NAG Library Routine Document
F07FAF (DPOSV)
1 Purpose
F07FAF (DPOSV) computes the solution to a real system of linear equations
where
is an
by
symmetric positive definite matrix and
and
are
by
matrices.
2 Specification
INTEGER |
N, NRHS, LDA, LDB, INFO |
REAL (KIND=nag_wp) |
A(LDA,*), B(LDB,*) |
CHARACTER(1) |
UPLO |
|
The routine may be called by its
LAPACK
name dposv.
3 Description
F07FAF (DPOSV) uses the Cholesky decomposition to factor as if or if , where is an upper triangular matrix and is a lower triangular matrix. The factored form of is then used to solve the system of equations .
4 References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lug
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: if
, the upper triangle of
is stored.
If , the lower triangle of is stored.
Constraint:
or .
- 2: N – INTEGERInput
On entry: , the number of linear equations, i.e., the order of the matrix .
Constraint:
.
- 3: NRHS – INTEGERInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
Constraint:
.
- 4: A(LDA,) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
.
On entry: the
by
symmetric matrix
.
- If , the upper triangular part of must be stored and the elements of the array below the diagonal are not referenced.
- If , the lower triangular part of must be stored and the elements of the array above the diagonal are not referenced.
On exit: if , the factor or from the Cholesky factorization or .
- 5: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F07FAF (DPOSV) is called.
Constraint:
.
- 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: if , the by solution matrix .
- 7: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F07FAF (DPOSV) 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 argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
If , the leading minor of order of is not positive definite, so the factorization could not be completed, and the solution has not been computed.
7 Accuracy
The computed solution for a single right-hand side,
, satisfies an equation of the form
where
and
is the
machine precision. An approximate error bound for the computed solution is given by
where
, the condition number of
with respect to the solution of the linear equations. See Section 4.4 of
Anderson et al. (1999) for further details.
F07FBF (DPOSVX) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively,
F04BDF solves
and returns a forward error bound and condition estimate.
F04BDF calls F07FAF (DPOSV) to solve the equations.
The total number of floating point operations is approximately , where is the number of right-hand sides.
The complex analogue of this routine is
F07FNF (ZPOSV).
9 Example
This example solves the equations
where
is the symmetric positive definite matrix
Details of the Cholesky factorization of are also output.
9.1 Program Text
Program Text (f07fafe.f90)
9.2 Program Data
Program Data (f07fafe.d)
9.3 Program Results
Program Results (f07fafe.r)