# NAG Library Function Document

## 1Purpose

nag_matop_real_gen_matrix_fun_num (f01elc) computes the matrix function, $f\left(A\right)$, of a real $n$ by $n$ matrix $A$. Numerical differentiation is used to evaluate the derivatives of $f$ when they are required.

## 2Specification

 #include #include
void  nag_matop_real_gen_matrix_fun_num (Integer n, double a[], Integer pda,
 void (*f)(Integer *iflag, Integer nz, const Complex z[], Complex fz[], Nag_Comm *comm),
Nag_Comm *comm, Integer *iflag, double *imnorm, NagError *fail)

## 3Description

$f\left(A\right)$ is computed using the Schur–Parlett algorithm described in Higham (2008) and Davies and Higham (2003). The coefficients of the Taylor series used in the algorithm are evaluated using the numerical differentiation algorithm of Lyness and Moler (1967).
The scalar function $f$ is supplied via function f which evaluates $f\left({z}_{i}\right)$ at a number of points ${z}_{i}$.

## 4References

Davies P I and Higham N J (2003) A Schur–Parlett algorithm for computing matrix functions. SIAM J. Matrix Anal. Appl. 25(2) 464–485
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
Lyness J N and Moler C B (1967) Numerical differentiation of analytic functions SIAM J. Numer. Anal. 4(2) 202–210

## 5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:    $\mathbf{a}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array a must be at least ${\mathbf{pda}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ matrix, $f\left(A\right)$.
3:    $\mathbf{pda}$IntegerInput
On entry: the stride separating matrix row elements in the array a.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
4:    $\mathbf{f}$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 *iflag, Integer nz, const Complex z[], Complex fz[], Nag_Comm *comm)
1:    $\mathbf{iflag}$Integer *Input/Output
On entry: iflag will be zero.
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 $f\left({z}_{i}\right)$; for instance $f\left({z}_{i}\right)$ may not be defined. If iflag is returned as nonzero then nag_matop_real_gen_matrix_fun_num (f01elc) will terminate the computation, with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
2:    $\mathbf{nz}$IntegerInput
On entry: ${n}_{z}$, the number of function values required.
3:    $\mathbf{z}\left[{\mathbf{nz}}\right]$const ComplexInput
On entry: the ${n}_{z}$ points ${z}_{1},{z}_{2},\dots ,{z}_{{n}_{z}}$ at which the function $f$ is to be evaluated.
4:    $\mathbf{fz}\left[{\mathbf{nz}}\right]$ComplexOutput
On exit: the ${n}_{z}$ function values. ${\mathbf{fz}}\left[\mathit{i}-1\right]$ should return the value $f\left({z}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{z}$. If ${z}_{i}$ lies on the real line, then so must $f\left({z}_{i}\right)$.
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_real_gen_matrix_fun_num (f01elc) you may allocate memory and initialize these pointers with various quantities for use by f when called from nag_matop_real_gen_matrix_fun_num (f01elc) (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_gen_matrix_fun_num (f01elc). If your code inadvertently does return any NaNs or infinities, nag_matop_real_gen_matrix_fun_num (f01elc) is likely to produce unexpected results.
5:    $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
6:    $\mathbf{iflag}$Integer *Output
On exit: ${\mathbf{iflag}}=0$, unless iflag has been set nonzero inside f, in which case iflag will be the value set and fail will be set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
7:    $\mathbf{imnorm}$double *Output
On exit: if $A$ has complex eigenvalues, nag_matop_real_gen_matrix_fun_num (f01elc) will use complex arithmetic to compute $f\left(A\right)$. The imaginary part is discarded at the end of the computation, because it will theoretically vanish. imnorm contains the $1$-norm of the imaginary part, which should be used to check that the routine has given a reliable answer.
If a has real eigenvalues, nag_matop_real_gen_matrix_fun_num (f01elc) uses real arithmetic and ${\mathbf{imnorm}}=0$.
8:    $\mathbf{fail}$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
A Taylor series failed to converge after $40$ terms. Further Taylor series coefficients can no longer reliably be obtained by numerical differentiation.
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.
An unexpected internal error occurred when ordering the eigenvalues of $A$. Please contact NAG.
The function was unable to compute the Schur decomposition of $A$.
Note:  this failure should not occur and suggests that the function has been called incorrectly.
There was an error whilst reordering the Schur form of $A$.
Note:  this failure should not occur and suggests that the function has been called incorrectly.
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
iflag has been set nonzero by the user.

## 7Accuracy

For a normal matrix $A$ (for which ${A}^{\mathrm{T}}A=A{A}^{\mathrm{T}}$) the Schur decomposition is diagonal and the algorithm reduces to evaluating $f$ at the eigenvalues of $A$ and then constructing $f\left(A\right)$ using the Schur vectors. See Section 9.4 of Higham (2008) for further discussion of the Schur–Parlett algorithm, and Lyness and Moler (1967) for a discussion of numerical differentiation.

## 8Parallelism and Performance

nag_matop_real_gen_matrix_fun_num (f01elc) 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.
nag_matop_real_gen_matrix_fun_num (f01elc) 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 $n$. If $A$ has real eigenvalues then up to $6{n}^{2}$ of double allocatable memory may be required. If $A$ has complex eigenvalues then up to $6{n}^{2}$ of Complex allocatable memory may be required.
The cost of the Schur–Parlett algorithm depends on the spectrum of $A$, but is roughly between $28{n}^{3}$ and ${n}^{4}/3$ floating-point operations. There is an additional cost in numerically differentiating $f$, in order to obtain the Taylor series coefficients. If the derivatives of $f$ are known analytically, then nag_matop_real_gen_matrix_fun_usd (f01emc) can be used to evaluate $f\left(A\right)$ more accurately. If $A$ is real symmetric then it is recommended that nag_matop_real_symm_matrix_fun (f01efc) be used as it is more efficient and, in general, more accurate than nag_matop_real_gen_matrix_fun_num (f01elc).
For any $z$ on the real line, $f\left(z\right)$ must be real. $f$ must also be complex analytic on the spectrum of $A$. These conditions ensure that $f\left(A\right)$ is real for real $A$.
For further information on matrix functions, see Higham (2008).
If estimates of the condition number of the matrix function are required then nag_matop_real_gen_matrix_cond_num (f01jbc) should be used.
nag_matop_complex_gen_matrix_fun_num (f01flc) can be used to find the matrix function $f\left(A\right)$ for a complex matrix $A$.

## 10Example

This example finds $\mathrm{cos}2A$ where
 $A= 3 0 1 2 -1 1 3 1 0 2 2 1 2 1 -1 1 .$

### 10.1Program Text

Program Text (f01elce.c)

### 10.2Program Data

Program Data (f01elce.d)

### 10.3Program Results

Program Results (f01elce.r)

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