NAG Library Routine Document
F07HSF (ZPBTRS)
1 Purpose
F07HSF (ZPBTRS) solves a complex Hermitian positive definite band system of linear equations with multiple right-hand sides,
where
has been factorized by
F07HRF (ZPBTRF).
2 Specification
INTEGER |
N, KD, NRHS, LDAB, LDB, INFO |
COMPLEX (KIND=nag_wp) |
AB(LDAB,*), B(LDB,*) |
CHARACTER(1) |
UPLO |
|
The routine may be called by its
LAPACK
name zpbtrs.
3 Description
F07HSF (ZPBTRS) is used to solve a complex Hermitian positive definite band system of linear equations
, the routine must be preceded by a call to
F07HRF (ZPBTRF) which computes the Cholesky factorization of
. The solution
is computed by forward and backward substitution.
If , , where is upper triangular; the solution is computed by solving and then .
If , , where is lower triangular; the solution is computed by solving and then .
4 References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: UPLO – CHARACTER(1)Input
On entry: specifies how
has been factorized.
- , where is upper triangular.
- , where is lower triangular.
Constraint:
or .
- 2: N – INTEGERInput
On entry: , the order of the matrix .
Constraint:
.
- 3: KD – INTEGERInput
On entry: , the number of superdiagonals or subdiagonals of the matrix .
Constraint:
.
- 4: NRHS – INTEGERInput
On entry: , the number of right-hand sides.
Constraint:
.
- 5: AB(LDAB,) – COMPLEX (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
AB
must be at least
.
On entry: the Cholesky factor of
, as returned by
F07HRF (ZPBTRF).
- 6: LDAB – INTEGERInput
On entry: the first dimension of the array
AB as declared in the (sub)program from which F07HSF (ZPBTRS) is called.
Constraint:
.
- 7: B(LDB,) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
B
must be at least
.
On entry: the by right-hand side matrix .
On exit: the by solution matrix .
- 8: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F07HSF (ZPBTRS) is called.
Constraint:
.
- 9: INFO – INTEGEROutput
On exit:
unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
If , the th 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
, the computed solution
is the exact solution of a perturbed system of equations
, where
- if , ;
- if , ,
is a modest linear function of
, and
is the
machine precision.
If
is the true solution, then the computed solution
satisfies a forward error bound of the form
where
. Note that
can be much smaller than
.
Forward and backward error bounds can be computed by calling
F07HVF (ZPBRFS), and an estimate for
(
) can be obtained by calling
F07HUF (ZPBCON).
The total number of real floating point operations is approximately , assuming .
This routine may be followed by a call to
F07HVF (ZPBRFS) to refine the solution and return an error estimate.
The real analogue of this routine is
F07HEF (DPBTRS).
9 Example
This example solves the system of equations
, where
and
Here
is Hermitian positive definite, and is treated as a band matrix, which must first be factorized by
F07HRF (ZPBTRF).
9.1 Program Text
Program Text (f07hsfe.f90)
9.2 Program Data
Program Data (f07hsfe.d)
9.3 Program Results
Program Results (f07hsfe.r)