The routine may be called by the names f07bgf, nagf_lapacklin_dgbcon or its LAPACK name dgbcon.
f07bgf estimates the condition number of a real 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 should be preceded by a call to f06rbf to compute or , and a call to f07bdf to compute the factorization of . The routine then 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: – IntegerInput
On entry: , the order of the matrix .
3: – IntegerInput
On entry: , the number of subdiagonals within the band of the matrix .
4: – IntegerInput
On entry: , the number of superdiagonals within the band of the matrix .
5: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array ab
must be at least
On entry: the factorization of , as returned by f07bdf.
6: – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07bgf is called.
7: – Integer arrayInput
Note: the dimension of the array ipiv
must be at least
On entry: the pivot indices, as returned by f07bdf.
8: – Real (Kind=nag_wp)Input
On entry: if or , the -norm of the original matrix .
If , the -norm of the original matrix .
anorm may be computed by calling f06rbf with the same value for the argument norm.
anorm must be computed either before calling f07bdf or else from a copy of the original matrix .
9: – 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.
10: – Real (Kind=nag_wp) arrayWorkspace
11: – Integer arrayWorkspace
12: – 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.
f07bgf 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 f07bgf 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 and ) but takes considerably longer than a call to f07bef with one right-hand side, because extra care is taken to avoid overflow when is approximately singular.