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comparison scripts/special-matrix/invhilb.m @ 20162:2645f9ef8c88 stable

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doc: Update more docstrings to have one sentence summary as first line. Reviewed specfun, special-matrix, testfun, and time script directories. * scripts/specfun/expint.m, scripts/specfun/isprime.m, scripts/specfun/legendre.m, scripts/specfun/primes.m, scripts/specfun/reallog.m, scripts/specfun/realsqrt.m, scripts/special-matrix/gallery.m, scripts/special-matrix/hadamard.m, scripts/special-matrix/hankel.m, scripts/special-matrix/hilb.m, scripts/special-matrix/invhilb.m, scripts/special-matrix/magic.m, scripts/special-matrix/pascal.m, scripts/special-matrix/rosser.m, scripts/special-matrix/toeplitz.m, scripts/special-matrix/vander.m, scripts/special-matrix/wilkinson.m, scripts/testfun/assert.m, scripts/testfun/demo.m, scripts/testfun/example.m, scripts/testfun/fail.m, scripts/testfun/rundemos.m, scripts/testfun/runtests.m, scripts/testfun/speed.m, scripts/time/asctime.m, scripts/time/calendar.m, scripts/time/clock.m, scripts/time/ctime.m, scripts/time/datenum.m, scripts/time/datestr.m, scripts/time/datevec.m, scripts/time/etime.m, scripts/time/is_leap_year.m, scripts/time/now.m, scripts/time/weekday.m: Update more docstrings to have one sentence summary as first line.
author Rik <rik@octave.org>
date Sun, 03 May 2015 17:00:11 -0700
parents 4197fc428c7d
children
comparison
equal deleted inserted replaced
20160:03b9d17a2d95 20162:2645f9ef8c88
16 ## along with Octave; see the file COPYING. If not, see 16 ## along with Octave; see the file COPYING. If not, see
17 ## <http://www.gnu.org/licenses/>. 17 ## <http://www.gnu.org/licenses/>.
18 18
19 ## -*- texinfo -*- 19 ## -*- texinfo -*-
20 ## @deftypefn {Function File} {} invhilb (@var{n}) 20 ## @deftypefn {Function File} {} invhilb (@var{n})
21 ## Return the inverse of the Hilbert matrix of order @var{n}. This can be 21 ## Return the inverse of the Hilbert matrix of order @var{n}.
22 ## computed exactly using 22 ##
23 ## This can be computed exactly using
23 ## @tex 24 ## @tex
24 ## $$\eqalign{ 25 ## $$\eqalign{
25 ## A_{ij} &= -1^{i+j} (i+j-1) 26 ## A_{ij} &= -1^{i+j} (i+j-1)
26 ## \left( \matrix{n+i-1 \cr n-j } \right) 27 ## \left( \matrix{n+i-1 \cr n-j } \right)
27 ## \left( \matrix{n+j-1 \cr n-i } \right) 28 ## \left( \matrix{n+j-1 \cr n-i } \right)
58 ## \ k-1 / \k/ 59 ## \ k-1 / \k/
59 ## @end group 60 ## @end group
60 ## @end example 61 ## @end example
61 ## 62 ##
62 ## @end ifnottex 63 ## @end ifnottex
63 ## The validity of this formula can easily be checked by expanding 64 ## The validity of this formula can easily be checked by expanding the binomial
64 ## the binomial coefficients in both formulas as factorials. It can 65 ## coefficients in both formulas as factorials. It can be derived more
65 ## be derived more directly via the theory of Cauchy matrices. 66 ## directly via the theory of Cauchy matrices. See @nospell{J. W. Demmel},
66 ## See @nospell{J. W. Demmel}, @cite{Applied Numerical Linear Algebra}, p. 92. 67 ## @cite{Applied Numerical Linear Algebra}, p. 92.
67 ## 68 ##
68 ## Compare this with the numerical calculation of @code{inverse (hilb (n))}, 69 ## Compare this with the numerical calculation of @code{inverse (hilb (n))},
69 ## which suffers from the ill-conditioning of the Hilbert matrix, and the 70 ## which suffers from the ill-conditioning of the Hilbert matrix, and the
70 ## finite precision of your computer's floating point arithmetic. 71 ## finite precision of your computer's floating point arithmetic.
71 ## @seealso{hilb} 72 ## @seealso{hilb}