The routine may be called by the names f07tsf, nagf_lapacklin_ztrtrs or its LAPACK name ztrtrs.
3Description
f07tsf solves a complex triangular system of linear equations , or .
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: – Character(1)Input
On entry: specifies whether is upper or lower triangular.
is upper triangular.
is lower triangular.
Constraint:
or .
2: – Character(1)Input
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 .
3: – Character(1)Input
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 .
4: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
5: – IntegerInput
On entry: , the number of right-hand sides.
Constraint:
.
6: – Complex (Kind=nag_wp) arrayInput
Note: the second dimension of the array a
must be at least
.
On entry: the triangular matrix .
If , is upper triangular and the elements of the array below the diagonal are not referenced.
If , is lower triangular and the elements of the array above the diagonal are not referenced.
If , the diagonal elements of are assumed to be , and are not referenced.
7: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07tsf is called.
Constraint:
.
8: – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b
must be at least
.
On entry: the right-hand side matrix .
On exit: the solution matrix .
9: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07tsf is called.
Constraint:
.
10: – 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.
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 f07tvf, and an estimate for can be obtained by calling f07tuf with .
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07tsf 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 real floating-point operations is approximately .