NAG Library Routine Document
solves a real symmetric positive definite system of linear equations with multiple right-hand sides,
has been factorized by f07gdf (dpptrf)
, using packed storage.
|Integer, Intent (In)||:: ||n, nrhs, ldb|
|Integer, Intent (Out)||:: ||info|
|Real (Kind=nag_wp), Intent (In)||:: ||ap(*)|
|Real (Kind=nag_wp), Intent (Inout)||:: ||b(ldb,*)|
|Character (1), Intent (In)||:: ||uplo|C Header Interface
f07gef_ (const char *uplo, const Integer *n, const Integer *nrhs, const double ap, double b, const Integer *ldb, Integer *info, const Charlen length_uplo)|
The routine may be called by its
is used to solve a real symmetric positive definite system of linear equations
, the routine must be preceded by a call to f07gdf (dpptrf)
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 .
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
- 1: – Character(1)Input
: specifies how
has been factorized.
- , where is upper triangular.
- , where is lower triangular.
- 2: – IntegerInput
On entry: , the order of the matrix .
- 3: – IntegerInput
On entry: , the number of right-hand sides.
- 4: – Real (Kind=nag_wp) arrayInput
the dimension of the array ap
must be at least
: the Cholesky factor of
stored in packed form, as returned by f07gdf (dpptrf)
- 5: – Real (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 .
- 6: – IntegerInput
: the first dimension of the array b
as declared in the (sub)program from which f07gef (dpptrs)
- 7: – IntegerOutput
unless the routine detects an error (see Section 6
Error 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
- 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
Note that can be much smaller than .
Forward and backward error bounds can be computed by calling f07ghf (dpprfs)
, and an estimate for
) can be obtained by calling f07ggf (dppcon)
Parallelism and Performance
f07gef (dpptrs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07gef (dpptrs) 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 floating-point operations is approximately .
This routine may be followed by a call to f07ghf (dpprfs)
to refine the solution and return an error estimate.
The complex analogue of this routine is f07gsf (zpptrs)
This example solves the system of equations
is symmetric positive definite, stored in packed form, and must first be factorized by f07gdf (dpptrf)
Program Text (f07gefe.f90)
Program Data (f07gefe.d)
Program Results (f07gefe.r)