NAG Library Function Document
nag_zpotrs (f07fsc)
1
Purpose
nag_zpotrs (f07fsc) solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides,
where
has been factorized by
nag_zpotrf (f07frc).
2
Specification
#include <nag.h> |
#include <nagf07.h> |
void |
nag_zpotrs (Nag_OrderType order,
Nag_UploType uplo,
Integer n,
Integer nrhs,
const Complex a[],
Integer pda,
Complex b[],
Integer pdb,
NagError *fail) |
|
3
Description
nag_zpotrs (f07fsc) is used to solve a complex Hermitian positive definite system of linear equations
, this function must be preceded by a call to
nag_zpotrf (f07frc) which computes the Cholesky factorization of
. 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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments
- 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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_UploTypeInput
-
On entry: specifies how
has been factorized.
- , where is upper triangular.
- , where is lower triangular.
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 ComplexInput
-
Note: the dimension,
dim, of the array
a
must be at least
.
On entry: the Cholesky factor of
, as returned by
nag_zpotrf (f07frc).
- 6:
– IntegerInput
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix in the array
a.
Constraint:
.
- 7:
– ComplexInput/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 by right-hand side matrix .
On exit: the by solution matrix .
- 8:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 9:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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
.
Note that can be much smaller than .
Forward and backward error bounds can be computed by calling
nag_zporfs (f07fvc), and an estimate for
(
) can be obtained by calling
nag_zpocon (f07fuc).
8
Parallelism and Performance
nag_zpotrs (f07fsc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zpotrs (f07fsc) 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 real floating-point operations is approximately .
This function may be followed by a call to
nag_zporfs (f07fvc) to refine the solution and return an error estimate.
The real analogue of this function is
nag_dpotrs (f07fec).
10
Example
This example solves the system of equations
, where
and
Here
is Hermitian positive definite and must first be factorized by
nag_zpotrf (f07frc).
10.1
Program Text
Program Text (f07fsce.c)
10.2
Program Data
Program Data (f07fsce.d)
10.3
Program Results
Program Results (f07fsce.r)