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	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. https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.7.9804

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

See Also

norm, rcond.

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