NAG CL Interface
f07gec (dpptrs)
1
Purpose
f07gec solves a real symmetric positive definite system of linear equations with multiple right-hand sides,
where
has been factorized by
f07gdc, using packed storage.
2
Specification
void |
f07gec (Nag_OrderType order,
Nag_UploType uplo,
Integer n,
Integer nrhs,
const double ap[],
double b[],
Integer pdb,
NagError *fail) |
|
The function may be called by the names: f07gec, nag_lapacklin_dpptrs or nag_dpptrs.
3
Description
f07gec is used to solve a real symmetric positive definite system of linear equations
, the function must be preceded by a call to
f07gdc which computes the Cholesky factorization of
, using packed storage. 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_OrderType
Input
-
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.
Constraint:
or .
-
2:
– Nag_UploType
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:
– const double
Input
-
Note: the dimension,
dim, of the array
ap
must be at least
.
On entry: the Cholesky factor of
stored in packed form, as returned by
f07gdc.
-
6:
– double
Input/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 .
-
7:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
-
8:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
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.
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
f07ghc, and an estimate for
(
) can be obtained by calling
f07ggc.
8
Parallelism and Performance
f07gec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07gec 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 .
This function may be followed by a call to
f07ghc to refine the solution and return an error estimate.
The complex analogue of this function is
f07gsc.
10
Example
This example solves the system of equations
, where
Here
is symmetric positive definite, stored in packed form, and must first be factorized by
f07gdc.
10.1
Program Text
10.2
Program Data
10.3
Program Results