The function may be called by the names: f07mgc, nag_lapacklin_dsycon or nag_dsycon.
f07mgc estimates the condition number (in the -norm) of a real symmetric indefinite matrix :
Since is symmetric, .
Because is infinite if is singular, the function actually returns an estimate of the reciprocal of .
The function should be preceded by a call to f16rcc to compute and a call to f07mdc to compute the Bunch–Kaufman factorization of . The function then 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. 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_UploTypeInput
On entry: specifies how has been factorized.
, where is upper triangular.
, where is lower triangular.
3: – IntegerInput
On entry: , the order of the matrix .
4: – const doubleInput
Note: the dimension, dim, of the array a
must be at least
On entry: details of the factorization of , as returned by f07mdc.
5: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array
6: – const IntegerInput
Note: the dimension, dim, of the array ipiv
must be at least
On entry: details of the interchanges and the block structure of , as returned by f07mdc.
7: – doubleInput
On entry: the -norm of the original matrix , which may be computed by calling f16rcc with its argument . anorm must be computed either before calling f07mdc or else from a copy of the original matrix .
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.
On entry, .
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.
f07mgc 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 f07mgc involves solving a number of systems of linear equations of the form ; the number is usually or and never more than . Each solution involves approximately floating-point operations but takes considerably longer than a call to f07mec with one right-hand side, because extra care is taken to avoid overflow when is approximately singular.
The complex analogues of this function are f07muc for Hermitian matrices and f07nuc for symmetric matrices.
This example estimates the condition number in the -norm (or -norm) of the matrix , where
Here is symmetric indefinite and must first be factorized by f07mdc. The true condition number in the -norm is .