NAG Library Routine Document
f07ugf (dtpcon) estimates the condition number of a real triangular matrix, using packed storage.
|Integer, Intent (In)||:: ||n|
|Integer, Intent (Out)||:: ||iwork(n), info|
|Real (Kind=nag_wp), Intent (In)||:: ||ap(*)|
|Real (Kind=nag_wp), Intent (Out)||:: ||rcond, work(3*n)|
|Character (1), Intent (In)||:: ||norm, uplo, diag|C Header Interface
f07ugf_ (const char *norm, const char *uplo, const char *diag, const Integer *n, const double ap, 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 matrix
, in either the
-norm or the
-norm, using packed storage:
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: – Real (Kind=nag_wp) arrayInput
the dimension of the array ap
must be at least
, packed by columns.
- if , the upper triangle of must be stored with element in for ;
- if , the lower triangle of must be stored with element in for .
If , the diagonal elements of are assumed to be , and are not referenced; the same storage scheme is used whether or ‘U’.
- 6: – 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.
- 7: – Real (Kind=nag_wp) arrayWorkspace
- 8: – Integer arrayWorkspace
- 9: – 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
f07ugf (dtpcon) 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 f07ugf (dtpcon)
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 but takes considerably longer than a call to f07uef (dtptrs)
with one right-hand side, because extra care is taken to avoid overflow when
is approximately singular.
The complex analogue of this routine is f07uuf (ztpcon)
This example estimates the condition number in the
-norm of the matrix
using packed storage. The true condition number in the
Program Text (f07ugfe.f90)
Program Data (f07ugfe.d)
Program Results (f07ugfe.r)