NAG Library Routine Document
F07BBF (DGBSVX)
1 Purpose
F07BBF (DGBSVX) computes the solution to a real system of linear equations
where
A is an
n by
n band matrix with
kl subdiagonals and
ku superdiagonals, and
X and
B are
n by
r matrices. Error bounds on the solution and a condition estimate are also provided.
2 Specification
| SUBROUTINE F07BBF ( | FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO) |
| INTEGER | N, KL, KU, NRHS, LDAB, LDAFB, IPIV(*), LDB, LDX, IWORK(N), INFO |
| double precision | AB(LDAB,*), AFB(LDAFB,*), R(*), C(*), B(LDB,*), X(LDX,*), RCOND, FERR(NRHS), BERR(NRHS), WORK(max(1,3*N)) |
| CHARACTER*1 | FACT, TRANS, EQUED |
The routine may be called by its
LAPACK
name dgbsvx.
3 Description
F07BBF (DGBSVX) performs the following steps:
- Equilibration
The linear system to be solved may be badly scaled. However, the system can be equilibrated as a first stage by setting
FACT='E'. In this case, real scaling factors are computed and these factors then determine whether the system is to be equilibrated. Equilibrated forms of the systems
AX=B
and
ATX=B
are
and
respectively, where
DR
and
DC
are diagonal matrices, with positive diagonal elements, formed from the computed scaling factors.
When equilibration is used, A will be overwritten by
DR
A
DC
and B will be overwritten by
DR
B
(or
DC
B
when the solution of
ATX=B
is sought).
- Factorization
The matrix
A, or its scaled form, is copied and factored using the
LU decomposition
where
P is a permutation matrix,
L is a unit lower triangular matrix, and
U is upper triangular.
This stage can be by-passed when a factored matrix (with scaled matrices and scaling factors) are supplied; for example, as provided by a previous call to F07BBF (DGBSVX) with the same matrix A.
- Condition Number Estimation
The LU factorization of A determines whether a solution to the linear system exists. If some diagonal element of U is zero, then U is exactly singular, no solution exists and the routine returns with a failure. Otherwise the factorized 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 then a warning code is returned on final exit.
- Solution
The (equilibrated) system is solved for X (
DC-1X
or
DR-1X
) using the factored form of A (
DRADC
).
- Iterative Refinement
Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for the computed solution.
- Construct Solution Matrix X
If equilibration was used, the matrix X is premultiplied by
DC
(if TRANS='N') or
DR
(if TRANS='T' or 'C') 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/lugGolub 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*1Input
-
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'
-
AFB and IPIV contain
the factorized form of A. If
EQUED≠'N', the matrix A has been equilibrated with scaling factors given by R and C.
AB, AFB and IPIV are not modified.
- FACT='N'
- The matrix A will be copied to
AFB and factorized.
- FACT='E'
- The matrix A will be equilibrated if necessary, then copied to
AFB and factorized.
Constraint:
FACT='F', 'N' or 'E'.
- 2: TRANS – CHARACTER*1Input
-
On entry: specifies the form of the system of equations.
- TRANS='N'
- AX=B (No transpose).
- TRANS='T'
- ATX=B (Transpose).
- TRANS='C'
- AHX=B (Transpose).
Constraint:
TRANS='N', 'T' or 'C'.
- 3: N – INTEGERInput
-
On entry:
n, the number of linear equations, i.e., the order of the matrix A.
Constraint:
N≥0.
- 4: KL – INTEGERInput
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint:
KL≥0.
- 5: KU – INTEGERInput
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint:
KU≥0.
- 6: NRHS – INTEGERInput
-
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint:
NRHS≥0.
- 7: AB(LDAB,*) – double precision arrayInput/Output
Note: the second dimension of the array
AB
must be at least
max1,N.
On entry: the
n by
n coefficient matrix
A.
The matrix is stored in rows
1 to
kl+ku+1, more precisely, the element
Aij must be stored in
See
Section 8 for further details.
If
FACT='F' and
EQUED≠'N',
A must have been equilibrated by the scaling factors in
R and/or
C.
On exit: if
FACT='F' or
'N', or if
FACT='E' and
EQUED='N',
AB is not modified.
If
EQUED≠'N' then, if no constraints are violated,
A is scaled as follows:
- if EQUED='R', A=DrA;
- if EQUED='C', A=ADc;
- if EQUED='B', A=DrADc.
- 8: LDAB – INTEGERInput
-
On entry: the first dimension of the array
AB as declared in the (sub)program from which F07BBF (DGBSVX) is called.
Constraint:
LDAB≥KL+KU+1.
- 9: AFB(LDAFB,*) – double precision arrayInput/Output
Note: the second dimension of the array
AFB
must be at least
max1,N.
On entry: if
FACT='N' or
'E',
AFB need not be set.
If
FACT='F', details of the
LU factorization of the
n by
n band matrix
A, as computed by
F07BDF (DGBTRF).
The upper triangular band matrix U, with kl+ku superdiagonals, is stored in rows 1 to kl+ku+1 of the array, and the multipliers used to form the matrix L are stored in rows kl+ku+2 to 2kl+ku+1.
If
EQUED≠'N',
AFB is the factorized form of the equilibrated matrix
A.
On exit: if
FACT='F',
AFB is unchanged from entry.
Otherwise, if no constraints are violated, then if
FACT='N',
AFB returns details of the
LU factorization of the band matrix
A, and if
FACT='E',
AFB returns details of the
LU factorization of the equilibrated band matrix
A (see the description of
AB for the form of the equilibrated matrix).
- 10: LDAFB – INTEGERInput
-
On entry: the first dimension of the array
AFB as declared in the (sub)program from which F07BBF (DGBSVX) is called.
Constraint:
LDAFB≥2×KL+KU+1.
- 11: IPIV(*) – INTEGER arrayInput/Output
-
Note: the dimension of the array
IPIV
must be at least
max1,N.
On entry: if
FACT='N' or
'E',
IPIV need not be set.
If
FACT='F',
IPIV contains the pivot indices from the factorization
A=LU, as computed by
F07BDF (DGBTRF); row
i of the matrix was interchanged with row
IPIVi.
On exit: if
FACT='F',
IPIV is unchanged from entry.
Otherwise, if no constraints are violated,
IPIV contains the pivot indices that define the permutation matrix
P; at the
ith step row
i of the matrix was interchanged with row
IPIVi.
IPIVi=i indicates a row interchange was not required.
If FACT='N', the pivot indices are those corresponding to the factorization A=LU of the original matrix A.
If FACT='E', the pivot indices are those corresponding to the factorization of A=LU of the equilibrated matrix A.
- 12: EQUED – CHARACTER*1Input/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='R', row equilibration, i.e., A has been premultiplied by DR;
- if EQUED='C', column equilibration, i.e., A has been postmultiplied by DC;
- if EQUED='B', both row and column equilibration, i.e., A has been replaced by DRADC.
On exit: if
FACT='F',
EQUED is unchanged from entry.
Otherwise, if no constraints are violated,
EQUED specifies the form of equilibration that was performed as specified above.
Constraint:
if FACT='F', EQUED='N', 'R', 'C' or 'B'.
- 13: R(*) – double precision arrayInput/Output
-
Note: the dimension of the array
R
must be at least
max1,N.
On entry: if
FACT='N' or
'E',
R need not be set.
If
FACT='F' and
EQUED='R' or
'B',
R must contain the row scale factors for
A,
DR; each element of
R must be positive.
On exit: if
FACT='F',
R is unchanged from entry.
Otherwise, if no constraints are violated and
EQUED='R' or
'B',
R contains the row scale factors for
A,
DR, such that
A is multiplied on the left by
DR; each element of
R is positive.
- 14: C(*) – double precision arrayInput/Output
-
Note: the dimension of the array
C
must be at least
max1,N.
On entry: if
FACT='N' or
'E',
C need not be set.
If
FACT='F' or
EQUED='C' or
'B',
C must contain the column scale factors for
A,
DC; each element of
C must be positive.
On exit: if
FACT='F',
C is unchanged from entry.
Otherwise, if no constraints are violated and
EQUED='C' or
'B',
C contains the row scale factors for
A,
DC; each element of
C is positive.
- 15: B(LDB,*) – double precision 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
TRANS='N' and
EQUED='R' or
'B',
B is overwritten by
DRB.
If
TRANS='T' or
'C' and
EQUED='C' or
'B',
B is overwritten by
DCB.
- 16: LDB – INTEGERInput
-
On entry: the first dimension of the array
B as declared in the (sub)program from which F07BBF (DGBSVX) is called.
Constraint:
LDB≥max1,N.
- 17: X(LDX,*) – double precision 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≠'N', and the solution to the equilibrated system is DC-1X if TRANS='N' and EQUED='C' or 'B', or DR-1X if TRANS='T' or 'C' and EQUED='R' or 'B'.
- 18: LDX – INTEGERInput
-
On entry: the first dimension of the array
X as declared in the (sub)program from which F07BBF (DGBSVX) is called.
Constraint:
LDX≥max1,N.
- 19: RCOND – double precisionOutput
-
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/A1
A-11
.
- 20: FERR(NRHS) – double precision 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.
- 21: BERR(NRHS) – double precision 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).
- 22: WORK(max1,3×N) – double precision arrayOutput
On exit: if
INFO=0,
WORK1 contains the reciprocal pivot growth factor
maxaij/maxuij. If
WORK1 is much less than
1, then the stability of the
LU factorization of the (equilibrated) matrix
A could be poor. This also means that the solution
X, condition estimator
RCOND, and forward error bound
FERR could be unreliable. If the factorization fails with
INFO>0 and INFO≤N,
WORK1 contains the reciprocal pivot growth factor for the leading
INFO columns of
A.
- 23: IWORK(N) – INTEGER arrayWorkspace
- 24: 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, uii is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCOND=0 is returned.
- INFO=N+1
The triangular matrix
U 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
cn is a modest linear function of
n, and
ε is the
machine precision. See Section 9.3 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 band storage scheme for the array
AB is illustrated by the following example, when
n=6
,
kl=1
, and
ku=2
. Storage of the band matrix
A
in the array
AB:
The total number of floating-point operations required to solve the equations
AX=B
depends upon the pivoting required, but if
n≫kl+ku
then it is approximately bounded by
O
n
kl
kl
+
ku
for the factorization and
O
n
2
kl
+
ku
r
for the solution following the factorization. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization. The solution is then refined, and the errors estimated, using iterative refinement; see
F07BHF (DGBRFS) for information on the floating-point operations required.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of
Higham (2002) for further details.
The complex analogue of this routine is
F07BPF (ZGBSVX).
9 Example
This example solves the equations
where
A
is the band matrix
Estimates for the backward errors, forward errors, condition number and pivot growth are also output, together with information on the equilibration of
A
.
9.1 Program Text
Program Text (f07bbfe.f)
9.2 Program Data
Program Data (f07bbfe.d)
9.3 Program Results
Program Results (f07bbfe.r)