NAG Library Function Document

nag_matop_real_gen_matrix_fun_usd (f01emc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint: $\mathbf{order}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.

2:     n – IntegerInput

On entry: $n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

3:     a[$\mathit{dim}$] – doubleInput/Output

Note: the dimension, dim, of the array a must be at least $\mathbf{pda}\times \mathbf{n}$.

The $\left(i,j\right)$th element of the matrix $A$ is stored in

• $\mathbf{a}\left[\left(j-1\right)\times \mathbf{pda}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{a}\left[\left(i-1\right)\times \mathbf{pda}+j-1\right]$ 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:     pda – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraint: $\mathbf{pda}\ge \mathbf{n}$.

5:     f – function, supplied by the userExternal Function

Given an integer $m$, the function f evaluates $f^{\left(m\right)}\left(z_i\right)$ at a number of points $z_i$.

The specification of f is:

void  f (Integer m, Integer *iflag, Integer nz, const Complex z[], Complex fz[], Nag_Comm *comm)

1:     m – IntegerInput

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:     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 $f\left(z\right)$; for instance $f\left(z_i\right)$ may not be defined for a particular $z_i$. If iflag is returned as nonzero then nag_matop_real_gen_matrix_fun_usd (f01emc) will terminate the computation, with $\mathbf{fail}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.

3:     nz – IntegerInput

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:     comm – 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_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 3.2.1.1 in the Essential Introduction).

6:     comm – Nag_Comm *Communication Structure

The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).

7:     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.

8:     imnorm – double *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:     fail – NagError *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.

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: $\mathbf{pda}\ge \mathbf{n}$.

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.

An unexpected internal error occurred 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

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (f01emce.c)

10.2  Program Data

Program Data (f01emce.d)

10.3  Program Results

Program Results (f01emce.r)