The function may be called by the names: f07tuc, nag_lapacklin_ztrcon or nag_ztrcon.
f07tuc estimates the condition number of a complex triangular matrix , in either the -norm or the -norm:
Note that .
Because the condition number is infinite if is singular, the function actually returns an estimate of the reciprocal of the condition number.
The function 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: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
2: – Nag_NormTypeInput
On entry: indicates whether or is estimated.
3: – Nag_UploTypeInput
On entry: specifies whether is upper or lower triangular.
is upper triangular.
is lower triangular.
4: – Nag_DiagTypeInput
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 .
5: – IntegerInput
On entry: , the order of the matrix .
6: – const ComplexInput
Note: the dimension, dim, of the array a
must be at least
On entry: the triangular matrix .
If , is stored in .
If , is stored in .
If , the upper triangular part of must be stored and the elements of the array below the diagonal are not referenced.
If , the lower triangular part of must be stored and the elements of the array above the diagonal are not referenced.
If , the diagonal elements of are assumed to be , and are not referenced.
7: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array
8: – double *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: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
On entry, . Constraint: .
On entry, and .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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.
f07tuc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
A call to f07tuc involves solving a number of systems of linear equations of the form or ; the number is usually and never more than . Each solution involves approximately real floating-point operations but takes considerably longer than a call to f07tsc with one right-hand side, because extra care is taken to avoid overflow when is approximately singular.