Peter, I have indeed worked with Gregory-Newton and divided differences in my very first numerical analysis course a couple of decades ago! However, I am perplexed by the particular form of this matrix where the differences are stored along the diagonals. I know that this is not the *same* as the Wronskian, but was just wondering whether it is an established matrix that is some kind of an *ian* like Hermitian, Jacobian, Hessian, Wronskian, Laplacian, ...
Best, Ravi. ------------------------------------------------------- Ravi Varadhan, Ph.D. Assistant Professor, Division of Geriatric Medicine and Gerontology School of Medicine Johns Hopkins University Ph. (410) 502-2619 email: rvarad...@jhmi.edu -----Original Message----- From: peter dalgaard [mailto:pda...@gmail.com] Sent: Wednesday, April 27, 2011 4:59 PM To: Ravi Varadhan Cc: R Help Subject: Re: [R] matrix of higher order differences On Apr 27, 2011, at 21:34 , Ravi Varadhan wrote: > My apologies in advance for being a bit off-topic, but I could not quell my > curiosity. > > What might one do with a matrix of all order finite differences? It seems > that such a matrix might be related to the Wronskian (its discrete analogue, > perhaps). > > http://en.wikipedia.org/wiki/Wronskian Not quite, I think. This is one function at different values of x, the Wronskian is about n different functions. Tables of higher-order differences were used fundamentally for interpolation and error detection in tables of function values (remember those?), but rarely computed to the full extent - usually only until the effects of truncation set in and the differences start alternating in sign. > > Ravi. > ------------------------------------------------------- > Ravi Varadhan, Ph.D. > Assistant Professor, > Division of Geriatric Medicine and Gerontology School of Medicine Johns > Hopkins University > > Ph. (410) 502-2619 > email: rvarad...@jhmi.edu > > > -----Original Message----- > From: r-help-boun...@r-project.org [mailto:r-help-boun...@r-project.org] On > Behalf Of Petr Savicky > Sent: Wednesday, April 27, 2011 11:01 AM > To: r-help@r-project.org > Subject: Re: [R] matrix of higher order differences > > On Wed, Apr 27, 2011 at 11:25:42AM +0000, Hans W Borchers wrote: >> Jeroen Ooms <jeroenooms <at> gmail.com> writes: >> >>> >>> Is there an easy way to turn a vector of length n into an n by n matrix, in >>> which the diagonal equals the vector, the first off diagonal equals the >>> first order differences, the second... etc. I.e. to do this more >>> efficiently: >>> >>> diffmatrix <- function(x){ >>> n <- length(x); >>> M <- diag(x); >>> for(i in 1:(n-1)){ >>> differences <- diff(x, dif=i); >>> for(j in 1:length(differences)){ >>> M[j,i+j] <- differences[j] >>> } >>> } >>> M[lower.tri(M)] <- t(M)[lower.tri(M)]; >>> return(M); >>> } >>> >>> x <- c(1,2,3,5,7,11,13,17,19); >>> diffmatrix(x); >>> >> >> I do not know whether you will call the appended version more elegant, >> but at least it is much faster -- up to ten times for length(x) = 1000, >> i.e. less than 2 secs for generating and filling a 1000-by-1000 matrix. >> I also considered col(), row() indexing: >> >> M[col(M) == row(M) + k] <- x >> >> Surprisingly (for me), this makes it even slower than your version with >> a double 'for' loop. >> >> -- Hans Werner >> >> # ---- >> diffmatrix <- function(x){ >> n <- length(x) >> if (n == 1) return(x) >> >> M <- diag(x) >> for(i in 1:(n-1)){ >> x <- diff(x) # use 'diff' in a loop >> for(j in 1:(n-i)){ # length is known >> M[j, i+j] <- x[j] # and reuse x >> } >> } >> M[lower.tri(M)] <- t(M)[lower.tri(M)] >> return(M) >> } >> # ---- > > Hi. > > The following avoids the inner loop and it was faster > for x of length 100 and 1000. > > diffmatrix2 <- function(x){ > n <- length(x) > if (n == 1) return(x) > A <- matrix(nrow=n+1, ncol=n) > for(i in 1:n){ > A[i, seq.int(along=x)] <- x > x <- diff(x) > } > M <- matrix(A, nrow=n, ncol=n) > M[upper.tri(M)] <- t(M)[upper.tri(M)] > return(M) > } > > Reorganizing an (n+1) x n matrix into an n x n matrix > shifts i-th column by (i-1) downwards. In particular, > the first row becomes the main diagonal. The initial > part of each of the remaining rows becomes a diagonal > starting at the first component of the original row. > > Petr Savicky. > > ______________________________________________ > R-help@r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code. > > ______________________________________________ > R-help@r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code. -- Peter Dalgaard Center for Statistics, Copenhagen Business School Solbjerg Plads 3, 2000 Frederiksberg, Denmark Phone: (+45)38153501 Email: pd....@cbs.dk Priv: pda...@gmail.com ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
