NAG Library Function Document
nag_matop_real_symm_matrix_fun (f01efc)
1
Purpose
nag_matop_real_symm_matrix_fun (f01efc) computes the matrix function, , of a real symmetric by matrix . must also be a real symmetric matrix.
2
Specification
#include <nag.h> |
#include <nagf01.h> |
void |
nag_matop_real_symm_matrix_fun (Nag_OrderType order,
Nag_UploType uplo,
Integer n,
double a[],
Integer pda,
void |
(*f)(Integer *flag,
Integer n,
const double x[],
double fx[],
Nag_Comm *comm),
|
|
Nag_Comm *comm, Integer *flag,
NagError *fail) |
|
3
Description
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.
4
References
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
5
Arguments
- 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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_UploTypeInput
-
On entry: if
, the upper triangle of the matrix
is stored.
If , the lower triangle of the matrix is stored.
Constraint:
or .
- 3:
– IntegerInput
-
On entry: , the order of the matrix .
Constraint:
.
- 4:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
a
must be at least
.
On entry: the
by
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 by 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
a.
Constraint:
.
- 6:
– function, supplied by the userExternal Function
-
The function
f evaluates
at a number of points
.
The specification of
f is:
void |
f (Integer *flag,
Integer n,
const double x[],
double fx[],
Nag_Comm *comm)
|
|
- 1:
– Integer *Input/Output
-
On entry:
flag will be zero.
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
nag_matop_real_symm_matrix_fun (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
nag_matop_real_symm_matrix_fun (f01efc) you may allocate memory and initialize these pointers with various quantities for use by
f when called from
nag_matop_real_symm_matrix_fun (f01efc) (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
nag_matop_real_symm_matrix_fun (f01efc). If your code inadvertently
does return any NaNs or infinities,
nag_matop_real_symm_matrix_fun (f01efc) is likely to produce unexpected results.
- 7:
– Nag_Comm *
-
The NAG communication argument (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
- 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 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_CONVERGENCE
-
The computation of the spectral factorization failed to converge.
- 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
An internal error occurred when computing the spectral factorization. Please contact
NAG.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_USER_STOP
-
flag was set to a nonzero value in
f.
7
Accuracy
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.
8
Parallelism and Performance
nag_matop_real_symm_matrix_fun (f01efc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_real_symm_matrix_fun (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
nag_dsyev (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
,
.
For further information on matrix functions, see
Higham (2008).
nag_matop_complex_herm_matrix_fun (f01ffc) can be used to find the matrix function
for a complex Hermitian matrix
.
10
Example
This example finds the matrix cosine,
, of the symmetric matrix
10.1
Program Text
Program Text (f01efce.c)
10.2
Program Data
Program Data (f01efce.d)
10.3
Program Results
Program Results (f01efce.r)