NAG Library Routine Document
solves a real symmetric positive definite system of linear equations with multiple right-hand sides,
using the Cholesky factorization computed by f07wdf (dpftrf)
stored in Rectangular Full Packed (RFP) format.
|Integer, Intent (In)||:: ||
|Integer, Intent (Out)||:: ||
|Real (Kind=nag_wp), Intent (In)||:: ||
|Real (Kind=nag_wp), Intent (Inout)||:: ||
|Character (1), Intent (In)||:: ||
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 f07wdf (dpftrf)
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.
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: – Character(1)Input
: specifies whether the RFP representation of
is normal or transposed.
- The matrix is stored in normal RFP format.
- The matrix is stored in transposed RFP format.
- 2: – Character(1)Input
: specifies how
has been factorized.
- , where is upper triangular.
- , where is lower triangular.
- 3: – IntegerInput
On entry: , the order of the matrix .
- 4: – IntegerInput
On entry: , the number of right-hand sides.
- 5: – Real (Kind=nag_wp) arrayInput
: the Cholesky factorization of
stored in RFP format, as returned by f07wdf (dpftrf)
- 6: – 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 .
- 7: – IntegerInput
: the first dimension of the array b
as declared in the (sub)program from which f07wef (dpftrs)
- 8: – 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
is the condition number when using the
Note that can be much smaller than .
Parallelism and Performance
f07wef (dpftrs) 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 .
The complex analogue of this routine is f07wsf (zpftrs)
This example solves the system of equations
is symmetric positive definite, stored in RFP format, and must first be factorized by f07wdf (dpftrf)
Program Text (f07wefe.f90)
Program Data (f07wefe.d)
Program Results (f07wefe.r)