kappa
Compute the Exact or Estimated Condition Number
Description
Computethe exact or estimated condition number of a
matrixbased on various norms.
Thisfunction is an S Version 3 generic. One can write methods
handlenew classes of data. Classes that already have methods
includematrix, lm, and qr.
Usage
#Generic function
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, ...)
Arguments
| z |
amatrix or an object, like a fitted model,
thatcontains information from a decomposition of some matrix.
|
| exact |
logicalvalue. If TRUE, the exact 2-norm condition number,
basedon the extrema of the singular values, is returned.
Otherwisea more quickly computed approximate condition number is returned.
|
| norm |
characterstring specifying the norm type of matrix. It is one
of"2", "1", "O" or "I". See the help file norm
fordetails.
Thedefault NULL is interpreted as 2-norm type in kappa.default,
or1-norm type in .kappa_tri.
|
| method |
Either"qr" or "direct". If "direct" the matrix
zwill be assumed to be upper triangular -- the values in its
lowertriangle will be taken to be 0.
If"qr", the default, then the condition number will be computed
fromthe the (upper triangula) R matrix of the QR decompostion of z.
Thisargument is ignored when exact is TRUE: only the
approximatemethods require an upper triangular matrix.
|
| LINPACK |
Thisargument is ignored and is here only for compatibility with R 2.15.1.
|
| ... |
other arguments are ignored by the methods described above. Methods defined
inother packages may use more arguments.
|
Details
Thecondition number of a matrix is the product of the matrix and the norm of
itsinverse (or pseudo-inverse if the matrix is not square).
Sinceit can take on values between 1 and infinity, inclusive,
itcan be viewed as a measure of how close a matrix is to being rank deficient.
Itcan also be viewed as a factor by which errors in solving linear systems
withthis matrix as coefficient matrix could be magnified.
Conditionnumbers are usually approximate, since exact computation is costly
interms of floating-point operations. The exact condition number of 2-norm type matrix
isthe ratio of the largest to the smallest non-zero singular value decomposition of the
matrix(callfunction svd) .
Methodsfor kappa generally use a cheap approximation to this.
Theycall the "internal" function .kappa_tri directly or indirectly to do the linear algebra.
.kappa_tricomputes the approximate condition number of a triangular square matrix
byusing the LINPACK subroutine "dtrco" ("ztrco" for complex matrices) for the 2-norm and
theLAPACK subroutine "dtrcon" ("ztrcon" for complex matrices) for the 1- and Infinity-norms.
(Whenexact is TRUE it uses svd to compute the 2-norm condition number.)
Value
anestimate of the condition number (large values indicate near-singularity
ofthe matrix) if exact is FALSE, or an exact condition number of 2-norm
typematrix if exact is TRUE.
SNextSpec
[xipan,2011-04-18]: Review and update the most of content to be compatible with R.
SNextS-PLUS Diff
[xipan,2011-04-18]: kappa.qr() and .kappa.tri() are not implemented in S-PLUS.
Arguments"exact", "norm", "method" and "LINPACK" are not supported in S-PLUS.
kappa.upper()and kappa.svd.righ() are removed from TIBCO Enterprise Runtime for R.
FileInfo
$Rev: 17391 $
$Date: 2014-05-12 11:18:31 -0700 (Mon, 12 May 2014) $
$Id: kappa.Rd 17391 2014-05-12 18:18:31Z skaluzny $
RVersion
R-2.13.0or earlier
References
Chambers, J. M. (1992)
Linear models.
Chapter 4 of Statistical Models in S
eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.
Anderson. E. and ten others (1999)
LAPACK Users' Guide. Third Edition. SIAM.
See Also
Examples
kappa(var(Sdatasets::fuel.frame[,1:4]))
kappa(var(Sdatasets::fuel.frame[,1:4]), norm = "I")
fitMpg <- lm(Fuel ~ Weight + Type, data=Sdatasets::fuel.frame)
kappa(fitMpg)
kappa(fitMpg, exact=TRUE)