NAG Library Function Document
nag_real_gen_matrix_exp (f01ecc) computes the matrix exponential, , of a real by matrix .
||nag_real_gen_matrix_exp (Nag_OrderType order,
is computed using a Padé approximant and the scaling and squaring method described in Al–Mohy and Higham (2009)
Al–Mohy A H and Higham N J (2009) A new scaling and squaring algorithm for the matrix exponential SIAM J. Matrix Anal. 31(3) 970–989
Higham N J (2005) The scaling and squaring method for the matrix exponential revisited SIAM J. Matrix Anal. Appl. 26(4) 1179–1193
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA
Moler C B and Van Loan C F (2003) Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later SIAM Rev. 45 3–49
order – Nag_OrderTypeInput
: 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
. See Section 188.8.131.52
in the Essential Introduction for a more detailed explanation of the use of this argument.
n – IntegerInput
On entry: , the order of the matrix .
a – doubleInput/Output
the dimension, dim
, of the array a
must be at least
th element of the matrix
is stored in
- when ;
- when .
On entry: the by matrix .
On exit: the by matrix exponential .
pda – IntegerInput
: the stride separating row or column elements (depending on the value of order
) in the array a
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, argument had an illegal value.
On entry, .
On entry, and .
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
An unexpected internal error has occurred. Please contact NAG
The linear equations to be solved are nearly singular and the Padé approximant probably has no correct figures; it is likely that this function has been called incorrectly.
The linear equations to be solved for the Padé approximant are singular; it is likely that this function has been called incorrectly.
has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
For a normal matrix
) the computed matrix,
, is guaranteed to be close to the exact matrix, that is, the method is forward stable. No such guarantee can be given for non-normal matrices. See Al–Mohy and Higham (2009)
and Section 10.3 of Higham (2008)
for details and further discussion.
If estimates of the condition number of the matrix exponential are required then nag_matop_real_gen_matrix_cond_exp (f01jgc)
should be used.
8 Parallelism and Performance
nag_real_gen_matrix_exp (f01ecc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_real_gen_matrix_exp (f01ecc) 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 Users' Note
for your implementation for any additional implementation-specific information.
The cost of the algorithm is
; see Section 5 of of Al–Mohy and Higham (2009)
. The real allocatable memory required is approximately
If the Fréchet derivative of the matrix exponential is required then nag_matop_real_gen_matrix_frcht_exp (f01jhc)
should be used.
As well as the excellent book cited above, the classic reference for the computation of the matrix exponential is Moler and Van Loan (2003)
This example finds the matrix exponential of the matrix
10.1 Program Text
Program Text (f01ecce.c)
10.2 Program Data
Program Data (f01ecce.d)
10.3 Program Results
Program Results (f01ecce.r)