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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_matop_complex_gen_matrix_actexp_rcomm (f01hb)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_matop_complex_gen_matrix_actexp_rcomm (f01hb) computes the action of the matrix exponential etA, on the matrix B, where A is a complex n by n matrix, B is a complex n by m matrix and t is a complex scalar. It uses reverse communication for evaluating matrix products, so that the matrix A is not accessed explicitly.

Syntax

[irevcm, b, b2, x, y, p, r, z, ccomm, comm, icomm, ifail] = f01hb(irevcm, m, b, t, b2, x, y, p, r, z, ccomm, comm, icomm, 'n', n, 'tr', tr)
[irevcm, b, b2, x, y, p, r, z, ccomm, comm, icomm, ifail] = nag_matop_complex_gen_matrix_actexp_rcomm(irevcm, m, b, t, b2, x, y, p, r, z, ccomm, comm, icomm, 'n', n, 'tr', tr)
Note: the interface to this routine has changed since earlier releases of the toolbox:
At Mark 25: m was made optional

Description

etAB is computed using the algorithm described in Al–Mohy and Higham (2011) which uses a truncated Taylor series to compute the etAB without explicitly forming etA.
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 AHY. A reverse communication interface is used, in which control is returned to the calling program whenever a matrix product is required.

References

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

Parameters

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.

Compulsory Input Parameters

1:     irevcm int64int32nag_int scalar
On initial entry: must be set to 0.
2:     m int64int32nag_int scalar
The number of columns of the matrix B.
Constraint: m0.
3:     bldb: – complex array
The first dimension of the array b must be at least n.
The second dimension of the array b must be at least m.
On initial entry: the n by m matrix B.
On intermediate re-entry: must not be changed.
4:     t – complex scalar
The scalar t.
5:     b2ldb2: – complex array
The first dimension of the array b2 must be at least n.
The second dimension of the array b2 must be at least m.
On initial entry: need not be set.
On intermediate re-entry: if irevcm=1, must contain AB.
6:     xldx: – complex array
The first dimension of the array x must be at least n.
The second dimension of the array x must be at least 2.
On initial entry: need not be set.
On intermediate re-entry: if irevcm=3, must contain AHY.
7:     yldy: – complex array
The first dimension of the array y must be at least n.
The second dimension of the array y must be at least 2.
On initial entry: need not be set.
On intermediate re-entry: if irevcm=2, must contain AX.
8:     pn – complex array
On initial entry: need not be set.
On intermediate re-entry: if irevcm=4, must contain Az.
9:     rn – complex array
On initial entry: need not be set.
On intermediate re-entry: if irevcm=5, must contain AHz.
10:   zn – complex array
On initial entry: need not be set.
On intermediate re-entry: must not be changed.
11:   ccommn×m+2 – complex array
12:   comm3×n+14 – double array
13:   icomm2×n+40 int64int32nag_int array

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the arrays b, b2, x, y, ccomm and the dimension of the arrays p, r, z. (An error is raised if these dimensions are not equal.)
n, the order of the matrix A.
Constraint: n0.
2:     tr – complex scalar
Default: 0
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 etrtB is immediately returned in the first row of B. See Further Comments.

Output Parameters

1:     irevcm int64int32nag_int scalar
On intermediate exit: irevcm=1, 2, 3, 4 or 5. The calling program must:
(a) if irevcm=1: evaluate B2=AB, where B2 is an n by m matrix, and store the result in b2;
if irevcm=2: evaluate Y=AX, where X and Y are n by 2 matrices, and store the result in y;
if irevcm=3: evaluate X=AHY and store the result in x;
if irevcm=4: evaluate p=Az and store the result in p;
if irevcm=5: evaluate r=AHz and store the result in r.
(b) call nag_matop_complex_gen_matrix_actexp_rcomm (f01hb) again with all other parameters unchanged.
On final exit: irevcm=0.
2:     bldb: – complex array
The first dimension of the array b will be n.
The second dimension of the array b will be m.
On intermediate exit: if irevcm=1, contains the n by m matrix B.
On final exit: the n by m matrix etAB.
3:     b2ldb2: – complex array
The first dimension of the array b2 will be n.
The second dimension of the array b2 will be m.
On final exit: the array is undefined.
4:     xldx: – complex array
The first dimension of the array x will be n.
The second dimension of the array x will be 2.
On intermediate exit: if irevcm=2, contains the current n by 2 matrix X.
On final exit: the array is undefined.
5:     yldy: – complex array
The first dimension of the array y will be n.
The second dimension of the array y will be 2.
On intermediate exit: if irevcm=3, contains the current n by 2 matrix Y.
On final exit: the array is undefined.
6:     pn – complex array
On final exit: the array is undefined.
7:     rn – complex array
On final exit: the array is undefined.
8:     zn – complex array
On intermediate exit: if irevcm=4 or 5, contains the vector z.
On final exit: the array is undefined.
9:     ccommn×m+2 – complex array
10:   comm3×n+14 – double array
11:   icomm2×n+40 int64int32nag_int array
12:   ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W  ifail=2
etAB has been computed using an IEEE double precision Taylor series, although the arithmetic precision is higher than IEEE double precision.
   ifail=-1
On initial entry, irevcm=_.
Constraint: irevcm=0.
On intermediate re-entry, irevcm=_.
Constraint: irevcm=1, 2, 3, 4 or 5.
   ifail=-2
On initial entry, n=_.
Constraint: n0.
   ifail=-3
On initial entry, m=_.
Constraint: m0.
   ifail=-5
On initial entry, ldb=_ and n=_.
Constraint: ldbn.
   ifail=-9
On initial entry, ldb2=_ and n=_.
Constraint: ldb2n.
   ifail=-11
On initial entry, ldx=_ and n=_.
Constraint: ldxn.
   ifail=-13
On initial entry, ldy=_ and n=_.
Constraint: ldyn.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

For an Hermitian matrix A (for which AH=A) the computed matrix etAB is guaranteed to be close to the exact matrix, that is, the method is forward stable. No such guarantee can be given for non-Hermitian matrices. See Section 4 of Al–Mohy and Higham (2011) for details and further discussion.

Further Comments

Use of TrA

The elements of A are not explicitly required by nag_matop_complex_gen_matrix_actexp_rcomm (f01hb). However, the trace of A is used in the preprocessing phase of the algorithm. If TrA 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.

When to use nag_matop_complex_gen_matrix_actexp_rcomm (f01hb)

nag_matop_complex_gen_matrix_actexp_rcomm (f01hb) 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 etAB will not, in general, be sparse even if A is sparse.
If A is small and dense then nag_matop_complex_gen_matrix_actexp (f01ha) can be used to compute etAB without the use of a reverse communication interface.
The real analog of nag_matop_complex_gen_matrix_actexp_rcomm (f01hb) is nag_matop_real_gen_matrix_actexp_rcomm (f01gb).

Use in Conjunction with NAG Toolbox Functions

To compute etAB, the following skeleton code can normally be used:
while irevcm ~= 0
  switch irevcm
    case {1}
      % Compute ab and store in b2
    case {2}
      % Compute ax and store in y
    case {3}
      % Compute a^Hy and store in x
    case {4}
      % compute az and store in p
    case {5}
      % compute a^Hz and store in r
  end

Example

This example computes etAB where
A = 0.7+0.8i -0.2+0.0i 1.0+0.0i 0.6+0.5i 0.3+0.7i 0.7+0.0i 0.9+3.0i 1.0+0.8i 0.3+3.0i -0.7+0.0i 0.2+0.6i 0.7+0.5i 0.0+0.9i 4.0+0.0i 0.0+0.0i 0.2+0.0i ,  
B = 0.1+0.0i 1.2+0.1i 1.3+0.9i -0.2+2.0i 4.0+0.6i -1.0+0.8i 0.4+0.0i -0.9+0.0i  
and
t=1.1+0.0i .  
A is stored in compressed column storage format (CCS) and matrix multiplications are performed using the function matmul.
function f01hb_example


fprintf('f01hb example results\n\n');

a = [0.7+0.8i, -0.2+0.0i, 1.0+0.0i, 0.6+0.5i;
     0.3+0.7i,  0.7+0.0i, 0.9+3.0i, 1.0+0.8i;
     0.3+3.0i, -7.0+0.0i, 0.2+0.6i, 0.7+0.5i;
     0.0+0.9i,  4.0+0.0i, 0.0+0.0i, 0.2+0.0i];
b = [0.1+0.0i,  1.2+0.1i;
     1.3+0.9i, -0.2+2.0i;
     4.0+0.6i, -1.0+0.8i;
     0.4+0.0i, -0.9+0.0i];
n = 4;
m = 2;

t = complex(1.1);
b2 = b; x = b; y = b;
p = complex(zeros(n, 1)); r = p; z = p;
comm =  zeros(3*n+14, 1);
icomm = zeros(2*n+40, 1, 'int64');
ccomm = complex(zeros(n*(m+2), 1));
tr = trace(a);

% Compute exp(ta)b by reverse communication
irevcm = int64(0);
done = false;

while ~done
  [irevcm, b, b2, x, y, p, r, z, ccomm, comm, icomm, ifail] = ...
  f01hb( ...
         irevcm, b, t, b2, x, y, p, r, z, ccomm, comm, icomm, 'tr', tr);
  switch irevcm
    case {0}
      done = true;
    case {1}
      % Compute ab and store in b2
      b2 = a*b;
    case {2}
      % Compute ax and store in y
      y = a*x;
    case {3}
      % Compute a^Hy and store in x
      x = a'*y;
    case {4}
      % compute az and store in p
      p = a*z;
    case {5}
      % compute a^Hz and store in r
      r = a'*z;
  end
end

fprintf('\n exp(tA)B\n');
disp(b);


f01hb example results


 exp(tA)B
 -15.3125 + 5.9123i  -4.5605 - 2.4288i
  12.3396 -50.6993i   9.2005 -10.3632i
 -65.4353 +34.3271i -17.6075 - 1.0019i
  45.6506 -28.3253i  11.3339 + 0.1127i


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