R: Compute Approximate CONDition number and 1-Norm of (Large)...

condest {Matrix}

R Documentation

Compute Approximate CONDition number and 1-Norm of (Large) Matrices

Description

“Estimate”, i.e. compute approximately the CONDition number of a (potentially large, often sparse) matrix A. It works by apply a fast randomized approximation of the 1-norm, norm(A,"1"), through onenormest(.).

Usage

condest(A, t = min(n, 5), normA = norm(A, "1"),
        silent = FALSE, quiet = TRUE)

onenormest(A, t = min(n, 5), A.x, At.x, n,
           silent = FALSE, quiet = silent,
           iter.max = 10, eps = 4 * .Machine$double.eps)

Arguments

`A`	a square matrix, optional for `onenormest()`, where instead of `A`, `A.x` and `At.x` can be specified, see there.
`t`	number of columns to use in the iterations.
`normA`	number; (an estimate of) the 1-norm of `A`, by default `norm(A, "1")`; may be replaced by an estimate.
`silent`	logical indicating if warning and (by default) convergence messages should be displayed.
`quiet`	logical indicating if convergence messages should be displayed.
`A.x, At.x`	when `A` is missing, these two must be given as functions which compute `A %% x`, or `t(A) %% x`, respectively.
`n`	`== nrow(A)`, only needed when `A` is not specified.
`iter.max`	maximal number of iterations for the 1-norm estimator.
`eps`	the relative change that is deemed irrelevant.

Details

condest() calls lu(A), and subsequently onenormest(A.x = , At.x = ) to compute an approximate norm of the inverse of A, A^{-1}, in a way which keeps using sparse matrices efficiently when A is sparse.

Note that onenormest() uses random vectors and hence both functions' results are random, i.e., depend on the random seed, see, e.g., set.seed().

Value

Both functions return a list; condest() with components,

`est`	a number > 0, the estimated (1-norm) condition number k.; when r :=`rcond(A)`, 1/k. ~= r.
`v`	the maximal A x column, scaled to norm(v) = 1. Consequently, norm(A v) = norm(A) / est; when `est` is large, `v` is an approximate null vector.

The function onenormest() returns a list with components,

`est`	a number > 0, the estimated `norm(A, "1")`.
`v`	0-1 integer vector length `n`, with an `1` at the index `j` with maximal column `A[,j]` in A.
`w`	numeric vector, the largest A x found.
`iter`	the number of iterations used.

Author(s)

This is based on octave's condest() and onenormest() implementations with original author Jason Riedy, U Berkeley; translation to R and adaption by Martin Maechler.

References

Nicholas J. Higham and Françoise Tisseur (2000). A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra. SIAM J. Matrix Anal. Appl. 21, 4, 1185–1201. http://dx.doi.org/10.1137/S0895479899356080

William W. Hager (1984). Condition Estimates. SIAM J. Sci. Stat. Comput. 5, 311–316.

Examples

data(KNex)
mtm <- with(KNex, crossprod(mm))
system.time(ce <- condest(mtm))
sum(abs(ce$v)) ## || v ||_1  == 1
## Prove that  || A v || = || A || / est  (as ||v|| = 1):
stopifnot(all.equal(norm(mtm %*% ce$v),
                    norm(mtm) / ce$est))

## reciprocal
1 / ce$est
system.time(rc <- rcond(mtm)) # takes ca  3 x  longer
rc
all.equal(rc, 1/ce$est) # TRUE -- the approxmation was good

one <- onenormest(mtm)
str(one) ## est = 12.3
## the maximal column:
which(one$v == 1) # mostly 4, rarely 1, depending on random seed

[Package Matrix version 1.2-12 Index]