NAG Library Function Document

1Purpose

nag_matop_complex_herm_matrix_fun (f01ffc) computes the matrix function, $f\left(A\right)$, of a complex Hermitian $n$ by $n$ matrix $A$. $f\left(A\right)$ must also be a complex Hermitian matrix.

2Specification

 #include #include
void  nag_matop_complex_herm_matrix_fun (Nag_OrderType order, Nag_UploType uplo, Integer n, Complex a[], Integer pda,
 void (*f)(Integer *flag, Integer n, const double x[], double fx[], Nag_Comm *comm),
Nag_Comm *comm, Integer *flag, NagError *fail)

3Description

$f\left(A\right)$ is computed using a spectral factorization of $A$
 $A = Q D QH ,$
where $D$ is the real diagonal matrix whose diagonal elements, ${d}_{i}$, are the eigenvalues of $A$, $Q$ is a unitary matrix whose columns are the eigenvectors of $A$. $f\left(A\right)$ is then given by
 $fA = Q fD QH ,$
where $f\left(D\right)$ is the diagonal matrix whose $i$th diagonal element is $f\left({d}_{i}\right)$. See for example Section 4.5 of Higham (2008). $f\left({d}_{i}\right)$ is assumed to be real.

4References

Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

5Arguments

1:    $\mathbf{order}$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. 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: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{uplo}$Nag_UploTypeInput
On entry: if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangle of the matrix $A$ is stored.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the lower triangle of the matrix $A$ is stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:    $\mathbf{a}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array a must be at least ${\mathbf{pda}}×{\mathbf{n}}$.
On entry: the $n$ by $n$ Hermitian matrix $A$.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${A}_{ij}$ is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${A}_{ij}$ is stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$.
If ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the upper or lower triangular part of the $n$ by $n$ matrix function, $f\left(A\right)$.
5:    $\mathbf{pda}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6:    $\mathbf{f}$function, supplied by the userExternal Function
The function f evaluates $f\left({z}_{i}\right)$ at a number of points ${z}_{i}$.
The specification of f is:
 void f (Integer *flag, Integer n, const double x[], double fx[], Nag_Comm *comm)
1:    $\mathbf{flag}$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 $f\left(x\right)$; for instance $f\left(x\right)$ may not be defined, or may be complex. If flag is returned as nonzero then nag_matop_complex_herm_matrix_fun (f01ffc) will terminate the computation, with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
2:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of function values required.
3:    $\mathbf{x}\left[{\mathbf{n}}\right]$const doubleInput
On entry: the $n$ points ${x}_{1},{x}_{2},\dots ,{x}_{n}$ at which the function $f$ is to be evaluated.
4:    $\mathbf{fx}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the $n$ function values. ${\mathbf{fx}}\left[\mathit{i}-1\right]$ should return the value $f\left({x}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,n$.
5:    $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to f.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_matop_complex_herm_matrix_fun (f01ffc) you may allocate memory and initialize these pointers with various quantities for use by f when called from nag_matop_complex_herm_matrix_fun (f01ffc) (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_complex_herm_matrix_fun (f01ffc). If your code inadvertently does return any NaNs or infinities, nag_matop_complex_herm_matrix_fun (f01ffc) is likely to produce unexpected results.
7:    $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
8:    $\mathbf{flag}$Integer *Output
On exit: ${\mathbf{flag}}=0$, unless you have set flag nonzero inside f, in which case flag will be the value you set and fail will be set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
9:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error 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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CONVERGENCE
The computation of the spectral factorization failed to converge.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
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.
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.

7Accuracy

Provided that $f\left(D\right)$ 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

nag_matop_complex_herm_matrix_fun (f01ffc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_complex_herm_matrix_fun (f01ffc) 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, the double allocatable memory required is $4×{\mathbf{n}}-2$ and the Complex allocatable memory required is approximately $\left({\mathbf{n}}+\mathit{nb}+1\right)×{\mathbf{n}}$, where nb is the block size required by nag_zheev (f08fnc).
The cost of the algorithm is $O\left({n}^{3}\right)$ plus the cost of evaluating $f\left(D\right)$. If ${\stackrel{^}{\lambda }}_{\mathit{i}}$ is the $\mathit{i}$th computed eigenvalue of $A$, then the user-supplied function f will be asked to evaluate the function $f$ at $f\left({\stackrel{^}{\lambda }}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,n$.
For further information on matrix functions, see Higham (2008).
nag_matop_real_symm_matrix_fun (f01efc) can be used to find the matrix function $f\left(A\right)$ for a real symmetric matrix $A$.

10Example

This example finds the matrix cosine, $\mathrm{cos}\left(A\right)$, of the Hermitian matrix
 $A= 1 2+i 3+2i 4+3i 2-i 1 2+i 3+2i 3-2i 2-i 1 2+i 4-3i 3-2i 2-i 1 .$

10.1Program Text

Program Text (f01ffce.c)

10.2Program Data

Program Data (f01ffce.d)

10.3Program Results

Program Results (f01ffce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017