NAG Library Routine Document
F07WSF (ZPFTRS) solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides,
using the Cholesky factorization computed by F07WRF (ZPFTRF)
stored in Rectangular Full Packed (RFP) format.
The RFP storage format is described in Section 3.3.3
in the F07 Chapter Introduction.
||N, NRHS, LDB, INFO
The routine may be called by its
F07WSF (ZPFTRS) is used to solve a complex Hermitian positive definite system of linear equations
, the routine must be preceded by a call to F07WRF (ZPFTRF)
which computes the Cholesky factorization of
, stored in RFP format. 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. Software 37, 2
- 1: TRANSR – CHARACTER(1)Input
: 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: UPLO – CHARACTER(1)Input
: specifies how
has been factorized.
- , where is upper triangular.
- , where is lower triangular.
- 3: N – INTEGERInput
On entry: , the order of the matrix .
- 4: NRHS – INTEGERInput
On entry: , the number of right-hand sides.
- 5: A() – COMPLEX (KIND=nag_wp) arrayInput
: the Cholesky factorization of
stored in RFP format, as returned by F07WRF (ZPFTRF)
- 6: B(LDB,) – COMPLEX (KIND=nag_wp) arrayInput/Output
the second dimension of the array B
must be at least
On entry: the by right-hand side matrix .
On exit: the by solution matrix .
- 7: LDB – INTEGERInput
: the first dimension of the array B
as declared in the (sub)program from which F07WSF (ZPFTRS) is called.
- 8: INFO – INTEGEROutput
unless the routine detects an error (see Section 6
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
If , the th parameter 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
- if , ;
- if , ,
is a modest linear function of
is the machine precision
is the true solution, then the computed solution
satisfies a forward error bound of the form
is the condition number when using the
Note that can be much smaller than .
The total number of real floating point operations is approximately .
The real analogue of this routine is F07WEF (DPFTRS)
This example solves the system of equations
is Hermitian positive definite, stored in RFP format, and must first be factorized by F07WRF (ZPFTRF)
9.1 Program Text
Program Text (f07wsfe.f90)
9.2 Program Data
Program Data (f07wsfe.d)
9.3 Program Results
Program Results (f07wsfe.r)