NAG Library Routine Document
f07vgf (dtbcon) estimates the condition number of a real triangular band matrix.
|Subroutine f07vgf (
||norm, uplo, diag, n, kd, ab, ldab, rcond, work, iwork, info)|
|Integer, Intent (In)||:: ||n, kd, ldab|
|Integer, Intent (Out)||:: ||iwork(n), info|
|Real (Kind=nag_wp), Intent (In)||:: ||ab(ldab,*)|
|Real (Kind=nag_wp), Intent (Out)||:: ||rcond, work(3*n)|
|Character (1), Intent (In)||:: ||norm, uplo, diag|C Header Interface
f07vgf_ (const char *norm, const char *uplo, const char *diag, const Integer *n, const Integer *kd, const double ab, const Integer *ldab, double *rcond, double work, Integer iwork, Integer *info, const Charlen length_norm, const Charlen length_uplo, const Charlen length_diag)|
The routine may be called by its
estimates the condition number of a real triangular band matrix
, in either the
-norm or the
Note that .
Because the condition number is infinite if is singular, the routine actually returns an estimate of the reciprocal of the condition number.
The routine computes
exactly, and uses Higham's implementation of Hager's method (see Higham (1988)
) to estimate
Higham N J (1988) FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396
- 1: – Character(1)Input
: indicates whether
- is estimated.
- is estimated.
, or .
- 2: – Character(1)Input
: specifies whether
is upper or lower triangular.
- is upper triangular.
- is lower triangular.
- 3: – Character(1)Input
: indicates whether
is a nonunit or unit triangular matrix.
- is a nonunit triangular matrix.
- is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be .
- 4: – IntegerInput
On entry: , the order of the matrix .
- 5: – IntegerInput
On entry: , the number of superdiagonals of the matrix if , or the number of subdiagonals if .
- 6: – Real (Kind=nag_wp) arrayInput
the second dimension of the array ab
must be at least
triangular band matrix
The matrix is stored in rows
, more precisely,
- if , the elements of the upper triangle of within the band must be stored with element in ;
- if , the elements of the lower triangle of within the band must be stored with element in
If , the diagonal elements of are assumed to be , and are not referenced.
- 7: – IntegerInput
: the first dimension of the array ab
as declared in the (sub)program from which f07vgf (dtbcon)
- 8: – Real (Kind=nag_wp)Output
: an estimate of the reciprocal of the condition number of
is set to zero if exact singularity is detected or the estimate underflows. If rcond
is less than machine precision
is singular to working precision.
- 9: – Real (Kind=nag_wp) arrayWorkspace
- 10: – Integer arrayWorkspace
- 11: – IntegerOutput
unless the routine detects an error (see Section 6
Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The computed estimate rcond
is never less than the true value
, and in practice is nearly always less than
, although examples can be constructed where rcond
is much larger.
Parallelism and Performance
f07vgf (dtbcon) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction
for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note
for your implementation for any additional implementation-specific information.
A call to f07vgf (dtbcon)
involves solving a number of systems of linear equations of the form
; the number is usually
and never more than
. Each solution involves approximately
floating-point operations (assuming
) but takes considerably longer than a call to f07vef (dtbtrs)
with one right-hand side, because extra care is taken to avoid overflow when
is approximately singular.
The complex analogue of this routine is f07vuf (ztbcon)
This example estimates the condition number in the
-norm of the matrix
is treated as a lower triangular band matrix with one subdiagonal. The true condition number in the
Program Text (f07vgfe.f90)
Program Data (f07vgfe.d)
Program Results (f07vgfe.r)