[R] Choleski and Choleski with pivoting of matrix fails
Hi Everyone, I need to take the square root of the following matrix: [,1] [,2][,3] [1,] 0.5401984 -0.3998675 -1.3785897 [2,] -0.3998675 1.0561872 0.8158639 [3,] -1.3785897 0.8158639 1.6073119 I tried Choleski which fails. I then tried Choleski with pivoting, but unfortunately the square root I get is not valid. I also tried eigen decomposition but i did no get far. Any clue on how to do it?! Thanks, Simon __ 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] R: chol( neg.def.matrix ) WAS: Re: Choleski and Choleski with pivoting of matrix fails
Thanks Chuck! It works! But Once I have the square root of this matrix, how do I convert it to a real (not imaginary) matrix which has the same property? Is that possible? Best, Simon >Messaggio originale >Da: cbe...@tajo.ucsd.edu >Data: 21-nov-2009 18.11 >A: "simona.racio...@libero.it" >Cc: >Ogg: chol( neg.def.matrix ) WAS: Re: [R] Choleski and Choleski with pivoting of matrix fails > >On Sat, 21 Nov 2009, simona.racio...@libero.it wrote: > >> Hi Everyone, >> >> I need to take the square root of the following matrix: >> >>[,1] [,2][,3] >> [1,] 0.5401984 -0.3998675 -1.3785897 >> [2,] -0.3998675 1.0561872 0.8158639 >> [3,] -1.3785897 0.8158639 1.6073119 >> >> I tried Choleski which fails. I then tried Choleski with pivoting, but >> unfortunately the square root I get is not valid. I also tried eigen >> decomposition but i did no get far. >> >> Any clue on how to do it?! > > >If you want to take the square root of a negative definite matrix, you >could use > > sqrtm( neg.def.mat ) > >from the expm package on rforge: > > http://r-forge.r-project.org/projects/expm/ > >HTH, > >Chuck > > >> >> Thanks, >> Simon >> >> __ >> 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. >> > >Charles C. Berry(858) 534-2098 > Dept of Family/Preventive Medicine >E mailto:cbe...@tajo.ucsd.edu UC San Diego >http://famprevmed.ucsd.edu/faculty/cberry/ La Jolla, San Diego 92093-0901 > > > __ 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] R: Re: chol( neg.def.matrix ) WAS: Re: Choleski and Choleski with pivoting of matrix fails
It works! But Once I have the square root of this matrix, how do I convert it to a real (not imaginary) matrix which has the same property? Is that possible? Best, Simon >Messaggio originale >Da: p.dalga...@biostat.ku.dk >Data: 21-nov-2009 18.56 >A: "Charles C. Berry" >Cc: "simona.racio...@libero.it", >Ogg: Re: [R] chol( neg.def.matrix ) WAS: Re: Choleski and Choleski with pivoting of matrix fails > >Charles C. Berry wrote: >> On Sat, 21 Nov 2009, simona.racio...@libero.it wrote: >> >>> Hi Everyone, >>> >>> I need to take the square root of the following matrix: >>> >>>[,1] [,2][,3] >>> [1,] 0.5401984 -0.3998675 -1.3785897 >>> [2,] -0.3998675 1.0561872 0.8158639 >>> [3,] -1.3785897 0.8158639 1.6073119 >>> >>> I tried Choleski which fails. I then tried Choleski with pivoting, but >>> unfortunately the square root I get is not valid. I also tried eigen >>> decomposition but i did no get far. >>> >>> Any clue on how to do it?! >> >> >> If you want to take the square root of a negative definite matrix, you >> could use >> >> sqrtm( neg.def.mat ) >> >> from the expm package on rforge: >> >> http://r-forge.r-project.org/projects/expm/ > >But that matrix is not negative definite! It has 2 positive and one >negative eigenvalue. It is non-positive definite. > >It is fairly easy in any case to get a matrix square root from the eigen >decomposition: > > > v%*%diag(sqrt(d+0i))%*%t(v) > [,1] [,2] [,3] >[1,] 0.5164499+0.4152591i -0.1247682-0.0562317i -0.7257079+0.3051868i >[2,] -0.1247682-0.0562317i 0.9618445+0.0076145i 0.3469916-0.0413264i >[3,] -0.7257079+0.3051868i 0.3469916-0.0413264i 1.0513849+0.2242912i > > ch <- v%*%diag(sqrt(d+0i))%*%t(v) > > t(ch)%*% ch > [,1] [,2] [,3] >[1,] 0.5401984+0i -0.3998675-0i -1.3785897-0i >[2,] -0.3998675-0i 1.0561872+0i 0.8158639-0i >[3,] -1.3785897-0i 0.8158639-0i 1.6073119-0i > >A triangular square root is, er, more difficult, but hardly impossible. > >-- >O__ Peter Dalgaard Øster Farimagsgade 5, Entr.B > c/ /'_ --- Dept. of Biostatistics PO Box 2099, 1014 Cph. K > (*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918 >~~ - (p.dalga...@biostat.ku.dk) FAX: (+45) 35327907 > __ 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] R: Re: R: Re: chol( neg.def.matrix ) WAS: Re: Choleski and Choleski with pivoting of matrix fails
Dear Peter, thank you very much for your answer. My problem is that I need to calculate the following quantity: solve(chol(A)%*%Y) Y is a 3*3 diagonal matrix and A is a 3*3 matrix. Unfortunately one eigenvalue of A is negative. I can anyway take the square root of A but when I multiply it by Y, the imaginary part of the square root of A is dropped, and I do not get the right answer. I tried to exploit the diagonal structure of Y by using 2*2 matrices for A and Y. In this way the problem mentioned above disappears (since all eigenvalues of A are positive) and when I perform the calculation above I get approximately the right answer. The approximation is quite good. However it is an approximation. Any suggestion? Thank you very much! Simon >Messaggio originale >Da: p.dalga...@biostat.ku.dk >Data: 23-nov-2009 14.09 >A: "simona.racio...@libero.it" >Cc: "Charles C. Berry", >Ogg: Re: R: Re: [R] chol( neg.def.matrix ) WAS: Re: Choleski and Choleski with pivoting of matrix fails > >simona.racio...@libero.it wrote: >> It works! But Once I have the square root of this matrix, how do I convert it >> to a real (not imaginary) matrix which has the same property? Is that >> possible? > >No. That is theoretically impossible. > >If A = B'B, then x'Ax = ||Bx||^2 >= 0 > >for any x, which implies in particular that all eigenvalues of A should >be nonnegative. > >> >> Best, >> Simon >> >>> Messaggio originale >>> Da: p.dalga...@biostat.ku.dk >>> Data: 21-nov-2009 18.56 >>> A: "Charles C. Berry" >>> Cc: "simona.racio...@libero.it", > project.org> >>> Ogg: Re: [R] chol( neg.def.matrix ) WAS: Re: Choleski and Choleski with >> pivoting of matrix fails >>> Charles C. Berry wrote: >>>> On Sat, 21 Nov 2009, simona.racio...@libero.it wrote: >>>> >>>>> Hi Everyone, >>>>> >>>>> I need to take the square root of the following matrix: >>>>> >>>>>[,1] [,2][,3] >>>>> [1,] 0.5401984 -0.3998675 -1.3785897 >>>>> [2,] -0.3998675 1.0561872 0.8158639 >>>>> [3,] -1.3785897 0.8158639 1.6073119 >>>>> >>>>> I tried Choleski which fails. I then tried Choleski with pivoting, but >>>>> unfortunately the square root I get is not valid. I also tried eigen >>>>> decomposition but i did no get far. >>>>> >>>>> Any clue on how to do it?! >>>> >>>> If you want to take the square root of a negative definite matrix, you >>>> could use >>>> >>>> sqrtm( neg.def.mat ) >>>> >>>> from the expm package on rforge: >>>> >>>> http://r-forge.r-project.org/projects/expm/ >>> But that matrix is not negative definite! It has 2 positive and one >>> negative eigenvalue. It is non-positive definite. >>> >>> It is fairly easy in any case to get a matrix square root from the eigen >>> decomposition: >>> >>>> v%*%diag(sqrt(d+0i))%*%t(v) >>> [,1] [,2] [,3] >>> [1,] 0.5164499+0.4152591i -0.1247682-0.0562317i -0.7257079+0.3051868i >>> [2,] -0.1247682-0.0562317i 0.9618445+0.0076145i 0.3469916-0.0413264i >>> [3,] -0.7257079+0.3051868i 0.3469916-0.0413264i 1.0513849+0.2242912i >>>> ch <- v%*%diag(sqrt(d+0i))%*%t(v) >>>> t(ch)%*% ch >>> [,1] [,2] [,3] >>> [1,] 0.5401984+0i -0.3998675-0i -1.3785897-0i >>> [2,] -0.3998675-0i 1.0561872+0i 0.8158639-0i >>> [3,] -1.3785897-0i 0.8158639-0i 1.6073119-0i >>> >>> A triangular square root is, er, more difficult, but hardly impossible. >>> >>> -- >>>O__ Peter Dalgaard Øster Farimagsgade 5, Entr.B >>> c/ /'_ --- Dept. of Biostatistics PO Box 2099, 1014 Cph. K >>> (*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918 >>> ~~ - (p.dalga...@biostat.ku.dk) FAX: (+45) 35327907 >>> >> >> > > >-- > O__ Peter Dalgaard Øster Farimagsgade 5, Entr.B > c/ /'_ --- Dept. of Biostatistics PO Box 2099, 1014 Cph. K > (*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918 >~~ - (p.dalga...@biostat.ku.dk) FAX: (+45) 35327907 > > __ 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] R: RE: R: Re: R: Re: chol( neg.def.matrix ) WAS: Re: Choleski and Choleski with pivoting of matrix fails
Thanks for your message! Actually it works quite well for me too. If I then take the trace of the final result below, I end up with a number made up of both a real and an imaginary part. This does not probably mean much if the trace of the matrix below givens me info about the degrees of freedom of a model... Simona >Messaggio originale >Da: rvarad...@jhmi.edu >Data: 25-nov-2009 18.55 >A: , >Cc: >Ogg: RE: [R] R: Re: R: Re: chol( neg.def.matrix ) WAS: Re: Choleski and Choleski with pivoting of matrix fails > >I do not understand what the problem is, as it works just fine for me: > >A <- matrix(c(0.5401984,-0.3998675,-1.3785897,-0.3998675,1.0561872, >0.8158639,-1.3785897, 0.8158639, 1.6073119), 3, 3, byrow=TRUE) > >eA <- eigen(A) > >chA <- eA$vec %*% diag(sqrt(eA$val+0i)) %*% t(eA$vec) > >all.equal(A, Re(chA %*% t(chA))) > >Y <- diag(c(1,2,3)) > >solve(chA %*% Y) > >Ravi. > >--- > >Ravi Varadhan, Ph.D. > >Assistant Professor, The Center on Aging and Health > >Division of Geriatric Medicine and Gerontology > >Johns Hopkins University > >Ph: (410) 502-2619 > >Fax: (410) 614-9625 > >Email: rvarad...@jhmi.edu > >Webpage: http://www.jhsph. edu/agingandhealth/People/Faculty_personal_pages/Varadhan.html > > > >---- > >-Original Message- >From: r-help-boun...@r-project.org [mailto:r-help-boun...@r-project.org] On Behalf Of simona.racio...@libero.it >Sent: Wednesday, November 25, 2009 9:59 AM >To: p.dalga...@biostat.ku.dk >Cc: r-help@r-project.org >Subject: [R] R: Re: R: Re: chol( neg.def.matrix ) WAS: Re: Choleski and Choleski with pivoting of matrix fails > >Dear Peter, >thank you very much for your answer. > >My problem is that I need to calculate the following quantity: > >solve(chol(A)%*%Y) > >Y is a 3*3 diagonal matrix and A is a 3*3 matrix. Unfortunately one >eigenvalue of A is negative. I can anyway take the square root of A but when I >multiply it by Y, the imaginary part of the square root of A is dropped, and I >do not get the right answer. > >I tried to exploit the diagonal structure of Y by using 2*2 matrices for A >and Y. In this way the problem mentioned above disappears (since all >eigenvalues of A are positive) and when I perform the calculation above I get >approximately the right answer. The approximation is quite good. However it is >an approximation. > >Any suggestion? >Thank you very much! >Simon > > > > >>Messaggio originale >>Da: p.dalga...@biostat.ku.dk >>Data: 23-nov-2009 14.09 >>A: "simona.racio...@libero.it" >>Cc: "Charles C. Berry", >>Ogg: Re: R: Re: [R] chol( neg.def.matrix ) WAS: Re: Choleski and Choleski >with pivoting of matrix fails >> >>simona.racio...@libero.it wrote: >>> It works! But Once I have the square root of this matrix, how do I convert >it >>> to a real (not imaginary) matrix which has the same property? Is that >>> possible? >> >>No. That is theoretically impossible. >> >>If A = B'B, then x'Ax = ||Bx||^2 >= 0 >> >>for any x, which implies in particular that all eigenvalues of A should >>be nonnegative. >> >>> >>> Best, >>> Simon >>> >>>> Messaggio originale >>>> Da: p.dalga...@biostat.ku.dk >>>> Data: 21-nov-2009 18.56 >>>> A: "Charles C. Berry" >>>> Cc: "simona.racio...@libero.it", >> project.org> >>>> Ogg: Re: [R] chol( neg.def.matrix ) WAS: Re: Choleski and Choleski with >>> pivoting of matrix fails >>>> Charles C. Berry wrote: >>>>> On Sat, 21 Nov 2009, simona.racio...@libero.it wrote: >>>>> >>>>>> Hi Everyone, >>>>>> >>>>>> I need to take the square root of the following matrix: >>>>>> >>>>>>[,1] [,2][,3] >>>>>> [1,] 0.5401984 -0.3998675 -1.3785897 >>>>>> [2,] -0.3998675 1.0561872 0.8158639 >>>>>> [3,] -1.3785897 0.8158639 1.6073119 >>>>>> >>>>>> I tried Choleski which fails. I then tried Choleski with pivoting, but >>>>>> unfortunately the square root I get is not valid. I also tried eigen >>>>>> decomposition but i did no get far. >>>>>> >>>>>>
