condest function

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

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(.).

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().

Returns

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.

References

Nicholas J. Higham and 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.

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

Author(s)

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

See Also

norm, rcond.

Examples

data(KNex, package = "Matrix")
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 approximation 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