NAG Library Routine Document
F07GBF (DPPSVX)
1 Purpose
F07GBF (DPPSVX) uses the Cholesky factorization
to compute the solution to a real system of linear equations
where
A is an
n by
n symmetric positive definite matrix stored in packed format and
X and
B are
n by
r matrices. Error bounds on the solution and a condition estimate are also provided.
2 Specification
SUBROUTINE F07GBF ( |
FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO) |
INTEGER |
N, NRHS, LDB, LDX, IWORK(N), INFO |
REAL (KIND=nag_wp) |
AP(*), AFP(*), S(*), B(LDB,*), X(LDX,*), RCOND, FERR(NRHS), BERR(NRHS), WORK(3*N) |
CHARACTER(1) |
FACT, UPLO, EQUED |
|
The routine may be called by its
LAPACK
name dppsvx.
3 Description
F07GBF (DPPSVX) performs the following steps:
- If FACT='E', real diagonal scaling factors,
DS
, are computed to equilibrate the system:
Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by
DS
A
DS
and B by
DS
B.
- If FACT='N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if FACT='E') as A=UTU if UPLO='U' or A=LLT if UPLO='L', where U is an upper triangular matrix and L is a lower triangular matrix.
- If the leading i by i principal minor of A is not positive definite, then the routine returns with INFO=i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO=N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.
- The system of equations is solved for X using the factored form of A.
- Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.
- If equilibration was used, the matrix X is premultiplied by
DS
so that it solves the original system before equilibration.
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
Higham N J (2002)
Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5 Parameters
- 1: FACT – CHARACTER(1)Input
On entry: specifies whether or not the factorized form of the matrix
A is supplied on entry, and if not, whether the matrix
A should be equilibrated before it is factorized.
- FACT='F'
- AFP contains the factorized form of A. If EQUED='Y', the matrix A has been equilibrated with scaling factors given by S. AP and AFP will not be modified.
- FACT='N'
- The matrix A will be copied to AFP and factorized.
- FACT='E'
- The matrix A will be equilibrated if necessary, then copied to AFP and factorized.
Constraint:
FACT='F', 'N' or 'E'.
- 2: UPLO – CHARACTER(1)Input
On entry: if
UPLO='U', the upper triangle of
A is stored.
If UPLO='L', the lower triangle of A is stored.
Constraint:
UPLO='U' or 'L'.
- 3: N – INTEGERInput
On entry: n, the number of linear equations, i.e., the order of the matrix A.
Constraint:
N≥0.
- 4: NRHS – INTEGERInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint:
NRHS≥0.
- 5: AP(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
AP
must be at least
max1,N×N+1/2.
On entry: if
FACT='F' and
EQUED='Y',
AP must contain the equilibrated matrix
DSADS; otherwise,
AP must contain the
n by
n symmetric matrix
A, packed by columns.
More precisely,
- if UPLO='U', the upper triangle of A must be stored with element Aij in APi+jj-1/2 for i≤j;
- if UPLO='L', the lower triangle of A must be stored with element Aij in APi+2n-jj-1/2 for i≥j.
On exit: if
FACT='F' or
'N', or if
FACT='E' and
EQUED='N',
AP is not modified.
If
FACT='E' and
EQUED='Y',
AP is overwritten by
DSADS.
- 6: AFP(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
AFP
must be at least
max1,N×N+1/2.
On entry: if
FACT='F',
AFP contains the triangular factor
U or
L from the Cholesky factorization
A=UTU or
A=LLT, in the same storage format as
AP. If
EQUED='Y',
AFP is the factorized form of the equilibrated matrix
DSADS.
On exit: if
FACT='N' or if
FACT='E' and
EQUED='N',
AFP returns the triangular factor
U or
L from the Cholesky factorization
A=UTU or
A=LLT of the original matrix
A.
If
FACT='E' and
EQUED='Y',
AFP returns the triangular factor
U or
L from the Cholesky factorization
A=UTU or
A=LLT of the equilibrated matrix
A (see the description of
AP for the form of the equilibrated matrix).
- 7: EQUED – CHARACTER(1)Input/Output
On entry: if
FACT='N' or
'E',
EQUED need not be set.
If
FACT='F',
EQUED must specify the form of the equilibration that was performed as follows:
- if EQUED='N', no equilibration;
- if EQUED='Y', equilibration was performed, i.e., A has been replaced by DSADS.
On exit: if
FACT='F',
EQUED is unchanged from entry.
Otherwise, if no constraints are violated,
EQUED specifies the form of the equilibration that was performed as specified above.
Constraint:
if FACT='F', EQUED='N' or 'Y'.
- 8: S(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
S
must be at least
max1,N.
On entry: if
FACT='N' or
'E',
S need not be set.
If
FACT='F' and
EQUED='Y',
S must contain the scale factors,
DS, for
A; each element of
S must be positive.
On exit: if
FACT='F',
S is unchanged from entry.
Otherwise, if no constraints are violated and
EQUED='Y',
S contains the scale factors,
DS, for
A; each element of
S is positive.
- 9: 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: if
EQUED='N',
B is not modified.
If
EQUED='Y',
B is overwritten by
DSB.
- 10: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F07GBF (DPPSVX) is called.
Constraint:
LDB≥max1,N.
- 11: X(LDX,*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
X
must be at least
max1,NRHS.
On exit: if INFO=0 or N+1, the n by r solution matrix X to the original system of equations. Note that the arrays A and B are modified on exit if EQUED='Y', and the solution to the equilibrated system is DS-1X.
- 12: LDX – INTEGERInput
On entry: the first dimension of the array
X as declared in the (sub)program from which F07GBF (DPPSVX) is called.
Constraint:
LDX≥max1,N.
- 13: RCOND – REAL (KIND=nag_wp)Output
On exit: if no constraints are violated, an estimate of the reciprocal condition number of the matrix A (after equilibration if that is performed), computed as RCOND=1.0/A1 A-11 .
- 14: FERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
On exit: if
INFO=0 or
N+1, an estimate of the forward error bound for each computed solution vector, such that
x^j-xj∞/xj∞≤FERRj where
x^j is the
jth column of the computed solution returned in the array
X and
xj is the corresponding column of the exact solution
X. The estimate is as reliable as the estimate for
RCOND, and is almost always a slight overestimate of the true error.
- 15: BERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
On exit: if INFO=0 or N+1, an estimate of the component-wise relative backward error of each computed solution vector x^j (i.e., the smallest relative change in any element of A or B that makes x^j an exact solution).
- 16: WORK(3×N) – REAL (KIND=nag_wp) arrayWorkspace
- 17: IWORK(N) – INTEGER arrayWorkspace
- 18: 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 argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
- INFO>0 and INFO≤N
If INFO=i and i≤N, the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been computed. RCOND=0.0 is returned.
- INFO=N+1
The triangular matrix
U (or
L) is nonsingular,
but
RCOND is less than
machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of
RCOND would suggest.
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. See Section 10.1 of
Higham (2002) for further details.
If
x^ is the true solution, then the computed solution
x satisfies a forward error bound of the form
where
condA,x^,b
=
A-1
A
x^
+
b
∞/
x^∞
≤
condA
=
A-1
A
∞≤κ∞
A
. If
x^
is the
j
th column of
X
, then
wc
is returned in
BERRj
and a bound on
x
-
x^
∞
/
x^
∞
is returned in
FERRj
. See Section 4.4 of
Anderson et al. (1999) for further details.
8 Further Comments
The factorization of
A
requires approximately
13
n3
floating point operations.
For each right-hand side, computation of the backward error involves a minimum of
4n2
floating point operations. Each step of iterative refinement involves an additional
6n2
operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form
Ax=b
; the number is usually 4 or 5 and never more than 11. Each solution involves approximately
2n2
operations.
The complex analogue of this routine is
F07GPF (ZPPSVX).
9 Example
This example solves the equations
where
A
is the symmetric positive definite matrix
and
Error estimates for the solutions, information on equilibration and an estimate of the reciprocal of the condition number of the scaled matrix
A
are also output.
9.1 Program Text
Program Text (f07gbfe.f90)
9.2 Program Data
Program Data (f07gbfe.d)
9.3 Program Results
Program Results (f07gbfe.r)