| kappa | R Documentation |
The condition number of a regular (square) matrix is the product of the norm of the matrix and the norm of its inverse (or pseudo-inverse), and hence depends on the kind of matrix-norm.
kappa() computes by default (an estimate of) the 2-norm
condition number of a matrix or of the R matrix of a QR
decomposition, perhaps of a linear fit. The 2-norm condition number
can be shown to be the ratio of the largest to the smallest
non-zero singular value of the matrix.
rcond() computes an approximation of the reciprocal
condition number, see the details.
kappa(z, ...)
## Default S3 method:
kappa(z, exact = FALSE,
norm = NULL, method = c("qr", "direct"), ...)
## S3 method for class 'lm'
kappa(z, ...)
## S3 method for class 'qr'
kappa(z, ...)
.kappa_tri(z, exact = FALSE, LINPACK = TRUE, norm = NULL, ...)
rcond(x, norm = c("O","I","1"), triangular = FALSE, ...)
z, x |
A matrix or a the result of |
exact |
logical. Should the result be exact? |
norm |
character string, specifying the matrix norm with respect
to which the condition number is to be computed, see also
|
method |
a partially matched character string specifying the method to be used;
|
triangular |
logical. If true, the matrix used is just the lower
triangular part of |
LINPACK |
logical. If true and |
... |
further arguments passed to or from other methods;
for |
For kappa(), if exact = FALSE (the default) the 2-norm
condition number is estimated by a cheap approximation. However, the
exact calculation (via svd) is also likely to be quick
enough.
Note that the 1- and Inf-norm condition numbers are much faster to
calculate, and rcond() computes these reciprocal
condition numbers, also for complex matrices, using standard LAPACK
routines.
kappa and rcond are different interfaces to
partly identical functionality.
.kappa_tri is an internal function called by kappa.qr and
kappa.default.
Unsuccessful results from the underlying LAPACK code will result in an error giving a positive error code: these can only be interpreted by detailed study of the FORTRAN code.
The condition number, kappa, or an approximation if
exact = FALSE.
The design was inspired by (but differs considerably from) the S function of the same name described in Chambers (1992).
The LAPACK routines DTRCON and ZTRCON and the LINPACK
routine DTRCO.
LAPACK and LINPACK are from https://www.netlib.org/lapack/ and https://www.netlib.org/linpack/ and their guides are listed in the references.
Anderson. E. and ten others (1999)
LAPACK Users' Guide. Third Edition. SIAM.
Available on-line at
https://www.netlib.org/lapack/lug/lapack_lug.html.
Chambers, J. M. (1992) Linear models. Chapter 4 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.
Dongarra, J. J., Bunch, J. R., Moler, C. B. and Stewart, G. W. (1978) LINPACK Users Guide. Philadelphia: SIAM Publications.
norm;
svd for the singular value decomposition and
qr for the QR one.
kappa(x1 <- cbind(1, 1:10)) # 15.71
kappa(x1, exact = TRUE) # 13.68
kappa(x2 <- cbind(x1, 2:11)) # high! [x2 is singular!]
hilbert <- function(n) { i <- 1:n; 1 / outer(i - 1, i, "+") }
sv9 <- svd(h9 <- hilbert(9))$ d
kappa(h9) # pretty high!
kappa(h9, exact = TRUE) == max(sv9) / min(sv9)
kappa(h9, exact = TRUE) / kappa(h9) # 0.677 (i.e., rel.error = 32%)
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