NAG Library Routine Document
F07BEF (DGBTRS)
1 Purpose
F07BEF (DGBTRS) solves a real band system of linear equations with multiple right-hand sides,
where
A has been factorized by
F07BDF (DGBTRF).
2 Specification
| SUBROUTINE F07BEF ( | TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO) |
| INTEGER | N, KL, KU, NRHS, LDAB, IPIV(*), LDB, INFO |
| double precision | AB(LDAB,*), B(LDB,*) |
| CHARACTER*1 | TRANS |
The routine may be called by its
LAPACK
name dgbtrs.
3 Description
F07BEF (DGBTRS) is used to solve a real band system of linear equations
AX=B or
ATX=B, the routine must be preceded by a call to
F07BDF (DGBTRF) which computes the
LU factorization of
A as
A=PLU. The solution is computed by forward and backward substitution.
If TRANS='N', the solution is computed by solving PLY=B and then UX=Y.
If TRANS='T' or 'C', the solution is computed by solving UTY=B and then LTPTX=Y.
4 References
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: TRANS – CHARACTER*1Input
-
On entry: indicates the form of the equations.
- TRANS='N'
- AX=B is solved for X.
- TRANS='T' or 'C'
- ATX=B is solved for X.
Constraint:
TRANS='N', 'T' or 'C'.
- 2: N – INTEGERInput
-
On entry:
n, the order of the matrix A.
Constraint:
N≥0.
- 3: KL – INTEGERInput
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint:
KL≥0.
- 4: KU – INTEGERInput
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint:
KU≥0.
- 5: NRHS – INTEGERInput
-
On entry: r, the number of right-hand sides.
Constraint:
NRHS≥0.
- 6: AB(LDAB,*) – double precision arrayInput
Note: the second dimension of the array
AB
must be at least
max1,N.
On entry: the
LU factorization of
A, as returned by
F07BDF (DGBTRF).
- 7: LDAB – INTEGERInput
-
On entry: the first dimension of the array
AB as declared in the (sub)program from which F07BEF (DGBTRS) is called.
Constraint:
LDAB≥2×KL+KU+1.
- 8: IPIV(*) – INTEGER arrayInput
-
Note: the dimension of the array
IPIV
must be at least
max1,N.
On entry: the pivot indices, as returned by
F07BDF (DGBTRF).
- 9: 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: the n by r solution matrix X.
- 10: LDB – INTEGERInput
-
On entry: the first dimension of the array
B as declared in the (sub)program from which F07BEF (DGBTRS) is called.
Constraint:
LDB≥max1,N.
- 11: 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
ck is a modest linear function of
k=kl+ku+1, and
ε is the
machine precision. This assumes
k≪n.
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, and condAT can be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling
F07BHF (DGBRFS), and an estimate for
κ∞A can be obtained by calling
F07BGF (DGBCON) with
NORM='I'.
8 Further Comments
The total number of floating-point operations is approximately 2n2kl+kur, assuming n≫kl and n≫ku.
This routine may be followed by a call to
F07BHF (DGBRFS) to refine the solution and return an error estimate.
The complex analogue of this routine is
F07BSF (ZGBTRS).
9 Example
This example solves the system of equations
AX=B, where
Here
A is nonsymmetric and is treated as a band matrix, which must first be factorized by
F07BDF (DGBTRF).
9.1 Program Text
Program Text (f07befe.f)
9.2 Program Data
Program Data (f07befe.d)
9.3 Program Results
Program Results (f07befe.r)