nag_matop_real_gen_matrix_fun_usd (f01emc) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_matop_real_gen_matrix_fun_usd (f01emc)

## 1  Purpose

nag_matop_real_gen_matrix_fun_usd (f01emc) computes the matrix function, $f\left(A\right)$, of a real $n$ by $n$ matrix $A$, using analytical derivatives of $f$ you have supplied.

## 2  Specification

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

## 3  Description

$f\left(A\right)$ is computed using the Schur–Parlett algorithm described in Higham (2008) and Davies and Higham (2003).
The scalar function $f$, and the derivatives of $f$, are returned by function f which, given an integer $m$, evaluates ${f}^{\left(m\right)}\left({z}_{\mathit{i}}\right)$ at a number of (generally complex) points ${z}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{n}_{z}$. For any $z$ on the real line, $f\left(z\right)$ must also be real. nag_matop_real_gen_matrix_fun_usd (f01emc) 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.

## 4  References

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

## 5  Arguments

1:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     a[${\mathbf{pda}}×{\mathbf{n}}$]doubleInput/Output
Note: the $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ matrix, $f\left(A\right)$.
4:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge {\mathbf{n}}$.
5:     ffunction, supplied by the userExternal Function
The specification of f is:
 void f (Integer m, Integer *iflag, Integer nz, const Complex z[], Complex fz[], Nag_Comm *comm)
1:     mIntegerInput
On entry: the order, $m$, of the derivative required.
If ${\mathbf{m}}=0$, $f\left({z}_{i}\right)$ should be returned. For ${\mathbf{m}}>0$, ${f}^{\left(m\right)}\left({z}_{i}\right)$ should be returned.
2:     iflagInteger *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 $f\left(z\right)$; for instance $f\left(z\right)$ may not be defined. If iflag is returned as nonzero then nag_matop_real_gen_matrix_fun_usd (f01emc) will terminate the computation, with NE_USER_STOP.
3:     nzIntegerInput
On entry: ${n}_{z}$, the number of function or derivative values required.
4:     z[nz]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.
5:     fz[nz]ComplexOutput
On exit: the ${n}_{z}$ function or derivative values. ${\mathbf{fz}}\left[\mathit{i}-1\right]$ should return the value ${f}^{\left(m\right)}\left({z}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{z}$. If ${z}_{i}$ lies on the real line, then so must ${f}^{\left(m\right)}\left({z}_{i}\right)$.
6:     commNag_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_usd (f01emc) you may allocate memory and initialize these pointers with various quantities for use by f when called from nag_matop_real_gen_matrix_fun_usd (f01emc) (see Section 3.2.1 in the Essential Introduction).
6:     commNag_Comm *Communication Structure
The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).
7:     iflagInteger *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 NE_USER_STOP.
8:     imnormdouble *Output
On exit: if $A$ has complex eigenvalues, nag_matop_real_gen_matrix_fun_usd (f01emc) 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 function has given a reliable answer.
If $A$ has real eigenvalues, nag_matop_real_gen_matrix_fun_usd (f01emc) uses real arithmetic and ${\mathbf{imnorm}}=0$.
9:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Allocation of memory failed. If $A$ has real eigenvalues then up to $6×{N}^{2}$ of double allocatable memory may be required. Otherwise up to $6×{N}^{2}$ of Complex allocatable memory may be required.
NE_BAD_PARAM
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CONVERGENCE
A Taylor series 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: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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.
An unexpected internal error occured when ordering the eigenvalues of $A$. Please contact NAG.
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.
The routine was unable to compute the Schur decomposition of $A$.
Note:  this failure should not occur and suggests that the function has been called incorrectly.
NE_USER_STOP
iflag has been set nonzero by the user.

## 7  Accuracy

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.

## 8  Further Comments

If $A$ has real eigenvalues then up to $6×{{\mathbf{n}}}^{2}$ of double allocatable memory may be required. If $A$ has complex eigenvalues then up to $6×{{\mathbf{n}}}^{2}$ 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 evaluating $f$ and its derivatives. 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_usd (f01emc).
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).
nag_matop_complex_gen_matrix_fun_usd (f01fmc) can be used to find the matrix function $f\left(A\right)$ for a complex matrix $A$.

## 9  Example

This example finds the ${e}^{2A}$ where
 $A= 1 0 -2 1 -1 2 0 1 2 0 1 0 1 0 -1 2 .$

### 9.1  Program Text

Program Text (f01emce.c)

### 9.2  Program Data

Program Data (f01emce.d)

### 9.3  Program Results

Program Results (f01emce.r)

nag_matop_real_gen_matrix_fun_usd (f01emc) (PDF version)
f01 Chapter Contents
f01 Chapter Introduction
NAG C Library Manual