NAG Library Routine Document
F07VSF (ZTBTRS)
1 Purpose
F07VSF (ZTBTRS) solves a complex triangular band system of linear equations with multiple right-hand sides, AX=B, ATX=B or AHX=B.
2 Specification
SUBROUTINE F07VSF ( |
UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, LDB, INFO) |
INTEGER |
N, KD, NRHS, LDAB, LDB, INFO |
COMPLEX (KIND=nag_wp) |
AB(LDAB,*), B(LDB,*) |
CHARACTER(1) |
UPLO, TRANS, DIAG |
|
The routine may be called by its
LAPACK
name ztbtrs.
3 Description
F07VSF (ZTBTRS) solves a complex triangular band system of linear equations AX=B, ATX=B or AHX=B.
4 References
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1989) The accuracy of solutions to triangular systems
SIAM J. Numer. Anal. 26 1252–1265
5 Parameters
- 1: UPLO – CHARACTER(1)Input
On entry: specifies whether
A is upper or lower triangular.
- UPLO='U'
- A is upper triangular.
- UPLO='L'
- A is lower triangular.
Constraint:
UPLO='U' or 'L'.
- 2: TRANS – CHARACTER(1)Input
On entry: indicates the form of the equations.
- TRANS='N'
- The equations are of the form AX=B.
- TRANS='T'
- The equations are of the form ATX=B.
- TRANS='C'
- The equations are of the form AHX=B.
Constraint:
TRANS='N', 'T' or 'C'.
- 3: DIAG – CHARACTER(1)Input
On entry: indicates whether
A is a nonunit or unit triangular matrix.
- DIAG='N'
- A is a nonunit triangular matrix.
- DIAG='U'
- A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint:
DIAG='N' or 'U'.
- 4: N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint:
N≥0.
- 5: KD – INTEGERInput
On entry: kd, the number of superdiagonals of the matrix A if UPLO='U', or the number of subdiagonals if UPLO='L'.
Constraint:
KD≥0.
- 6: NRHS – INTEGERInput
On entry: r, the number of right-hand sides.
Constraint:
NRHS≥0.
- 7: AB(LDAB,*) – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
AB
must be at least
max1,N.
On entry: the
n by
n triangular band matrix
A.
The matrix is stored in rows
1 to
kd+1, more precisely,
- if UPLO='U', the elements of the upper triangle of A within the band must be stored with element Aij in ABkd+1+i-jj for max1,j-kd≤i≤j;
- if UPLO='L', the elements of the lower triangle of A within the band must be stored with element Aij in AB1+i-jj for j≤i≤minn,j+kd.
If DIAG='U', the diagonal elements of A are assumed to be 1, and are not referenced.
- 8: LDAB – INTEGERInput
On entry: the first dimension of the array
AB as declared in the (sub)program from which F07VSF (ZTBTRS) is called.
Constraint:
LDAB≥KD+1.
- 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 F07VSF (ZTBTRS) 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.
- INFO>0
If INFO=i, ai,i is exactly zero; A is singular and the solution has not been computed.
7 Accuracy
The solutions of triangular systems of equations are usually computed to high accuracy. See
Higham (1989).
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, and
ε is the
machine precision.
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∞.
Note that condA,x≤condA=A-1A∞≤κ∞A; condA,x can be much smaller than condA and it is also possible for condAH, which is the same as condAT, to be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling
F07VVF (ZTBRFS), and an estimate for
κ∞A can be obtained by calling
F07VUF (ZTBCON) with
NORM='I'.
8 Further Comments
The total number of real floating point operations is approximately 8nkr if k≪n.
The real analogue of this routine is
F07VEF (DTBTRS).
9 Example
This example solves the system of equations
AX=B, where
and
Here
A is treated as a lower triangular band matrix with two subdiagonals.
9.1 Program Text
Program Text (f07vsfe.f90)
9.2 Program Data
Program Data (f07vsfe.d)
9.3 Program Results
Program Results (f07vsfe.r)