NAG FL Interface
f07wsf (zpftrs)
1
Purpose
f07wsf solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides,
using the Cholesky factorization computed by
f07wrf stored in Rectangular Full Packed (RFP) format.
2
Specification
Fortran Interface
Integer, Intent (In) |
:: |
n, nrhs, ldb |
Integer, Intent (Out) |
:: |
info |
Complex (Kind=nag_wp), Intent (In) |
:: |
ar(n*(n+1)/2) |
Complex (Kind=nag_wp), Intent (Inout) |
:: |
b(ldb,*) |
Character (1), Intent (In) |
:: |
transr, uplo |
|
C++ Header Interface
#include <nag.h> extern "C" {
}
|
The routine may be called by the names f07wsf, nagf_lapacklin_zpftrs or its LAPACK name zpftrs.
3
Description
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 .
4
References
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
5
Arguments
-
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.
Constraint:
or .
-
2:
– Character(1)
Input
-
On entry: specifies how
has been factorized.
- , where is upper triangular.
- , where is lower triangular.
Constraint:
or .
-
3:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
4:
– Integer
Input
-
On entry: , the number of right-hand sides.
Constraint:
.
-
5:
– Complex (Kind=nag_wp) array
Input
-
On entry: the Cholesky factorization of
stored in RFP format, as returned by
f07wrf.
-
6:
– Complex (Kind=nag_wp) array
Input/Output
-
Note: 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:
– Integer
Input
-
On entry: the first dimension of the array
b as declared in the (sub)program from which
f07wsf is called.
Constraint:
.
-
8:
– Integer
Output
-
On exit:
unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
7
Accuracy
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 .
8
Parallelism and Performance
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 .
The real analogue of this routine is
f07wef.
10
Example
This example solves the system of equations
, where
and
Here
is Hermitian positive definite, stored in RFP format, and must first be factorized by
f07wrf.
10.1
Program Text
10.2
Program Data
10.3
Program Results