The routine may be called by the names f07vgf, nagf_lapacklin_dtbcon or its LAPACK name dtbcon.
f07vgf estimates the condition number of a real triangular band matrix , in either the -norm or the -norm:
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 or exactly, and uses Higham's implementation of Hager's method (see Higham (1988)) to estimate or .
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. Software14 381–396
1: – Character(1)Input
On entry: indicates whether or is estimated.
, or .
2: – Character(1)Input
On entry: specifies whether is upper or lower triangular.
is upper triangular.
is lower triangular.
3: – Character(1)Input
On entry: 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
Note: the second dimension of the array ab
must be at least
On entry: the triangular band matrix .
The matrix is stored in rows to , 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
On entry: the first dimension of the array ab as declared in the (sub)program from which f07vgf is called.
8: – Real (Kind=nag_wp)Output
On exit: an estimate of the reciprocal of the condition number of . rcond 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
On exit: unless the routine detects an error (see Section 6).
6Error 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.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07vgf 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 involves solving a number of systems of linear equations of the form or ; the number is usually or and never more than . Each solution involves approximately floating-point operations (assuming ) but takes considerably longer than a call to f07vef with one right-hand side, because extra care is taken to avoid overflow when is approximately singular.