R: Exponential Condition Number of a Matrix

expmCond {expm} | R Documentation

## Exponential Condition Number of a Matrix

### Description

Compute the exponential condition number of a matrix, either with approximation methods, or exactly and very slowly.

### Usage

    expmCond(A, method = c("1.est", "F.est", "exact"),
             expm = TRUE, abstol = 0.1, reltol = 1e-6,
             give.exact = c("both", "1.norm", "F.norm"))

### Arguments

`A` | a square matrix
`method` | a string; either compute 1-norm or F-norm approximations, or compte these exactly.
`expm` | logical indicating if the matrix exponential itself, which is computed anyway, should be returned as well.
`abstol, reltol` | for `method = "F.est"`, numerical >= 0, as absolute and relative error tolerance.
`give.exact` | for `method = "exact"`, specify if only the 1-norm, the Frobenius norm, or both are to be computed.

### Details

`method = "exact"`, aka Kronecker-Sylvester algorithm, computes a Kronecker matrix of dimension n^2 x n^2 and hence, with O(n^5) complexity, is prohibitely slow for non-small n. It computes the exact exponential-condition numbers for both the Frobenius and/or the 1-norm, depending on `give.exact`.

The two other methods compute approximations, to these norms, i.e., estimate them, using algorithms from Higham, chapt.~3.4, both with complexity O(n^3).

### Value

when `expm = TRUE`, for `method = "exact"`, a `list` with components

`expm` | containing the matrix exponential, `expm.Higham08(A)`.
`expmCond(F|1)` | numeric scalar, (an approximation to) the (matrix exponential) condition number, for either the 1-norm (`expmCond1`) or the Frobenius-norm (`expmCondF`).

When `expm` is false and `method` one of the approximations (`"*.est"`), the condition number is returned directly (i.e., `numeric` of length one).

### Author(s)

Michael Stadelmann (final polish by Martin Maechler).

### References

Awad H. Al-Mohy and Nicholas J. Higham (2009). Computing Fréchet Derivative of the Matrix Exponential, with an application to Condition Number Estimation; MIMS EPrint 2008.26; Manchester Institute for Mathematical Sciences, U. Manchester, UK. http://eprints.ma.man.ac.uk/1218/01/covered/MIMS_ep2008_26.pdf

Higham, N.~J. (2008). Functions of Matrices: Theory and Computation; Society for Industrial and Applied Mathematics, Philadelphia, PA, USA.

Michael Stadelmann (2009) Matrixfunktionen ... Master's thesis; see reference in `expm.Higham08`.

### See Also

`expm.Higham08` for the matrix exponential.

### Examples

    set.seed(101)
    (A <- matrix(round(rnorm(3^2),1), 3,3))

    eA <- expm.Higham08(A)
    stopifnot(all.equal(eA, expm::expm(A), tolerance= 1e-15))

    C1 <- expmCond(A, "exact")
    C2 <- expmCond(A, "1.est")
    C3 <- expmCond(A, "F.est")
    all.equal(C1$expmCond1, C2$expmCond, tolerance= 1e-15)# TRUE
    all.equal(C1$expmCondF, C3$expmCond)# relative difference of 0.001...

[Package expm version 0.999-2 Index]