The function may be called by the names: f07aec, nag_lapacklin_dgetrs or nag_dgetrs.
3Description
f07aec is used to solve a real system of linear equations or , the function must be preceded by a call to f07adc which computes the factorization of as . The solution is computed by forward and backward substitution.
If , the solution is computed by solving and then .
If or , the solution is computed by solving and then .
4References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
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_TransTypeInput
On entry: indicates the form of the equations.
is solved for .
or
is solved for .
Constraint:
, or .
3: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
4: – IntegerInput
On entry: , the number of right-hand sides.
Constraint:
.
5: – const doubleInput
Note: the dimension, dim, of the array a
must be at least
.
The th element of the matrix is stored in
when ;
when .
On entry: the factorization of , as returned by f07adc.
6: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint:
.
7: – const IntegerInput
Note: the dimension, dim, of the array ipiv
must be at least
.
On entry: the pivot indices, as returned by f07adc.
8: – doubleInput/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: .
On entry, . Constraint: .
NE_INT_2
On entry, and .
Constraint: .
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.
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.
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 can be much larger (or smaller) than .
Forward and backward error bounds can be computed by calling f07ahc, and an estimate for can be obtained by calling f07agc with .
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07aec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07aec 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 is approximately .
This function may be followed by a call to f07ahc to refine the solution and return an error estimate.