NAG Library Routine Document
F07BSF (ZGBTRS)
1 Purpose
F07BSF (ZGBTRS) solves a complex band system of linear equations with multiple right-hand sides,
where
A has been factorized by
F07BRF (ZGBTRF).
2 Specification
SUBROUTINE F07BSF ( |
TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO) |
INTEGER |
N, KL, KU, NRHS, LDAB, IPIV(*), LDB, INFO |
COMPLEX (KIND=nag_wp) |
AB(LDAB,*), B(LDB,*) |
CHARACTER(1) |
TRANS |
|
The routine may be called by its
LAPACK
name zgbtrs.
3 Description
F07BSF (ZGBTRS) is used to solve a complex band system of linear equations
AX=B,
ATX=B or
AHX=B, the routine must be preceded by a call to
F07BRF (ZGBTRF) 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', the solution is computed by solving UTY=B and then LTPTX=Y.
If TRANS='C', the solution is computed by solving UHY=B and then LHPTX=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(1)Input
On entry: indicates the form of the equations.
- TRANS='N'
- AX=B is solved for X.
- TRANS='T'
- ATX=B is solved for X.
- TRANS='C'
- AHX=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,*) – COMPLEX (KIND=nag_wp) 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
F07BRF (ZGBTRF).
- 7: LDAB – INTEGERInput
On entry: the first dimension of the array
AB as declared in the (sub)program from which F07BSF (ZGBTRS) 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
F07BRF (ZGBTRF).
- 9: B(LDB,*) – COMPLEX (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: 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 F07BSF (ZGBTRS) 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 condAH (which is the same as condAT) can be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling
F07BVF (ZGBRFS), and an estimate for
κ∞A can be obtained by calling
F07BUF (ZGBCON) with
NORM='I'.
8 Further Comments
The total number of real floating point operations is approximately 8n2kl+kur, assuming n≫kl and n≫ku.
This routine may be followed by a call to
F07BVF (ZGBRFS) to refine the solution and return an error estimate.
The real analogue of this routine is
F07BEF (DGBTRS).
9 Example
This example solves the system of equations
AX=B, where
and
Here
A is nonsymmetric and is treated as a band matrix, which must first be factorized by
F07BRF (ZGBTRF).
9.1 Program Text
Program Text (f07bsfe.f90)
9.2 Program Data
Program Data (f07bsfe.d)
9.3 Program Results
Program Results (f07bsfe.r)