NAG Library Function Document
nag_matop_complex_gen_matrix_cond_exp (f01kgc) computes an estimate of the relative condition number of the exponential of a complex by matrix , in the -norm. The matrix exponential is also returned.
||nag_matop_complex_gen_matrix_cond_exp (Integer n,
The Fréchet derivative of the matrix exponential of
is the unique linear mapping
such that for any matrix
The derivative describes the first-order effect of perturbations in on the exponential .
The relative condition number of the matrix exponential can be defined by
is the norm of the Fréchet derivative of the matrix exponential at
To obtain the estimate of , nag_matop_complex_gen_matrix_cond_exp (f01kgc) first estimates by computing an estimate of a quantity , such that .
The algorithms used to compute
are detailed in the Al–Mohy and Higham (2009a)
and Al–Mohy and Higham (2009b)
The matrix exponential is computed using a Padé approximant and the scaling and squaring method. The Padé approximant is differentiated to obtain the Fréchet derivatives which are used to estimate the condition number.
Al–Mohy A H and Higham N J (2009a) A new scaling and squaring algorithm for the matrix exponential SIAM J. Matrix Anal. 31(3) 970–989
Al–Mohy A H and Higham N J (2009b) Computing the Fréchet derivative of the matrix exponential, with an application to condition number estimation SIAM J. Matrix Anal. Appl. 30(4) 1639–1657
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
n – IntegerInput
On entry: , the order of the matrix .
a – ComplexInput/Output
the dimension, dim
, of the array a
must be at least
The th element of the matrix is stored in .
On entry: the by matrix .
On exit: the by matrix exponential .
pda – IntegerInput
: the stride separating matrix row elements in the array a
condea – double *Output
On exit: an estimate of the relative condition number of the matrix exponential .
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
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.
nag_matop_complex_gen_matrix_cond_exp (f01kgc) uses the norm estimation function nag_linsys_complex_gen_norm_rcomm (f04zdc)
to produce an estimate
of a quantity
, such that
. For further details on the accuracy of norm estimation, see the documentation for nag_linsys_complex_gen_norm_rcomm (f04zdc)
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 Section 10.3 of Higham (2008)
for details and further discussion.
For further discussion of the condition of the matrix exponential see Section 10.2 of Higham (2008)
8 Parallelism and Performance
nag_matop_complex_gen_matrix_cond_exp (f01kgc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_complex_gen_matrix_cond_exp (f01kgc) 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.
uses a similar algorithm to nag_matop_complex_gen_matrix_cond_exp (f01kgc) to compute an estimate of the absolute
condition number (which is related to the relative condition number by a factor of
). However, the required Fréchet derivatives are computed in a more efficient and stable manner by nag_matop_complex_gen_matrix_cond_exp (f01kgc) and so its use is recommended over nag_matop_complex_gen_matrix_cond_std (f01kac)
The cost of the algorithm is
and the complex allocatable memory required is approximately
; see Al–Mohy and Higham (2009a)
and Al–Mohy and Higham (2009b)
for further details.
If the matrix exponential alone is required, without an estimate of the condition number, then nag_matop_complex_gen_matrix_exp (f01fcc)
should be used. If the Fréchet derivative of the matrix exponential is required then nag_matop_complex_gen_matrix_frcht_exp (f01khc)
should be used.
As well as the excellent book Higham (2008)
, the classic reference for the computation of the matrix exponential is Moler and Van Loan (2003)
This example estimates the relative condition number of the matrix exponential
10.1 Program Text
Program Text (f01kgce.c)
10.2 Program Data
Program Data (f01kgce.d)
10.3 Program Results
Program Results (f01kgce.r)