends on the order and uplo arguments as follows:
if and ,
is stored in , for ;
if and ,
is stored in , for ;
if and ,
is stored in , for ;
if and ,
is stored in , for .
If , the diagonal elements of are assumed to be , and are not referenced; the same storage scheme is used whether or .
8: – 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 right-hand side matrix .
On exit: the solution matrix .
9: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
if ,
;
if , .
10: – 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.
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_SINGULAR
Element of the diagonal is exactly zero.
is singular and the solution has not been computed.
7Accuracy
The solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).
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.
If is the true solution, then the computed solution satisfies a forward error bound of the form
where .
Note that ; can be much smaller than and it is also possible for , which is the same as , to be much larger (or smaller) than .
Forward and backward error bounds can be computed by calling f07uvc, and an estimate for can be obtained by calling f07uuc with .
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07usc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07usc 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 real floating-point operations is approximately .