f01kgc computes an estimate of the relative condition number ${\kappa}_{\mathrm{exp}}\left(A\right)$ of the exponential of a complex $n\times n$ matrix $A$, in the $1$-norm. The matrix exponential ${e}^{A}$ is also returned.
where $\Vert L\left(A\right)\Vert $ is the norm of the Fréchet derivative of the matrix exponential at $A$.
To obtain the estimate of ${\kappa}_{\mathrm{exp}}\left(A\right)$, f01kgc first estimates $\Vert L\left(A\right)\Vert $ by computing an estimate $\gamma $ of a quantity $K\in [{n}^{-1}{\Vert L\left(A\right)\Vert}_{1},n{\Vert L\left(A\right)\Vert}_{1}]$, such that $\gamma \le K$.
The matrix exponential ${e}^{A}$ is computed using a Padé approximant and the scaling and squaring method. The Padé approximant is differentiated to obtain the Fréchet derivatives $L(A,E)$ which are used to estimate the condition number.
4References
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
Note: the dimension, dim, of the array a
must be at least
${\mathbf{pda}}\times {\mathbf{n}}$.
The $(i,j)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[(j-1)\times {\mathbf{pda}}+i-1\right]$.
On entry: the $n\times n$ matrix $A$.
On exit: the $n\times n$ matrix exponential ${e}^{A}$.
3: $\mathbf{pda}$ – IntegerInput
On entry: the stride separating matrix row elements in the array a.
Constraint:
${\mathbf{pda}}\ge {\mathbf{n}}$.
4: $\mathbf{condea}$ – double *Output
On exit: an estimate of the relative condition number of the matrix exponential ${\kappa}_{\mathrm{exp}}\left(A\right)$.
5: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{pda}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR
The linear equations to be solved for the Padé approximant are singular; it is likely that this function has been called incorrectly.
NW_SOME_PRECISION_LOSS
${e}^{A}$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
7Accuracy
f01kgc uses the norm estimation function f04zdc to produce an estimate $\gamma $ of a quantity $K\in [{n}^{-1}{\Vert L\left(A\right)\Vert}_{1},n{\Vert L\left(A\right)\Vert}_{1}]$, such that $\gamma \le K$. For further details on the accuracy of norm estimation, see the documentation for f04zdc.
For a normal matrix $A$ (for which ${A}^{\mathrm{H}}A=A{A}^{\mathrm{H}}$) the computed matrix, ${e}^{A}$, 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).
8Parallelism and Performance
f01kgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 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.
9Further Comments
f01kac uses a similar algorithm to f01kgc to compute an estimate of the absolute condition number (which is related to the relative condition number by a factor of $\Vert A\Vert /\Vert \mathrm{exp}\left(A\right)\Vert $). However, the required Fréchet derivatives are computed in a more efficient and stable manner by f01kgc and so its use is recommended over f01kac.
The cost of the algorithm is $O\left({n}^{3}\right)$ and the complex allocatable memory required is approximately $15{n}^{2}$; 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 f01fcc should be used. If the Fréchet derivative of the matrix exponential is required then f01khc should be used.