NAG Library Routine Document
F07GEF (DPPTRS)
1 Purpose
F07GEF (DPPTRS) solves a real symmetric positive definite system of linear equations with multiple right-hand sides,
where
A has been factorized by
F07GDF (DPPTRF), using packed storage.
2 Specification
INTEGER |
N, NRHS, LDB, INFO |
REAL (KIND=nag_wp) |
AP(*), B(LDB,*) |
CHARACTER(1) |
UPLO |
|
The routine may be called by its
LAPACK
name dpptrs.
3 Description
F07GEF (DPPTRS) is used to solve a real symmetric positive definite system of linear equations
AX=B, the routine must be preceded by a call to
F07GDF (DPPTRF) which computes the Cholesky factorization of
A, using packed storage. The solution
X is computed by forward and backward substitution.
If UPLO='U', A=UTU, where U is upper triangular; the solution X is computed by solving UTY=B and then UX=Y.
If UPLO='L', A=LLT, where L is lower triangular; the solution X is computed by solving LY=B and then LTX=Y.
4 References
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: UPLO – CHARACTER(1)Input
On entry: specifies how
A has been factorized.
- UPLO='U'
- A=UTU, where U is upper triangular.
- UPLO='L'
- A=LLT, where L is lower triangular.
Constraint:
UPLO='U' or 'L'.
- 2: N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint:
N≥0.
- 3: NRHS – INTEGERInput
On entry: r, the number of right-hand sides.
Constraint:
NRHS≥0.
- 4: AP(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
AP
must be at least
max1,N×N+1/2.
On entry: the Cholesky factor of
A stored in packed form, as returned by
F07GDF (DPPTRF).
- 5: B(LDB,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
B
must be at least
max1,NRHS.
On entry: the n by r right-hand side matrix B.
On exit: the n by r solution matrix X.
- 6: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F07GEF (DPPTRS) is called.
Constraint:
LDB≥max1,N.
- 7: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, the ith 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
b, the computed solution
x is the exact solution of a perturbed system of equations
A+Ex=b, where
- if UPLO='U', E≤cnεUTU;
- if UPLO='L', E≤cnεLLT,
cn is a modest linear function of
n, and
ε is the
machine precision.
If
x^ is the true solution, then the computed solution
x satisfies a forward error bound of the form
where
condA,x=A-1Ax∞/x∞≤condA=A-1A∞≤κ∞A.
Note that condA,x can be much smaller than condA.
Forward and backward error bounds can be computed by calling
F07GHF (DPPRFS), and an estimate for
κ∞A (
=κ1A) can be obtained by calling
F07GGF (DPPCON).
8 Further Comments
The total number of floating point operations is approximately 2n2r.
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).
9 Example
This example solves the system of equations
AX=B, where
Here
A is symmetric positive definite, stored in packed form, and must first be factorized by
F07GDF (DPPTRF).
9.1 Program Text
Program Text (f07gefe.f90)
9.2 Program Data
Program Data (f07gefe.d)
9.3 Program Results
Program Results (f07gefe.r)