The function may be called by the names: f01efc or nag_matop_real_symm_matrix_fun.
is computed using a spectral factorization of
where is the diagonal matrix whose diagonal elements, , are the eigenvalues of , and is an orthogonal matrix whose columns are the eigenvectors of . is then given by
where is the diagonal matrix whose th diagonal element is . See for example Section 4.5 of Higham (2008). is assumed to be real.
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
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: if , the upper triangle of the matrix is stored.
If , the lower triangle of the matrix is stored.
3: – IntegerInput
On entry: , the order of the matrix .
4: – doubleInput/Output
Note: the dimension, dim, of the array a
must be at least
On entry: the symmetric 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.
On exit: if NE_NOERROR, the upper or lower triangular part of the matrix function, .
5: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array
6: – function, supplied by the userExternal Function
On exit: flag 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, or may be complex. If flag is returned as nonzero then f01efc will terminate the computation, with NE_USER_STOP.
2: – IntegerInput
On entry: , the number of function values required.
3: – const doubleInput
On entry: the points at which the function is to be evaluated.
4: – doubleOutput
On exit: the function values.
should return the value , for .
5: – 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 f01efc you may allocate memory and initialize these pointers with various quantities for use by f when called from f01efc (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 f01efc. If your code inadvertently does return any NaNs or infinities, f01efc is likely to produce unexpected results.
7: – Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
8: – Integer *Output
On exit: , unless you have set flag nonzero inside f, in which case flag will be the value you set and fail will be set to NE_USER_STOP.
9: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
The value of fail gives the number of off-diagonal elements of an intermediate tridiagonal form that did not converge to zero (see f08fac).
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.
The computation of the spectral factorization failed to converge.
On entry, .
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.
An internal error occurred when computing the spectral factorization. Please contact NAG.
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.
Provided that can be computed accurately then the computed matrix function will be close to the exact matrix function. See Section 10.2 of Higham (2008) for details and further discussion.
8Parallelism and Performance
f01efc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f01efc 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.
The Integer allocatable memory required is n, and the double allocatable memory required is approximately , where nb is the block size required by f08fac.
The cost of the algorithm is plus the cost of evaluating . If is the th computed eigenvalue of , then the user-supplied function f will be asked to evaluate the function at , .