The condition number of a regular (square) matrix is the product of the norm of the matrix and the norm of its inverse (or pseudoinverse), and hence depends on the kind of matrixnorm.
kappa()
computes by default (an estimate of) the 2norm
condition number of a matrix or of the R matrix of a QR
decomposition, perhaps of a linear fit. The 2norm condition number
can be shown to be the ratio of the largest to the smallest
nonzero singular value of the matrix.
rcond()
computes an approximation of the reciprocal
condition number, see the details.
1 2 3 4 5 6 7 8 9 10 11 12  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 2norm
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 Infnorm 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 http://www.netlib.org/lapack and http://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 online at
http://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.
1 2 3 4 5 6 7 8 9  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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