# NAG Library Function Document

## 1Purpose

nag_matop_real_gen_matrix_actexp_rcomm (f01gbc) computes the action of the matrix exponential ${e}^{tA}$, on the matrix $B$, where $A$ is a real $n$ by $n$ matrix, $B$ is a real $n$ by $m$ matrix and $t$ is a real scalar. It uses reverse communication for evaluating matrix products, so that the matrix $A$ is not accessed explicitly.

## 2Specification

 #include #include
 void nag_matop_real_gen_matrix_actexp_rcomm (Integer *irevcm, Integer n, Integer m, double b[], Integer pdb, double t, double tr, double b2[], Integer pdb2, double x[], Integer pdx, double y[], Integer pdy, double p[], double r[], double z[], double comm[], Integer icomm[], NagError *fail)

## 3Description

${e}^{tA}B$ is computed using the algorithm described in Al–Mohy and Higham (2011) which uses a truncated Taylor series to compute the ${e}^{tA}B$ without explicitly forming ${e}^{tA}$.
The algorithm does not explicity need to access the elements of $A$; it only requires the result of matrix multiplications of the form $AX$ or ${A}^{\mathrm{T}}Y$. A reverse communication interface is used, in which control is returned to the calling program whenever a matrix product is required.
Al–Mohy A H and Higham N J (2011) Computing the action of the matrix exponential, with an application to exponential integrators SIAM J. Sci. Statist. Comput. 33(2) 488-511
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

## 5Arguments

Note: this function uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument irevcm. Between intermediate exits and re-entries, all arguments other than b2, x, y, p and r must remain unchanged.
1:    $\mathbf{irevcm}$Integer *Input/Output
On initial entry: must be set to $0$.
On intermediate exit: ${\mathbf{irevcm}}=1$, $2$, $3$, $4$ or $5$. The calling program must:
 (a) if ${\mathbf{irevcm}}=1$: evaluate ${B}_{2}=AB$, where ${B}_{2}$ is an $n$ by $m$ matrix, and store the result in b2; if ${\mathbf{irevcm}}=2$: evaluate $Y=AX$, where $X$ and $Y$ are $n$ by $2$ matrices, and store the result in y; if ${\mathbf{irevcm}}=3$: evaluate $X={A}^{\mathrm{T}}Y$ and store the result in x; if ${\mathbf{irevcm}}=4$: evaluate $p=Az$ and store the result in p; if ${\mathbf{irevcm}}=5$: evaluate $r={A}^{\mathrm{T}}z$ and store the result in r. (b) call nag_matop_real_gen_matrix_actexp_rcomm (f01gbc) again with all other parameters unchanged.
On final exit: ${\mathbf{irevcm}}=0$.
Note: any values you return to nag_matop_real_gen_matrix_actexp_rcomm (f01gbc) as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_matop_real_gen_matrix_actexp_rcomm (f01gbc). If your code inadvertently does return any NaNs or infinities, nag_matop_real_gen_matrix_actexp_rcomm (f01gbc) is likely to produce unexpected results.
2:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:    $\mathbf{m}$IntegerInput
On entry: the number of columns of the matrix $B$.
Constraint: ${\mathbf{m}}\ge 0$.
4:    $\mathbf{b}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array b must be at least ${\mathbf{pdb}}×{\mathbf{m}}$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$.
On initial entry: the $n$ by $m$ matrix $B$.
On intermediate exit: if ${\mathbf{irevcm}}=1$, contains the $n$ by $m$ matrix $B$.
On intermediate re-entry: must not be changed.
On final exit: the $n$ by $m$ matrix ${e}^{tA}B$.
5:    $\mathbf{pdb}$IntegerInput
On entry: the stride separating matrix row elements in the array b.
Constraint: ${\mathbf{pdb}}\ge {\mathbf{n}}$.
6:    $\mathbf{t}$doubleInput
On entry: the scalar $t$.
7:    $\mathbf{tr}$doubleInput
On entry: the trace of $A$. If this is not available then any number can be supplied ($0$ is a reasonable default); however, in the trivial case, $n=1$, the result ${e}^{{\mathbf{tr}}t}B$ is immediately returned in the first row of $B$. See Section 9.
8:    $\mathbf{b2}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array b2 must be at least ${\mathbf{pdb2}}×{\mathbf{m}}$.
The $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{b2}}\left[\left(j-1\right)×{\mathbf{pdb2}}+i-1\right]$.
On initial entry: need not be set.
On intermediate re-entry: if ${\mathbf{irevcm}}=1$, must contain $AB$.
On final exit: the array is undefined.
9:    $\mathbf{pdb2}$IntegerInput
On entry: the stride separating matrix row elements in the array b2.
Constraint: ${\mathbf{pdb2}}\ge {\mathbf{n}}$.
10:  $\mathbf{x}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array x must be at least ${\mathbf{pdx}}×2$.
The $\left(i,j\right)$th element of the matrix $X$ is stored in ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$.
On initial entry: need not be set.
On intermediate exit: if ${\mathbf{irevcm}}=2$, contains the current $n$ by $2$ matrix $X$.
On intermediate re-entry: if ${\mathbf{irevcm}}=3$, must contain ${A}^{\mathrm{T}}Y$.
On final exit: the array is undefined.
11:  $\mathbf{pdx}$IntegerInput
On entry: the stride separating matrix row elements in the array x.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
12:  $\mathbf{y}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array y must be at least ${\mathbf{pdy}}×2$.
The $\left(i,j\right)$th element of the matrix $Y$ is stored in ${\mathbf{y}}\left[\left(j-1\right)×{\mathbf{pdy}}+i-1\right]$.
On initial entry: need not be set.
On intermediate exit: if ${\mathbf{irevcm}}=3$, contains the current $n$ by $2$ matrix $Y$.
On intermediate re-entry: if ${\mathbf{irevcm}}=2$, must contain $AX$.
On final exit: the array is undefined.
13:  $\mathbf{pdy}$IntegerInput
On entry: the stride separating matrix row elements in the array y.
Constraint: ${\mathbf{pdy}}\ge {\mathbf{n}}$.
14:  $\mathbf{p}\left[{\mathbf{n}}\right]$doubleInput/Output
On initial entry: need not be set.
On intermediate re-entry: if ${\mathbf{irevcm}}=4$, must contain $Az$.
On final exit: the array is undefined.
15:  $\mathbf{r}\left[{\mathbf{n}}\right]$doubleInput/Output
On initial entry: need not be set.
On intermediate re-entry: if ${\mathbf{irevcm}}=5$, must contain ${A}^{\mathrm{T}}z$.
On final exit: the array is undefined.
16:  $\mathbf{z}\left[{\mathbf{n}}\right]$doubleInput/Output
On initial entry: need not be set.
On intermediate exit: if ${\mathbf{irevcm}}=4$ or $5$, contains the vector $z$.
On intermediate re-entry: must not be changed.
On final exit: the array is undefined.
17:  $\mathbf{comm}\left[{\mathbf{n}}×{\mathbf{m}}+3×{\mathbf{n}}+12\right]$doubleCommunication Array
18:  $\mathbf{icomm}\left[2×{\mathbf{n}}+40\right]$IntegerCommunication Array
19:  $\mathbf{fail}$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_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On initial entry, ${\mathbf{irevcm}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{irevcm}}=0$.
On intermediate re-entry, ${\mathbf{irevcm}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{irevcm}}=1$, $2$, $3$, $4$ or $5$.
NE_INT_2
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdb2}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb2}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdy}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdy}}\ge {\mathbf{n}}$.
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.
NW_SOME_PRECISION_LOSS
${e}^{tA}B$ has been computed using an IEEE double precision Taylor series, although the arithmetic precision is higher than IEEE double precision.

## 7Accuracy

For a symmetric matrix $A$ (for which ${A}^{\mathrm{T}}=A$) the computed matrix ${e}^{tA}B$ is guaranteed to be close to the exact matrix, that is, the method is forward stable. No such guarantee can be given for non-symmetric matrices. See Section 4 of Al–Mohy and Higham (2011) for details and further discussion.

## 8Parallelism and Performance

nag_matop_real_gen_matrix_actexp_rcomm (f01gbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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.

### 9.1Use of $Tr\left(A\right)$

The elements of $A$ are not explicitly required by nag_matop_real_gen_matrix_actexp_rcomm (f01gbc). However, the trace of $A$ is used in the preprocessing phase of the algorithm. If $Tr\left(A\right)$ is not available to the calling function then any number can be supplied ($0$ is recommended). This will not affect the stability of the algorithm, but it may reduce its efficiency.

### 9.2When to use nag_matop_real_gen_matrix_actexp_rcomm (f01gbc)

nag_matop_real_gen_matrix_actexp_rcomm (f01gbc) is designed to be used when $A$ is large and sparse. Whenever a matrix multiplication is required, the function will return control to the calling program so that the multiplication can be done in the most efficient way possible. Note that ${e}^{tA}B$ will not, in general, be sparse even if $A$ is sparse.
If $A$ is small and dense then nag_matop_real_gen_matrix_actexp (f01gac) can be used to compute ${e}^{tA}B$ without the use of a reverse communication interface.
The complex analog of nag_matop_real_gen_matrix_actexp_rcomm (f01gbc) is nag_matop_complex_gen_matrix_actexp_rcomm (f01hbc).

### 9.3Use in Conjunction with NAG C Library Functions

To compute ${e}^{tA}B$, the following skeleton code can normally be used:
```do {
f01gbc(&irevcm,n,m,b,tdb,t,tr,b2,tdb2,x,tdx,y,tdy,p,r,z,comm,icomm,&fail);
if (irevcm == 1) {
.. Code to compute B2=AB ..
}
else if (irevcm == 2){
.. Code to compute Y=AX ..
}
else if (irevcm == 3){
.. Code to compute X=A^T Y ..
}
else if (irevcm == 4){
.. Code to compute P=AZ ..
}
else if (irevcm == 5){
.. Code to compute R=A^T Z ..
}
} (while irevcm !=0)```
The code used to compute the matrix products will vary depending on the way $A$ is stored. If all the elements of $A$ are stored explicitly, then nag_dgemm (f16yac)) can be used. If $A$ is triangular then nag_dtrmm (f16yfc) should be used. If $A$ is symmetric, then nag_dsymm (f16ycc) should be used. For sparse $A$ stored in coordinate storage format nag_sparse_nsym_matvec (f11xac) and nag_sparse_sym_matvec (f11xec) can be used. Alternatively if $A$ is stored in compressed column format nag_superlu_matrix_product (f11mkc) can be used.

## 10Example

This example computes ${e}^{tA}B$, where
 $A = 0.4 -0.2 1.3 0.6 0.3 0.8 1.0 1.0 3.0 4.8 0.2 0.7 0.5 0.0 -5.0 0.7 ,$
 $B = 0.1 1.1 1.7 -0.2 0.5 1.0 0.4 -0.2 ,$
and
 $t=-0.2 .$

### 10.1Program Text

Program Text (f01gbce.c)

### 10.2Program Data

Program Data (f01gbce.d)

### 10.3Program Results

Program Results (f01gbce.r)

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