f01jcc computes an estimate of the absolute condition number of a matrix function at a real matrix in the -norm, using analytical derivatives of you have supplied.
The function may be called by the names: f01jcc or nag_matop_real_gen_matrix_cond_usd.
3Description
The absolute condition number of at , is given by the norm of the Fréchet derivative of , , which is defined by
where is the Fréchet derivative in the direction . is linear in and can, therefore, be written as
where the operator stacks the columns of a matrix into one vector, so that is . f01jcc computes an estimate such that , where . The relative condition number can then be computed via
The algorithm used to find is detailed in Section 3.4 of Higham (2008).
The function , and the derivatives of , are returned by function f which, given an integer , evaluates at a number of (generally complex) points , for . For any on the real line, must also be real. f01jcc is, therefore, appropriate for functions that can be evaluated on the complex plane and whose derivatives, of arbitrary order, can also be evaluated on the complex plane.
4References
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
5Arguments
1: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
2: – doubleInput/Output
Note: the dimension, dim, of the array a
must be at least
.
The th element of the matrix is stored in .
On entry: the matrix .
On exit: the matrix, .
3: – IntegerInput
On entry: the stride separating matrix row elements in the array a.
Constraint:
.
4: – function, supplied by the userExternal Function
Given an integer , the function f evaluates at a number of points .
On exit: iflag should either be unchanged from its entry value of zero, or may be set nonzero to indicate that there is a problem in evaluating the function ; for instance may not be defined. If iflag is returned as nonzero then f01jcc will terminate the computation, with NE_INT, NE_INT_2 or NE_USER_STOP.
3: – IntegerInput
On entry: , the number of function or derivative values required.
4: – const ComplexInput
On entry: the points at which the function is to be evaluated.
5: – ComplexOutput
On exit: the function or derivative values.
should return the value , for . If lies on the real line, then so must .
6: – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to f.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling f01jcc you may allocate memory and initialize these pointers with various quantities for use by f when called from f01jcc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note:f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by f01jcc. If your code inadvertently does return any NaNs or infinities, f01jcc is likely to produce unexpected results.
5: – Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
On exit: an estimate of the absolute condition number of at .
8: – double *Output
On exit: the -norm of .
9: – double *Output
On exit: the -norm of .
10: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_INT
On entry, .
Constraint: .
NE_INT_2
On entry, and .
Constraint: .
NE_INTERNAL_ERROR
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.
An internal error occurred when estimating the norm of the Fréchet derivative of at . Please contact NAG.
An internal error occurred when evaluating the matrix function . You can investigate further by calling f01emc with the matrix and the function .
NE_NO_LICENCE
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.
f01jcc uses the norm estimation routine f04ydc to estimate a quantity , where and . For further details on the accuracy of norm estimation, see the documentation for f04ydc.
8Parallelism and Performance
f01jcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library. In these implementations, this function may make calls to the user-supplied functions from within an OpenMP parallel region. Thus OpenMP pragmas within the user functions can only be used if you are compiling the user-supplied function and linking the executable in accordance with the instructions in the Users' Note for your implementation. You must also ensure that you use the NAG communication argument comm in a thread safe manner, which is best achieved by only using it to supply read-only data to the user functions.
f01jcc 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.
9Further Comments
The matrix function is computed using the underlying matrix function routine f01emc. Approximately of real allocatable memory is required by the routine, in addition to the memory used by the underlying matrix function routine.
If only is required, without an estimate of the condition number, then it is far more efficient to use the underlying matrix function routine directly.