NAG Library Routine Document
F07JHF (DPTRFS)
1 Purpose
F07JHF (DPTRFS) computes error bounds and refines the solution to a real system of linear equations
AX=B
, where
A
is an
n
by
n
symmetric positive definite tridiagonal matrix and
X
and
B
are
n
by
r
matrices, using the modified Cholesky factorization returned by
F07JDF (DPTTRF) and an initial solution returned by
F07JEF (DPTTRS). Iterative refinement is used to reduce the backward error as much as possible.
2 Specification
SUBROUTINE F07JHF ( |
N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, INFO) |
INTEGER |
N, NRHS, LDB, LDX, INFO |
REAL (KIND=nag_wp) |
D(*), E(*), DF(*), EF(*), B(LDB,*), X(LDX,*), FERR(NRHS), BERR(NRHS), WORK(2*N) |
|
The routine may be called by its
LAPACK
name dptrfs.
3 Description
F07JHF (DPTRFS) should normally be preceded by calls to
F07JDF (DPTTRF) and
F07JEF (DPTTRS).
F07JDF (DPTTRF) computes a modified Cholesky factorization of the matrix
A
as
where
L
is a unit lower bidiagonal matrix and
D
is a diagonal matrix, with positive diagonal elements.
F07JEF (DPTTRS) then utilizes the factorization to compute a solution,
X^
, to the required equations. Letting
x^
denote a column of
X^
, F07JHF (DPTRFS) computes a
component-wise backward error,
β
, the smallest relative perturbation in each element of
A
and
b
such that
x^
is the exact solution of a perturbed system
The routine also estimates a bound for the component-wise forward error in the computed solution defined by
max
xi
-
xi^
/
max
xi^
, where
x
is the corresponding column of the exact solution,
X
.
Note that the modified Cholesky factorization of
A
can also be expressed as
where
U
is unit upper bidiagonal.
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/lug5 Parameters
- 1: N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint:
N≥0.
- 2: NRHS – INTEGERInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint:
NRHS≥0.
- 3: D(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
D
must be at least
max1,N.
On entry: must contain the n diagonal elements of the matrix of A.
- 4: E(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
E
must be at least
max1,N-1.
On entry: must contain the n-1 subdiagonal elements of the matrix A.
- 5: DF(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
DF
must be at least
max1,N.
On entry: must contain the n diagonal elements of the diagonal matrix D from the LDLT factorization of A.
- 6: EF(*) – REAL (KIND=nag_wp) arrayInput
-
Note: the dimension of the array
EF
must be at least
max1,N.
On entry: must contain the n-1 subdiagonal elements of the unit bidiagonal matrix L from the LDLT factorization of A.
- 7: B(LDB,*) – REAL (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
B
must be at least
max1,NRHS.
On entry: the n by r matrix of right-hand sides B.
- 8: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F07JHF (DPTRFS) is called.
Constraint:
LDB≥max1,N.
- 9: X(LDX,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
X
must be at least
max1,NRHS.
On entry: the n by r initial solution matrix X.
On exit: the n by r refined solution matrix X.
- 10: LDX – INTEGERInput
On entry: the first dimension of the array
X as declared in the (sub)program from which F07JHF (DPTRFS) is called.
Constraint:
LDX≥max1,N.
- 11: FERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
On exit: estimate of the forward error bound for each computed solution vector, such that
x^j-xj∞/x^j∞≤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 almost always a slight overestimate of the true error.
- 12: BERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
On exit: 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).
- 13: WORK(2×N) – REAL (KIND=nag_wp) arrayWorkspace
- 14: 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.
7 Accuracy
The computed solution for a single right-hand side,
x^
, satisfies an equation of the form
where
and
ε
is the
machine precision. An approximate error bound for the computed solution is given by
where
κA=A-1∞ A∞
, the condition number of
A
with respect to the solution of the linear equations. See Section 4.4 of
Anderson et al. (1999) for further details.
Routine
F07JGF (DPTCON) can be used to compute the condition number of
A
.
8 Further Comments
The total number of floating point operations required to solve the equations
AX=B
is proportional to
nr
. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
The complex analogue of this routine is
F07JVF (ZPTRFS).
9 Example
This example solves the equations
where
A
is the symmetric positive definite tridiagonal matrix
Estimates for the backward errors and forward errors are also output.
9.1 Program Text
Program Text (f07jhfe.f90)
9.2 Program Data
Program Data (f07jhfe.d)
9.3 Program Results
Program Results (f07jhfe.r)