condest: 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 square matrix, optional for onenormest(), where instead of A, A.x and At.x can be specified, see there.

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.

== 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.

Value

list; condest() with components,The function onenormest() returns a list with components,

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().

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

Run this code

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

Run the code above in your browser using DataLab

Get 50% off unlimited learning