The function may be called by the names: f07wec, nag_lapacklin_dpftrs or nag_dpftrs.
f07wec is used to solve a real symmetric positive definite system of linear equations , the function must be preceded by a call to f07wdc which computes the Cholesky factorization of , stored in RFP format.
The RFP storage format is described in Section 3.4.3 in the F07 Chapter Introduction.
The solution is computed by forward and backward substitution.
If , , where is upper triangular; the solution is computed by solving and then .
If , , where is lower triangular; the solution is computed by solving and then .
Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software37, 2
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.
2: – Nag_RFP_StoreInput
On entry: specifies whether the RFP representation of is normal or transposed.
The matrix is stored in normal RFP format.
The matrix is stored in transposed RFP format.
3: – Nag_UploTypeInput
On entry: specifies how has been factorized.
, where is upper triangular.
, where is lower triangular.
4: – IntegerInput
On entry: , the order of the matrix .
5: – IntegerInput
On entry: , the number of right-hand sides.
6: – const doubleInput
On entry: the Cholesky factorization of stored in RFP format, as returned by f07wdc.
7: – doubleInput/Output
Note: the dimension, dim, of the array b
must be at least
The th element of the matrix is stored in
On entry: the right-hand side matrix .
On exit: the solution matrix .
8: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
if , .
9: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
On entry, .
On entry, and .
On entry, and .
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.
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.
For each right-hand side vector , the computed solution is the exact solution of a perturbed system of equations , where
if , ;
if , ,
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 and is the condition number when using the -norm.
Note that can be much smaller than .
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07wec 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.
The total number of floating-point operations is approximately .