NAG Library Routine Document
F07WEF (DPFTRS)
1 Purpose
F07WEF (DPFTRS) solves a real symmetric positive definite system of linear equations with multiple right-hand sides,
where
has been factorized by
F07WDF (DPFTRF), stored in Rectangular Full Packed (RFP) format.
The RFP storage format is described in
Section 3.3.3 in the F07 Chapter Introduction.
2 Specification
INTEGER |
N, NRHS, LDB, INFO |
REAL (KIND=nag_wp) |
A(N*(N+1)/2), B(LDB,*) |
CHARACTER(1) |
TRANSR, UPLO |
|
The routine may be called by its
LAPACK
name dpftrs.
3 Description
F07WEF (DPFTRS) 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
, where
is stored in RFP format. 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 Parameters
- 1: TRANSR – CHARACTER(1)Input
On entry: 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.
Constraint:
or .
- 2: UPLO – CHARACTER(1)Input
On entry: specifies how
has been factorized.
- , where is upper triangular.
- , where is lower triangular.
Constraint:
or .
- 3: N – INTEGERInput
On entry: , the order of the matrix .
Constraint:
.
- 4: NRHS – INTEGERInput
On entry: , the number of right-hand sides.
Constraint:
.
- 5: A() – REAL (KIND=nag_wp) arrayInput
On entry: the Cholesky factorization of
stored in RFP format, as returned by
F07WDF (DPFTRF).
- 6: B(LDB,) – REAL (KIND=nag_wp) arrayInput/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: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F07WEF (DPFTRS) is called.
Constraint:
.
- 8: INFO – INTEGEROutput
On exit:
unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
If , the th parameter 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 .
The total number of floating point operations is approximately .
The complex analogue of this routine is
F07WSF (ZPFTRS).
9 Example
This example solves the system of equations
, where
Here
is symmetric positive definite, stored in RFP format, and must first be factorized by
F07WDF (DPFTRF).
9.1 Program Text
Program Text (f07wefe.f90)
9.2 Program Data
Program Data (f07wefe.d)
9.3 Program Results
Program Results (f07wefe.r)