The routine may be called by the names f07wsf, nagf_lapacklin_zpftrs or its LAPACK name zpftrs.
f07wsf is used to solve a complex Hermitian positive definite system of linear equations , the routine must be preceded by a call to f07wrf which computes the Cholesky factorization of , stored in RFP format.
The RFP storage format is described in Section 3.3.3 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: – Character(1)Input
On entry: specifies whether the normal RFP representation of or its conjugate transpose is stored.
The matrix is stored in normal RFP format.
The conjugate transpose of the RFP representation of the matrix is stored.
2: – Character(1)Input
On entry: specifies how has been factorized.
, where is upper triangular.
, where is lower triangular.
3: – IntegerInput
On entry: , the order of the matrix .
4: – IntegerInput
On entry: , the number of right-hand sides.
5: – Complex (Kind=nag_wp) arrayInput
On entry: the Cholesky factorization of stored in RFP format, as returned by f07wrf.
6: – 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 .
7: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07wsf is called.
8: – 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.
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.
f07wsf 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.
The total number of real floating-point operations is approximately .